Research ArticleHelical Phase Inflation and Monodromy in Supergravity Theory

Tianjun Li12 Zhijin Li3 and Dimitri V Nanopoulos345

1State Key Laboratory ofTheoretical Physics and Kavli Institute forTheoretical Physics China (KITPC) Institute ofTheoretical PhysicsChinese Academy of Sciences Beijing 100190 China2School of Physical Electronics University of Electronic Science and Technology of China Chengdu 610054 China3George P and Cynthia W Mitchell Institute for Fundamental Physics and Astronomy Texas AampM University College Station TX77843 USA4Astroparticle Physics Group Houston Advanced Research Center (HARC) Mitchell Campus The Woodlands TX 77381 USA5Division of Natural Sciences Academy of Athens 28 Panepistimiou Avenue 10679 Athens Greece

Correspondence should be addressed to Zhijin Li lizhijinphysicstamuedu

Received 3 July 2015 Accepted 17 November 2015

Academic Editor Ignatios Antoniadis

Copyright copy 2015 Tianjun Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited Thepublication of this article was funded by SCOAP3

We study helical phase inflation which realizes ldquomonodromy inflationrdquo in supergravity theory In the model inflation is drivenby the phase component of a complex field whose potential possesses helicoid structure We construct phase monodromy basedon explicitly breaking global 119880(1) symmetry in the superpotential By integrating out heavy fields the phase monodromy fromsingle complex scalar field is realized and the model fulfills natural inflationThe phase-axion alignment is achieved from explicitlysymmetry breaking and gives super-Planckian phase decay constantThe 119865-term scalar potential provides strong field stabilizationfor all the scalars except inflaton which is protected by the approximate global119880(1) symmetry Besides we show that helical phaseinflation can be naturally realized in no-scale supergravity with 119878119880(2 1)119878119880(2) times 119880(1) symmetry since the supergravity setupneeded for phase monodromy is automatically provided in the no-scale Kahler potential We also demonstrate that helical phaseinflation can be reduced to another well-known supergravity inflation model with shift symmetry Helical phase inflation is freefrom the UV-sensitivity problem although there is super-Planckian field excursion and it suggests that inflation can be effectivelystudied based on supersymmetric field theory while a UV-completed framework is not prerequisite

1 Introduction

Inflation plays a crucial role in the early stage of our universe[1ndash3] and supersymmetry was found to be necessary forinflation soon after its discovery A simple argument is thatthe inflation process is triggered close to the unification scalein Grand Unified Theory (GUT) [4 5] At this scale physicstheory is widely believed to be supersymmetric To realizethe slow-roll inflation it requires strict flat conditions on thepotential 119881(120601) of inflaton 120601 The mass of inflaton 119898120601 shouldbe significantly smaller than the inflation energy scale due tothe slow-roll parameter

120578 equiv 1198722119875

11988110158401015840

119881≃

1198982120601

31198672≪ 1 (1)

where 119872119875 is the reduced Planck mass otherwise inflationcannot be triggered or last for a sufficient long periodHowever as a scalar field the inflaton is expected to obtainlarge quantum loop-corrections on the potential which canbreak the slow-roll conditions unless there is extremely finetuning Supersymmetry is a natural way to eliminate suchquantum corrections By introducing supersymmetry theflatness problem can be partially relaxed but not completelysolved since supersymmetry is broken during inflationMoreover gravity plays an important role in inflation soit is natural to study inflation within supergravity the-ory

Once combining the supersymmetry and gravity theorytogether the flatness problem reappears known as 120578 prob-lem 119873 = 1 supergravity in four-dimensional space-time

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 397410 12 pageshttpdxdoiorg1011552015397410

2 Advances in High Energy Physics

is determined by three functions Kahler potential 119870 super-potential 119882 and gauge kinetic function The 119865-term scalarpotential contains an exponential factor 119890119870 In the minimalsupergravity with 119870 = ΦΦ the exponential factor 119890

119870

introduces a term on the inflaton mass at Hubble scalewhich breaks the slow-roll condition (1) To realize inflationin supergravity the large contribution from 119890

119870 to scalarmass should be suppressed which needs a symmetry in119870 In the minimal supergravity 120578 problem can be solvedby introducing shift symmetry in the Kahler potential asproposed by Kawasaki Yamaguchi and Yanagida (KYY) [6]119870 is invariant under the shift Φ rarr Φ + 119894119862 Consequently119870 is independent of Im(Φ) so is the factor 119890

119870 in the 119865-term potential By employing Im(Φ) as inflaton its mass isnot affected by 119890

119870 and then there is no 120578 problem anymoreThe shift symmetry can be slightly broken in this case thereis still no 120578 problem and the model gives a broad range oftensor-to-scalar ratios r [7 8] 120578 problem is automaticallysolved in no-scale supergravity because of 119878119880(119873 1)119878119880(119873)times

119880(1) symmetry in the Kahler potential Historically the no-scale supergravity was proposed to get vanishing cosmol-ogy constant At classical level the potential is strictly flatguaranteed by 119878119880(119873 1)119878119880(119873) times 119880(1) symmetry of theKahler potential whichmeanwhile protects the no-scale typeinflation away from 120578 problemMoreover 119878119880(119873 1)119878119880(119873)times

119880(1) symmetry has rich structure that allows different typesof inflation Thus inflation based on no-scale supergravityhas been extensively studied [9ndash19] In thisworkwewill showthat in no-scale supergravity with 119878119880(2 1)119878119880(2) times 119880(1)

symmetry one can pick up 119880(1) subsector together withthe superpotential phase monodromy to realize helical phaseinflation

Recently it was shown that 120578 problem can be naturallysolved in helical phase inflation [20] This solution employs aglobal 119880(1) symmetry which is a trivial fact in the minimalsupergravity with 119870 = ΦΦ Using the phase of a complexfield Φ as an inflaton 120578 problem is solved due to the global119880(1) symmetry The norm of Φ needs to be stabilizedotherwise it will generate notable isocurvature perturbationsthat contradict the observations However it is a nontrivialtask to stabilize the norm of Φ while keeping the phase lightas the norm and phase couple with each other In that workthe field stabilization and quadratic inflation are realizedvia a helicoid type potential The inflationary trajectory is ahelix line and this is the reason for the name ldquohelical phaseinflationrdquo In addition the superpotential of helical phaseinflation realizesmonodromy in supersymmetric field theoryFurthermore helical phase inflation gives a method to avoidthe dangerous quantum gravity effect on inflation

The single field slow-roll inflation agrees with recentobservations [4 5] Such kind of inflation admits a relation-ship between the inflaton field excursion and the tensor-to-scalar ratio which is known as the Lyth bound [21] It suggeststhat to get large tensor-to-scalar ratio the field excursionduring inflation should be much larger than the Planck massThe super-Planckian field excursion challenges the validityin the Wilsonian sense of inflationary models described byeffective field theory At Planck scale the quantum gravity

effect is likely to introduce extra terms which are suppressedby the Planck mass and then irrelevant in the low energyscale while for a super-Planckian field the irrelevant termsbecome important and may introduce significant correctionsor even destroy the inflation process In this sense thepredictions just based on the effective field theory are nottrustable A more detailed discussion on the ultraviolet(UV) sensitivity of the inflation process is provided in[22]

A lot of works have been proposed to realize inflationbased on the UV-completed theory for example in [23ndash30] However to realize inflation in string theory it needsto address several difficult problems such as moduli stabi-lization Minkowski or de Sitter vacuum and 120572

1015840- and higherstring loop-corrections on theKahler potential However onemay doubt whether such difficult UV-completed frameworkis necessary for inflation In certain scenario the super-Planckian field excursion does not necessarily lead to thephysical field above the Planck scale A simple example isthe phase of a complex field The phase factor like a pseudo-Nambu-Goldstone boson (PNGB) can be shifted to any valuewithout any effect on the energy scale By employing thephase as an inflaton the super-Planckian field excursionis not problematic at all as there is no polynomial higherorder quantum gravity correction for the phase componentBesides helical phase inflation inflationary models usingPNGB as an inflaton have been studied [31ndash39] For naturalinflation it requires super-Planckian axion decay constantwhich can be obtained by aligned axions [34] (the axionalignment relates to 119878119899 symmetry amongKahlermoduli [40])or anomalous119880(1) gauge symmetry with large condensationgauge group [41] In helical phase inflation as will beshown later the phase monodromy in superpotential canbe easily modified to generate natural inflation and alsorealize the super-Planckian phase decay constant whichis from the phase-axion alignment hidden in the processof integrating out heavy fields Furthermore all the extrafields are consistently stabilized based on the helicoid poten-tial

Like helical phase inflation ldquomonodromy inflationrdquo wasproposed to solve the UV sensitivity of large field inflation[42 43] In such model the inflaton is identified as an axionobtained from 119901-form field after string compactificationsThe inflaton potential arises from the DBI action of branesor coupling between axion and fluxes During inflation theaxion rotates along internal cycles and reduces the axionpotential slowly while all the other physical parameters areunaffected by the axion rotation Interesting realization ofmonodromy inflation is the axion alignment [34] which wasproposed to get super-Planckian axion decay constant fornatural inflation and it was noticed that this mechanismactually provides an axion monodromy in [44ndash46] Actuallya similar name ldquohelical inflationrdquo was firstly introduced in[45] for an inflationmodel with axionmonodromy Howevera major difference should be noted the ldquohelicalrdquo structurein [45] is to describe the alignment structure of two axionswhile the ldquohelicalrdquo structure in our model is from a singlecomplex field with stabilized field normThe physical pictureof axionmonodromy is analogical to the superpotential119882 in

Advances in High Energy Physics 3

helical phase inflation For 119882 there is monodromy aroundthe singularityΦ = 0

Φ 997888rarr Φ1198902120587119894

119882 997888rarr 119882 + 2120587119894119882

logΦ

(2)

The phase monodromy together with 119880(1) symmetry inthe Kahler potential provides flat direction for inflationIn the following we will show that this monodromy iscorresponding to the global119880(1) symmetry explicitly brokenby the inflation term

In this work we will study helical phase inflation fromseveral aspects in detail Firstly we will show that the phasemonodromy in the superpotential which leads to the helicoidstructure of inflaton potential can be effectively generatedby integrating out heavy fields in supersymmetric fieldtheory Besides quadratic inflation the phase monodromyfor helical phase inflation can be easily modified to realizenatural inflation in which the process of integrating outheavy fields fulfills the phase-axion alignment indirectlyand leads to super-Planckian phase decay constant withconsistent field stabilization as well We also show that helicalphase inflation can be reduced to the KYY inflation by fieldredefinition however there is no such field transformationthat can map the KYY model back to helical phase inflationFurthermore we show that the no-scale supergravity with119878119880(2 1)119878119880(2)times119880(1) symmetry provides a natural frame forhelical phase inflation as 119878119880(2 1)119878119880(2) times 119880(1) symmetryof no-scale Kahler potential already combines the symmetryfactors needed for phase monodromy Moreover we arguethat helical phase inflation is free from the UV-sensitivityproblem

This paper is organized as follows In Section 2 we reviewthe minimal supergravity construction of helical phase infla-tion In Section 3 we present the realization of phase mon-odromy based on supersymmetric field theory In Section 4natural inflation as a special type of helical phase inflation isstudied In Section 5 the relationship between helical phaseinflation and the KYY model is discussed In Section 6 westudy helical phase inflation in no-scale supergravity with119878119880(2 1)119878119880(2)times119880(1) symmetry In Section 7 we discuss howhelical phase inflation dodges the UV-sensitivity problem oflarge field inflation Conclusion is given in Section 8

2 Helical Phase Inflation

In four dimensions 119873 = 1 supergravity is determined bythe Kahler potential 119870 superpotential 119882 and gauge kineticfunction The 119865-term scalar potential is given by

119881 = 119890119870(119870119894119895119863119894119882119863119895119882 minus 3119882119882) (3)

To realize inflation in supergravity the factor 119890119870 in the aboveformula is an obstacle as it makes the potential too steep fora sufficient long slow-roll process This is the well-known 120578

problem To solve 120578 problem usually one needs a symmetryin the Kahler potential In the minimal supergravity there is

a global119880(1) symmetry in the Kahler potential119870 = ΦΦThisglobal 119880(1) symmetry is employed in helical phase inflationAs the Kahler potential is independent of the phase 120579 thepotential of phase 120579 is not affected by the exponential factor119890119870 Consequently there is no 120578 problem for phase inflationHowever the field stabilization becomes more subtle All theextra fields except inflaton have to be stabilized for single fieldinflation but normally the phase and norm of a complex fieldcouple with each other and then it is very difficult to stabilizenorm while keeping phase light

The physical picture of helical phase inflation is thatthe phase evolves along a flat circular path with constantor almost constant radiusmdashthe field magnitude and thepotential decreases slowly So even before writing down theexplicit supergravity formula one can deduce that phaseinflation if realizable should be particular realization ofcomplex phase monodromy and there exists a singularity inthe superpotential that generates the phasemonodromy Suchsingularity further indicates that themodel is described by aneffective theory

Helical phase inflation is realized in the minimal super-gravity with the Kahler potential

119870 = ΦΦ + 119883119883 minus 119892 (119883119883)2 (4)

and superpotential

119882 = 119886119883

Φln (Φ) (5)

The global119880(1) symmetry in119870 is broken by the superpoten-tial with a small factor 119886 when 119886 rarr 0 119880(1) symmetry isrestored Therefore the superpotential with small coefficientis technically natural [47] whichmakes themodel technicallystable against radiative corrections As discussed before thesuperpotential 119882 is singular at Φ = 0 and exhibits a phasemonodromy

Φ 997888rarr Φ1198902120587119894

119882 997888rarr 119882 + 2120587119886119894119883

Φ

(6)

The theory is well defined only forΦ away from the singular-ity

During inflation the field 119883 is stabilized at 119883 = 0 andthe scalar potential is simplified as

119881 = 119890ΦΦ

119882119883119882119883 = 11988621198901199032 1

1199032((ln 119903)

2+ 1205792) (7)

where Φ = 119903119890119894120579 and the kinetic term is 119871119870 = 120597120583119903120597

120583119903 +

1199032120597120583120579120597120583120579 Interestingly in the potential (7) both the norm-

dependent factor 1198901199032

(11199032) and (ln 119903)

2 reach the minimum at119903 = 1 The physical mass of norm 119903 is

1198982119903 =

1

2

1205972119881

1205971199032

100381610038161003816100381610038161003816100381610038161003816119903=1

= (2 +1

1205792)119881119868 (8)

4 Advances in High Energy Physics

therefore the norm is strongly stabilized at 119903 = 1 during infla-tion and the Lagrangian for the inflaton is

119871 = 120597120583120579120597120583120579 minus 119890119886

21205792 (9)

which gives quadratic inflation driven by the phase ofcomplex field Φ

In the above simple example given by (4) and (5)the field stabilization is obtained from the combination ofsupergravity correction 119890

119870 and the pole 1Φ in119882 besides anaccidental agreement that both the factor (1119903

2)1198901199032 and the

term (ln 119903)2 obtain their minima at 119903 = 1 while for more

general helical phase inflation such accidental agreementis not guaranteed For example one may get the followinginflaton potential

119881 = 119890119870(119903) 1

1199032((ln 119903

Λ)

2

+ 1205792) (10)

in which the coefficient 119890119870(119903)

(11199032) admits a minimum at

1199030 sim Λ but 1199030 = Λ In this case the coefficient 119890119870(119903)(11199032)still gives a mass above the Hubble scale for 119903 while ⟨119903⟩ isslightly shifted away from 1199030 in the early stage of inflation andafter inflation 119903 evolves toΛ rapidly Also the term (ln(119903Λ))

2

gives a small correction to the potential and inflationaryobservables so this correction is ignorable comparing withthe contributions from the super-Planckian valued phaseunless it is unexpectedly large

Potential Deformations In the Kahler potential there are cor-rections from the quantum loop effect while the superpoten-tial119882 is nonrenormalized Besides when coupledwith heavyfields the Kahler potential ofΦ receives corrections throughintegrating out the heavy fields Nevertheless because ofthe global 119880(1) symmetry in the Kahler potential thesecorrections can only affect the field stabilization while phaseinflation is not sensitive to these corrections

Given a higher order correction on the Kahler potential

119870 = ΦΦ + 119887 (ΦΦ)2+ 119883119883 minus 119892 (119883119883)

2 (11)

onemay introduce an extra parameterΛ in the superpotential

119882 = 119886119883

Φln Φ

Λ (12)

Based on the same argument it is easy to see the scalarpotential reduces to

119881 = 11988621198901199032+1198871199034 1

1199032((ln 119903 minus lnΛ)

2+ 1205792) (13)

The factor 1198901199032+1198871199034

(11199032) reaches its minimum at 11990320 = 2(1 +

radic1 + 8119887) below 119872119875 for 119887 gt 0 To get the ldquoaccidentalagreementrdquo it needs the parameter Λ = 1199030 and then inflationis still driven by the phase with exact quadratic potential

Without Λ the superpotential comes back to (5) and thescalar potential is shown in Figure 1 with 119887 = 01 Duringinflation ⟨119903⟩ ≃ 1199030 for small 119887 the term (ln 119903)

2 contribution to

1

1

0

0

minus1

minus1

V(102a2)

15

10

5

0

120601x

120601y

Figure 1 The helicoid structure of potential (13) scaled by 1021198862

In the graph parameters 119887 and Λ are set to be 119887 = 01 and Λ = 1respectively

the potential at the lowest order is proportional to 1198872 After

canonical field normalization the inflaton potential takes theform

119881 (120579) =1

21198982120579 (21198872+ 1205792) (14)

in which the higher order terms proportional to 1198873+119894 are

ignored With regard to the inflationary observations takethe tensor-to-scalar ratio r for example as

r =321205792119894

(1205792119894 + 21198872)

2asymp

8

119873(1 +

1198872

2119873)

minus2

(15)

where 120579119894 is the phase when inflation starts and 119873 isin (50 60)

is the 119890-folding number So the correction from higher orderterm is insignificant for 119887 lt 1

3 Monodromy in SupersymmetricField Theory

As discussed before phase inflation naturally leads to thephase monodromy (in mathematical sense) in the superpo-tential The phase monodromy requires singularity whichmeans the superpotential proposed for phase inflation shouldbe an effective theory It is preferred to show how such phasemonodromy appears from a more ldquofundamentalrdquo theory athigher scale In [20] the monodromy needed for phaseinflation is realized based on the supersymmetric field theoryin which the monodromy relates to the soft breaking of aglobal 119880(1) symmetry

Historically the monodromy inflation as an attractivemethod to realize super-Planckian field excursion was firstproposed in a more physical sense for axions arising from

Advances in High Energy Physics 5

string compactifications [42] In the inflaton potential theonly factor that changes during axion circular rotation isfrom the DBI action of branes In [44ndash46] the generalizedaxion alignment mechanisms are considered as particularrealization of axion monodromy with the potential fromnonperturbative effects We will show that such kind ofaxion monodromy can also be fulfilled by the superpotentialphase monodromy [20] even though it is not shown in theeffective superpotential after integrating out the heavy fieldsFurthermore all the extra fields can be consistently stabilized

The more ldquofundamentalrdquo field theory for the superpoten-tial in (5) is

1198820 = 120590119883Ψ (119879 minus 120575) + 119884 (119890minus120572119879

minus 120573Ψ) + 119885 (ΨΦ minus 120582) (16)

where the coupling constants for the second and third termsare taken to be 1 for simplicity and a small hierarchy isassumed between the first term and the last two terms thatis 120590 ≪ 1 The coupling 119884119890

minus120572119879 is assumed to be an effectivedescription of certain nonperturbative effects Similar formscan be obtained from 119863-brane instanton effect in type stringtheory (for a review see [48]) besides the coefficient 120572 prop

1119891 ≫ 1 in Planck unit since 119891 ≪ 1 is the decay constantand should be significantly lower than the Planck scale Forthe last two terms of1198820 there is a global 119880(1) symmetry

Ψ 997888rarr Ψ119890minus119894119902120579

Φ 997888rarr Φ119890119894119902120579

119884 997888rarr 119884119890119894119902120579

119879 997888rarr 119879 +119894119902120579

120572

(17)

which is anomalous and explicitly broken by the first termPhase rotation of the stabilizer field 119883 has no effect oninflation and so is ignored here The phase monodromy ofsuperpotential 119882 in (6) originates from the 119880(1) rotation of1198820

Ψ 997888rarr Ψ119890minus1198942120587

1198820 997888rarr 1198820 + 11989421205871205901

120572119883Ψ

(18)

As shown in [20] the supersymmetric field theory withsuperpotential1198820 admits the Minkowski vacuum at

⟨119883⟩ = ⟨119884⟩ = ⟨119885⟩ = 0

⟨119879⟩ = 120575

⟨Ψ⟩ =1

120573119890minus120572120575

⟨Φ⟩ = 120582120573119890120572120575

(19)

with ⟨Φ⟩ ≫ ⟨Ψ⟩ so that near the vacuum the masses of 119884119885 and Ψ are much larger than Φ besides the effective massof 119879 is also large near the vacuum due to large 120572 Large 120572

was an obstacle for natural inflation as it leads to the axiondecays too small for inflation while in this scenario large 120572

is helpful for phase inflation to stabilize the axion Thereforefor the physical process at scale below the mass scale of threeheavy fields the only unfixed degrees of freedom are 119883 andΦ which can be described by an effective field theory withthree heavy fields integrated out The coupling 120590119883Ψ(119879 minus 120575)

is designed for inflation and hierarchically smaller than theextra terms in1198820Therefore to describe inflation process theheavy fields need to be integrated out

To integrate out heavy fields we need to consider the 119865-terms again The 119865-term flatness of fields 119884 and 119885 gives

119865119884 = 119890minus120572119879

minus 120573Ψ + 1198701198841198820 = 0

119865119885 = ΨΦ minus 120582 + 1198701198851198820 = 0

(20)

Near the vacuum 119884 = 119885 asymp 0 ≪ 119872119875 the abovesupergravity corrections 119870119884(119885)1198820 are ignorable and thenthe 119865-term flatness conditions reduce to these for globalsupersymmetryThis is gained from the fact that although theinflation dynamics are subtle the inflation energy density isclose to the GUT scale far below the Planck scale Solvingthe 119865-term flatness equations in (20) we obtain the effectivesuperpotential119882 in (5)

Based on the above construction it is clear that the phasemonodromy in 119882 is from the 119880(1) transformation of 1198820and the pole of superpotential (5) at Φ = 0 arises from theintegration process The heavy fieldΨ is integrated out basedon the 119865-term flatness conditions when ⟨Φ⟩ ≫ ⟨Ψ⟩ whileif Φ rarr 0 Ψ becomes massless from |119865119885|

2 and it is illegalto integrate out a ldquomasslessrdquo field For inflation the condition⟨Φ⟩ ≫ ⟨Ψ⟩ is satisfied so the theory with superpotential (5)is reliable

As to the inflation term a question appears as global119880(1) is explicitly broken by the first term in (16) at inflationscale why is the phase light while the norm is much heavierThe supergravity correction to the scalar potential plays acrucial role at this stage The coefficient 119890119870 appears in thescalar potential and because of119880(1) symmetry in the Kahlerpotential the factor 119890119870 is invariant under119880(1) symmetry butincreases exponentially for a large norm Here the Kahlerpotential of 119879 should be shift invariant that is 119870 = 119870(119879 +

119879) instead of the minimal type Otherwise the exponentialfactor 119890119870 depends on phase as well and the phase rotationwillbe strongly fixed like the norm component or Re(119879)

When integrating out the heavy fields they should bereplaced both in superpotential and in Kahler potential bythe solutions from vanishing 119865-term equations So differentfrom the superpotential the Kahler potential obtained in thisway is slightly different from the minimal case given in (4)There are extra terms like

ΨΨ =1205822

1199032

119870 (119879 + 119879) = 119870(1

1205722(ln 119903)2)

(21)

where |Φ| = 119903 Nevertheless since 120582 ≪ 1 and 120572 ≫ 1 theseterms are rather small and have little effect on phase inflation

6 Advances in High Energy Physics

as shown in the last section Furthermore the quantumloop effects during integrating out heavy fields can introducecorrections to the Kahler potential as well However becauseof 119880(1) symmetry built in the Kahler potential these termstogether with (21) can only mildly affect the field stabiliza-tion and phase inflation is not sensitive to the correctionsin Kahler potential As to the superpotential it is protectedby the nonrenormalized theorem and free from radiativecorrections

4 Natural Inflation in Helical Phase Inflation

In the superpotential 1198820 the inflation term is perturbativecoupling of complex field119879which shifts under global119880(1) aninterestingmodification is to consider inflation given by non-perturbative coupling of 119879 Such term gives a modified 119880(1)

phase monodromy in the superpotential And it leads to nat-ural inflation as a special type of helical phase inflation withphase-axion alignment which is similar to the axion-axionalignment mechanism proposed in [34] for natural inflationwith super-Planckian axion decay constant Specifically itcan be shown that the phase monodromy realized in super-symmetric field theory has similar physical picture to themodified axion alignment mechanism provided in [44 45]

To realize natural inflation the superpotential 1198820 in (16)just needs to be slightly modified

1198821 = 120590119883Ψ(119890minus120572119879

minus 120575) + 119884 (119890minus120573119879

minus 120583Ψ)

+ 119885 (ΨΦ minus 120582)

(22)

in which 1 ≪ 120572 ≪ 120573 Again there is a global 119880(1)

symmetry in the last two terms of 1198821 and the fields transferunder119880(1) like in (17) The first term which is hierarchicallysmaller breaks the 119880(1) symmetry explicitly besides ashift symmetry of 119879 is needed in the Kahler potential Themonodromy of1198821 under a circular 119880(1) rotation is

Ψ 997888rarr Ψ119890minus1198942120587

1198821 997888rarr 1198821 + 120590119883Ψ119890minus120572119879

(119890minus2120587119894(120572120573)

minus 1)

(23)

The supersymmetric field theory given by (22) admits thefollowing supersymmetric Minkowski vacuum

⟨119883⟩ = ⟨119884⟩ = ⟨119885⟩ = 0

⟨119879⟩ = minus1

120572ln 120575

⟨Ψ⟩ =1

120583120575120573120572

⟨Φ⟩ = 120582120583120575minus120573120572

(24)

The parameters are set to satisfy conditions

120575120573120572

sim 120582120583

120582 ≪ 1

120575 lt 1

(25)

so that ⟨Ψ⟩ ≪ ⟨Φ⟩ and ⟨119879⟩ gt 0

Near vacuum fields 119884119885Ψ and 119879 obtain large effectivemasses above inflation scale while 119883Φ are much lighter Atthe inflation scale the heavy fields should be integrated outThe 119865-term flatness conditions for fields 119884 and 119885 are

119865119884 = 119890minus120573119879

minus 120583Ψ = 0

119865119885 = ΨΦ minus 120582 = 0

(26)

in which the supergravity corrections 1198701198841198851198821 are neglectedas both119884 and119885 get close to zero during inflation IntegratingoutΨ and 119879 from (26) we obtain the effective superpotential1198821015840 from1198821

1198821015840= 120590120582

119883

Φ((

120583120582

Φ)

120572120573

minus 120575) (27)

Given 120572 ≪ 120573 the effective superpotential contains a termwith fractional power Inflation driven by complex potentialwith fractional power was considered in [49] to get sufficientlarge axion decay constant Here the supersymmetric fieldmonodromy naturally leads to the superpotential with frac-tional power which arises from the small hierarchy of axiondecay constants in two nonperturbative terms

Helical phase inflation is described by the effective super-potential 1198821015840 the role of phase-axion alignment is not clearfrom119882

1015840 since it is hidden in the procedure of integrating outheavy fields

The 119865-term flatness conditions (26) fix four degrees offreedom for the extra degree of freedom they correspond tothe transformations free from constraints (26)

Ψ 997888rarr Ψ119890minus119906minus119894V

Φ 997888rarr Φ119890119906+119894V

119879 997888rarr 119879 +119906

120573+

119894V120573

(28)

Parameter 119906 corresponds to the norm variation of complexfield Φ which is fixed by the supergravity correction on thescalar potential 119890119870(ΦΦ) Parameter V relates to 119880(1) transfor-mation which leads to the phase monodromy from the firstterm of1198821The scalar potential including the inflation termdepends on the superpositions among phases ofΨΦ and theaxion Im(119879) which are constrained as in (28) Among thesefields the phase ofΦ has the lightestmass after canonical fieldnormalization Similar physical picture appears in the axion-axion alignment where inflation is triggered by the axionsuperposition along the flat direction

After integrating out the heavy fields the Kahler potentialis

119870 = ΦΦ +1205822

ΦΦ+ sdot sdot sdot (29)

As 120582 ≪ 1 the Kahler potential is dominated by ΦΦ asdiscussed before the extra terms like 1205822ΦΦ only give small

Advances in High Energy Physics 7

corrections to the field stabilization Helical phase inflationcan be simply described by the supergravity

119870 = ΦΦ + 119883119883 +

119882 = 119886119883

Φ(Φminus119887

minus 119888)

(30)

where 119887 = 120572120573 ≪ 1 and 119888 asymp 1 The scalar potential given bythe above Kahler potential and superpotential is

119881 = 1198901199032 1198862

1199032(119903minus2119887

+ 1198882minus 2119888119903minus119887 cos (119887120579)) (31)

in which we have usedΦ equiv 119903119890119894120579 As usual the norm 119903 couples

with the phase in the scalar potential To generate singlefield inflation the norm component needs to be stabilizedwith heavy mass above the Hubble scale while the phasecomponent remains light Due to the phase-norm couplingthe phase and norm are likely to obtain comparable effectivemasses during inflation so it is difficult to realize singlefield inflation However it is a bit different in helical phaseinflation The above scalar potential can be rewritten asfollows

119881 = 1198901199032 1198862

1199032(119903minus119887

minus 119888)2+ 1198901199032 41198862119888

1199032+119887(sin 119887

2120579)

2

(32)

So its vacuum locates at ⟨119903⟩ = 1199030 = 119888minus1119887 and 120579 = 0 Besides

the coefficient of the phase term 1198901199032(41198881199032+119887

) reaches itsminimal value at 1199031 = radic1 + 1198872 For 119888 asymp 1 we have theapproximation 1199030 asymp 1199031 asymp 1 The extra terms in 119870 givesmall corrections to 1199030 and 1199031 but the approximation is stillvalid So with the parameters in (25) the norm 119903 in thetwo terms of scalar potential 119881 can be stabilized at the closeregion 119903 asymp 1 separately If the parameters are tuned so that1199030 = 1199031 then the norm of complex field is strictly stabilizedat the vacuum value during inflation Without such tuninga small difference between 1199030 and 1199031 is expected but theshift of 119903 during inflation is rather small and inflation is stillapproximate to the single field inflation driven by the phaseterm prop (sin(1198872)120579)2 The helicoid structure of the potential(31) is shown in Figure 2

It is known that to realize aligned axion mechanism insupergravity [50] it is very difficult to stabilize the moduli asthey couple with the axions In [40] the moduli stabilizationis fulfilled with gauged anomalous 119880(1) symmetries since119880(1) 119863-terms only depend on the norm |Φ| or Re(119879) andthen directly separate the norms and phases of matter fieldsIn helical phase inflation the modulus and matter fieldsexcept the phase are stabilized at higher scale or by thesupergravity scalar potential Only the phase can be aninflaton candidate because of the protection from the global119880(1) symmetry in Kahler potential and approximate 119880(1)

symmetry in superpotential

5 Helical Phase Inflation and the KYY Model

120578 problem in supergravity inflation can be solved both inhelical phase inflation and in the shift symmetry in KYY

1

1

00

minus1

minus1

10

8

6

4

V(a

2)

120601x120601y

Figure 2 The helicoid structure of potential (31) scaled by 1198862 The

parameters with 119888 = 096 and 119887 = 01 are adopted in the graph Itis shown that the local valley locates around 119903 asymp 1 Note that thepotential gets flatter at the top of the graph

model The physics in these two solutions are obviously dif-ferent For helical phase inflation the solution employs 119880(1)

symmetry in Kahler potential of the minimal supergravityand the superpotential admits a phase monodromy arisingfrom the global 119880(1) transformation

119870 = ΦΦ + 119883119883 + sdot sdot sdot

119882 = 119886119883

ΦlnΦ

(33)

The inflation is driven by the phase of complex field Φ

and several special virtues appear in the model For theKYY model with shift symmetry [6] the Kahler potential isadjusted so that it admits a shift symmetry along the directionof Im(119879)

119870 =1

2(119879 + 119879)

2+ 119883119883 + sdot sdot sdot

119882 = 119886119883119879

(34)

and inflation is driven by Im(119879) The shift symmetry isendowedwith axions so thismechanism is attractive for axioninflation

Here we will show that although the physical pictures aremuch different in helical phase inflation andKYY typemodeljust considering the lower order terms in the Kahler potentialof redefined complex field helical phase inflation can reduceto the KYY model

Because the phase of Φ rotates under the global 119880(1)

transformation to connect helical phase inflation with theKYY model a natural guess is to take the following fieldredefinitionΦ = 119890

119879 then helical phase inflation (33) becomes

119870 = 119890119879+119879

+ sdot sdot sdot = 1 + 119879 + 119879 +1

2(119879 + 119879)

2+ sdot sdot sdot

119882 = 119886119883119879119890minus119879

(35)

8 Advances in High Energy Physics

TheKahler manifold of complex field119879 is invariant under theholomorphic Kahler transformation

119870(119879 119879) 997888rarr 119870(119879 119879) + 119865 (119879) + 119865 (119879) (36)

in which 119865(119879) is a holomorphic function of 119879 To keepthe whole Lagrangian also invariant under the Kahler trans-formation the superpotential transforms under the Kahlertransformation

119882 997888rarr 119890minus119865(119879)

119882 (37)

For the supergravity model in (35) taking the Kahler trans-formation with 119865(119879) = minus(12) minus 119879 the Kahler potential andsuperpotential become

119870 =1

2(119879 + 119879)

2+ sdot sdot sdot

119882 = 119886radic119890119883119879

(38)

which is just the KYY model (34) with higher order cor-rections in the Kahler potential These higher order termsvanish after field stabilization and have no effect on inflationprocess The field relation Φ = 119890

119879 also gives a map betweenthe simplest helical phase inflation and the KYY model suchas the inflaton arg(Φ) rarr Im(119879) and field stabilization|Φ| = 1 rarr Re(119879) = 0

Nevertheless helical phase inflation is not equivalent tothe KYY model There are higher order corrections to theKahler potential in the map from helical phase inflation toKYY model which have no effect on inflation after fieldstabilization but indicate different physics in two models Bydropping these terms certain information is lost so the mapis irreversible Specifically the inverse function 119879 = lnΦ ofthe field redefinition Φ = 119890

119879 cannot reproduce helical phaseinflation from the KYY model unless one introduces higherdimensional operators in the Kahler potential exactly thesame as given in the expansion (35) Such strict constraint oneach higher dimensional operator is not guaranteed by anyknown rule and so is unphysical As there is no pole or singu-larity in the KYY model it is unlikely to introduce pole andsingularity at origin with phase monodromy through well-defined field redefinition Actually the singularity with phasemonodromy in the superpotential indicates rich physics inthe scale above inflation

6 Helical Phase Inflation inNo-Scale Supergravity

The no-scale supergravity is an attractive frame for GUTscale phenomenology it is interesting to realize helicalphase inflation in no-scale supergravity Generally theKahler manifold of the no-scale supergravity is equippedwith 119878119880(119873 1)119878119880(119873) times 119880(1) symmetry For the no-scalesupergravity with exact 119878119880(1 1)119880(1) symmetry withoutextra fields no inflation can be realized so the case with119878119880(2 1)119878119880(2) times 119880(1) symmetry is the simplest one thatadmits inflation

The Kahler potential with 119878119880(2 1)119878119880(2)times119880(1) symme-try is

119870 = minus3 ln(119879 + 119879 minusΦΦ

3) (39)

In the symmetry of the Kahler manifold there is 119880(1)

subsector the phase rotation of complex fieldΦ which can beemployed for helical phase inflation Besides the modulus 119879should be stabilized during inflation which can be fulfilled byintroducing extra terms on119879 As a simple example of no-scalehelical phase inflation here we follow the simplification in [9]that the modulus 119879 has already been stabilized at ⟨119879⟩ = 119888Different from the minimal supergravity the kinetic termgiven by the no-scale Kahler potential is noncanonical

119871119870 = 119870ΦΦ120597120583Φ120597120583Φ

=2119888

(2119888 minus 11990323)2(120597120583119903120597

120583119903 + 1199032120597120583120579120597120583120579)

(40)

in which Φ equiv 119903119890119894120579 is used The 119865-term scalar potential is

119881 = 119890minus(23)119870 1003816100381610038161003816119882Φ

1003816100381610038161003816

2=

1003816100381610038161003816119882Φ1003816100381610038161003816

2

(119879 + 119879 minus ΦΦ3)2 (41)

where the superpotential 119882 is a holomorphic function ofsuperfieldΦ and119879 = ⟨119879⟩ = 119888 It requires a phasemonodromyin superpotential119882 for phase inflation the simple choice is

119882 =119886

Φln Φ

Λ (42)

The scalar potential given by this superpotential is

119881 =91198862

(6119888 minus 1199032)21199034

((ln 119903

119890Λ)

2

+ 1205792) (43)

As in the minimal supergravity the norm and phase ofcomplex field Φ are separated in the scalar potential For the119903-dependent coefficient factor 1(6119888 minus 119903

2)21199034 its minimum

locates at 1199030 = radic3119888 and another term (ln(119903119890Λ))2 reaches its

minimum at 1199031 = 119890Λ Given that the parameter Λ is tuned sothat 1199030 = 1199031 in the radial direction the potential has a globalminimum at 119903 = 1199030 Similar to helical phase inflation in theminimal supergravity the potential (43) also shows helicoidstructure as presented in Figure 3

The physical mass of 119903 in the region near the vacuum is

1198982119903 =

(2119888 minus 11990323)2

4119888

1205972119881

1205971199032

100381610038161003816100381610038161003816100381610038161003816100381610038161003816119903=radic3119888

= (4 +1

21205792)1198672 (44)

where119867 is theHubble constant during inflation So the norm119903 is strongly stabilized at 1199030 If 1199030 and 1199031 are not equal butclose to each other 119903 will slightly shift during inflation butits mass remains above the Hubble scale and inflation is stillapproximately the single field inflation

Advances in High Energy Physics 9

30

20

10

0

minus10

minus10

minus05

minus05

0000

05

05

10

10

120601x (radic

3c) 120601 y(radic3

c)

V(a

2 c4 )

Figure 3The helicoid structure of potential (43) scaled by 11988621198884 Inthe graph the parameter Λ is tuned so that 1199030 = 1199031 The complexfieldΦ has been rescaled byradic3119888 and the scale of field norm at localvalley is determined by the parameter 119888 instead of the Planck mass

Instead of stabilizing 119879 independently with the inflationprocess we can realize helical phase inflation from the no-scale supergravity with dynamical119879There is a natural reasonfor such consideration The phase monodromy requiresmatter fields transforming as rotations under 119880(1) and alsoa modulus 119879 as a shift under 119880(1) The Kahler potential of 119879is shift invariant Interestingly for the no-scale supergravitywith 119878119880(2 1)119878119880(2) times 119880(1) symmetry as shown in (39) theKahler potential 119870 is automatically endowed with the shiftsymmetry and global 119880(1) symmetry

119879 997888rarr 119879 + 119894119862

Φ 997888rarr Φ119890119894120579

(45)

Therefore the no-scale Kahler potential fits with the phasemonodromy in (16) and (22) initiatively

The Kahler potentials of superfields 119911 isin 119883 119884 119885 areof the minimal type 119911119911 while for Ψ its Kahler potentialcan be the minimal type ΨΨ or the no-scale type 119870 =

minus3 ln(119879+119879minus (ΦΦ+ΨΨ)3) which extends the symmetry ofKahler manifold to 119878119880(3 1)119878119880(3) times 119880(1) In this scenariothe process to integrate out heavy fields is the same asbefore Besides the potential of phase proportional to |119882119883|

2

is insensitive to the formula of Kahler potential due to119880(1) symmetry The major difference appears in the fieldstabilization The scalar potential from phase monodromy in(16) is

119881 = 119890119870 1003816100381610038161003816119882119883

1003816100381610038161003816

2=

1

(ln 1199032120572 + 119888 minus 11990323 minus 120582211990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792) asymp

1

(119888 minus 11990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792)

(46)

with 120572 ≫ 1 and 120582 ≪ 1 The field norm 119903 is stabilized byminimizing the coefficient 11199032(119888 minus 119903

23)3 and the parameter

119888 from ⟨119879⟩ determines the scale of ⟨119903⟩ = radic31198882 This isdifferent from helical phase inflation in the minimal super-gravity which minimizes the norm based on the coefficient1198901199032(11199032) and then the scale of ⟨119903⟩ is close to119872119875 the unique

energy scale of supergravity corrections Similarly combiningthe no-scale Kahlermanifold with phasemonodromy in (22)we can obtain natural inflation

7 UV-Sensitivity and Helical Phase Inflation

For the slow-roll inflation the Lyth bound [21] provides arelationship between the tensor-to-scalar ratio r and the fieldexcursion Δ120601 Roughly it requires

Δ120601

119872119875

⩾ (r

001)

12

(47)

To get large tensor-to-scalar ratio r ⩾ 001 such as in chaoticinflation or natural inflation the field excursion should bemuch larger than the Planck mass The super-Planckianfield excursion makes the description based on the effectivefield theory questionable In the Wilsonian sense the lowenergy field theory is an effective theory with higher ordercorrections introduced by the physics at the cut-off scalelike quantum gravity and these terms are irrelevant in theeffective field theory since they are suppressed by the cut-offenergy scale However for the inflation process the inflatonhas super-Planckian field excursion which is much largerthan the cut-off scaleThus the higher order terms cannot besuppressed by the Planckmass andmay affect the inflationaryobservations significantly And then they may significantlyaffect inflation or even destroy the inflation process Forexample considering the corrections

Δ119881 = 119888119894119881(120601

119872119875

)

119894

+ sdot sdot sdot (48)

to the original inflaton potential 119881 as long as 119888119894 are of theorder 10minus119894 in the initial stage of inflation 120601 sim 119874(10)119872119875 thehigher order terms can be as large as the original potential119881 So for large field inflation it is sensitive to the physics atthe cut-off scale and the predictions of inflation just based oneffective field theory are questionable

In consideration of the UV sensitivity of large fieldinflation a possible choice is to realize inflation in UV-completed theory like string theory (for a review see [22])To realize inflation in string theory there are a lot of problemsto solve besides inflation such as the moduli stabilizationMinkowskide Sitter vacua and effects of 1205721015840- and string loop-corrections Alternatively in the bottom-up approach onemay avoid the higher order corrections by introducing anextra shift symmetry in the theory The shift symmetry istechnically natural and safe under quantum loop-correctionsHowever the global symmetry can be broken by the quantumgravity effect So it is still questionable whether the shiftsymmetry can safely evade the higher corrections like in (48)

10 Advances in High Energy Physics

The UV-completion problem is dodged in helical phaseinflation Since the super-Planckian field excursion is thephase of a complex field and the phase component is notdirectly involved in the gravity interaction there are nodangerous high-order corrections like in (48) for the phasepotentialThe inflaton evolves along the helical trajectory anddoes not relate to the physics in the region above the Planckscale For helical phase inflation in the minimal supergravitythe norm of field is stabilized at the marginal point of thePlanck scale where the supergravity correction on the scalarpotential gets important based onwhich the normof complexfield can be strongly stabilized The extra corrections arelikely to appear in the Kahler potential however they canonly slightly affect the field stabilization while the phaseinflation is protected by the global 119880(1) symmetry in theKahler potential in consequence helical phase inflation isnot sensitive to these corrections at all For the helical phaseinflation in no-scale supergravity the normof complex field isstabilized at the scale of themodulus ⟨119879⟩ instead of the Planckscale one can simply adjust the scale of ⟨119879⟩ to keep themodelaway from super-Planckian region

Helical phase inflation is free from the UV-sensitivityproblem and it is just a typical physical process at theGUT scale with special superpotential that admits phasemonodromy So it provides an inflationary model that can bereliably studied just in supersymmetric field theory

8 Discussions and Conclusion

In this work we have studied the details of helical phase infla-tion from several aspects Helical phase inflation is realizedin supergravity with global 119880(1) symmetry 119880(1) symmetryis built in the Kahler potential so that helical phase inflationcan be realized by the ordinary Kahler potentials such as theminimal or no-scale types Helical phase inflation directlyleads to the phasemonodromy in the superpotential which issingular and an effective field theory arising from integratingout heavy fieldsThe phasemonodromy originates from119880(1)

rotation of the superpotential at higher scale Generically thesuperpotential can be separated into two parts119882 = 119882119868 +119882119878where119882119878 admits the global119880(1) symmetrywhile119882119868 breaks itexplicitly at scale much lower than 119882119878 Under 119880(1) rotationthe inflation term 119882119868 is slightly changed which realizes thephase monodromy in the effective theory and introducesa flat potential along the direction of phase rotation Bybreaking the global119880(1) symmetry in different ways we mayget different kinds of inflation such as quadratic inflationnatural inflation or the other types of inflation that are notpresented in this work

An amazing fact of helical phase inflation is that itdeeply relates to several interesting points of inflation andnaturally combines them in a rather simple potential withhelicoid structure The features of helical phase inflation canbe summarized as follows

(i) The global 119880(1) symmetry is built in the Kahlerpotential so helical phase inflation provides a naturalsolution to 120578 problem

(ii) The phase excursion requires phase monodromy inthe superpotential So helical phase inflation providesin the mathematical sense a new type of monodromyin supersymmetric field theory

(iii) The singularity in the superpotential together withthe supergravity scalar potential provides strong fieldstabilization which is consistent with phase inflation

(iv) The super-Planckian field excursion is realized bythe phase of a complex field instead of any otherldquophysicalrdquo fields that directly couple with gravity Sothere are no polynomial higher order corrections forthe phase and thus inflation is not sensitive to thequantum gravity corrections

To summarize helical phase inflation introduces a newtype of inflation that can be effectively described by super-symmetric field theory at the GUT scale Generically thesuper-Planckian field excursion makes the inflationary pre-dictions based on effective field theory questionable sincethe higher order corrections from quantum gravity arelikely to affect the inflation process significantly One of thesolutions is to realize inflation in a UV-completed theory likestring theory nevertheless there are many difficult issues instring theory to resolve before realizing inflation completelyHelical phase inflation is another simple solution to the UV-sensitivity problem It is based on the supersymmetric fieldtheory and the physics are clear and much easier to controlFurthermore helical phase inflation makes the unification ofinflation theory with GUT more natural since both of themare triggered at the scale of 1016 GeV and can be effectivelystudied based on supersymmetric field theory

Besides we have shown that helical phase inflation alsorelates to several interesting developments in inflation theoryIt can be easily modified for natural inflation and realizethe phase-axion alignment indirectly which is similar tothe axion-axion alignment mechanism for super-Planckianaxion decay constant [34] The phase-axion alignment isnot shown in the final supergravity model which exhibitsexplicit phase monodromy only However the phase-axionalignment is hidden in the process when the heavy fields areintegrating out For 120578 problem in the supergravity inflationthere is another well-known solution the KYY model withshift symmetry We showed that through field redefinitionhelical phase inflation given by (4) and (5) reduces to theKYYmodel where the higher order corrections in the Kahlerpotential have no effect on inflation process However thereis no inverse transformation from the KYY model to helicalphase inflation since no well-defined field redefinition canintroduce the pole and phase singularity needed for phasemonodromy Helical phase inflation can be realized in no-scale supergravity The no-scale Kahler potential automati-cally provides the symmetry needed by phase monodromyIn the no-scale supergravity the norm of complex field isstabilized at the scale of the modulus

Our inflation models are constructed within the super-gravity theory with global 119880(1) symmetry broken explicitlyby the subleading order superpotential term So it is justtypical GUT scale physics and indicates that a UV-completed

Advances in High Energy Physics 11

framework seems to be not prerequisite to effectively describesuch inflation process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work of DVN was supported in part by the DOEGrant DE-FG03-95-ER-40917 The work of Tianjun Li issupported in part by theNatural Science Foundation of ChinaunderGrants nos 10821504 11075194 11135003 11275246 and11475238 and by the National Basic Research Program ofChina (973 Program) under Grant no 2010CB833000

References

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[2] A D Linde ldquoColeman-Weinberg theory and the new inflation-ary universe scenariordquo Physics Letters B vol 114 no 6 pp 431ndash435 1982

[3] A Albrecht and P J Steinhardt ldquoCosmology for grand unifiedtheories with radiatively induced symmetry breakingrdquo PhysicalReview Letters vol 48 no 17 pp 1220ndash1223 1982

[4] P A R Ade N Aghanim C Armitage-Caplan et al ldquoPlanck2013 results XXII Constraints on inflationrdquo Astronomy ampAstrophysics vol 571 article A22 42 pages 2014

[5] P A R Ade R W Aikin D Barkats et al ldquoDetection of B-mode polarization at degree angular scales by BICEP2rdquo PhysicalReview Letters vol 112 no 24 Article ID 241101 2014

[6] M Kawasaki M Yamaguchi and T Yanagida ldquoNatural chaoticinflation in supergravityrdquo Physical Review Letters vol 85 no 17pp 3572ndash3575 2000

[7] T Li Z Li and D V Nanopoulos ldquoSupergravity inflation withbroken shift symmetry and large tensor-to-scalar ratiordquo Journalof Cosmology and Astroparticle Physics vol 2014 no 2 article028 2014

[8] K Harigaya and T T Yanagida ldquoDiscovery of large scale tensormode and chaotic inflation in supergravityrdquo Physics Letters Bvol 734 pp 13ndash16 2014

[9] J Ellis D V Nanopoulos and K A Olive ldquoNo-scale supergrav-ity realization of the starobinsky model of inflationrdquo PhysicalReview Letters vol 111 no 11 Article ID 111301 2013

[10] T Li Z Li and D V Nanopoulos ldquoNo-scale ripple inflationrevisitedrdquo Journal of Cosmology and Astroparticle Physics vol04 article 018 2014

[11] C Pallis ldquoInduced-gravity inflation in no-scale supergravityand beyondrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 8 article 057 2014

[12] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky modelrdquo Fortschritte der Physik vol 62 no 7 pp 573ndash5832014

[13] J Ellis M A G Garca D V Nanopoulos and K A OliveldquoResurrecting quadratic inflation in no-scale supergravity inlight of BICEP2rdquo Journal of Cosmology andAstroparticle Physicsvol 2014 no 5 article 037 2014

[14] T Li Z Li and D V Nanopoulos ldquoChaotic inflation in no-scale supergravitywith string inspiredmoduli stabilizationrdquoTheEuropean Physical Journal C vol 75 no 2 article 55 2015

[15] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky model and higher curvature correctionsrdquo Fortschritte derPhysik vol 63 no 1 pp 2ndash11 2015

[16] S V Ketov andT Terada ldquoInflation in supergravity with a singlechiral superfieldrdquo Physics Letters B vol 736 pp 272ndash277 2014

[17] C Kounnas D Lust and N Toumbas ldquoR2 inflation fromscale invariant supergravity and anomaly free superstrings withfluxesrdquo Fortschritte der Physik vol 63 no 1 pp 12ndash35 2015

[18] G A Diamandis B C Georgalas K Kaskavelis PKouroumalou A B Lahanas and G Pavlopoulos ldquoInflationin R2 supergravity with non-minimal superpotentialsrdquo PhysicsLetters B vol 744 pp 74ndash81 2015

[19] C Pallis and Q Shafi ldquoGravity waves from non-minimalquadratic inflationrdquo Journal of Cosmology and AstroparticlePhysics vol 2015 no 3 article 023 2015

[20] T Li Z Li and D V Nanopoulos ldquoHelical phase inflationrdquoPhysical Review D vol 91 no 6 Article ID 061303 2015

[21] D H Lyth ldquoWhat would we learn by detecting a gravitationalwave signal in the cosmic microwave background anisotropyrdquoPhysical Review Letters vol 78 no 10 article 1861 1997

[22] D Baumann and L McAllister ldquoInflation and stringtheoryrdquohttparxivorgabs14042601

[23] F Marchesano G Shiu and A M Uranga ldquoF-term axionmonodromy inflationrdquo Journal of High Energy Physics vol 2014no 9 article 184 2014

[24] C Long L McAllister and P McGuirk ldquoAligned naturalinflation in string theoryrdquo Physical Review D vol 90 no 2Article ID 023501 2014

[25] X Gao T Li and P Shukla ldquoCombining universal and odd RRaxions for aligned natural inflationrdquo Journal of Cosmology andAstroparticle Physics vol 2014 no 10 article 048 2014

[26] I Ben-Dayan F G Pedro and A Westphal ldquoTowards naturalinflation in string theoryrdquo Physical Review D vol 92 no 2Article ID 023515 2015

[27] R Blumenhagen D Herschmann and E Plauschinn ldquoThechallenge of realizing F-term axion monodromy inflation instring theoryrdquo Journal of High Energy Physics vol 2015 article7 2015

[28] H Abe T Kobayashi and H Otsuka ldquoTowards natural infla-tion from weakly coupled heterotic string theoryrdquo Progress ofTheoretical and Experimental Physics Article ID 063E02 2015

[29] T Ali S S Haque and V Jejjala ldquoNatural inflation from nearalignment in heterotic string theoryrdquo Physical Review D vol 91no 8 Article ID 083516 9 pages 2015

[30] R Flauger LMcAllister E Silverstein and AWestphal ldquoDrift-ing oscillations in axionmonodromyrdquo httparxivorgabs14121814

[31] K Freese J A Frieman andAVOlinto ldquoNatural inflationwithpseudo nambu-goldstone bosonsrdquo Physical Review Letters vol65 no 26 pp 3233ndash3236 1990

[32] F C Adams J R Bond K Freese J A Frieman and AV Olinto ldquoNatural inflation particle physics models power-law spectra for large-scale structure and constraints from thecosmic background explorerrdquo Physical Review D vol 47 no 2pp 426ndash455 1993

[33] G German A Mazumdar and A Perez-Lorenzana ldquoAngularinflation from supergravityrdquo Modern Physics Letters A vol 17no 25 pp 1627ndash1634 2002

12 Advances in High Energy Physics

[34] J E Kim H P Nilles and M Peloso ldquoCompleting naturalinflationrdquo Journal of Cosmology and Astroparticle Physics vol0501 article 005 2005

[35] D Baumann and D Green ldquoInflating with baryonsrdquo Journal ofHigh Energy Physics vol 2011 no 4 article 071 2011

[36] JMcDonald ldquoSub-Planckian two-field inflation consistent withthe Lyth boundrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 9 article 027 2014

[37] J McDonald ldquoA minimal sub-Planckian axion inflation modelwith large tensor-to-scalar ratiordquo Journal of Cosmology andAstroparticle Physics vol 2015 no 1 article 018 2015

[38] C D Carone J Erlich A Sensharma and Z Wang ldquoTwo-fieldaxion-monodromy hybrid inflation model Dantersquos WaterfallrdquoPhysical ReviewD vol 91 no 4 Article ID 043512 8 pages 2015

[39] G Barenboim and W-I Park ldquoSpiral Inflationrdquo Physics LettersB vol 741 pp 252ndash255 2015

[40] T Li Z Li andDVNanopoulos ldquoAligned natural inflation andmoduli stabilization from anomalous U(1) gauge symmetriesrdquoJournal of High Energy Physics vol 2014 no 11 article 012 2014

[41] T Li Z Li and D V Nanopoulos ldquoNatural inflation withnatural trans-planckian axion decay constant from anomalousU(1)119883rdquo Journal of High Energy Physics vol 2014 no 7 article052 2014

[42] E Silverstein and A Westphal ldquoMonodromy in the CMBgravity waves and string inflationrdquo Physical Review D vol 78no 10 Article ID 106003 21 pages 2008

[43] L McAllister E Silverstein and A Westphal ldquoGravity wavesand linear inflation from axion monodromyrdquo Physical ReviewD vol 82 no 4 Article ID 046003 2010

[44] K Choi H Kim and S Yun ldquoNatural inflation with multiplesub-Planckian axionsrdquo Physical Review D vol 90 no 2 ArticleID 023545 2014

[45] S-H H Tye and S S C Wong ldquoHelical inflation and cosmic-stringsrdquo httparxivorgabs14046988

[46] R Kappl S Krippendorf and H P Nilles ldquoAligned naturalinflation monodromies of two axionsrdquo Physics Letters B vol737 pp 124ndash128 2014

[47] G rsquotHooft ldquoNaturalness chiral symmetry and spontaneouschiral symmetry breakingrdquo in Recent Developments in GaugeTheories G rsquotHooft C Itzykson A Jaffe et al Eds vol 59of NATO Advanced Study Institutes Series pp 135ndash157 PlenumPress New York NY USA 1980

[48] R Blumenhagen M Cvetic S Kachru and T Weigand ldquoD-brane instantons in type II string theoryrdquo Annual Review ofNuclear and Particle Science vol 59 pp 269ndash296 2009

[49] K Harigaya andM Ibe ldquoSimple realization of inflaton potentialon a Riemann surfacerdquo Physics Letters B vol 738 pp 301ndash3042014

[50] R Kallosh A Linde and B Vercnocke ldquoNatural inflation insupergravity and beyondrdquo Physical Review D vol 90 Article ID041303(R) 2014

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2 Advances in High Energy Physics

is determined by three functions Kahler potential 119870 super-potential 119882 and gauge kinetic function The 119865-term scalarpotential contains an exponential factor 119890119870 In the minimalsupergravity with 119870 = ΦΦ the exponential factor 119890

119870

introduces a term on the inflaton mass at Hubble scalewhich breaks the slow-roll condition (1) To realize inflationin supergravity the large contribution from 119890

119870 to scalarmass should be suppressed which needs a symmetry in119870 In the minimal supergravity 120578 problem can be solvedby introducing shift symmetry in the Kahler potential asproposed by Kawasaki Yamaguchi and Yanagida (KYY) [6]119870 is invariant under the shift Φ rarr Φ + 119894119862 Consequently119870 is independent of Im(Φ) so is the factor 119890

119870 in the 119865-term potential By employing Im(Φ) as inflaton its mass isnot affected by 119890

119870 and then there is no 120578 problem anymoreThe shift symmetry can be slightly broken in this case thereis still no 120578 problem and the model gives a broad range oftensor-to-scalar ratios r [7 8] 120578 problem is automaticallysolved in no-scale supergravity because of 119878119880(119873 1)119878119880(119873)times

119880(1) symmetry in the Kahler potential Historically the no-scale supergravity was proposed to get vanishing cosmol-ogy constant At classical level the potential is strictly flatguaranteed by 119878119880(119873 1)119878119880(119873) times 119880(1) symmetry of theKahler potential whichmeanwhile protects the no-scale typeinflation away from 120578 problemMoreover 119878119880(119873 1)119878119880(119873)times

119880(1) symmetry has rich structure that allows different typesof inflation Thus inflation based on no-scale supergravityhas been extensively studied [9ndash19] In thisworkwewill showthat in no-scale supergravity with 119878119880(2 1)119878119880(2) times 119880(1)

symmetry one can pick up 119880(1) subsector together withthe superpotential phase monodromy to realize helical phaseinflation

Recently it was shown that 120578 problem can be naturallysolved in helical phase inflation [20] This solution employs aglobal 119880(1) symmetry which is a trivial fact in the minimalsupergravity with 119870 = ΦΦ Using the phase of a complexfield Φ as an inflaton 120578 problem is solved due to the global119880(1) symmetry The norm of Φ needs to be stabilizedotherwise it will generate notable isocurvature perturbationsthat contradict the observations However it is a nontrivialtask to stabilize the norm of Φ while keeping the phase lightas the norm and phase couple with each other In that workthe field stabilization and quadratic inflation are realizedvia a helicoid type potential The inflationary trajectory is ahelix line and this is the reason for the name ldquohelical phaseinflationrdquo In addition the superpotential of helical phaseinflation realizesmonodromy in supersymmetric field theoryFurthermore helical phase inflation gives a method to avoidthe dangerous quantum gravity effect on inflation

The single field slow-roll inflation agrees with recentobservations [4 5] Such kind of inflation admits a relation-ship between the inflaton field excursion and the tensor-to-scalar ratio which is known as the Lyth bound [21] It suggeststhat to get large tensor-to-scalar ratio the field excursionduring inflation should be much larger than the Planck massThe super-Planckian field excursion challenges the validityin the Wilsonian sense of inflationary models described byeffective field theory At Planck scale the quantum gravity

effect is likely to introduce extra terms which are suppressedby the Planck mass and then irrelevant in the low energyscale while for a super-Planckian field the irrelevant termsbecome important and may introduce significant correctionsor even destroy the inflation process In this sense thepredictions just based on the effective field theory are nottrustable A more detailed discussion on the ultraviolet(UV) sensitivity of the inflation process is provided in[22]

A lot of works have been proposed to realize inflationbased on the UV-completed theory for example in [23ndash30] However to realize inflation in string theory it needsto address several difficult problems such as moduli stabi-lization Minkowski or de Sitter vacuum and 120572

1015840- and higherstring loop-corrections on theKahler potential However onemay doubt whether such difficult UV-completed frameworkis necessary for inflation In certain scenario the super-Planckian field excursion does not necessarily lead to thephysical field above the Planck scale A simple example isthe phase of a complex field The phase factor like a pseudo-Nambu-Goldstone boson (PNGB) can be shifted to any valuewithout any effect on the energy scale By employing thephase as an inflaton the super-Planckian field excursionis not problematic at all as there is no polynomial higherorder quantum gravity correction for the phase componentBesides helical phase inflation inflationary models usingPNGB as an inflaton have been studied [31ndash39] For naturalinflation it requires super-Planckian axion decay constantwhich can be obtained by aligned axions [34] (the axionalignment relates to 119878119899 symmetry amongKahlermoduli [40])or anomalous119880(1) gauge symmetry with large condensationgauge group [41] In helical phase inflation as will beshown later the phase monodromy in superpotential canbe easily modified to generate natural inflation and alsorealize the super-Planckian phase decay constant whichis from the phase-axion alignment hidden in the processof integrating out heavy fields Furthermore all the extrafields are consistently stabilized based on the helicoid poten-tial

Like helical phase inflation ldquomonodromy inflationrdquo wasproposed to solve the UV sensitivity of large field inflation[42 43] In such model the inflaton is identified as an axionobtained from 119901-form field after string compactificationsThe inflaton potential arises from the DBI action of branesor coupling between axion and fluxes During inflation theaxion rotates along internal cycles and reduces the axionpotential slowly while all the other physical parameters areunaffected by the axion rotation Interesting realization ofmonodromy inflation is the axion alignment [34] which wasproposed to get super-Planckian axion decay constant fornatural inflation and it was noticed that this mechanismactually provides an axion monodromy in [44ndash46] Actuallya similar name ldquohelical inflationrdquo was firstly introduced in[45] for an inflationmodel with axionmonodromy Howevera major difference should be noted the ldquohelicalrdquo structurein [45] is to describe the alignment structure of two axionswhile the ldquohelicalrdquo structure in our model is from a singlecomplex field with stabilized field normThe physical pictureof axionmonodromy is analogical to the superpotential119882 in

Advances in High Energy Physics 3

helical phase inflation For 119882 there is monodromy aroundthe singularityΦ = 0

Φ 997888rarr Φ1198902120587119894

119882 997888rarr 119882 + 2120587119894119882

logΦ

(2)

The phase monodromy together with 119880(1) symmetry inthe Kahler potential provides flat direction for inflationIn the following we will show that this monodromy iscorresponding to the global119880(1) symmetry explicitly brokenby the inflation term

In this work we will study helical phase inflation fromseveral aspects in detail Firstly we will show that the phasemonodromy in the superpotential which leads to the helicoidstructure of inflaton potential can be effectively generatedby integrating out heavy fields in supersymmetric fieldtheory Besides quadratic inflation the phase monodromyfor helical phase inflation can be easily modified to realizenatural inflation in which the process of integrating outheavy fields fulfills the phase-axion alignment indirectlyand leads to super-Planckian phase decay constant withconsistent field stabilization as well We also show that helicalphase inflation can be reduced to the KYY inflation by fieldredefinition however there is no such field transformationthat can map the KYY model back to helical phase inflationFurthermore we show that the no-scale supergravity with119878119880(2 1)119878119880(2)times119880(1) symmetry provides a natural frame forhelical phase inflation as 119878119880(2 1)119878119880(2) times 119880(1) symmetryof no-scale Kahler potential already combines the symmetryfactors needed for phase monodromy Moreover we arguethat helical phase inflation is free from the UV-sensitivityproblem

This paper is organized as follows In Section 2 we reviewthe minimal supergravity construction of helical phase infla-tion In Section 3 we present the realization of phase mon-odromy based on supersymmetric field theory In Section 4natural inflation as a special type of helical phase inflation isstudied In Section 5 the relationship between helical phaseinflation and the KYY model is discussed In Section 6 westudy helical phase inflation in no-scale supergravity with119878119880(2 1)119878119880(2)times119880(1) symmetry In Section 7 we discuss howhelical phase inflation dodges the UV-sensitivity problem oflarge field inflation Conclusion is given in Section 8

2 Helical Phase Inflation

In four dimensions 119873 = 1 supergravity is determined bythe Kahler potential 119870 superpotential 119882 and gauge kineticfunction The 119865-term scalar potential is given by

119881 = 119890119870(119870119894119895119863119894119882119863119895119882 minus 3119882119882) (3)

To realize inflation in supergravity the factor 119890119870 in the aboveformula is an obstacle as it makes the potential too steep fora sufficient long slow-roll process This is the well-known 120578

problem To solve 120578 problem usually one needs a symmetryin the Kahler potential In the minimal supergravity there is

a global119880(1) symmetry in the Kahler potential119870 = ΦΦThisglobal 119880(1) symmetry is employed in helical phase inflationAs the Kahler potential is independent of the phase 120579 thepotential of phase 120579 is not affected by the exponential factor119890119870 Consequently there is no 120578 problem for phase inflationHowever the field stabilization becomes more subtle All theextra fields except inflaton have to be stabilized for single fieldinflation but normally the phase and norm of a complex fieldcouple with each other and then it is very difficult to stabilizenorm while keeping phase light

The physical picture of helical phase inflation is thatthe phase evolves along a flat circular path with constantor almost constant radiusmdashthe field magnitude and thepotential decreases slowly So even before writing down theexplicit supergravity formula one can deduce that phaseinflation if realizable should be particular realization ofcomplex phase monodromy and there exists a singularity inthe superpotential that generates the phasemonodromy Suchsingularity further indicates that themodel is described by aneffective theory

Helical phase inflation is realized in the minimal super-gravity with the Kahler potential

119870 = ΦΦ + 119883119883 minus 119892 (119883119883)2 (4)

and superpotential

119882 = 119886119883

Φln (Φ) (5)

The global119880(1) symmetry in119870 is broken by the superpoten-tial with a small factor 119886 when 119886 rarr 0 119880(1) symmetry isrestored Therefore the superpotential with small coefficientis technically natural [47] whichmakes themodel technicallystable against radiative corrections As discussed before thesuperpotential 119882 is singular at Φ = 0 and exhibits a phasemonodromy

Φ 997888rarr Φ1198902120587119894

119882 997888rarr 119882 + 2120587119886119894119883

Φ

(6)

The theory is well defined only forΦ away from the singular-ity

During inflation the field 119883 is stabilized at 119883 = 0 andthe scalar potential is simplified as

119881 = 119890ΦΦ

119882119883119882119883 = 11988621198901199032 1

1199032((ln 119903)

2+ 1205792) (7)

where Φ = 119903119890119894120579 and the kinetic term is 119871119870 = 120597120583119903120597

120583119903 +

1199032120597120583120579120597120583120579 Interestingly in the potential (7) both the norm-

dependent factor 1198901199032

(11199032) and (ln 119903)

2 reach the minimum at119903 = 1 The physical mass of norm 119903 is

1198982119903 =

1

2

1205972119881

1205971199032

100381610038161003816100381610038161003816100381610038161003816119903=1

= (2 +1

1205792)119881119868 (8)

4 Advances in High Energy Physics

therefore the norm is strongly stabilized at 119903 = 1 during infla-tion and the Lagrangian for the inflaton is

119871 = 120597120583120579120597120583120579 minus 119890119886

21205792 (9)

which gives quadratic inflation driven by the phase ofcomplex field Φ

In the above simple example given by (4) and (5)the field stabilization is obtained from the combination ofsupergravity correction 119890

119870 and the pole 1Φ in119882 besides anaccidental agreement that both the factor (1119903

2)1198901199032 and the

term (ln 119903)2 obtain their minima at 119903 = 1 while for more

general helical phase inflation such accidental agreementis not guaranteed For example one may get the followinginflaton potential

119881 = 119890119870(119903) 1

1199032((ln 119903

Λ)

2

+ 1205792) (10)

in which the coefficient 119890119870(119903)

(11199032) admits a minimum at

1199030 sim Λ but 1199030 = Λ In this case the coefficient 119890119870(119903)(11199032)still gives a mass above the Hubble scale for 119903 while ⟨119903⟩ isslightly shifted away from 1199030 in the early stage of inflation andafter inflation 119903 evolves toΛ rapidly Also the term (ln(119903Λ))

2

gives a small correction to the potential and inflationaryobservables so this correction is ignorable comparing withthe contributions from the super-Planckian valued phaseunless it is unexpectedly large

Potential Deformations In the Kahler potential there are cor-rections from the quantum loop effect while the superpoten-tial119882 is nonrenormalized Besides when coupledwith heavyfields the Kahler potential ofΦ receives corrections throughintegrating out the heavy fields Nevertheless because ofthe global 119880(1) symmetry in the Kahler potential thesecorrections can only affect the field stabilization while phaseinflation is not sensitive to these corrections

Given a higher order correction on the Kahler potential

119870 = ΦΦ + 119887 (ΦΦ)2+ 119883119883 minus 119892 (119883119883)

2 (11)

onemay introduce an extra parameterΛ in the superpotential

119882 = 119886119883

Φln Φ

Λ (12)

Based on the same argument it is easy to see the scalarpotential reduces to

119881 = 11988621198901199032+1198871199034 1

1199032((ln 119903 minus lnΛ)

2+ 1205792) (13)

The factor 1198901199032+1198871199034

(11199032) reaches its minimum at 11990320 = 2(1 +

radic1 + 8119887) below 119872119875 for 119887 gt 0 To get the ldquoaccidentalagreementrdquo it needs the parameter Λ = 1199030 and then inflationis still driven by the phase with exact quadratic potential

Without Λ the superpotential comes back to (5) and thescalar potential is shown in Figure 1 with 119887 = 01 Duringinflation ⟨119903⟩ ≃ 1199030 for small 119887 the term (ln 119903)

2 contribution to

1

1

0

0

minus1

minus1

V(102a2)

15

10

5

0

120601x

120601y

Figure 1 The helicoid structure of potential (13) scaled by 1021198862

In the graph parameters 119887 and Λ are set to be 119887 = 01 and Λ = 1respectively

the potential at the lowest order is proportional to 1198872 After

canonical field normalization the inflaton potential takes theform

119881 (120579) =1

21198982120579 (21198872+ 1205792) (14)

in which the higher order terms proportional to 1198873+119894 are

ignored With regard to the inflationary observations takethe tensor-to-scalar ratio r for example as

r =321205792119894

(1205792119894 + 21198872)

2asymp

8

119873(1 +

1198872

2119873)

minus2

(15)

where 120579119894 is the phase when inflation starts and 119873 isin (50 60)

is the 119890-folding number So the correction from higher orderterm is insignificant for 119887 lt 1

3 Monodromy in SupersymmetricField Theory

As discussed before phase inflation naturally leads to thephase monodromy (in mathematical sense) in the superpo-tential The phase monodromy requires singularity whichmeans the superpotential proposed for phase inflation shouldbe an effective theory It is preferred to show how such phasemonodromy appears from a more ldquofundamentalrdquo theory athigher scale In [20] the monodromy needed for phaseinflation is realized based on the supersymmetric field theoryin which the monodromy relates to the soft breaking of aglobal 119880(1) symmetry

Historically the monodromy inflation as an attractivemethod to realize super-Planckian field excursion was firstproposed in a more physical sense for axions arising from

Advances in High Energy Physics 5

string compactifications [42] In the inflaton potential theonly factor that changes during axion circular rotation isfrom the DBI action of branes In [44ndash46] the generalizedaxion alignment mechanisms are considered as particularrealization of axion monodromy with the potential fromnonperturbative effects We will show that such kind ofaxion monodromy can also be fulfilled by the superpotentialphase monodromy [20] even though it is not shown in theeffective superpotential after integrating out the heavy fieldsFurthermore all the extra fields can be consistently stabilized

The more ldquofundamentalrdquo field theory for the superpoten-tial in (5) is

1198820 = 120590119883Ψ (119879 minus 120575) + 119884 (119890minus120572119879

minus 120573Ψ) + 119885 (ΨΦ minus 120582) (16)

where the coupling constants for the second and third termsare taken to be 1 for simplicity and a small hierarchy isassumed between the first term and the last two terms thatis 120590 ≪ 1 The coupling 119884119890

minus120572119879 is assumed to be an effectivedescription of certain nonperturbative effects Similar formscan be obtained from 119863-brane instanton effect in type stringtheory (for a review see [48]) besides the coefficient 120572 prop

1119891 ≫ 1 in Planck unit since 119891 ≪ 1 is the decay constantand should be significantly lower than the Planck scale Forthe last two terms of1198820 there is a global 119880(1) symmetry

Ψ 997888rarr Ψ119890minus119894119902120579

Φ 997888rarr Φ119890119894119902120579

119884 997888rarr 119884119890119894119902120579

119879 997888rarr 119879 +119894119902120579

120572

(17)

which is anomalous and explicitly broken by the first termPhase rotation of the stabilizer field 119883 has no effect oninflation and so is ignored here The phase monodromy ofsuperpotential 119882 in (6) originates from the 119880(1) rotation of1198820

Ψ 997888rarr Ψ119890minus1198942120587

1198820 997888rarr 1198820 + 11989421205871205901

120572119883Ψ

(18)

As shown in [20] the supersymmetric field theory withsuperpotential1198820 admits the Minkowski vacuum at

⟨119883⟩ = ⟨119884⟩ = ⟨119885⟩ = 0

⟨119879⟩ = 120575

⟨Ψ⟩ =1

120573119890minus120572120575

⟨Φ⟩ = 120582120573119890120572120575

(19)

with ⟨Φ⟩ ≫ ⟨Ψ⟩ so that near the vacuum the masses of 119884119885 and Ψ are much larger than Φ besides the effective massof 119879 is also large near the vacuum due to large 120572 Large 120572

was an obstacle for natural inflation as it leads to the axiondecays too small for inflation while in this scenario large 120572

is helpful for phase inflation to stabilize the axion Thereforefor the physical process at scale below the mass scale of threeheavy fields the only unfixed degrees of freedom are 119883 andΦ which can be described by an effective field theory withthree heavy fields integrated out The coupling 120590119883Ψ(119879 minus 120575)

is designed for inflation and hierarchically smaller than theextra terms in1198820Therefore to describe inflation process theheavy fields need to be integrated out

To integrate out heavy fields we need to consider the 119865-terms again The 119865-term flatness of fields 119884 and 119885 gives

119865119884 = 119890minus120572119879

minus 120573Ψ + 1198701198841198820 = 0

119865119885 = ΨΦ minus 120582 + 1198701198851198820 = 0

(20)

Near the vacuum 119884 = 119885 asymp 0 ≪ 119872119875 the abovesupergravity corrections 119870119884(119885)1198820 are ignorable and thenthe 119865-term flatness conditions reduce to these for globalsupersymmetryThis is gained from the fact that although theinflation dynamics are subtle the inflation energy density isclose to the GUT scale far below the Planck scale Solvingthe 119865-term flatness equations in (20) we obtain the effectivesuperpotential119882 in (5)

Based on the above construction it is clear that the phasemonodromy in 119882 is from the 119880(1) transformation of 1198820and the pole of superpotential (5) at Φ = 0 arises from theintegration process The heavy fieldΨ is integrated out basedon the 119865-term flatness conditions when ⟨Φ⟩ ≫ ⟨Ψ⟩ whileif Φ rarr 0 Ψ becomes massless from |119865119885|

2 and it is illegalto integrate out a ldquomasslessrdquo field For inflation the condition⟨Φ⟩ ≫ ⟨Ψ⟩ is satisfied so the theory with superpotential (5)is reliable

As to the inflation term a question appears as global119880(1) is explicitly broken by the first term in (16) at inflationscale why is the phase light while the norm is much heavierThe supergravity correction to the scalar potential plays acrucial role at this stage The coefficient 119890119870 appears in thescalar potential and because of119880(1) symmetry in the Kahlerpotential the factor 119890119870 is invariant under119880(1) symmetry butincreases exponentially for a large norm Here the Kahlerpotential of 119879 should be shift invariant that is 119870 = 119870(119879 +

119879) instead of the minimal type Otherwise the exponentialfactor 119890119870 depends on phase as well and the phase rotationwillbe strongly fixed like the norm component or Re(119879)

When integrating out the heavy fields they should bereplaced both in superpotential and in Kahler potential bythe solutions from vanishing 119865-term equations So differentfrom the superpotential the Kahler potential obtained in thisway is slightly different from the minimal case given in (4)There are extra terms like

ΨΨ =1205822

1199032

119870 (119879 + 119879) = 119870(1

1205722(ln 119903)2)

(21)

where |Φ| = 119903 Nevertheless since 120582 ≪ 1 and 120572 ≫ 1 theseterms are rather small and have little effect on phase inflation

6 Advances in High Energy Physics

as shown in the last section Furthermore the quantumloop effects during integrating out heavy fields can introducecorrections to the Kahler potential as well However becauseof 119880(1) symmetry built in the Kahler potential these termstogether with (21) can only mildly affect the field stabiliza-tion and phase inflation is not sensitive to the correctionsin Kahler potential As to the superpotential it is protectedby the nonrenormalized theorem and free from radiativecorrections

4 Natural Inflation in Helical Phase Inflation

In the superpotential 1198820 the inflation term is perturbativecoupling of complex field119879which shifts under global119880(1) aninterestingmodification is to consider inflation given by non-perturbative coupling of 119879 Such term gives a modified 119880(1)

phase monodromy in the superpotential And it leads to nat-ural inflation as a special type of helical phase inflation withphase-axion alignment which is similar to the axion-axionalignment mechanism proposed in [34] for natural inflationwith super-Planckian axion decay constant Specifically itcan be shown that the phase monodromy realized in super-symmetric field theory has similar physical picture to themodified axion alignment mechanism provided in [44 45]

To realize natural inflation the superpotential 1198820 in (16)just needs to be slightly modified

1198821 = 120590119883Ψ(119890minus120572119879

minus 120575) + 119884 (119890minus120573119879

minus 120583Ψ)

+ 119885 (ΨΦ minus 120582)

(22)

in which 1 ≪ 120572 ≪ 120573 Again there is a global 119880(1)

symmetry in the last two terms of 1198821 and the fields transferunder119880(1) like in (17) The first term which is hierarchicallysmaller breaks the 119880(1) symmetry explicitly besides ashift symmetry of 119879 is needed in the Kahler potential Themonodromy of1198821 under a circular 119880(1) rotation is

Ψ 997888rarr Ψ119890minus1198942120587

1198821 997888rarr 1198821 + 120590119883Ψ119890minus120572119879

(119890minus2120587119894(120572120573)

minus 1)

(23)

The supersymmetric field theory given by (22) admits thefollowing supersymmetric Minkowski vacuum

⟨119883⟩ = ⟨119884⟩ = ⟨119885⟩ = 0

⟨119879⟩ = minus1

120572ln 120575

⟨Ψ⟩ =1

120583120575120573120572

⟨Φ⟩ = 120582120583120575minus120573120572

(24)

The parameters are set to satisfy conditions

120575120573120572

sim 120582120583

120582 ≪ 1

120575 lt 1

(25)

so that ⟨Ψ⟩ ≪ ⟨Φ⟩ and ⟨119879⟩ gt 0

Near vacuum fields 119884119885Ψ and 119879 obtain large effectivemasses above inflation scale while 119883Φ are much lighter Atthe inflation scale the heavy fields should be integrated outThe 119865-term flatness conditions for fields 119884 and 119885 are

119865119884 = 119890minus120573119879

minus 120583Ψ = 0

119865119885 = ΨΦ minus 120582 = 0

(26)

in which the supergravity corrections 1198701198841198851198821 are neglectedas both119884 and119885 get close to zero during inflation IntegratingoutΨ and 119879 from (26) we obtain the effective superpotential1198821015840 from1198821

1198821015840= 120590120582

119883

Φ((

120583120582

Φ)

120572120573

minus 120575) (27)

Given 120572 ≪ 120573 the effective superpotential contains a termwith fractional power Inflation driven by complex potentialwith fractional power was considered in [49] to get sufficientlarge axion decay constant Here the supersymmetric fieldmonodromy naturally leads to the superpotential with frac-tional power which arises from the small hierarchy of axiondecay constants in two nonperturbative terms

Helical phase inflation is described by the effective super-potential 1198821015840 the role of phase-axion alignment is not clearfrom119882

1015840 since it is hidden in the procedure of integrating outheavy fields

The 119865-term flatness conditions (26) fix four degrees offreedom for the extra degree of freedom they correspond tothe transformations free from constraints (26)

Ψ 997888rarr Ψ119890minus119906minus119894V

Φ 997888rarr Φ119890119906+119894V

119879 997888rarr 119879 +119906

120573+

119894V120573

(28)

Parameter 119906 corresponds to the norm variation of complexfield Φ which is fixed by the supergravity correction on thescalar potential 119890119870(ΦΦ) Parameter V relates to 119880(1) transfor-mation which leads to the phase monodromy from the firstterm of1198821The scalar potential including the inflation termdepends on the superpositions among phases ofΨΦ and theaxion Im(119879) which are constrained as in (28) Among thesefields the phase ofΦ has the lightestmass after canonical fieldnormalization Similar physical picture appears in the axion-axion alignment where inflation is triggered by the axionsuperposition along the flat direction

After integrating out the heavy fields the Kahler potentialis

119870 = ΦΦ +1205822

ΦΦ+ sdot sdot sdot (29)

As 120582 ≪ 1 the Kahler potential is dominated by ΦΦ asdiscussed before the extra terms like 1205822ΦΦ only give small

Advances in High Energy Physics 7

corrections to the field stabilization Helical phase inflationcan be simply described by the supergravity

119870 = ΦΦ + 119883119883 +

119882 = 119886119883

Φ(Φminus119887

minus 119888)

(30)

where 119887 = 120572120573 ≪ 1 and 119888 asymp 1 The scalar potential given bythe above Kahler potential and superpotential is

119881 = 1198901199032 1198862

1199032(119903minus2119887

+ 1198882minus 2119888119903minus119887 cos (119887120579)) (31)

in which we have usedΦ equiv 119903119890119894120579 As usual the norm 119903 couples

with the phase in the scalar potential To generate singlefield inflation the norm component needs to be stabilizedwith heavy mass above the Hubble scale while the phasecomponent remains light Due to the phase-norm couplingthe phase and norm are likely to obtain comparable effectivemasses during inflation so it is difficult to realize singlefield inflation However it is a bit different in helical phaseinflation The above scalar potential can be rewritten asfollows

119881 = 1198901199032 1198862

1199032(119903minus119887

minus 119888)2+ 1198901199032 41198862119888

1199032+119887(sin 119887

2120579)

2

(32)

So its vacuum locates at ⟨119903⟩ = 1199030 = 119888minus1119887 and 120579 = 0 Besides

the coefficient of the phase term 1198901199032(41198881199032+119887

) reaches itsminimal value at 1199031 = radic1 + 1198872 For 119888 asymp 1 we have theapproximation 1199030 asymp 1199031 asymp 1 The extra terms in 119870 givesmall corrections to 1199030 and 1199031 but the approximation is stillvalid So with the parameters in (25) the norm 119903 in thetwo terms of scalar potential 119881 can be stabilized at the closeregion 119903 asymp 1 separately If the parameters are tuned so that1199030 = 1199031 then the norm of complex field is strictly stabilizedat the vacuum value during inflation Without such tuninga small difference between 1199030 and 1199031 is expected but theshift of 119903 during inflation is rather small and inflation is stillapproximate to the single field inflation driven by the phaseterm prop (sin(1198872)120579)2 The helicoid structure of the potential(31) is shown in Figure 2

It is known that to realize aligned axion mechanism insupergravity [50] it is very difficult to stabilize the moduli asthey couple with the axions In [40] the moduli stabilizationis fulfilled with gauged anomalous 119880(1) symmetries since119880(1) 119863-terms only depend on the norm |Φ| or Re(119879) andthen directly separate the norms and phases of matter fieldsIn helical phase inflation the modulus and matter fieldsexcept the phase are stabilized at higher scale or by thesupergravity scalar potential Only the phase can be aninflaton candidate because of the protection from the global119880(1) symmetry in Kahler potential and approximate 119880(1)

symmetry in superpotential

5 Helical Phase Inflation and the KYY Model

120578 problem in supergravity inflation can be solved both inhelical phase inflation and in the shift symmetry in KYY

1

1

00

minus1

minus1

10

8

6

4

V(a

2)

120601x120601y

Figure 2 The helicoid structure of potential (31) scaled by 1198862 The

parameters with 119888 = 096 and 119887 = 01 are adopted in the graph Itis shown that the local valley locates around 119903 asymp 1 Note that thepotential gets flatter at the top of the graph

model The physics in these two solutions are obviously dif-ferent For helical phase inflation the solution employs 119880(1)

symmetry in Kahler potential of the minimal supergravityand the superpotential admits a phase monodromy arisingfrom the global 119880(1) transformation

119870 = ΦΦ + 119883119883 + sdot sdot sdot

119882 = 119886119883

ΦlnΦ

(33)

The inflation is driven by the phase of complex field Φ

and several special virtues appear in the model For theKYY model with shift symmetry [6] the Kahler potential isadjusted so that it admits a shift symmetry along the directionof Im(119879)

119870 =1

2(119879 + 119879)

2+ 119883119883 + sdot sdot sdot

119882 = 119886119883119879

(34)

and inflation is driven by Im(119879) The shift symmetry isendowedwith axions so thismechanism is attractive for axioninflation

Here we will show that although the physical pictures aremuch different in helical phase inflation andKYY typemodeljust considering the lower order terms in the Kahler potentialof redefined complex field helical phase inflation can reduceto the KYY model

Because the phase of Φ rotates under the global 119880(1)

transformation to connect helical phase inflation with theKYY model a natural guess is to take the following fieldredefinitionΦ = 119890

119879 then helical phase inflation (33) becomes

119870 = 119890119879+119879

+ sdot sdot sdot = 1 + 119879 + 119879 +1

2(119879 + 119879)

2+ sdot sdot sdot

119882 = 119886119883119879119890minus119879

(35)

8 Advances in High Energy Physics

TheKahler manifold of complex field119879 is invariant under theholomorphic Kahler transformation

119870(119879 119879) 997888rarr 119870(119879 119879) + 119865 (119879) + 119865 (119879) (36)

in which 119865(119879) is a holomorphic function of 119879 To keepthe whole Lagrangian also invariant under the Kahler trans-formation the superpotential transforms under the Kahlertransformation

119882 997888rarr 119890minus119865(119879)

119882 (37)

For the supergravity model in (35) taking the Kahler trans-formation with 119865(119879) = minus(12) minus 119879 the Kahler potential andsuperpotential become

119870 =1

2(119879 + 119879)

2+ sdot sdot sdot

119882 = 119886radic119890119883119879

(38)

which is just the KYY model (34) with higher order cor-rections in the Kahler potential These higher order termsvanish after field stabilization and have no effect on inflationprocess The field relation Φ = 119890

119879 also gives a map betweenthe simplest helical phase inflation and the KYY model suchas the inflaton arg(Φ) rarr Im(119879) and field stabilization|Φ| = 1 rarr Re(119879) = 0

Nevertheless helical phase inflation is not equivalent tothe KYY model There are higher order corrections to theKahler potential in the map from helical phase inflation toKYY model which have no effect on inflation after fieldstabilization but indicate different physics in two models Bydropping these terms certain information is lost so the mapis irreversible Specifically the inverse function 119879 = lnΦ ofthe field redefinition Φ = 119890

119879 cannot reproduce helical phaseinflation from the KYY model unless one introduces higherdimensional operators in the Kahler potential exactly thesame as given in the expansion (35) Such strict constraint oneach higher dimensional operator is not guaranteed by anyknown rule and so is unphysical As there is no pole or singu-larity in the KYY model it is unlikely to introduce pole andsingularity at origin with phase monodromy through well-defined field redefinition Actually the singularity with phasemonodromy in the superpotential indicates rich physics inthe scale above inflation

6 Helical Phase Inflation inNo-Scale Supergravity

The no-scale supergravity is an attractive frame for GUTscale phenomenology it is interesting to realize helicalphase inflation in no-scale supergravity Generally theKahler manifold of the no-scale supergravity is equippedwith 119878119880(119873 1)119878119880(119873) times 119880(1) symmetry For the no-scalesupergravity with exact 119878119880(1 1)119880(1) symmetry withoutextra fields no inflation can be realized so the case with119878119880(2 1)119878119880(2) times 119880(1) symmetry is the simplest one thatadmits inflation

The Kahler potential with 119878119880(2 1)119878119880(2)times119880(1) symme-try is

119870 = minus3 ln(119879 + 119879 minusΦΦ

3) (39)

In the symmetry of the Kahler manifold there is 119880(1)

subsector the phase rotation of complex fieldΦ which can beemployed for helical phase inflation Besides the modulus 119879should be stabilized during inflation which can be fulfilled byintroducing extra terms on119879 As a simple example of no-scalehelical phase inflation here we follow the simplification in [9]that the modulus 119879 has already been stabilized at ⟨119879⟩ = 119888Different from the minimal supergravity the kinetic termgiven by the no-scale Kahler potential is noncanonical

119871119870 = 119870ΦΦ120597120583Φ120597120583Φ

=2119888

(2119888 minus 11990323)2(120597120583119903120597

120583119903 + 1199032120597120583120579120597120583120579)

(40)

in which Φ equiv 119903119890119894120579 is used The 119865-term scalar potential is

119881 = 119890minus(23)119870 1003816100381610038161003816119882Φ

1003816100381610038161003816

2=

1003816100381610038161003816119882Φ1003816100381610038161003816

2

(119879 + 119879 minus ΦΦ3)2 (41)

where the superpotential 119882 is a holomorphic function ofsuperfieldΦ and119879 = ⟨119879⟩ = 119888 It requires a phasemonodromyin superpotential119882 for phase inflation the simple choice is

119882 =119886

Φln Φ

Λ (42)

The scalar potential given by this superpotential is

119881 =91198862

(6119888 minus 1199032)21199034

((ln 119903

119890Λ)

2

+ 1205792) (43)

As in the minimal supergravity the norm and phase ofcomplex field Φ are separated in the scalar potential For the119903-dependent coefficient factor 1(6119888 minus 119903

2)21199034 its minimum

locates at 1199030 = radic3119888 and another term (ln(119903119890Λ))2 reaches its

minimum at 1199031 = 119890Λ Given that the parameter Λ is tuned sothat 1199030 = 1199031 in the radial direction the potential has a globalminimum at 119903 = 1199030 Similar to helical phase inflation in theminimal supergravity the potential (43) also shows helicoidstructure as presented in Figure 3

The physical mass of 119903 in the region near the vacuum is

1198982119903 =

(2119888 minus 11990323)2

4119888

1205972119881

1205971199032

100381610038161003816100381610038161003816100381610038161003816100381610038161003816119903=radic3119888

= (4 +1

21205792)1198672 (44)

where119867 is theHubble constant during inflation So the norm119903 is strongly stabilized at 1199030 If 1199030 and 1199031 are not equal butclose to each other 119903 will slightly shift during inflation butits mass remains above the Hubble scale and inflation is stillapproximately the single field inflation

Advances in High Energy Physics 9

30

20

10

0

minus10

minus10

minus05

minus05

0000

05

05

10

10

120601x (radic

3c) 120601 y(radic3

c)

V(a

2 c4 )

Figure 3The helicoid structure of potential (43) scaled by 11988621198884 Inthe graph the parameter Λ is tuned so that 1199030 = 1199031 The complexfieldΦ has been rescaled byradic3119888 and the scale of field norm at localvalley is determined by the parameter 119888 instead of the Planck mass

Instead of stabilizing 119879 independently with the inflationprocess we can realize helical phase inflation from the no-scale supergravity with dynamical119879There is a natural reasonfor such consideration The phase monodromy requiresmatter fields transforming as rotations under 119880(1) and alsoa modulus 119879 as a shift under 119880(1) The Kahler potential of 119879is shift invariant Interestingly for the no-scale supergravitywith 119878119880(2 1)119878119880(2) times 119880(1) symmetry as shown in (39) theKahler potential 119870 is automatically endowed with the shiftsymmetry and global 119880(1) symmetry

119879 997888rarr 119879 + 119894119862

Φ 997888rarr Φ119890119894120579

(45)

Therefore the no-scale Kahler potential fits with the phasemonodromy in (16) and (22) initiatively

The Kahler potentials of superfields 119911 isin 119883 119884 119885 areof the minimal type 119911119911 while for Ψ its Kahler potentialcan be the minimal type ΨΨ or the no-scale type 119870 =

minus3 ln(119879+119879minus (ΦΦ+ΨΨ)3) which extends the symmetry ofKahler manifold to 119878119880(3 1)119878119880(3) times 119880(1) In this scenariothe process to integrate out heavy fields is the same asbefore Besides the potential of phase proportional to |119882119883|

2

is insensitive to the formula of Kahler potential due to119880(1) symmetry The major difference appears in the fieldstabilization The scalar potential from phase monodromy in(16) is

119881 = 119890119870 1003816100381610038161003816119882119883

1003816100381610038161003816

2=

1

(ln 1199032120572 + 119888 minus 11990323 minus 120582211990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792) asymp

1

(119888 minus 11990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792)

(46)

with 120572 ≫ 1 and 120582 ≪ 1 The field norm 119903 is stabilized byminimizing the coefficient 11199032(119888 minus 119903

23)3 and the parameter

119888 from ⟨119879⟩ determines the scale of ⟨119903⟩ = radic31198882 This isdifferent from helical phase inflation in the minimal super-gravity which minimizes the norm based on the coefficient1198901199032(11199032) and then the scale of ⟨119903⟩ is close to119872119875 the unique

energy scale of supergravity corrections Similarly combiningthe no-scale Kahlermanifold with phasemonodromy in (22)we can obtain natural inflation

7 UV-Sensitivity and Helical Phase Inflation

For the slow-roll inflation the Lyth bound [21] provides arelationship between the tensor-to-scalar ratio r and the fieldexcursion Δ120601 Roughly it requires

Δ120601

119872119875

⩾ (r

001)

12

(47)

To get large tensor-to-scalar ratio r ⩾ 001 such as in chaoticinflation or natural inflation the field excursion should bemuch larger than the Planck mass The super-Planckianfield excursion makes the description based on the effectivefield theory questionable In the Wilsonian sense the lowenergy field theory is an effective theory with higher ordercorrections introduced by the physics at the cut-off scalelike quantum gravity and these terms are irrelevant in theeffective field theory since they are suppressed by the cut-offenergy scale However for the inflation process the inflatonhas super-Planckian field excursion which is much largerthan the cut-off scaleThus the higher order terms cannot besuppressed by the Planckmass andmay affect the inflationaryobservations significantly And then they may significantlyaffect inflation or even destroy the inflation process Forexample considering the corrections

Δ119881 = 119888119894119881(120601

119872119875

)

119894

+ sdot sdot sdot (48)

to the original inflaton potential 119881 as long as 119888119894 are of theorder 10minus119894 in the initial stage of inflation 120601 sim 119874(10)119872119875 thehigher order terms can be as large as the original potential119881 So for large field inflation it is sensitive to the physics atthe cut-off scale and the predictions of inflation just based oneffective field theory are questionable

In consideration of the UV sensitivity of large fieldinflation a possible choice is to realize inflation in UV-completed theory like string theory (for a review see [22])To realize inflation in string theory there are a lot of problemsto solve besides inflation such as the moduli stabilizationMinkowskide Sitter vacua and effects of 1205721015840- and string loop-corrections Alternatively in the bottom-up approach onemay avoid the higher order corrections by introducing anextra shift symmetry in the theory The shift symmetry istechnically natural and safe under quantum loop-correctionsHowever the global symmetry can be broken by the quantumgravity effect So it is still questionable whether the shiftsymmetry can safely evade the higher corrections like in (48)

10 Advances in High Energy Physics

The UV-completion problem is dodged in helical phaseinflation Since the super-Planckian field excursion is thephase of a complex field and the phase component is notdirectly involved in the gravity interaction there are nodangerous high-order corrections like in (48) for the phasepotentialThe inflaton evolves along the helical trajectory anddoes not relate to the physics in the region above the Planckscale For helical phase inflation in the minimal supergravitythe norm of field is stabilized at the marginal point of thePlanck scale where the supergravity correction on the scalarpotential gets important based onwhich the normof complexfield can be strongly stabilized The extra corrections arelikely to appear in the Kahler potential however they canonly slightly affect the field stabilization while the phaseinflation is protected by the global 119880(1) symmetry in theKahler potential in consequence helical phase inflation isnot sensitive to these corrections at all For the helical phaseinflation in no-scale supergravity the normof complex field isstabilized at the scale of themodulus ⟨119879⟩ instead of the Planckscale one can simply adjust the scale of ⟨119879⟩ to keep themodelaway from super-Planckian region

Helical phase inflation is free from the UV-sensitivityproblem and it is just a typical physical process at theGUT scale with special superpotential that admits phasemonodromy So it provides an inflationary model that can bereliably studied just in supersymmetric field theory

8 Discussions and Conclusion

In this work we have studied the details of helical phase infla-tion from several aspects Helical phase inflation is realizedin supergravity with global 119880(1) symmetry 119880(1) symmetryis built in the Kahler potential so that helical phase inflationcan be realized by the ordinary Kahler potentials such as theminimal or no-scale types Helical phase inflation directlyleads to the phasemonodromy in the superpotential which issingular and an effective field theory arising from integratingout heavy fieldsThe phasemonodromy originates from119880(1)

rotation of the superpotential at higher scale Generically thesuperpotential can be separated into two parts119882 = 119882119868 +119882119878where119882119878 admits the global119880(1) symmetrywhile119882119868 breaks itexplicitly at scale much lower than 119882119878 Under 119880(1) rotationthe inflation term 119882119868 is slightly changed which realizes thephase monodromy in the effective theory and introducesa flat potential along the direction of phase rotation Bybreaking the global119880(1) symmetry in different ways we mayget different kinds of inflation such as quadratic inflationnatural inflation or the other types of inflation that are notpresented in this work

An amazing fact of helical phase inflation is that itdeeply relates to several interesting points of inflation andnaturally combines them in a rather simple potential withhelicoid structure The features of helical phase inflation canbe summarized as follows

(i) The global 119880(1) symmetry is built in the Kahlerpotential so helical phase inflation provides a naturalsolution to 120578 problem

(ii) The phase excursion requires phase monodromy inthe superpotential So helical phase inflation providesin the mathematical sense a new type of monodromyin supersymmetric field theory

(iii) The singularity in the superpotential together withthe supergravity scalar potential provides strong fieldstabilization which is consistent with phase inflation

(iv) The super-Planckian field excursion is realized bythe phase of a complex field instead of any otherldquophysicalrdquo fields that directly couple with gravity Sothere are no polynomial higher order corrections forthe phase and thus inflation is not sensitive to thequantum gravity corrections

To summarize helical phase inflation introduces a newtype of inflation that can be effectively described by super-symmetric field theory at the GUT scale Generically thesuper-Planckian field excursion makes the inflationary pre-dictions based on effective field theory questionable sincethe higher order corrections from quantum gravity arelikely to affect the inflation process significantly One of thesolutions is to realize inflation in a UV-completed theory likestring theory nevertheless there are many difficult issues instring theory to resolve before realizing inflation completelyHelical phase inflation is another simple solution to the UV-sensitivity problem It is based on the supersymmetric fieldtheory and the physics are clear and much easier to controlFurthermore helical phase inflation makes the unification ofinflation theory with GUT more natural since both of themare triggered at the scale of 1016 GeV and can be effectivelystudied based on supersymmetric field theory

Besides we have shown that helical phase inflation alsorelates to several interesting developments in inflation theoryIt can be easily modified for natural inflation and realizethe phase-axion alignment indirectly which is similar tothe axion-axion alignment mechanism for super-Planckianaxion decay constant [34] The phase-axion alignment isnot shown in the final supergravity model which exhibitsexplicit phase monodromy only However the phase-axionalignment is hidden in the process when the heavy fields areintegrating out For 120578 problem in the supergravity inflationthere is another well-known solution the KYY model withshift symmetry We showed that through field redefinitionhelical phase inflation given by (4) and (5) reduces to theKYYmodel where the higher order corrections in the Kahlerpotential have no effect on inflation process However thereis no inverse transformation from the KYY model to helicalphase inflation since no well-defined field redefinition canintroduce the pole and phase singularity needed for phasemonodromy Helical phase inflation can be realized in no-scale supergravity The no-scale Kahler potential automati-cally provides the symmetry needed by phase monodromyIn the no-scale supergravity the norm of complex field isstabilized at the scale of the modulus

Our inflation models are constructed within the super-gravity theory with global 119880(1) symmetry broken explicitlyby the subleading order superpotential term So it is justtypical GUT scale physics and indicates that a UV-completed

Advances in High Energy Physics 11

framework seems to be not prerequisite to effectively describesuch inflation process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work of DVN was supported in part by the DOEGrant DE-FG03-95-ER-40917 The work of Tianjun Li issupported in part by theNatural Science Foundation of ChinaunderGrants nos 10821504 11075194 11135003 11275246 and11475238 and by the National Basic Research Program ofChina (973 Program) under Grant no 2010CB833000

References

[1] A H Guth ldquoInflationary universe a possible solution to thehorizon and flatness problemsrdquo Physical Review D vol 23 no2 pp 347ndash356 1981

[2] A D Linde ldquoColeman-Weinberg theory and the new inflation-ary universe scenariordquo Physics Letters B vol 114 no 6 pp 431ndash435 1982

[3] A Albrecht and P J Steinhardt ldquoCosmology for grand unifiedtheories with radiatively induced symmetry breakingrdquo PhysicalReview Letters vol 48 no 17 pp 1220ndash1223 1982

[4] P A R Ade N Aghanim C Armitage-Caplan et al ldquoPlanck2013 results XXII Constraints on inflationrdquo Astronomy ampAstrophysics vol 571 article A22 42 pages 2014

[5] P A R Ade R W Aikin D Barkats et al ldquoDetection of B-mode polarization at degree angular scales by BICEP2rdquo PhysicalReview Letters vol 112 no 24 Article ID 241101 2014

[6] M Kawasaki M Yamaguchi and T Yanagida ldquoNatural chaoticinflation in supergravityrdquo Physical Review Letters vol 85 no 17pp 3572ndash3575 2000

[7] T Li Z Li and D V Nanopoulos ldquoSupergravity inflation withbroken shift symmetry and large tensor-to-scalar ratiordquo Journalof Cosmology and Astroparticle Physics vol 2014 no 2 article028 2014

[8] K Harigaya and T T Yanagida ldquoDiscovery of large scale tensormode and chaotic inflation in supergravityrdquo Physics Letters Bvol 734 pp 13ndash16 2014

[9] J Ellis D V Nanopoulos and K A Olive ldquoNo-scale supergrav-ity realization of the starobinsky model of inflationrdquo PhysicalReview Letters vol 111 no 11 Article ID 111301 2013

[10] T Li Z Li and D V Nanopoulos ldquoNo-scale ripple inflationrevisitedrdquo Journal of Cosmology and Astroparticle Physics vol04 article 018 2014

[11] C Pallis ldquoInduced-gravity inflation in no-scale supergravityand beyondrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 8 article 057 2014

[12] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky modelrdquo Fortschritte der Physik vol 62 no 7 pp 573ndash5832014

[13] J Ellis M A G Garca D V Nanopoulos and K A OliveldquoResurrecting quadratic inflation in no-scale supergravity inlight of BICEP2rdquo Journal of Cosmology andAstroparticle Physicsvol 2014 no 5 article 037 2014

[14] T Li Z Li and D V Nanopoulos ldquoChaotic inflation in no-scale supergravitywith string inspiredmoduli stabilizationrdquoTheEuropean Physical Journal C vol 75 no 2 article 55 2015

[15] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky model and higher curvature correctionsrdquo Fortschritte derPhysik vol 63 no 1 pp 2ndash11 2015

[16] S V Ketov andT Terada ldquoInflation in supergravity with a singlechiral superfieldrdquo Physics Letters B vol 736 pp 272ndash277 2014

[17] C Kounnas D Lust and N Toumbas ldquoR2 inflation fromscale invariant supergravity and anomaly free superstrings withfluxesrdquo Fortschritte der Physik vol 63 no 1 pp 12ndash35 2015

[18] G A Diamandis B C Georgalas K Kaskavelis PKouroumalou A B Lahanas and G Pavlopoulos ldquoInflationin R2 supergravity with non-minimal superpotentialsrdquo PhysicsLetters B vol 744 pp 74ndash81 2015

[19] C Pallis and Q Shafi ldquoGravity waves from non-minimalquadratic inflationrdquo Journal of Cosmology and AstroparticlePhysics vol 2015 no 3 article 023 2015

[20] T Li Z Li and D V Nanopoulos ldquoHelical phase inflationrdquoPhysical Review D vol 91 no 6 Article ID 061303 2015

[21] D H Lyth ldquoWhat would we learn by detecting a gravitationalwave signal in the cosmic microwave background anisotropyrdquoPhysical Review Letters vol 78 no 10 article 1861 1997

[22] D Baumann and L McAllister ldquoInflation and stringtheoryrdquohttparxivorgabs14042601

[23] F Marchesano G Shiu and A M Uranga ldquoF-term axionmonodromy inflationrdquo Journal of High Energy Physics vol 2014no 9 article 184 2014

[24] C Long L McAllister and P McGuirk ldquoAligned naturalinflation in string theoryrdquo Physical Review D vol 90 no 2Article ID 023501 2014

[25] X Gao T Li and P Shukla ldquoCombining universal and odd RRaxions for aligned natural inflationrdquo Journal of Cosmology andAstroparticle Physics vol 2014 no 10 article 048 2014

[26] I Ben-Dayan F G Pedro and A Westphal ldquoTowards naturalinflation in string theoryrdquo Physical Review D vol 92 no 2Article ID 023515 2015

[27] R Blumenhagen D Herschmann and E Plauschinn ldquoThechallenge of realizing F-term axion monodromy inflation instring theoryrdquo Journal of High Energy Physics vol 2015 article7 2015

[28] H Abe T Kobayashi and H Otsuka ldquoTowards natural infla-tion from weakly coupled heterotic string theoryrdquo Progress ofTheoretical and Experimental Physics Article ID 063E02 2015

[29] T Ali S S Haque and V Jejjala ldquoNatural inflation from nearalignment in heterotic string theoryrdquo Physical Review D vol 91no 8 Article ID 083516 9 pages 2015

[30] R Flauger LMcAllister E Silverstein and AWestphal ldquoDrift-ing oscillations in axionmonodromyrdquo httparxivorgabs14121814

[31] K Freese J A Frieman andAVOlinto ldquoNatural inflationwithpseudo nambu-goldstone bosonsrdquo Physical Review Letters vol65 no 26 pp 3233ndash3236 1990

[32] F C Adams J R Bond K Freese J A Frieman and AV Olinto ldquoNatural inflation particle physics models power-law spectra for large-scale structure and constraints from thecosmic background explorerrdquo Physical Review D vol 47 no 2pp 426ndash455 1993

[33] G German A Mazumdar and A Perez-Lorenzana ldquoAngularinflation from supergravityrdquo Modern Physics Letters A vol 17no 25 pp 1627ndash1634 2002

12 Advances in High Energy Physics

[34] J E Kim H P Nilles and M Peloso ldquoCompleting naturalinflationrdquo Journal of Cosmology and Astroparticle Physics vol0501 article 005 2005

[35] D Baumann and D Green ldquoInflating with baryonsrdquo Journal ofHigh Energy Physics vol 2011 no 4 article 071 2011

[36] JMcDonald ldquoSub-Planckian two-field inflation consistent withthe Lyth boundrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 9 article 027 2014

[37] J McDonald ldquoA minimal sub-Planckian axion inflation modelwith large tensor-to-scalar ratiordquo Journal of Cosmology andAstroparticle Physics vol 2015 no 1 article 018 2015

[38] C D Carone J Erlich A Sensharma and Z Wang ldquoTwo-fieldaxion-monodromy hybrid inflation model Dantersquos WaterfallrdquoPhysical ReviewD vol 91 no 4 Article ID 043512 8 pages 2015

[39] G Barenboim and W-I Park ldquoSpiral Inflationrdquo Physics LettersB vol 741 pp 252ndash255 2015

[40] T Li Z Li andDVNanopoulos ldquoAligned natural inflation andmoduli stabilization from anomalous U(1) gauge symmetriesrdquoJournal of High Energy Physics vol 2014 no 11 article 012 2014

[41] T Li Z Li and D V Nanopoulos ldquoNatural inflation withnatural trans-planckian axion decay constant from anomalousU(1)119883rdquo Journal of High Energy Physics vol 2014 no 7 article052 2014

[42] E Silverstein and A Westphal ldquoMonodromy in the CMBgravity waves and string inflationrdquo Physical Review D vol 78no 10 Article ID 106003 21 pages 2008

[43] L McAllister E Silverstein and A Westphal ldquoGravity wavesand linear inflation from axion monodromyrdquo Physical ReviewD vol 82 no 4 Article ID 046003 2010

[44] K Choi H Kim and S Yun ldquoNatural inflation with multiplesub-Planckian axionsrdquo Physical Review D vol 90 no 2 ArticleID 023545 2014

[45] S-H H Tye and S S C Wong ldquoHelical inflation and cosmic-stringsrdquo httparxivorgabs14046988

[46] R Kappl S Krippendorf and H P Nilles ldquoAligned naturalinflation monodromies of two axionsrdquo Physics Letters B vol737 pp 124ndash128 2014

[47] G rsquotHooft ldquoNaturalness chiral symmetry and spontaneouschiral symmetry breakingrdquo in Recent Developments in GaugeTheories G rsquotHooft C Itzykson A Jaffe et al Eds vol 59of NATO Advanced Study Institutes Series pp 135ndash157 PlenumPress New York NY USA 1980

[48] R Blumenhagen M Cvetic S Kachru and T Weigand ldquoD-brane instantons in type II string theoryrdquo Annual Review ofNuclear and Particle Science vol 59 pp 269ndash296 2009

[49] K Harigaya andM Ibe ldquoSimple realization of inflaton potentialon a Riemann surfacerdquo Physics Letters B vol 738 pp 301ndash3042014

[50] R Kallosh A Linde and B Vercnocke ldquoNatural inflation insupergravity and beyondrdquo Physical Review D vol 90 Article ID041303(R) 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances in High Energy Physics 3

helical phase inflation For 119882 there is monodromy aroundthe singularityΦ = 0

Φ 997888rarr Φ1198902120587119894

119882 997888rarr 119882 + 2120587119894119882

logΦ

(2)

The phase monodromy together with 119880(1) symmetry inthe Kahler potential provides flat direction for inflationIn the following we will show that this monodromy iscorresponding to the global119880(1) symmetry explicitly brokenby the inflation term

In this work we will study helical phase inflation fromseveral aspects in detail Firstly we will show that the phasemonodromy in the superpotential which leads to the helicoidstructure of inflaton potential can be effectively generatedby integrating out heavy fields in supersymmetric fieldtheory Besides quadratic inflation the phase monodromyfor helical phase inflation can be easily modified to realizenatural inflation in which the process of integrating outheavy fields fulfills the phase-axion alignment indirectlyand leads to super-Planckian phase decay constant withconsistent field stabilization as well We also show that helicalphase inflation can be reduced to the KYY inflation by fieldredefinition however there is no such field transformationthat can map the KYY model back to helical phase inflationFurthermore we show that the no-scale supergravity with119878119880(2 1)119878119880(2)times119880(1) symmetry provides a natural frame forhelical phase inflation as 119878119880(2 1)119878119880(2) times 119880(1) symmetryof no-scale Kahler potential already combines the symmetryfactors needed for phase monodromy Moreover we arguethat helical phase inflation is free from the UV-sensitivityproblem

This paper is organized as follows In Section 2 we reviewthe minimal supergravity construction of helical phase infla-tion In Section 3 we present the realization of phase mon-odromy based on supersymmetric field theory In Section 4natural inflation as a special type of helical phase inflation isstudied In Section 5 the relationship between helical phaseinflation and the KYY model is discussed In Section 6 westudy helical phase inflation in no-scale supergravity with119878119880(2 1)119878119880(2)times119880(1) symmetry In Section 7 we discuss howhelical phase inflation dodges the UV-sensitivity problem oflarge field inflation Conclusion is given in Section 8

2 Helical Phase Inflation

In four dimensions 119873 = 1 supergravity is determined bythe Kahler potential 119870 superpotential 119882 and gauge kineticfunction The 119865-term scalar potential is given by

119881 = 119890119870(119870119894119895119863119894119882119863119895119882 minus 3119882119882) (3)

To realize inflation in supergravity the factor 119890119870 in the aboveformula is an obstacle as it makes the potential too steep fora sufficient long slow-roll process This is the well-known 120578

problem To solve 120578 problem usually one needs a symmetryin the Kahler potential In the minimal supergravity there is

a global119880(1) symmetry in the Kahler potential119870 = ΦΦThisglobal 119880(1) symmetry is employed in helical phase inflationAs the Kahler potential is independent of the phase 120579 thepotential of phase 120579 is not affected by the exponential factor119890119870 Consequently there is no 120578 problem for phase inflationHowever the field stabilization becomes more subtle All theextra fields except inflaton have to be stabilized for single fieldinflation but normally the phase and norm of a complex fieldcouple with each other and then it is very difficult to stabilizenorm while keeping phase light

The physical picture of helical phase inflation is thatthe phase evolves along a flat circular path with constantor almost constant radiusmdashthe field magnitude and thepotential decreases slowly So even before writing down theexplicit supergravity formula one can deduce that phaseinflation if realizable should be particular realization ofcomplex phase monodromy and there exists a singularity inthe superpotential that generates the phasemonodromy Suchsingularity further indicates that themodel is described by aneffective theory

Helical phase inflation is realized in the minimal super-gravity with the Kahler potential

119870 = ΦΦ + 119883119883 minus 119892 (119883119883)2 (4)

and superpotential

119882 = 119886119883

Φln (Φ) (5)

The global119880(1) symmetry in119870 is broken by the superpoten-tial with a small factor 119886 when 119886 rarr 0 119880(1) symmetry isrestored Therefore the superpotential with small coefficientis technically natural [47] whichmakes themodel technicallystable against radiative corrections As discussed before thesuperpotential 119882 is singular at Φ = 0 and exhibits a phasemonodromy

Φ 997888rarr Φ1198902120587119894

119882 997888rarr 119882 + 2120587119886119894119883

Φ

(6)

The theory is well defined only forΦ away from the singular-ity

During inflation the field 119883 is stabilized at 119883 = 0 andthe scalar potential is simplified as

119881 = 119890ΦΦ

119882119883119882119883 = 11988621198901199032 1

1199032((ln 119903)

2+ 1205792) (7)

where Φ = 119903119890119894120579 and the kinetic term is 119871119870 = 120597120583119903120597

120583119903 +

1199032120597120583120579120597120583120579 Interestingly in the potential (7) both the norm-

dependent factor 1198901199032

(11199032) and (ln 119903)

2 reach the minimum at119903 = 1 The physical mass of norm 119903 is

1198982119903 =

1

2

1205972119881

1205971199032

100381610038161003816100381610038161003816100381610038161003816119903=1

= (2 +1

1205792)119881119868 (8)

4 Advances in High Energy Physics

therefore the norm is strongly stabilized at 119903 = 1 during infla-tion and the Lagrangian for the inflaton is

119871 = 120597120583120579120597120583120579 minus 119890119886

21205792 (9)

which gives quadratic inflation driven by the phase ofcomplex field Φ

In the above simple example given by (4) and (5)the field stabilization is obtained from the combination ofsupergravity correction 119890

119870 and the pole 1Φ in119882 besides anaccidental agreement that both the factor (1119903

2)1198901199032 and the

term (ln 119903)2 obtain their minima at 119903 = 1 while for more

general helical phase inflation such accidental agreementis not guaranteed For example one may get the followinginflaton potential

119881 = 119890119870(119903) 1

1199032((ln 119903

Λ)

2

+ 1205792) (10)

in which the coefficient 119890119870(119903)

(11199032) admits a minimum at

1199030 sim Λ but 1199030 = Λ In this case the coefficient 119890119870(119903)(11199032)still gives a mass above the Hubble scale for 119903 while ⟨119903⟩ isslightly shifted away from 1199030 in the early stage of inflation andafter inflation 119903 evolves toΛ rapidly Also the term (ln(119903Λ))

2

gives a small correction to the potential and inflationaryobservables so this correction is ignorable comparing withthe contributions from the super-Planckian valued phaseunless it is unexpectedly large

Potential Deformations In the Kahler potential there are cor-rections from the quantum loop effect while the superpoten-tial119882 is nonrenormalized Besides when coupledwith heavyfields the Kahler potential ofΦ receives corrections throughintegrating out the heavy fields Nevertheless because ofthe global 119880(1) symmetry in the Kahler potential thesecorrections can only affect the field stabilization while phaseinflation is not sensitive to these corrections

Given a higher order correction on the Kahler potential

119870 = ΦΦ + 119887 (ΦΦ)2+ 119883119883 minus 119892 (119883119883)

2 (11)

onemay introduce an extra parameterΛ in the superpotential

119882 = 119886119883

Φln Φ

Λ (12)

Based on the same argument it is easy to see the scalarpotential reduces to

119881 = 11988621198901199032+1198871199034 1

1199032((ln 119903 minus lnΛ)

2+ 1205792) (13)

The factor 1198901199032+1198871199034

(11199032) reaches its minimum at 11990320 = 2(1 +

radic1 + 8119887) below 119872119875 for 119887 gt 0 To get the ldquoaccidentalagreementrdquo it needs the parameter Λ = 1199030 and then inflationis still driven by the phase with exact quadratic potential

Without Λ the superpotential comes back to (5) and thescalar potential is shown in Figure 1 with 119887 = 01 Duringinflation ⟨119903⟩ ≃ 1199030 for small 119887 the term (ln 119903)

2 contribution to

1

1

0

0

minus1

minus1

V(102a2)

15

10

5

0

120601x

120601y

Figure 1 The helicoid structure of potential (13) scaled by 1021198862

In the graph parameters 119887 and Λ are set to be 119887 = 01 and Λ = 1respectively

the potential at the lowest order is proportional to 1198872 After

canonical field normalization the inflaton potential takes theform

119881 (120579) =1

21198982120579 (21198872+ 1205792) (14)

in which the higher order terms proportional to 1198873+119894 are

ignored With regard to the inflationary observations takethe tensor-to-scalar ratio r for example as

r =321205792119894

(1205792119894 + 21198872)

2asymp

8

119873(1 +

1198872

2119873)

minus2

(15)

where 120579119894 is the phase when inflation starts and 119873 isin (50 60)

is the 119890-folding number So the correction from higher orderterm is insignificant for 119887 lt 1

3 Monodromy in SupersymmetricField Theory

As discussed before phase inflation naturally leads to thephase monodromy (in mathematical sense) in the superpo-tential The phase monodromy requires singularity whichmeans the superpotential proposed for phase inflation shouldbe an effective theory It is preferred to show how such phasemonodromy appears from a more ldquofundamentalrdquo theory athigher scale In [20] the monodromy needed for phaseinflation is realized based on the supersymmetric field theoryin which the monodromy relates to the soft breaking of aglobal 119880(1) symmetry

Historically the monodromy inflation as an attractivemethod to realize super-Planckian field excursion was firstproposed in a more physical sense for axions arising from

Advances in High Energy Physics 5

string compactifications [42] In the inflaton potential theonly factor that changes during axion circular rotation isfrom the DBI action of branes In [44ndash46] the generalizedaxion alignment mechanisms are considered as particularrealization of axion monodromy with the potential fromnonperturbative effects We will show that such kind ofaxion monodromy can also be fulfilled by the superpotentialphase monodromy [20] even though it is not shown in theeffective superpotential after integrating out the heavy fieldsFurthermore all the extra fields can be consistently stabilized

The more ldquofundamentalrdquo field theory for the superpoten-tial in (5) is

1198820 = 120590119883Ψ (119879 minus 120575) + 119884 (119890minus120572119879

minus 120573Ψ) + 119885 (ΨΦ minus 120582) (16)

where the coupling constants for the second and third termsare taken to be 1 for simplicity and a small hierarchy isassumed between the first term and the last two terms thatis 120590 ≪ 1 The coupling 119884119890

minus120572119879 is assumed to be an effectivedescription of certain nonperturbative effects Similar formscan be obtained from 119863-brane instanton effect in type stringtheory (for a review see [48]) besides the coefficient 120572 prop

1119891 ≫ 1 in Planck unit since 119891 ≪ 1 is the decay constantand should be significantly lower than the Planck scale Forthe last two terms of1198820 there is a global 119880(1) symmetry

Ψ 997888rarr Ψ119890minus119894119902120579

Φ 997888rarr Φ119890119894119902120579

119884 997888rarr 119884119890119894119902120579

119879 997888rarr 119879 +119894119902120579

120572

(17)

which is anomalous and explicitly broken by the first termPhase rotation of the stabilizer field 119883 has no effect oninflation and so is ignored here The phase monodromy ofsuperpotential 119882 in (6) originates from the 119880(1) rotation of1198820

Ψ 997888rarr Ψ119890minus1198942120587

1198820 997888rarr 1198820 + 11989421205871205901

120572119883Ψ

(18)

As shown in [20] the supersymmetric field theory withsuperpotential1198820 admits the Minkowski vacuum at

⟨119883⟩ = ⟨119884⟩ = ⟨119885⟩ = 0

⟨119879⟩ = 120575

⟨Ψ⟩ =1

120573119890minus120572120575

⟨Φ⟩ = 120582120573119890120572120575

(19)

with ⟨Φ⟩ ≫ ⟨Ψ⟩ so that near the vacuum the masses of 119884119885 and Ψ are much larger than Φ besides the effective massof 119879 is also large near the vacuum due to large 120572 Large 120572

was an obstacle for natural inflation as it leads to the axiondecays too small for inflation while in this scenario large 120572

is helpful for phase inflation to stabilize the axion Thereforefor the physical process at scale below the mass scale of threeheavy fields the only unfixed degrees of freedom are 119883 andΦ which can be described by an effective field theory withthree heavy fields integrated out The coupling 120590119883Ψ(119879 minus 120575)

is designed for inflation and hierarchically smaller than theextra terms in1198820Therefore to describe inflation process theheavy fields need to be integrated out

To integrate out heavy fields we need to consider the 119865-terms again The 119865-term flatness of fields 119884 and 119885 gives

119865119884 = 119890minus120572119879

minus 120573Ψ + 1198701198841198820 = 0

119865119885 = ΨΦ minus 120582 + 1198701198851198820 = 0

(20)

Near the vacuum 119884 = 119885 asymp 0 ≪ 119872119875 the abovesupergravity corrections 119870119884(119885)1198820 are ignorable and thenthe 119865-term flatness conditions reduce to these for globalsupersymmetryThis is gained from the fact that although theinflation dynamics are subtle the inflation energy density isclose to the GUT scale far below the Planck scale Solvingthe 119865-term flatness equations in (20) we obtain the effectivesuperpotential119882 in (5)

Based on the above construction it is clear that the phasemonodromy in 119882 is from the 119880(1) transformation of 1198820and the pole of superpotential (5) at Φ = 0 arises from theintegration process The heavy fieldΨ is integrated out basedon the 119865-term flatness conditions when ⟨Φ⟩ ≫ ⟨Ψ⟩ whileif Φ rarr 0 Ψ becomes massless from |119865119885|

2 and it is illegalto integrate out a ldquomasslessrdquo field For inflation the condition⟨Φ⟩ ≫ ⟨Ψ⟩ is satisfied so the theory with superpotential (5)is reliable

As to the inflation term a question appears as global119880(1) is explicitly broken by the first term in (16) at inflationscale why is the phase light while the norm is much heavierThe supergravity correction to the scalar potential plays acrucial role at this stage The coefficient 119890119870 appears in thescalar potential and because of119880(1) symmetry in the Kahlerpotential the factor 119890119870 is invariant under119880(1) symmetry butincreases exponentially for a large norm Here the Kahlerpotential of 119879 should be shift invariant that is 119870 = 119870(119879 +

119879) instead of the minimal type Otherwise the exponentialfactor 119890119870 depends on phase as well and the phase rotationwillbe strongly fixed like the norm component or Re(119879)

When integrating out the heavy fields they should bereplaced both in superpotential and in Kahler potential bythe solutions from vanishing 119865-term equations So differentfrom the superpotential the Kahler potential obtained in thisway is slightly different from the minimal case given in (4)There are extra terms like

ΨΨ =1205822

1199032

119870 (119879 + 119879) = 119870(1

1205722(ln 119903)2)

(21)

where |Φ| = 119903 Nevertheless since 120582 ≪ 1 and 120572 ≫ 1 theseterms are rather small and have little effect on phase inflation

6 Advances in High Energy Physics

as shown in the last section Furthermore the quantumloop effects during integrating out heavy fields can introducecorrections to the Kahler potential as well However becauseof 119880(1) symmetry built in the Kahler potential these termstogether with (21) can only mildly affect the field stabiliza-tion and phase inflation is not sensitive to the correctionsin Kahler potential As to the superpotential it is protectedby the nonrenormalized theorem and free from radiativecorrections

4 Natural Inflation in Helical Phase Inflation

In the superpotential 1198820 the inflation term is perturbativecoupling of complex field119879which shifts under global119880(1) aninterestingmodification is to consider inflation given by non-perturbative coupling of 119879 Such term gives a modified 119880(1)

phase monodromy in the superpotential And it leads to nat-ural inflation as a special type of helical phase inflation withphase-axion alignment which is similar to the axion-axionalignment mechanism proposed in [34] for natural inflationwith super-Planckian axion decay constant Specifically itcan be shown that the phase monodromy realized in super-symmetric field theory has similar physical picture to themodified axion alignment mechanism provided in [44 45]

To realize natural inflation the superpotential 1198820 in (16)just needs to be slightly modified

1198821 = 120590119883Ψ(119890minus120572119879

minus 120575) + 119884 (119890minus120573119879

minus 120583Ψ)

+ 119885 (ΨΦ minus 120582)

(22)

in which 1 ≪ 120572 ≪ 120573 Again there is a global 119880(1)

symmetry in the last two terms of 1198821 and the fields transferunder119880(1) like in (17) The first term which is hierarchicallysmaller breaks the 119880(1) symmetry explicitly besides ashift symmetry of 119879 is needed in the Kahler potential Themonodromy of1198821 under a circular 119880(1) rotation is

Ψ 997888rarr Ψ119890minus1198942120587

1198821 997888rarr 1198821 + 120590119883Ψ119890minus120572119879

(119890minus2120587119894(120572120573)

minus 1)

(23)

The supersymmetric field theory given by (22) admits thefollowing supersymmetric Minkowski vacuum

⟨119883⟩ = ⟨119884⟩ = ⟨119885⟩ = 0

⟨119879⟩ = minus1

120572ln 120575

⟨Ψ⟩ =1

120583120575120573120572

⟨Φ⟩ = 120582120583120575minus120573120572

(24)

The parameters are set to satisfy conditions

120575120573120572

sim 120582120583

120582 ≪ 1

120575 lt 1

(25)

so that ⟨Ψ⟩ ≪ ⟨Φ⟩ and ⟨119879⟩ gt 0

Near vacuum fields 119884119885Ψ and 119879 obtain large effectivemasses above inflation scale while 119883Φ are much lighter Atthe inflation scale the heavy fields should be integrated outThe 119865-term flatness conditions for fields 119884 and 119885 are

119865119884 = 119890minus120573119879

minus 120583Ψ = 0

119865119885 = ΨΦ minus 120582 = 0

(26)

in which the supergravity corrections 1198701198841198851198821 are neglectedas both119884 and119885 get close to zero during inflation IntegratingoutΨ and 119879 from (26) we obtain the effective superpotential1198821015840 from1198821

1198821015840= 120590120582

119883

Φ((

120583120582

Φ)

120572120573

minus 120575) (27)

Given 120572 ≪ 120573 the effective superpotential contains a termwith fractional power Inflation driven by complex potentialwith fractional power was considered in [49] to get sufficientlarge axion decay constant Here the supersymmetric fieldmonodromy naturally leads to the superpotential with frac-tional power which arises from the small hierarchy of axiondecay constants in two nonperturbative terms

Helical phase inflation is described by the effective super-potential 1198821015840 the role of phase-axion alignment is not clearfrom119882

1015840 since it is hidden in the procedure of integrating outheavy fields

The 119865-term flatness conditions (26) fix four degrees offreedom for the extra degree of freedom they correspond tothe transformations free from constraints (26)

Ψ 997888rarr Ψ119890minus119906minus119894V

Φ 997888rarr Φ119890119906+119894V

119879 997888rarr 119879 +119906

120573+

119894V120573

(28)

Parameter 119906 corresponds to the norm variation of complexfield Φ which is fixed by the supergravity correction on thescalar potential 119890119870(ΦΦ) Parameter V relates to 119880(1) transfor-mation which leads to the phase monodromy from the firstterm of1198821The scalar potential including the inflation termdepends on the superpositions among phases ofΨΦ and theaxion Im(119879) which are constrained as in (28) Among thesefields the phase ofΦ has the lightestmass after canonical fieldnormalization Similar physical picture appears in the axion-axion alignment where inflation is triggered by the axionsuperposition along the flat direction

After integrating out the heavy fields the Kahler potentialis

119870 = ΦΦ +1205822

ΦΦ+ sdot sdot sdot (29)

As 120582 ≪ 1 the Kahler potential is dominated by ΦΦ asdiscussed before the extra terms like 1205822ΦΦ only give small

Advances in High Energy Physics 7

corrections to the field stabilization Helical phase inflationcan be simply described by the supergravity

119870 = ΦΦ + 119883119883 +

119882 = 119886119883

Φ(Φminus119887

minus 119888)

(30)

where 119887 = 120572120573 ≪ 1 and 119888 asymp 1 The scalar potential given bythe above Kahler potential and superpotential is

119881 = 1198901199032 1198862

1199032(119903minus2119887

+ 1198882minus 2119888119903minus119887 cos (119887120579)) (31)

in which we have usedΦ equiv 119903119890119894120579 As usual the norm 119903 couples

with the phase in the scalar potential To generate singlefield inflation the norm component needs to be stabilizedwith heavy mass above the Hubble scale while the phasecomponent remains light Due to the phase-norm couplingthe phase and norm are likely to obtain comparable effectivemasses during inflation so it is difficult to realize singlefield inflation However it is a bit different in helical phaseinflation The above scalar potential can be rewritten asfollows

119881 = 1198901199032 1198862

1199032(119903minus119887

minus 119888)2+ 1198901199032 41198862119888

1199032+119887(sin 119887

2120579)

2

(32)

So its vacuum locates at ⟨119903⟩ = 1199030 = 119888minus1119887 and 120579 = 0 Besides

the coefficient of the phase term 1198901199032(41198881199032+119887

) reaches itsminimal value at 1199031 = radic1 + 1198872 For 119888 asymp 1 we have theapproximation 1199030 asymp 1199031 asymp 1 The extra terms in 119870 givesmall corrections to 1199030 and 1199031 but the approximation is stillvalid So with the parameters in (25) the norm 119903 in thetwo terms of scalar potential 119881 can be stabilized at the closeregion 119903 asymp 1 separately If the parameters are tuned so that1199030 = 1199031 then the norm of complex field is strictly stabilizedat the vacuum value during inflation Without such tuninga small difference between 1199030 and 1199031 is expected but theshift of 119903 during inflation is rather small and inflation is stillapproximate to the single field inflation driven by the phaseterm prop (sin(1198872)120579)2 The helicoid structure of the potential(31) is shown in Figure 2

It is known that to realize aligned axion mechanism insupergravity [50] it is very difficult to stabilize the moduli asthey couple with the axions In [40] the moduli stabilizationis fulfilled with gauged anomalous 119880(1) symmetries since119880(1) 119863-terms only depend on the norm |Φ| or Re(119879) andthen directly separate the norms and phases of matter fieldsIn helical phase inflation the modulus and matter fieldsexcept the phase are stabilized at higher scale or by thesupergravity scalar potential Only the phase can be aninflaton candidate because of the protection from the global119880(1) symmetry in Kahler potential and approximate 119880(1)

symmetry in superpotential

5 Helical Phase Inflation and the KYY Model

120578 problem in supergravity inflation can be solved both inhelical phase inflation and in the shift symmetry in KYY

1

1

00

minus1

minus1

10

8

6

4

V(a

2)

120601x120601y

Figure 2 The helicoid structure of potential (31) scaled by 1198862 The

parameters with 119888 = 096 and 119887 = 01 are adopted in the graph Itis shown that the local valley locates around 119903 asymp 1 Note that thepotential gets flatter at the top of the graph

model The physics in these two solutions are obviously dif-ferent For helical phase inflation the solution employs 119880(1)

symmetry in Kahler potential of the minimal supergravityand the superpotential admits a phase monodromy arisingfrom the global 119880(1) transformation

119870 = ΦΦ + 119883119883 + sdot sdot sdot

119882 = 119886119883

ΦlnΦ

(33)

The inflation is driven by the phase of complex field Φ

and several special virtues appear in the model For theKYY model with shift symmetry [6] the Kahler potential isadjusted so that it admits a shift symmetry along the directionof Im(119879)

119870 =1

2(119879 + 119879)

2+ 119883119883 + sdot sdot sdot

119882 = 119886119883119879

(34)

and inflation is driven by Im(119879) The shift symmetry isendowedwith axions so thismechanism is attractive for axioninflation

Here we will show that although the physical pictures aremuch different in helical phase inflation andKYY typemodeljust considering the lower order terms in the Kahler potentialof redefined complex field helical phase inflation can reduceto the KYY model

Because the phase of Φ rotates under the global 119880(1)

transformation to connect helical phase inflation with theKYY model a natural guess is to take the following fieldredefinitionΦ = 119890

119879 then helical phase inflation (33) becomes

119870 = 119890119879+119879

+ sdot sdot sdot = 1 + 119879 + 119879 +1

2(119879 + 119879)

2+ sdot sdot sdot

119882 = 119886119883119879119890minus119879

(35)

8 Advances in High Energy Physics

TheKahler manifold of complex field119879 is invariant under theholomorphic Kahler transformation

119870(119879 119879) 997888rarr 119870(119879 119879) + 119865 (119879) + 119865 (119879) (36)

in which 119865(119879) is a holomorphic function of 119879 To keepthe whole Lagrangian also invariant under the Kahler trans-formation the superpotential transforms under the Kahlertransformation

119882 997888rarr 119890minus119865(119879)

119882 (37)

For the supergravity model in (35) taking the Kahler trans-formation with 119865(119879) = minus(12) minus 119879 the Kahler potential andsuperpotential become

119870 =1

2(119879 + 119879)

2+ sdot sdot sdot

119882 = 119886radic119890119883119879

(38)

which is just the KYY model (34) with higher order cor-rections in the Kahler potential These higher order termsvanish after field stabilization and have no effect on inflationprocess The field relation Φ = 119890

119879 also gives a map betweenthe simplest helical phase inflation and the KYY model suchas the inflaton arg(Φ) rarr Im(119879) and field stabilization|Φ| = 1 rarr Re(119879) = 0

Nevertheless helical phase inflation is not equivalent tothe KYY model There are higher order corrections to theKahler potential in the map from helical phase inflation toKYY model which have no effect on inflation after fieldstabilization but indicate different physics in two models Bydropping these terms certain information is lost so the mapis irreversible Specifically the inverse function 119879 = lnΦ ofthe field redefinition Φ = 119890

119879 cannot reproduce helical phaseinflation from the KYY model unless one introduces higherdimensional operators in the Kahler potential exactly thesame as given in the expansion (35) Such strict constraint oneach higher dimensional operator is not guaranteed by anyknown rule and so is unphysical As there is no pole or singu-larity in the KYY model it is unlikely to introduce pole andsingularity at origin with phase monodromy through well-defined field redefinition Actually the singularity with phasemonodromy in the superpotential indicates rich physics inthe scale above inflation

6 Helical Phase Inflation inNo-Scale Supergravity

The no-scale supergravity is an attractive frame for GUTscale phenomenology it is interesting to realize helicalphase inflation in no-scale supergravity Generally theKahler manifold of the no-scale supergravity is equippedwith 119878119880(119873 1)119878119880(119873) times 119880(1) symmetry For the no-scalesupergravity with exact 119878119880(1 1)119880(1) symmetry withoutextra fields no inflation can be realized so the case with119878119880(2 1)119878119880(2) times 119880(1) symmetry is the simplest one thatadmits inflation

The Kahler potential with 119878119880(2 1)119878119880(2)times119880(1) symme-try is

119870 = minus3 ln(119879 + 119879 minusΦΦ

3) (39)

In the symmetry of the Kahler manifold there is 119880(1)

subsector the phase rotation of complex fieldΦ which can beemployed for helical phase inflation Besides the modulus 119879should be stabilized during inflation which can be fulfilled byintroducing extra terms on119879 As a simple example of no-scalehelical phase inflation here we follow the simplification in [9]that the modulus 119879 has already been stabilized at ⟨119879⟩ = 119888Different from the minimal supergravity the kinetic termgiven by the no-scale Kahler potential is noncanonical

119871119870 = 119870ΦΦ120597120583Φ120597120583Φ

=2119888

(2119888 minus 11990323)2(120597120583119903120597

120583119903 + 1199032120597120583120579120597120583120579)

(40)

in which Φ equiv 119903119890119894120579 is used The 119865-term scalar potential is

119881 = 119890minus(23)119870 1003816100381610038161003816119882Φ

1003816100381610038161003816

2=

1003816100381610038161003816119882Φ1003816100381610038161003816

2

(119879 + 119879 minus ΦΦ3)2 (41)

where the superpotential 119882 is a holomorphic function ofsuperfieldΦ and119879 = ⟨119879⟩ = 119888 It requires a phasemonodromyin superpotential119882 for phase inflation the simple choice is

119882 =119886

Φln Φ

Λ (42)

The scalar potential given by this superpotential is

119881 =91198862

(6119888 minus 1199032)21199034

((ln 119903

119890Λ)

2

+ 1205792) (43)

As in the minimal supergravity the norm and phase ofcomplex field Φ are separated in the scalar potential For the119903-dependent coefficient factor 1(6119888 minus 119903

2)21199034 its minimum

locates at 1199030 = radic3119888 and another term (ln(119903119890Λ))2 reaches its

minimum at 1199031 = 119890Λ Given that the parameter Λ is tuned sothat 1199030 = 1199031 in the radial direction the potential has a globalminimum at 119903 = 1199030 Similar to helical phase inflation in theminimal supergravity the potential (43) also shows helicoidstructure as presented in Figure 3

The physical mass of 119903 in the region near the vacuum is

1198982119903 =

(2119888 minus 11990323)2

4119888

1205972119881

1205971199032

100381610038161003816100381610038161003816100381610038161003816100381610038161003816119903=radic3119888

= (4 +1

21205792)1198672 (44)

where119867 is theHubble constant during inflation So the norm119903 is strongly stabilized at 1199030 If 1199030 and 1199031 are not equal butclose to each other 119903 will slightly shift during inflation butits mass remains above the Hubble scale and inflation is stillapproximately the single field inflation

Advances in High Energy Physics 9

30

20

10

0

minus10

minus10

minus05

minus05

0000

05

05

10

10

120601x (radic

3c) 120601 y(radic3

c)

V(a

2 c4 )

Figure 3The helicoid structure of potential (43) scaled by 11988621198884 Inthe graph the parameter Λ is tuned so that 1199030 = 1199031 The complexfieldΦ has been rescaled byradic3119888 and the scale of field norm at localvalley is determined by the parameter 119888 instead of the Planck mass

Instead of stabilizing 119879 independently with the inflationprocess we can realize helical phase inflation from the no-scale supergravity with dynamical119879There is a natural reasonfor such consideration The phase monodromy requiresmatter fields transforming as rotations under 119880(1) and alsoa modulus 119879 as a shift under 119880(1) The Kahler potential of 119879is shift invariant Interestingly for the no-scale supergravitywith 119878119880(2 1)119878119880(2) times 119880(1) symmetry as shown in (39) theKahler potential 119870 is automatically endowed with the shiftsymmetry and global 119880(1) symmetry

119879 997888rarr 119879 + 119894119862

Φ 997888rarr Φ119890119894120579

(45)

Therefore the no-scale Kahler potential fits with the phasemonodromy in (16) and (22) initiatively

The Kahler potentials of superfields 119911 isin 119883 119884 119885 areof the minimal type 119911119911 while for Ψ its Kahler potentialcan be the minimal type ΨΨ or the no-scale type 119870 =

minus3 ln(119879+119879minus (ΦΦ+ΨΨ)3) which extends the symmetry ofKahler manifold to 119878119880(3 1)119878119880(3) times 119880(1) In this scenariothe process to integrate out heavy fields is the same asbefore Besides the potential of phase proportional to |119882119883|

2

is insensitive to the formula of Kahler potential due to119880(1) symmetry The major difference appears in the fieldstabilization The scalar potential from phase monodromy in(16) is

119881 = 119890119870 1003816100381610038161003816119882119883

1003816100381610038161003816

2=

1

(ln 1199032120572 + 119888 minus 11990323 minus 120582211990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792) asymp

1

(119888 minus 11990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792)

(46)

with 120572 ≫ 1 and 120582 ≪ 1 The field norm 119903 is stabilized byminimizing the coefficient 11199032(119888 minus 119903

23)3 and the parameter

119888 from ⟨119879⟩ determines the scale of ⟨119903⟩ = radic31198882 This isdifferent from helical phase inflation in the minimal super-gravity which minimizes the norm based on the coefficient1198901199032(11199032) and then the scale of ⟨119903⟩ is close to119872119875 the unique

energy scale of supergravity corrections Similarly combiningthe no-scale Kahlermanifold with phasemonodromy in (22)we can obtain natural inflation

7 UV-Sensitivity and Helical Phase Inflation

For the slow-roll inflation the Lyth bound [21] provides arelationship between the tensor-to-scalar ratio r and the fieldexcursion Δ120601 Roughly it requires

Δ120601

119872119875

⩾ (r

001)

12

(47)

To get large tensor-to-scalar ratio r ⩾ 001 such as in chaoticinflation or natural inflation the field excursion should bemuch larger than the Planck mass The super-Planckianfield excursion makes the description based on the effectivefield theory questionable In the Wilsonian sense the lowenergy field theory is an effective theory with higher ordercorrections introduced by the physics at the cut-off scalelike quantum gravity and these terms are irrelevant in theeffective field theory since they are suppressed by the cut-offenergy scale However for the inflation process the inflatonhas super-Planckian field excursion which is much largerthan the cut-off scaleThus the higher order terms cannot besuppressed by the Planckmass andmay affect the inflationaryobservations significantly And then they may significantlyaffect inflation or even destroy the inflation process Forexample considering the corrections

Δ119881 = 119888119894119881(120601

119872119875

)

119894

+ sdot sdot sdot (48)

to the original inflaton potential 119881 as long as 119888119894 are of theorder 10minus119894 in the initial stage of inflation 120601 sim 119874(10)119872119875 thehigher order terms can be as large as the original potential119881 So for large field inflation it is sensitive to the physics atthe cut-off scale and the predictions of inflation just based oneffective field theory are questionable

In consideration of the UV sensitivity of large fieldinflation a possible choice is to realize inflation in UV-completed theory like string theory (for a review see [22])To realize inflation in string theory there are a lot of problemsto solve besides inflation such as the moduli stabilizationMinkowskide Sitter vacua and effects of 1205721015840- and string loop-corrections Alternatively in the bottom-up approach onemay avoid the higher order corrections by introducing anextra shift symmetry in the theory The shift symmetry istechnically natural and safe under quantum loop-correctionsHowever the global symmetry can be broken by the quantumgravity effect So it is still questionable whether the shiftsymmetry can safely evade the higher corrections like in (48)

10 Advances in High Energy Physics

The UV-completion problem is dodged in helical phaseinflation Since the super-Planckian field excursion is thephase of a complex field and the phase component is notdirectly involved in the gravity interaction there are nodangerous high-order corrections like in (48) for the phasepotentialThe inflaton evolves along the helical trajectory anddoes not relate to the physics in the region above the Planckscale For helical phase inflation in the minimal supergravitythe norm of field is stabilized at the marginal point of thePlanck scale where the supergravity correction on the scalarpotential gets important based onwhich the normof complexfield can be strongly stabilized The extra corrections arelikely to appear in the Kahler potential however they canonly slightly affect the field stabilization while the phaseinflation is protected by the global 119880(1) symmetry in theKahler potential in consequence helical phase inflation isnot sensitive to these corrections at all For the helical phaseinflation in no-scale supergravity the normof complex field isstabilized at the scale of themodulus ⟨119879⟩ instead of the Planckscale one can simply adjust the scale of ⟨119879⟩ to keep themodelaway from super-Planckian region

Helical phase inflation is free from the UV-sensitivityproblem and it is just a typical physical process at theGUT scale with special superpotential that admits phasemonodromy So it provides an inflationary model that can bereliably studied just in supersymmetric field theory

8 Discussions and Conclusion

In this work we have studied the details of helical phase infla-tion from several aspects Helical phase inflation is realizedin supergravity with global 119880(1) symmetry 119880(1) symmetryis built in the Kahler potential so that helical phase inflationcan be realized by the ordinary Kahler potentials such as theminimal or no-scale types Helical phase inflation directlyleads to the phasemonodromy in the superpotential which issingular and an effective field theory arising from integratingout heavy fieldsThe phasemonodromy originates from119880(1)

rotation of the superpotential at higher scale Generically thesuperpotential can be separated into two parts119882 = 119882119868 +119882119878where119882119878 admits the global119880(1) symmetrywhile119882119868 breaks itexplicitly at scale much lower than 119882119878 Under 119880(1) rotationthe inflation term 119882119868 is slightly changed which realizes thephase monodromy in the effective theory and introducesa flat potential along the direction of phase rotation Bybreaking the global119880(1) symmetry in different ways we mayget different kinds of inflation such as quadratic inflationnatural inflation or the other types of inflation that are notpresented in this work

An amazing fact of helical phase inflation is that itdeeply relates to several interesting points of inflation andnaturally combines them in a rather simple potential withhelicoid structure The features of helical phase inflation canbe summarized as follows

(i) The global 119880(1) symmetry is built in the Kahlerpotential so helical phase inflation provides a naturalsolution to 120578 problem

(ii) The phase excursion requires phase monodromy inthe superpotential So helical phase inflation providesin the mathematical sense a new type of monodromyin supersymmetric field theory

(iii) The singularity in the superpotential together withthe supergravity scalar potential provides strong fieldstabilization which is consistent with phase inflation

(iv) The super-Planckian field excursion is realized bythe phase of a complex field instead of any otherldquophysicalrdquo fields that directly couple with gravity Sothere are no polynomial higher order corrections forthe phase and thus inflation is not sensitive to thequantum gravity corrections

To summarize helical phase inflation introduces a newtype of inflation that can be effectively described by super-symmetric field theory at the GUT scale Generically thesuper-Planckian field excursion makes the inflationary pre-dictions based on effective field theory questionable sincethe higher order corrections from quantum gravity arelikely to affect the inflation process significantly One of thesolutions is to realize inflation in a UV-completed theory likestring theory nevertheless there are many difficult issues instring theory to resolve before realizing inflation completelyHelical phase inflation is another simple solution to the UV-sensitivity problem It is based on the supersymmetric fieldtheory and the physics are clear and much easier to controlFurthermore helical phase inflation makes the unification ofinflation theory with GUT more natural since both of themare triggered at the scale of 1016 GeV and can be effectivelystudied based on supersymmetric field theory

Besides we have shown that helical phase inflation alsorelates to several interesting developments in inflation theoryIt can be easily modified for natural inflation and realizethe phase-axion alignment indirectly which is similar tothe axion-axion alignment mechanism for super-Planckianaxion decay constant [34] The phase-axion alignment isnot shown in the final supergravity model which exhibitsexplicit phase monodromy only However the phase-axionalignment is hidden in the process when the heavy fields areintegrating out For 120578 problem in the supergravity inflationthere is another well-known solution the KYY model withshift symmetry We showed that through field redefinitionhelical phase inflation given by (4) and (5) reduces to theKYYmodel where the higher order corrections in the Kahlerpotential have no effect on inflation process However thereis no inverse transformation from the KYY model to helicalphase inflation since no well-defined field redefinition canintroduce the pole and phase singularity needed for phasemonodromy Helical phase inflation can be realized in no-scale supergravity The no-scale Kahler potential automati-cally provides the symmetry needed by phase monodromyIn the no-scale supergravity the norm of complex field isstabilized at the scale of the modulus

Our inflation models are constructed within the super-gravity theory with global 119880(1) symmetry broken explicitlyby the subleading order superpotential term So it is justtypical GUT scale physics and indicates that a UV-completed

Advances in High Energy Physics 11

framework seems to be not prerequisite to effectively describesuch inflation process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work of DVN was supported in part by the DOEGrant DE-FG03-95-ER-40917 The work of Tianjun Li issupported in part by theNatural Science Foundation of ChinaunderGrants nos 10821504 11075194 11135003 11275246 and11475238 and by the National Basic Research Program ofChina (973 Program) under Grant no 2010CB833000

References

[1] A H Guth ldquoInflationary universe a possible solution to thehorizon and flatness problemsrdquo Physical Review D vol 23 no2 pp 347ndash356 1981

[2] A D Linde ldquoColeman-Weinberg theory and the new inflation-ary universe scenariordquo Physics Letters B vol 114 no 6 pp 431ndash435 1982

[3] A Albrecht and P J Steinhardt ldquoCosmology for grand unifiedtheories with radiatively induced symmetry breakingrdquo PhysicalReview Letters vol 48 no 17 pp 1220ndash1223 1982

[4] P A R Ade N Aghanim C Armitage-Caplan et al ldquoPlanck2013 results XXII Constraints on inflationrdquo Astronomy ampAstrophysics vol 571 article A22 42 pages 2014

[5] P A R Ade R W Aikin D Barkats et al ldquoDetection of B-mode polarization at degree angular scales by BICEP2rdquo PhysicalReview Letters vol 112 no 24 Article ID 241101 2014

[6] M Kawasaki M Yamaguchi and T Yanagida ldquoNatural chaoticinflation in supergravityrdquo Physical Review Letters vol 85 no 17pp 3572ndash3575 2000

[7] T Li Z Li and D V Nanopoulos ldquoSupergravity inflation withbroken shift symmetry and large tensor-to-scalar ratiordquo Journalof Cosmology and Astroparticle Physics vol 2014 no 2 article028 2014

[8] K Harigaya and T T Yanagida ldquoDiscovery of large scale tensormode and chaotic inflation in supergravityrdquo Physics Letters Bvol 734 pp 13ndash16 2014

[9] J Ellis D V Nanopoulos and K A Olive ldquoNo-scale supergrav-ity realization of the starobinsky model of inflationrdquo PhysicalReview Letters vol 111 no 11 Article ID 111301 2013

[10] T Li Z Li and D V Nanopoulos ldquoNo-scale ripple inflationrevisitedrdquo Journal of Cosmology and Astroparticle Physics vol04 article 018 2014

[11] C Pallis ldquoInduced-gravity inflation in no-scale supergravityand beyondrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 8 article 057 2014

[12] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky modelrdquo Fortschritte der Physik vol 62 no 7 pp 573ndash5832014

[13] J Ellis M A G Garca D V Nanopoulos and K A OliveldquoResurrecting quadratic inflation in no-scale supergravity inlight of BICEP2rdquo Journal of Cosmology andAstroparticle Physicsvol 2014 no 5 article 037 2014

[14] T Li Z Li and D V Nanopoulos ldquoChaotic inflation in no-scale supergravitywith string inspiredmoduli stabilizationrdquoTheEuropean Physical Journal C vol 75 no 2 article 55 2015

[15] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky model and higher curvature correctionsrdquo Fortschritte derPhysik vol 63 no 1 pp 2ndash11 2015

[16] S V Ketov andT Terada ldquoInflation in supergravity with a singlechiral superfieldrdquo Physics Letters B vol 736 pp 272ndash277 2014

[17] C Kounnas D Lust and N Toumbas ldquoR2 inflation fromscale invariant supergravity and anomaly free superstrings withfluxesrdquo Fortschritte der Physik vol 63 no 1 pp 12ndash35 2015

[18] G A Diamandis B C Georgalas K Kaskavelis PKouroumalou A B Lahanas and G Pavlopoulos ldquoInflationin R2 supergravity with non-minimal superpotentialsrdquo PhysicsLetters B vol 744 pp 74ndash81 2015

[19] C Pallis and Q Shafi ldquoGravity waves from non-minimalquadratic inflationrdquo Journal of Cosmology and AstroparticlePhysics vol 2015 no 3 article 023 2015

[20] T Li Z Li and D V Nanopoulos ldquoHelical phase inflationrdquoPhysical Review D vol 91 no 6 Article ID 061303 2015

[21] D H Lyth ldquoWhat would we learn by detecting a gravitationalwave signal in the cosmic microwave background anisotropyrdquoPhysical Review Letters vol 78 no 10 article 1861 1997

[22] D Baumann and L McAllister ldquoInflation and stringtheoryrdquohttparxivorgabs14042601

[23] F Marchesano G Shiu and A M Uranga ldquoF-term axionmonodromy inflationrdquo Journal of High Energy Physics vol 2014no 9 article 184 2014

[24] C Long L McAllister and P McGuirk ldquoAligned naturalinflation in string theoryrdquo Physical Review D vol 90 no 2Article ID 023501 2014

[25] X Gao T Li and P Shukla ldquoCombining universal and odd RRaxions for aligned natural inflationrdquo Journal of Cosmology andAstroparticle Physics vol 2014 no 10 article 048 2014

[26] I Ben-Dayan F G Pedro and A Westphal ldquoTowards naturalinflation in string theoryrdquo Physical Review D vol 92 no 2Article ID 023515 2015

[27] R Blumenhagen D Herschmann and E Plauschinn ldquoThechallenge of realizing F-term axion monodromy inflation instring theoryrdquo Journal of High Energy Physics vol 2015 article7 2015

[28] H Abe T Kobayashi and H Otsuka ldquoTowards natural infla-tion from weakly coupled heterotic string theoryrdquo Progress ofTheoretical and Experimental Physics Article ID 063E02 2015

[29] T Ali S S Haque and V Jejjala ldquoNatural inflation from nearalignment in heterotic string theoryrdquo Physical Review D vol 91no 8 Article ID 083516 9 pages 2015

[30] R Flauger LMcAllister E Silverstein and AWestphal ldquoDrift-ing oscillations in axionmonodromyrdquo httparxivorgabs14121814

[31] K Freese J A Frieman andAVOlinto ldquoNatural inflationwithpseudo nambu-goldstone bosonsrdquo Physical Review Letters vol65 no 26 pp 3233ndash3236 1990

[32] F C Adams J R Bond K Freese J A Frieman and AV Olinto ldquoNatural inflation particle physics models power-law spectra for large-scale structure and constraints from thecosmic background explorerrdquo Physical Review D vol 47 no 2pp 426ndash455 1993

[33] G German A Mazumdar and A Perez-Lorenzana ldquoAngularinflation from supergravityrdquo Modern Physics Letters A vol 17no 25 pp 1627ndash1634 2002

12 Advances in High Energy Physics

[34] J E Kim H P Nilles and M Peloso ldquoCompleting naturalinflationrdquo Journal of Cosmology and Astroparticle Physics vol0501 article 005 2005

[35] D Baumann and D Green ldquoInflating with baryonsrdquo Journal ofHigh Energy Physics vol 2011 no 4 article 071 2011

[36] JMcDonald ldquoSub-Planckian two-field inflation consistent withthe Lyth boundrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 9 article 027 2014

[37] J McDonald ldquoA minimal sub-Planckian axion inflation modelwith large tensor-to-scalar ratiordquo Journal of Cosmology andAstroparticle Physics vol 2015 no 1 article 018 2015

[38] C D Carone J Erlich A Sensharma and Z Wang ldquoTwo-fieldaxion-monodromy hybrid inflation model Dantersquos WaterfallrdquoPhysical ReviewD vol 91 no 4 Article ID 043512 8 pages 2015

[39] G Barenboim and W-I Park ldquoSpiral Inflationrdquo Physics LettersB vol 741 pp 252ndash255 2015

[40] T Li Z Li andDVNanopoulos ldquoAligned natural inflation andmoduli stabilization from anomalous U(1) gauge symmetriesrdquoJournal of High Energy Physics vol 2014 no 11 article 012 2014

[41] T Li Z Li and D V Nanopoulos ldquoNatural inflation withnatural trans-planckian axion decay constant from anomalousU(1)119883rdquo Journal of High Energy Physics vol 2014 no 7 article052 2014

[42] E Silverstein and A Westphal ldquoMonodromy in the CMBgravity waves and string inflationrdquo Physical Review D vol 78no 10 Article ID 106003 21 pages 2008

[43] L McAllister E Silverstein and A Westphal ldquoGravity wavesand linear inflation from axion monodromyrdquo Physical ReviewD vol 82 no 4 Article ID 046003 2010

[44] K Choi H Kim and S Yun ldquoNatural inflation with multiplesub-Planckian axionsrdquo Physical Review D vol 90 no 2 ArticleID 023545 2014

[45] S-H H Tye and S S C Wong ldquoHelical inflation and cosmic-stringsrdquo httparxivorgabs14046988

[46] R Kappl S Krippendorf and H P Nilles ldquoAligned naturalinflation monodromies of two axionsrdquo Physics Letters B vol737 pp 124ndash128 2014

[47] G rsquotHooft ldquoNaturalness chiral symmetry and spontaneouschiral symmetry breakingrdquo in Recent Developments in GaugeTheories G rsquotHooft C Itzykson A Jaffe et al Eds vol 59of NATO Advanced Study Institutes Series pp 135ndash157 PlenumPress New York NY USA 1980

[48] R Blumenhagen M Cvetic S Kachru and T Weigand ldquoD-brane instantons in type II string theoryrdquo Annual Review ofNuclear and Particle Science vol 59 pp 269ndash296 2009

[49] K Harigaya andM Ibe ldquoSimple realization of inflaton potentialon a Riemann surfacerdquo Physics Letters B vol 738 pp 301ndash3042014

[50] R Kallosh A Linde and B Vercnocke ldquoNatural inflation insupergravity and beyondrdquo Physical Review D vol 90 Article ID041303(R) 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

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Journal of

Biophysics

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ThermodynamicsJournal of

4 Advances in High Energy Physics

therefore the norm is strongly stabilized at 119903 = 1 during infla-tion and the Lagrangian for the inflaton is

119871 = 120597120583120579120597120583120579 minus 119890119886

21205792 (9)

which gives quadratic inflation driven by the phase ofcomplex field Φ

In the above simple example given by (4) and (5)the field stabilization is obtained from the combination ofsupergravity correction 119890

119870 and the pole 1Φ in119882 besides anaccidental agreement that both the factor (1119903

2)1198901199032 and the

term (ln 119903)2 obtain their minima at 119903 = 1 while for more

general helical phase inflation such accidental agreementis not guaranteed For example one may get the followinginflaton potential

119881 = 119890119870(119903) 1

1199032((ln 119903

Λ)

2

+ 1205792) (10)

in which the coefficient 119890119870(119903)

(11199032) admits a minimum at

1199030 sim Λ but 1199030 = Λ In this case the coefficient 119890119870(119903)(11199032)still gives a mass above the Hubble scale for 119903 while ⟨119903⟩ isslightly shifted away from 1199030 in the early stage of inflation andafter inflation 119903 evolves toΛ rapidly Also the term (ln(119903Λ))

2

gives a small correction to the potential and inflationaryobservables so this correction is ignorable comparing withthe contributions from the super-Planckian valued phaseunless it is unexpectedly large

Potential Deformations In the Kahler potential there are cor-rections from the quantum loop effect while the superpoten-tial119882 is nonrenormalized Besides when coupledwith heavyfields the Kahler potential ofΦ receives corrections throughintegrating out the heavy fields Nevertheless because ofthe global 119880(1) symmetry in the Kahler potential thesecorrections can only affect the field stabilization while phaseinflation is not sensitive to these corrections

Given a higher order correction on the Kahler potential

119870 = ΦΦ + 119887 (ΦΦ)2+ 119883119883 minus 119892 (119883119883)

2 (11)

onemay introduce an extra parameterΛ in the superpotential

119882 = 119886119883

Φln Φ

Λ (12)

Based on the same argument it is easy to see the scalarpotential reduces to

119881 = 11988621198901199032+1198871199034 1

1199032((ln 119903 minus lnΛ)

2+ 1205792) (13)

The factor 1198901199032+1198871199034

(11199032) reaches its minimum at 11990320 = 2(1 +

radic1 + 8119887) below 119872119875 for 119887 gt 0 To get the ldquoaccidentalagreementrdquo it needs the parameter Λ = 1199030 and then inflationis still driven by the phase with exact quadratic potential

Without Λ the superpotential comes back to (5) and thescalar potential is shown in Figure 1 with 119887 = 01 Duringinflation ⟨119903⟩ ≃ 1199030 for small 119887 the term (ln 119903)

2 contribution to

1

1

0

0

minus1

minus1

V(102a2)

15

10

5

0

120601x

120601y

Figure 1 The helicoid structure of potential (13) scaled by 1021198862

In the graph parameters 119887 and Λ are set to be 119887 = 01 and Λ = 1respectively

the potential at the lowest order is proportional to 1198872 After

canonical field normalization the inflaton potential takes theform

119881 (120579) =1

21198982120579 (21198872+ 1205792) (14)

in which the higher order terms proportional to 1198873+119894 are

ignored With regard to the inflationary observations takethe tensor-to-scalar ratio r for example as

r =321205792119894

(1205792119894 + 21198872)

2asymp

8

119873(1 +

1198872

2119873)

minus2

(15)

where 120579119894 is the phase when inflation starts and 119873 isin (50 60)

is the 119890-folding number So the correction from higher orderterm is insignificant for 119887 lt 1

3 Monodromy in SupersymmetricField Theory

As discussed before phase inflation naturally leads to thephase monodromy (in mathematical sense) in the superpo-tential The phase monodromy requires singularity whichmeans the superpotential proposed for phase inflation shouldbe an effective theory It is preferred to show how such phasemonodromy appears from a more ldquofundamentalrdquo theory athigher scale In [20] the monodromy needed for phaseinflation is realized based on the supersymmetric field theoryin which the monodromy relates to the soft breaking of aglobal 119880(1) symmetry

Historically the monodromy inflation as an attractivemethod to realize super-Planckian field excursion was firstproposed in a more physical sense for axions arising from

Advances in High Energy Physics 5

string compactifications [42] In the inflaton potential theonly factor that changes during axion circular rotation isfrom the DBI action of branes In [44ndash46] the generalizedaxion alignment mechanisms are considered as particularrealization of axion monodromy with the potential fromnonperturbative effects We will show that such kind ofaxion monodromy can also be fulfilled by the superpotentialphase monodromy [20] even though it is not shown in theeffective superpotential after integrating out the heavy fieldsFurthermore all the extra fields can be consistently stabilized

The more ldquofundamentalrdquo field theory for the superpoten-tial in (5) is

1198820 = 120590119883Ψ (119879 minus 120575) + 119884 (119890minus120572119879

minus 120573Ψ) + 119885 (ΨΦ minus 120582) (16)

where the coupling constants for the second and third termsare taken to be 1 for simplicity and a small hierarchy isassumed between the first term and the last two terms thatis 120590 ≪ 1 The coupling 119884119890

minus120572119879 is assumed to be an effectivedescription of certain nonperturbative effects Similar formscan be obtained from 119863-brane instanton effect in type stringtheory (for a review see [48]) besides the coefficient 120572 prop

1119891 ≫ 1 in Planck unit since 119891 ≪ 1 is the decay constantand should be significantly lower than the Planck scale Forthe last two terms of1198820 there is a global 119880(1) symmetry

Ψ 997888rarr Ψ119890minus119894119902120579

Φ 997888rarr Φ119890119894119902120579

119884 997888rarr 119884119890119894119902120579

119879 997888rarr 119879 +119894119902120579

120572

(17)

which is anomalous and explicitly broken by the first termPhase rotation of the stabilizer field 119883 has no effect oninflation and so is ignored here The phase monodromy ofsuperpotential 119882 in (6) originates from the 119880(1) rotation of1198820

Ψ 997888rarr Ψ119890minus1198942120587

1198820 997888rarr 1198820 + 11989421205871205901

120572119883Ψ

(18)

As shown in [20] the supersymmetric field theory withsuperpotential1198820 admits the Minkowski vacuum at

⟨119883⟩ = ⟨119884⟩ = ⟨119885⟩ = 0

⟨119879⟩ = 120575

⟨Ψ⟩ =1

120573119890minus120572120575

⟨Φ⟩ = 120582120573119890120572120575

(19)

with ⟨Φ⟩ ≫ ⟨Ψ⟩ so that near the vacuum the masses of 119884119885 and Ψ are much larger than Φ besides the effective massof 119879 is also large near the vacuum due to large 120572 Large 120572

was an obstacle for natural inflation as it leads to the axiondecays too small for inflation while in this scenario large 120572

is helpful for phase inflation to stabilize the axion Thereforefor the physical process at scale below the mass scale of threeheavy fields the only unfixed degrees of freedom are 119883 andΦ which can be described by an effective field theory withthree heavy fields integrated out The coupling 120590119883Ψ(119879 minus 120575)

is designed for inflation and hierarchically smaller than theextra terms in1198820Therefore to describe inflation process theheavy fields need to be integrated out

To integrate out heavy fields we need to consider the 119865-terms again The 119865-term flatness of fields 119884 and 119885 gives

119865119884 = 119890minus120572119879

minus 120573Ψ + 1198701198841198820 = 0

119865119885 = ΨΦ minus 120582 + 1198701198851198820 = 0

(20)

Near the vacuum 119884 = 119885 asymp 0 ≪ 119872119875 the abovesupergravity corrections 119870119884(119885)1198820 are ignorable and thenthe 119865-term flatness conditions reduce to these for globalsupersymmetryThis is gained from the fact that although theinflation dynamics are subtle the inflation energy density isclose to the GUT scale far below the Planck scale Solvingthe 119865-term flatness equations in (20) we obtain the effectivesuperpotential119882 in (5)

Based on the above construction it is clear that the phasemonodromy in 119882 is from the 119880(1) transformation of 1198820and the pole of superpotential (5) at Φ = 0 arises from theintegration process The heavy fieldΨ is integrated out basedon the 119865-term flatness conditions when ⟨Φ⟩ ≫ ⟨Ψ⟩ whileif Φ rarr 0 Ψ becomes massless from |119865119885|

2 and it is illegalto integrate out a ldquomasslessrdquo field For inflation the condition⟨Φ⟩ ≫ ⟨Ψ⟩ is satisfied so the theory with superpotential (5)is reliable

As to the inflation term a question appears as global119880(1) is explicitly broken by the first term in (16) at inflationscale why is the phase light while the norm is much heavierThe supergravity correction to the scalar potential plays acrucial role at this stage The coefficient 119890119870 appears in thescalar potential and because of119880(1) symmetry in the Kahlerpotential the factor 119890119870 is invariant under119880(1) symmetry butincreases exponentially for a large norm Here the Kahlerpotential of 119879 should be shift invariant that is 119870 = 119870(119879 +

119879) instead of the minimal type Otherwise the exponentialfactor 119890119870 depends on phase as well and the phase rotationwillbe strongly fixed like the norm component or Re(119879)

When integrating out the heavy fields they should bereplaced both in superpotential and in Kahler potential bythe solutions from vanishing 119865-term equations So differentfrom the superpotential the Kahler potential obtained in thisway is slightly different from the minimal case given in (4)There are extra terms like

ΨΨ =1205822

1199032

119870 (119879 + 119879) = 119870(1

1205722(ln 119903)2)

(21)

where |Φ| = 119903 Nevertheless since 120582 ≪ 1 and 120572 ≫ 1 theseterms are rather small and have little effect on phase inflation

6 Advances in High Energy Physics

as shown in the last section Furthermore the quantumloop effects during integrating out heavy fields can introducecorrections to the Kahler potential as well However becauseof 119880(1) symmetry built in the Kahler potential these termstogether with (21) can only mildly affect the field stabiliza-tion and phase inflation is not sensitive to the correctionsin Kahler potential As to the superpotential it is protectedby the nonrenormalized theorem and free from radiativecorrections

4 Natural Inflation in Helical Phase Inflation

In the superpotential 1198820 the inflation term is perturbativecoupling of complex field119879which shifts under global119880(1) aninterestingmodification is to consider inflation given by non-perturbative coupling of 119879 Such term gives a modified 119880(1)

phase monodromy in the superpotential And it leads to nat-ural inflation as a special type of helical phase inflation withphase-axion alignment which is similar to the axion-axionalignment mechanism proposed in [34] for natural inflationwith super-Planckian axion decay constant Specifically itcan be shown that the phase monodromy realized in super-symmetric field theory has similar physical picture to themodified axion alignment mechanism provided in [44 45]

To realize natural inflation the superpotential 1198820 in (16)just needs to be slightly modified

1198821 = 120590119883Ψ(119890minus120572119879

minus 120575) + 119884 (119890minus120573119879

minus 120583Ψ)

+ 119885 (ΨΦ minus 120582)

(22)

in which 1 ≪ 120572 ≪ 120573 Again there is a global 119880(1)

symmetry in the last two terms of 1198821 and the fields transferunder119880(1) like in (17) The first term which is hierarchicallysmaller breaks the 119880(1) symmetry explicitly besides ashift symmetry of 119879 is needed in the Kahler potential Themonodromy of1198821 under a circular 119880(1) rotation is

Ψ 997888rarr Ψ119890minus1198942120587

1198821 997888rarr 1198821 + 120590119883Ψ119890minus120572119879

(119890minus2120587119894(120572120573)

minus 1)

(23)

The supersymmetric field theory given by (22) admits thefollowing supersymmetric Minkowski vacuum

⟨119883⟩ = ⟨119884⟩ = ⟨119885⟩ = 0

⟨119879⟩ = minus1

120572ln 120575

⟨Ψ⟩ =1

120583120575120573120572

⟨Φ⟩ = 120582120583120575minus120573120572

(24)

The parameters are set to satisfy conditions

120575120573120572

sim 120582120583

120582 ≪ 1

120575 lt 1

(25)

so that ⟨Ψ⟩ ≪ ⟨Φ⟩ and ⟨119879⟩ gt 0

Near vacuum fields 119884119885Ψ and 119879 obtain large effectivemasses above inflation scale while 119883Φ are much lighter Atthe inflation scale the heavy fields should be integrated outThe 119865-term flatness conditions for fields 119884 and 119885 are

119865119884 = 119890minus120573119879

minus 120583Ψ = 0

119865119885 = ΨΦ minus 120582 = 0

(26)

in which the supergravity corrections 1198701198841198851198821 are neglectedas both119884 and119885 get close to zero during inflation IntegratingoutΨ and 119879 from (26) we obtain the effective superpotential1198821015840 from1198821

1198821015840= 120590120582

119883

Φ((

120583120582

Φ)

120572120573

minus 120575) (27)

Given 120572 ≪ 120573 the effective superpotential contains a termwith fractional power Inflation driven by complex potentialwith fractional power was considered in [49] to get sufficientlarge axion decay constant Here the supersymmetric fieldmonodromy naturally leads to the superpotential with frac-tional power which arises from the small hierarchy of axiondecay constants in two nonperturbative terms

Helical phase inflation is described by the effective super-potential 1198821015840 the role of phase-axion alignment is not clearfrom119882

1015840 since it is hidden in the procedure of integrating outheavy fields

The 119865-term flatness conditions (26) fix four degrees offreedom for the extra degree of freedom they correspond tothe transformations free from constraints (26)

Ψ 997888rarr Ψ119890minus119906minus119894V

Φ 997888rarr Φ119890119906+119894V

119879 997888rarr 119879 +119906

120573+

119894V120573

(28)

Parameter 119906 corresponds to the norm variation of complexfield Φ which is fixed by the supergravity correction on thescalar potential 119890119870(ΦΦ) Parameter V relates to 119880(1) transfor-mation which leads to the phase monodromy from the firstterm of1198821The scalar potential including the inflation termdepends on the superpositions among phases ofΨΦ and theaxion Im(119879) which are constrained as in (28) Among thesefields the phase ofΦ has the lightestmass after canonical fieldnormalization Similar physical picture appears in the axion-axion alignment where inflation is triggered by the axionsuperposition along the flat direction

After integrating out the heavy fields the Kahler potentialis

119870 = ΦΦ +1205822

ΦΦ+ sdot sdot sdot (29)

As 120582 ≪ 1 the Kahler potential is dominated by ΦΦ asdiscussed before the extra terms like 1205822ΦΦ only give small

Advances in High Energy Physics 7

corrections to the field stabilization Helical phase inflationcan be simply described by the supergravity

119870 = ΦΦ + 119883119883 +

119882 = 119886119883

Φ(Φminus119887

minus 119888)

(30)

where 119887 = 120572120573 ≪ 1 and 119888 asymp 1 The scalar potential given bythe above Kahler potential and superpotential is

119881 = 1198901199032 1198862

1199032(119903minus2119887

+ 1198882minus 2119888119903minus119887 cos (119887120579)) (31)

in which we have usedΦ equiv 119903119890119894120579 As usual the norm 119903 couples

with the phase in the scalar potential To generate singlefield inflation the norm component needs to be stabilizedwith heavy mass above the Hubble scale while the phasecomponent remains light Due to the phase-norm couplingthe phase and norm are likely to obtain comparable effectivemasses during inflation so it is difficult to realize singlefield inflation However it is a bit different in helical phaseinflation The above scalar potential can be rewritten asfollows

119881 = 1198901199032 1198862

1199032(119903minus119887

minus 119888)2+ 1198901199032 41198862119888

1199032+119887(sin 119887

2120579)

2

(32)

So its vacuum locates at ⟨119903⟩ = 1199030 = 119888minus1119887 and 120579 = 0 Besides

the coefficient of the phase term 1198901199032(41198881199032+119887

) reaches itsminimal value at 1199031 = radic1 + 1198872 For 119888 asymp 1 we have theapproximation 1199030 asymp 1199031 asymp 1 The extra terms in 119870 givesmall corrections to 1199030 and 1199031 but the approximation is stillvalid So with the parameters in (25) the norm 119903 in thetwo terms of scalar potential 119881 can be stabilized at the closeregion 119903 asymp 1 separately If the parameters are tuned so that1199030 = 1199031 then the norm of complex field is strictly stabilizedat the vacuum value during inflation Without such tuninga small difference between 1199030 and 1199031 is expected but theshift of 119903 during inflation is rather small and inflation is stillapproximate to the single field inflation driven by the phaseterm prop (sin(1198872)120579)2 The helicoid structure of the potential(31) is shown in Figure 2

It is known that to realize aligned axion mechanism insupergravity [50] it is very difficult to stabilize the moduli asthey couple with the axions In [40] the moduli stabilizationis fulfilled with gauged anomalous 119880(1) symmetries since119880(1) 119863-terms only depend on the norm |Φ| or Re(119879) andthen directly separate the norms and phases of matter fieldsIn helical phase inflation the modulus and matter fieldsexcept the phase are stabilized at higher scale or by thesupergravity scalar potential Only the phase can be aninflaton candidate because of the protection from the global119880(1) symmetry in Kahler potential and approximate 119880(1)

symmetry in superpotential

5 Helical Phase Inflation and the KYY Model

120578 problem in supergravity inflation can be solved both inhelical phase inflation and in the shift symmetry in KYY

1

1

00

minus1

minus1

10

8

6

4

V(a

2)

120601x120601y

Figure 2 The helicoid structure of potential (31) scaled by 1198862 The

parameters with 119888 = 096 and 119887 = 01 are adopted in the graph Itis shown that the local valley locates around 119903 asymp 1 Note that thepotential gets flatter at the top of the graph

model The physics in these two solutions are obviously dif-ferent For helical phase inflation the solution employs 119880(1)

symmetry in Kahler potential of the minimal supergravityand the superpotential admits a phase monodromy arisingfrom the global 119880(1) transformation

119870 = ΦΦ + 119883119883 + sdot sdot sdot

119882 = 119886119883

ΦlnΦ

(33)

The inflation is driven by the phase of complex field Φ

and several special virtues appear in the model For theKYY model with shift symmetry [6] the Kahler potential isadjusted so that it admits a shift symmetry along the directionof Im(119879)

119870 =1

2(119879 + 119879)

2+ 119883119883 + sdot sdot sdot

119882 = 119886119883119879

(34)

and inflation is driven by Im(119879) The shift symmetry isendowedwith axions so thismechanism is attractive for axioninflation

Here we will show that although the physical pictures aremuch different in helical phase inflation andKYY typemodeljust considering the lower order terms in the Kahler potentialof redefined complex field helical phase inflation can reduceto the KYY model

Because the phase of Φ rotates under the global 119880(1)

transformation to connect helical phase inflation with theKYY model a natural guess is to take the following fieldredefinitionΦ = 119890

119879 then helical phase inflation (33) becomes

119870 = 119890119879+119879

+ sdot sdot sdot = 1 + 119879 + 119879 +1

2(119879 + 119879)

2+ sdot sdot sdot

119882 = 119886119883119879119890minus119879

(35)

8 Advances in High Energy Physics

TheKahler manifold of complex field119879 is invariant under theholomorphic Kahler transformation

119870(119879 119879) 997888rarr 119870(119879 119879) + 119865 (119879) + 119865 (119879) (36)

in which 119865(119879) is a holomorphic function of 119879 To keepthe whole Lagrangian also invariant under the Kahler trans-formation the superpotential transforms under the Kahlertransformation

119882 997888rarr 119890minus119865(119879)

119882 (37)

For the supergravity model in (35) taking the Kahler trans-formation with 119865(119879) = minus(12) minus 119879 the Kahler potential andsuperpotential become

119870 =1

2(119879 + 119879)

2+ sdot sdot sdot

119882 = 119886radic119890119883119879

(38)

which is just the KYY model (34) with higher order cor-rections in the Kahler potential These higher order termsvanish after field stabilization and have no effect on inflationprocess The field relation Φ = 119890

119879 also gives a map betweenthe simplest helical phase inflation and the KYY model suchas the inflaton arg(Φ) rarr Im(119879) and field stabilization|Φ| = 1 rarr Re(119879) = 0

Nevertheless helical phase inflation is not equivalent tothe KYY model There are higher order corrections to theKahler potential in the map from helical phase inflation toKYY model which have no effect on inflation after fieldstabilization but indicate different physics in two models Bydropping these terms certain information is lost so the mapis irreversible Specifically the inverse function 119879 = lnΦ ofthe field redefinition Φ = 119890

119879 cannot reproduce helical phaseinflation from the KYY model unless one introduces higherdimensional operators in the Kahler potential exactly thesame as given in the expansion (35) Such strict constraint oneach higher dimensional operator is not guaranteed by anyknown rule and so is unphysical As there is no pole or singu-larity in the KYY model it is unlikely to introduce pole andsingularity at origin with phase monodromy through well-defined field redefinition Actually the singularity with phasemonodromy in the superpotential indicates rich physics inthe scale above inflation

6 Helical Phase Inflation inNo-Scale Supergravity

The no-scale supergravity is an attractive frame for GUTscale phenomenology it is interesting to realize helicalphase inflation in no-scale supergravity Generally theKahler manifold of the no-scale supergravity is equippedwith 119878119880(119873 1)119878119880(119873) times 119880(1) symmetry For the no-scalesupergravity with exact 119878119880(1 1)119880(1) symmetry withoutextra fields no inflation can be realized so the case with119878119880(2 1)119878119880(2) times 119880(1) symmetry is the simplest one thatadmits inflation

The Kahler potential with 119878119880(2 1)119878119880(2)times119880(1) symme-try is

119870 = minus3 ln(119879 + 119879 minusΦΦ

3) (39)

In the symmetry of the Kahler manifold there is 119880(1)

subsector the phase rotation of complex fieldΦ which can beemployed for helical phase inflation Besides the modulus 119879should be stabilized during inflation which can be fulfilled byintroducing extra terms on119879 As a simple example of no-scalehelical phase inflation here we follow the simplification in [9]that the modulus 119879 has already been stabilized at ⟨119879⟩ = 119888Different from the minimal supergravity the kinetic termgiven by the no-scale Kahler potential is noncanonical

119871119870 = 119870ΦΦ120597120583Φ120597120583Φ

=2119888

(2119888 minus 11990323)2(120597120583119903120597

120583119903 + 1199032120597120583120579120597120583120579)

(40)

in which Φ equiv 119903119890119894120579 is used The 119865-term scalar potential is

119881 = 119890minus(23)119870 1003816100381610038161003816119882Φ

1003816100381610038161003816

2=

1003816100381610038161003816119882Φ1003816100381610038161003816

2

(119879 + 119879 minus ΦΦ3)2 (41)

where the superpotential 119882 is a holomorphic function ofsuperfieldΦ and119879 = ⟨119879⟩ = 119888 It requires a phasemonodromyin superpotential119882 for phase inflation the simple choice is

119882 =119886

Φln Φ

Λ (42)

The scalar potential given by this superpotential is

119881 =91198862

(6119888 minus 1199032)21199034

((ln 119903

119890Λ)

2

+ 1205792) (43)

As in the minimal supergravity the norm and phase ofcomplex field Φ are separated in the scalar potential For the119903-dependent coefficient factor 1(6119888 minus 119903

2)21199034 its minimum

locates at 1199030 = radic3119888 and another term (ln(119903119890Λ))2 reaches its

minimum at 1199031 = 119890Λ Given that the parameter Λ is tuned sothat 1199030 = 1199031 in the radial direction the potential has a globalminimum at 119903 = 1199030 Similar to helical phase inflation in theminimal supergravity the potential (43) also shows helicoidstructure as presented in Figure 3

The physical mass of 119903 in the region near the vacuum is

1198982119903 =

(2119888 minus 11990323)2

4119888

1205972119881

1205971199032

100381610038161003816100381610038161003816100381610038161003816100381610038161003816119903=radic3119888

= (4 +1

21205792)1198672 (44)

where119867 is theHubble constant during inflation So the norm119903 is strongly stabilized at 1199030 If 1199030 and 1199031 are not equal butclose to each other 119903 will slightly shift during inflation butits mass remains above the Hubble scale and inflation is stillapproximately the single field inflation

Advances in High Energy Physics 9

30

20

10

0

minus10

minus10

minus05

minus05

0000

05

05

10

10

120601x (radic

3c) 120601 y(radic3

c)

V(a

2 c4 )

Figure 3The helicoid structure of potential (43) scaled by 11988621198884 Inthe graph the parameter Λ is tuned so that 1199030 = 1199031 The complexfieldΦ has been rescaled byradic3119888 and the scale of field norm at localvalley is determined by the parameter 119888 instead of the Planck mass

Instead of stabilizing 119879 independently with the inflationprocess we can realize helical phase inflation from the no-scale supergravity with dynamical119879There is a natural reasonfor such consideration The phase monodromy requiresmatter fields transforming as rotations under 119880(1) and alsoa modulus 119879 as a shift under 119880(1) The Kahler potential of 119879is shift invariant Interestingly for the no-scale supergravitywith 119878119880(2 1)119878119880(2) times 119880(1) symmetry as shown in (39) theKahler potential 119870 is automatically endowed with the shiftsymmetry and global 119880(1) symmetry

119879 997888rarr 119879 + 119894119862

Φ 997888rarr Φ119890119894120579

(45)

Therefore the no-scale Kahler potential fits with the phasemonodromy in (16) and (22) initiatively

The Kahler potentials of superfields 119911 isin 119883 119884 119885 areof the minimal type 119911119911 while for Ψ its Kahler potentialcan be the minimal type ΨΨ or the no-scale type 119870 =

minus3 ln(119879+119879minus (ΦΦ+ΨΨ)3) which extends the symmetry ofKahler manifold to 119878119880(3 1)119878119880(3) times 119880(1) In this scenariothe process to integrate out heavy fields is the same asbefore Besides the potential of phase proportional to |119882119883|

2

is insensitive to the formula of Kahler potential due to119880(1) symmetry The major difference appears in the fieldstabilization The scalar potential from phase monodromy in(16) is

119881 = 119890119870 1003816100381610038161003816119882119883

1003816100381610038161003816

2=

1

(ln 1199032120572 + 119888 minus 11990323 minus 120582211990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792) asymp

1

(119888 minus 11990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792)

(46)

with 120572 ≫ 1 and 120582 ≪ 1 The field norm 119903 is stabilized byminimizing the coefficient 11199032(119888 minus 119903

23)3 and the parameter

119888 from ⟨119879⟩ determines the scale of ⟨119903⟩ = radic31198882 This isdifferent from helical phase inflation in the minimal super-gravity which minimizes the norm based on the coefficient1198901199032(11199032) and then the scale of ⟨119903⟩ is close to119872119875 the unique

energy scale of supergravity corrections Similarly combiningthe no-scale Kahlermanifold with phasemonodromy in (22)we can obtain natural inflation

7 UV-Sensitivity and Helical Phase Inflation

For the slow-roll inflation the Lyth bound [21] provides arelationship between the tensor-to-scalar ratio r and the fieldexcursion Δ120601 Roughly it requires

Δ120601

119872119875

⩾ (r

001)

12

(47)

To get large tensor-to-scalar ratio r ⩾ 001 such as in chaoticinflation or natural inflation the field excursion should bemuch larger than the Planck mass The super-Planckianfield excursion makes the description based on the effectivefield theory questionable In the Wilsonian sense the lowenergy field theory is an effective theory with higher ordercorrections introduced by the physics at the cut-off scalelike quantum gravity and these terms are irrelevant in theeffective field theory since they are suppressed by the cut-offenergy scale However for the inflation process the inflatonhas super-Planckian field excursion which is much largerthan the cut-off scaleThus the higher order terms cannot besuppressed by the Planckmass andmay affect the inflationaryobservations significantly And then they may significantlyaffect inflation or even destroy the inflation process Forexample considering the corrections

Δ119881 = 119888119894119881(120601

119872119875

)

119894

+ sdot sdot sdot (48)

to the original inflaton potential 119881 as long as 119888119894 are of theorder 10minus119894 in the initial stage of inflation 120601 sim 119874(10)119872119875 thehigher order terms can be as large as the original potential119881 So for large field inflation it is sensitive to the physics atthe cut-off scale and the predictions of inflation just based oneffective field theory are questionable

In consideration of the UV sensitivity of large fieldinflation a possible choice is to realize inflation in UV-completed theory like string theory (for a review see [22])To realize inflation in string theory there are a lot of problemsto solve besides inflation such as the moduli stabilizationMinkowskide Sitter vacua and effects of 1205721015840- and string loop-corrections Alternatively in the bottom-up approach onemay avoid the higher order corrections by introducing anextra shift symmetry in the theory The shift symmetry istechnically natural and safe under quantum loop-correctionsHowever the global symmetry can be broken by the quantumgravity effect So it is still questionable whether the shiftsymmetry can safely evade the higher corrections like in (48)

10 Advances in High Energy Physics

The UV-completion problem is dodged in helical phaseinflation Since the super-Planckian field excursion is thephase of a complex field and the phase component is notdirectly involved in the gravity interaction there are nodangerous high-order corrections like in (48) for the phasepotentialThe inflaton evolves along the helical trajectory anddoes not relate to the physics in the region above the Planckscale For helical phase inflation in the minimal supergravitythe norm of field is stabilized at the marginal point of thePlanck scale where the supergravity correction on the scalarpotential gets important based onwhich the normof complexfield can be strongly stabilized The extra corrections arelikely to appear in the Kahler potential however they canonly slightly affect the field stabilization while the phaseinflation is protected by the global 119880(1) symmetry in theKahler potential in consequence helical phase inflation isnot sensitive to these corrections at all For the helical phaseinflation in no-scale supergravity the normof complex field isstabilized at the scale of themodulus ⟨119879⟩ instead of the Planckscale one can simply adjust the scale of ⟨119879⟩ to keep themodelaway from super-Planckian region

Helical phase inflation is free from the UV-sensitivityproblem and it is just a typical physical process at theGUT scale with special superpotential that admits phasemonodromy So it provides an inflationary model that can bereliably studied just in supersymmetric field theory

8 Discussions and Conclusion

In this work we have studied the details of helical phase infla-tion from several aspects Helical phase inflation is realizedin supergravity with global 119880(1) symmetry 119880(1) symmetryis built in the Kahler potential so that helical phase inflationcan be realized by the ordinary Kahler potentials such as theminimal or no-scale types Helical phase inflation directlyleads to the phasemonodromy in the superpotential which issingular and an effective field theory arising from integratingout heavy fieldsThe phasemonodromy originates from119880(1)

rotation of the superpotential at higher scale Generically thesuperpotential can be separated into two parts119882 = 119882119868 +119882119878where119882119878 admits the global119880(1) symmetrywhile119882119868 breaks itexplicitly at scale much lower than 119882119878 Under 119880(1) rotationthe inflation term 119882119868 is slightly changed which realizes thephase monodromy in the effective theory and introducesa flat potential along the direction of phase rotation Bybreaking the global119880(1) symmetry in different ways we mayget different kinds of inflation such as quadratic inflationnatural inflation or the other types of inflation that are notpresented in this work

An amazing fact of helical phase inflation is that itdeeply relates to several interesting points of inflation andnaturally combines them in a rather simple potential withhelicoid structure The features of helical phase inflation canbe summarized as follows

(i) The global 119880(1) symmetry is built in the Kahlerpotential so helical phase inflation provides a naturalsolution to 120578 problem

(ii) The phase excursion requires phase monodromy inthe superpotential So helical phase inflation providesin the mathematical sense a new type of monodromyin supersymmetric field theory

(iii) The singularity in the superpotential together withthe supergravity scalar potential provides strong fieldstabilization which is consistent with phase inflation

(iv) The super-Planckian field excursion is realized bythe phase of a complex field instead of any otherldquophysicalrdquo fields that directly couple with gravity Sothere are no polynomial higher order corrections forthe phase and thus inflation is not sensitive to thequantum gravity corrections

To summarize helical phase inflation introduces a newtype of inflation that can be effectively described by super-symmetric field theory at the GUT scale Generically thesuper-Planckian field excursion makes the inflationary pre-dictions based on effective field theory questionable sincethe higher order corrections from quantum gravity arelikely to affect the inflation process significantly One of thesolutions is to realize inflation in a UV-completed theory likestring theory nevertheless there are many difficult issues instring theory to resolve before realizing inflation completelyHelical phase inflation is another simple solution to the UV-sensitivity problem It is based on the supersymmetric fieldtheory and the physics are clear and much easier to controlFurthermore helical phase inflation makes the unification ofinflation theory with GUT more natural since both of themare triggered at the scale of 1016 GeV and can be effectivelystudied based on supersymmetric field theory

Besides we have shown that helical phase inflation alsorelates to several interesting developments in inflation theoryIt can be easily modified for natural inflation and realizethe phase-axion alignment indirectly which is similar tothe axion-axion alignment mechanism for super-Planckianaxion decay constant [34] The phase-axion alignment isnot shown in the final supergravity model which exhibitsexplicit phase monodromy only However the phase-axionalignment is hidden in the process when the heavy fields areintegrating out For 120578 problem in the supergravity inflationthere is another well-known solution the KYY model withshift symmetry We showed that through field redefinitionhelical phase inflation given by (4) and (5) reduces to theKYYmodel where the higher order corrections in the Kahlerpotential have no effect on inflation process However thereis no inverse transformation from the KYY model to helicalphase inflation since no well-defined field redefinition canintroduce the pole and phase singularity needed for phasemonodromy Helical phase inflation can be realized in no-scale supergravity The no-scale Kahler potential automati-cally provides the symmetry needed by phase monodromyIn the no-scale supergravity the norm of complex field isstabilized at the scale of the modulus

Our inflation models are constructed within the super-gravity theory with global 119880(1) symmetry broken explicitlyby the subleading order superpotential term So it is justtypical GUT scale physics and indicates that a UV-completed

Advances in High Energy Physics 11

framework seems to be not prerequisite to effectively describesuch inflation process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work of DVN was supported in part by the DOEGrant DE-FG03-95-ER-40917 The work of Tianjun Li issupported in part by theNatural Science Foundation of ChinaunderGrants nos 10821504 11075194 11135003 11275246 and11475238 and by the National Basic Research Program ofChina (973 Program) under Grant no 2010CB833000

References

[1] A H Guth ldquoInflationary universe a possible solution to thehorizon and flatness problemsrdquo Physical Review D vol 23 no2 pp 347ndash356 1981

[2] A D Linde ldquoColeman-Weinberg theory and the new inflation-ary universe scenariordquo Physics Letters B vol 114 no 6 pp 431ndash435 1982

[3] A Albrecht and P J Steinhardt ldquoCosmology for grand unifiedtheories with radiatively induced symmetry breakingrdquo PhysicalReview Letters vol 48 no 17 pp 1220ndash1223 1982

[4] P A R Ade N Aghanim C Armitage-Caplan et al ldquoPlanck2013 results XXII Constraints on inflationrdquo Astronomy ampAstrophysics vol 571 article A22 42 pages 2014

[5] P A R Ade R W Aikin D Barkats et al ldquoDetection of B-mode polarization at degree angular scales by BICEP2rdquo PhysicalReview Letters vol 112 no 24 Article ID 241101 2014

[6] M Kawasaki M Yamaguchi and T Yanagida ldquoNatural chaoticinflation in supergravityrdquo Physical Review Letters vol 85 no 17pp 3572ndash3575 2000

[7] T Li Z Li and D V Nanopoulos ldquoSupergravity inflation withbroken shift symmetry and large tensor-to-scalar ratiordquo Journalof Cosmology and Astroparticle Physics vol 2014 no 2 article028 2014

[8] K Harigaya and T T Yanagida ldquoDiscovery of large scale tensormode and chaotic inflation in supergravityrdquo Physics Letters Bvol 734 pp 13ndash16 2014

[9] J Ellis D V Nanopoulos and K A Olive ldquoNo-scale supergrav-ity realization of the starobinsky model of inflationrdquo PhysicalReview Letters vol 111 no 11 Article ID 111301 2013

[10] T Li Z Li and D V Nanopoulos ldquoNo-scale ripple inflationrevisitedrdquo Journal of Cosmology and Astroparticle Physics vol04 article 018 2014

[11] C Pallis ldquoInduced-gravity inflation in no-scale supergravityand beyondrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 8 article 057 2014

[12] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky modelrdquo Fortschritte der Physik vol 62 no 7 pp 573ndash5832014

[13] J Ellis M A G Garca D V Nanopoulos and K A OliveldquoResurrecting quadratic inflation in no-scale supergravity inlight of BICEP2rdquo Journal of Cosmology andAstroparticle Physicsvol 2014 no 5 article 037 2014

[14] T Li Z Li and D V Nanopoulos ldquoChaotic inflation in no-scale supergravitywith string inspiredmoduli stabilizationrdquoTheEuropean Physical Journal C vol 75 no 2 article 55 2015

[15] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky model and higher curvature correctionsrdquo Fortschritte derPhysik vol 63 no 1 pp 2ndash11 2015

[16] S V Ketov andT Terada ldquoInflation in supergravity with a singlechiral superfieldrdquo Physics Letters B vol 736 pp 272ndash277 2014

[17] C Kounnas D Lust and N Toumbas ldquoR2 inflation fromscale invariant supergravity and anomaly free superstrings withfluxesrdquo Fortschritte der Physik vol 63 no 1 pp 12ndash35 2015

[18] G A Diamandis B C Georgalas K Kaskavelis PKouroumalou A B Lahanas and G Pavlopoulos ldquoInflationin R2 supergravity with non-minimal superpotentialsrdquo PhysicsLetters B vol 744 pp 74ndash81 2015

[19] C Pallis and Q Shafi ldquoGravity waves from non-minimalquadratic inflationrdquo Journal of Cosmology and AstroparticlePhysics vol 2015 no 3 article 023 2015

[20] T Li Z Li and D V Nanopoulos ldquoHelical phase inflationrdquoPhysical Review D vol 91 no 6 Article ID 061303 2015

[21] D H Lyth ldquoWhat would we learn by detecting a gravitationalwave signal in the cosmic microwave background anisotropyrdquoPhysical Review Letters vol 78 no 10 article 1861 1997

[22] D Baumann and L McAllister ldquoInflation and stringtheoryrdquohttparxivorgabs14042601

[23] F Marchesano G Shiu and A M Uranga ldquoF-term axionmonodromy inflationrdquo Journal of High Energy Physics vol 2014no 9 article 184 2014

[24] C Long L McAllister and P McGuirk ldquoAligned naturalinflation in string theoryrdquo Physical Review D vol 90 no 2Article ID 023501 2014

[25] X Gao T Li and P Shukla ldquoCombining universal and odd RRaxions for aligned natural inflationrdquo Journal of Cosmology andAstroparticle Physics vol 2014 no 10 article 048 2014

[26] I Ben-Dayan F G Pedro and A Westphal ldquoTowards naturalinflation in string theoryrdquo Physical Review D vol 92 no 2Article ID 023515 2015

[27] R Blumenhagen D Herschmann and E Plauschinn ldquoThechallenge of realizing F-term axion monodromy inflation instring theoryrdquo Journal of High Energy Physics vol 2015 article7 2015

[28] H Abe T Kobayashi and H Otsuka ldquoTowards natural infla-tion from weakly coupled heterotic string theoryrdquo Progress ofTheoretical and Experimental Physics Article ID 063E02 2015

[29] T Ali S S Haque and V Jejjala ldquoNatural inflation from nearalignment in heterotic string theoryrdquo Physical Review D vol 91no 8 Article ID 083516 9 pages 2015

[30] R Flauger LMcAllister E Silverstein and AWestphal ldquoDrift-ing oscillations in axionmonodromyrdquo httparxivorgabs14121814

[31] K Freese J A Frieman andAVOlinto ldquoNatural inflationwithpseudo nambu-goldstone bosonsrdquo Physical Review Letters vol65 no 26 pp 3233ndash3236 1990

[32] F C Adams J R Bond K Freese J A Frieman and AV Olinto ldquoNatural inflation particle physics models power-law spectra for large-scale structure and constraints from thecosmic background explorerrdquo Physical Review D vol 47 no 2pp 426ndash455 1993

[33] G German A Mazumdar and A Perez-Lorenzana ldquoAngularinflation from supergravityrdquo Modern Physics Letters A vol 17no 25 pp 1627ndash1634 2002

12 Advances in High Energy Physics

[34] J E Kim H P Nilles and M Peloso ldquoCompleting naturalinflationrdquo Journal of Cosmology and Astroparticle Physics vol0501 article 005 2005

[35] D Baumann and D Green ldquoInflating with baryonsrdquo Journal ofHigh Energy Physics vol 2011 no 4 article 071 2011

[36] JMcDonald ldquoSub-Planckian two-field inflation consistent withthe Lyth boundrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 9 article 027 2014

[37] J McDonald ldquoA minimal sub-Planckian axion inflation modelwith large tensor-to-scalar ratiordquo Journal of Cosmology andAstroparticle Physics vol 2015 no 1 article 018 2015

[38] C D Carone J Erlich A Sensharma and Z Wang ldquoTwo-fieldaxion-monodromy hybrid inflation model Dantersquos WaterfallrdquoPhysical ReviewD vol 91 no 4 Article ID 043512 8 pages 2015

[39] G Barenboim and W-I Park ldquoSpiral Inflationrdquo Physics LettersB vol 741 pp 252ndash255 2015

[40] T Li Z Li andDVNanopoulos ldquoAligned natural inflation andmoduli stabilization from anomalous U(1) gauge symmetriesrdquoJournal of High Energy Physics vol 2014 no 11 article 012 2014

[41] T Li Z Li and D V Nanopoulos ldquoNatural inflation withnatural trans-planckian axion decay constant from anomalousU(1)119883rdquo Journal of High Energy Physics vol 2014 no 7 article052 2014

[42] E Silverstein and A Westphal ldquoMonodromy in the CMBgravity waves and string inflationrdquo Physical Review D vol 78no 10 Article ID 106003 21 pages 2008

[43] L McAllister E Silverstein and A Westphal ldquoGravity wavesand linear inflation from axion monodromyrdquo Physical ReviewD vol 82 no 4 Article ID 046003 2010

[44] K Choi H Kim and S Yun ldquoNatural inflation with multiplesub-Planckian axionsrdquo Physical Review D vol 90 no 2 ArticleID 023545 2014

[45] S-H H Tye and S S C Wong ldquoHelical inflation and cosmic-stringsrdquo httparxivorgabs14046988

[46] R Kappl S Krippendorf and H P Nilles ldquoAligned naturalinflation monodromies of two axionsrdquo Physics Letters B vol737 pp 124ndash128 2014

[47] G rsquotHooft ldquoNaturalness chiral symmetry and spontaneouschiral symmetry breakingrdquo in Recent Developments in GaugeTheories G rsquotHooft C Itzykson A Jaffe et al Eds vol 59of NATO Advanced Study Institutes Series pp 135ndash157 PlenumPress New York NY USA 1980

[48] R Blumenhagen M Cvetic S Kachru and T Weigand ldquoD-brane instantons in type II string theoryrdquo Annual Review ofNuclear and Particle Science vol 59 pp 269ndash296 2009

[49] K Harigaya andM Ibe ldquoSimple realization of inflaton potentialon a Riemann surfacerdquo Physics Letters B vol 738 pp 301ndash3042014

[50] R Kallosh A Linde and B Vercnocke ldquoNatural inflation insupergravity and beyondrdquo Physical Review D vol 90 Article ID041303(R) 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

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Soft MatterJournal of

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AerodynamicsJournal of

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PhotonicsJournal of

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ThermodynamicsJournal of

Advances in High Energy Physics 5

string compactifications [42] In the inflaton potential theonly factor that changes during axion circular rotation isfrom the DBI action of branes In [44ndash46] the generalizedaxion alignment mechanisms are considered as particularrealization of axion monodromy with the potential fromnonperturbative effects We will show that such kind ofaxion monodromy can also be fulfilled by the superpotentialphase monodromy [20] even though it is not shown in theeffective superpotential after integrating out the heavy fieldsFurthermore all the extra fields can be consistently stabilized

The more ldquofundamentalrdquo field theory for the superpoten-tial in (5) is

1198820 = 120590119883Ψ (119879 minus 120575) + 119884 (119890minus120572119879

minus 120573Ψ) + 119885 (ΨΦ minus 120582) (16)

where the coupling constants for the second and third termsare taken to be 1 for simplicity and a small hierarchy isassumed between the first term and the last two terms thatis 120590 ≪ 1 The coupling 119884119890

minus120572119879 is assumed to be an effectivedescription of certain nonperturbative effects Similar formscan be obtained from 119863-brane instanton effect in type stringtheory (for a review see [48]) besides the coefficient 120572 prop

1119891 ≫ 1 in Planck unit since 119891 ≪ 1 is the decay constantand should be significantly lower than the Planck scale Forthe last two terms of1198820 there is a global 119880(1) symmetry

Ψ 997888rarr Ψ119890minus119894119902120579

Φ 997888rarr Φ119890119894119902120579

119884 997888rarr 119884119890119894119902120579

119879 997888rarr 119879 +119894119902120579

120572

(17)

which is anomalous and explicitly broken by the first termPhase rotation of the stabilizer field 119883 has no effect oninflation and so is ignored here The phase monodromy ofsuperpotential 119882 in (6) originates from the 119880(1) rotation of1198820

Ψ 997888rarr Ψ119890minus1198942120587

1198820 997888rarr 1198820 + 11989421205871205901

120572119883Ψ

(18)

As shown in [20] the supersymmetric field theory withsuperpotential1198820 admits the Minkowski vacuum at

⟨119883⟩ = ⟨119884⟩ = ⟨119885⟩ = 0

⟨119879⟩ = 120575

⟨Ψ⟩ =1

120573119890minus120572120575

⟨Φ⟩ = 120582120573119890120572120575

(19)

with ⟨Φ⟩ ≫ ⟨Ψ⟩ so that near the vacuum the masses of 119884119885 and Ψ are much larger than Φ besides the effective massof 119879 is also large near the vacuum due to large 120572 Large 120572

was an obstacle for natural inflation as it leads to the axiondecays too small for inflation while in this scenario large 120572

is helpful for phase inflation to stabilize the axion Thereforefor the physical process at scale below the mass scale of threeheavy fields the only unfixed degrees of freedom are 119883 andΦ which can be described by an effective field theory withthree heavy fields integrated out The coupling 120590119883Ψ(119879 minus 120575)

is designed for inflation and hierarchically smaller than theextra terms in1198820Therefore to describe inflation process theheavy fields need to be integrated out

To integrate out heavy fields we need to consider the 119865-terms again The 119865-term flatness of fields 119884 and 119885 gives

119865119884 = 119890minus120572119879

minus 120573Ψ + 1198701198841198820 = 0

119865119885 = ΨΦ minus 120582 + 1198701198851198820 = 0

(20)

Near the vacuum 119884 = 119885 asymp 0 ≪ 119872119875 the abovesupergravity corrections 119870119884(119885)1198820 are ignorable and thenthe 119865-term flatness conditions reduce to these for globalsupersymmetryThis is gained from the fact that although theinflation dynamics are subtle the inflation energy density isclose to the GUT scale far below the Planck scale Solvingthe 119865-term flatness equations in (20) we obtain the effectivesuperpotential119882 in (5)

Based on the above construction it is clear that the phasemonodromy in 119882 is from the 119880(1) transformation of 1198820and the pole of superpotential (5) at Φ = 0 arises from theintegration process The heavy fieldΨ is integrated out basedon the 119865-term flatness conditions when ⟨Φ⟩ ≫ ⟨Ψ⟩ whileif Φ rarr 0 Ψ becomes massless from |119865119885|

2 and it is illegalto integrate out a ldquomasslessrdquo field For inflation the condition⟨Φ⟩ ≫ ⟨Ψ⟩ is satisfied so the theory with superpotential (5)is reliable

As to the inflation term a question appears as global119880(1) is explicitly broken by the first term in (16) at inflationscale why is the phase light while the norm is much heavierThe supergravity correction to the scalar potential plays acrucial role at this stage The coefficient 119890119870 appears in thescalar potential and because of119880(1) symmetry in the Kahlerpotential the factor 119890119870 is invariant under119880(1) symmetry butincreases exponentially for a large norm Here the Kahlerpotential of 119879 should be shift invariant that is 119870 = 119870(119879 +

119879) instead of the minimal type Otherwise the exponentialfactor 119890119870 depends on phase as well and the phase rotationwillbe strongly fixed like the norm component or Re(119879)

When integrating out the heavy fields they should bereplaced both in superpotential and in Kahler potential bythe solutions from vanishing 119865-term equations So differentfrom the superpotential the Kahler potential obtained in thisway is slightly different from the minimal case given in (4)There are extra terms like

ΨΨ =1205822

1199032

119870 (119879 + 119879) = 119870(1

1205722(ln 119903)2)

(21)

where |Φ| = 119903 Nevertheless since 120582 ≪ 1 and 120572 ≫ 1 theseterms are rather small and have little effect on phase inflation

6 Advances in High Energy Physics

as shown in the last section Furthermore the quantumloop effects during integrating out heavy fields can introducecorrections to the Kahler potential as well However becauseof 119880(1) symmetry built in the Kahler potential these termstogether with (21) can only mildly affect the field stabiliza-tion and phase inflation is not sensitive to the correctionsin Kahler potential As to the superpotential it is protectedby the nonrenormalized theorem and free from radiativecorrections

4 Natural Inflation in Helical Phase Inflation

In the superpotential 1198820 the inflation term is perturbativecoupling of complex field119879which shifts under global119880(1) aninterestingmodification is to consider inflation given by non-perturbative coupling of 119879 Such term gives a modified 119880(1)

phase monodromy in the superpotential And it leads to nat-ural inflation as a special type of helical phase inflation withphase-axion alignment which is similar to the axion-axionalignment mechanism proposed in [34] for natural inflationwith super-Planckian axion decay constant Specifically itcan be shown that the phase monodromy realized in super-symmetric field theory has similar physical picture to themodified axion alignment mechanism provided in [44 45]

To realize natural inflation the superpotential 1198820 in (16)just needs to be slightly modified

1198821 = 120590119883Ψ(119890minus120572119879

minus 120575) + 119884 (119890minus120573119879

minus 120583Ψ)

+ 119885 (ΨΦ minus 120582)

(22)

in which 1 ≪ 120572 ≪ 120573 Again there is a global 119880(1)

symmetry in the last two terms of 1198821 and the fields transferunder119880(1) like in (17) The first term which is hierarchicallysmaller breaks the 119880(1) symmetry explicitly besides ashift symmetry of 119879 is needed in the Kahler potential Themonodromy of1198821 under a circular 119880(1) rotation is

Ψ 997888rarr Ψ119890minus1198942120587

1198821 997888rarr 1198821 + 120590119883Ψ119890minus120572119879

(119890minus2120587119894(120572120573)

minus 1)

(23)

The supersymmetric field theory given by (22) admits thefollowing supersymmetric Minkowski vacuum

⟨119883⟩ = ⟨119884⟩ = ⟨119885⟩ = 0

⟨119879⟩ = minus1

120572ln 120575

⟨Ψ⟩ =1

120583120575120573120572

⟨Φ⟩ = 120582120583120575minus120573120572

(24)

The parameters are set to satisfy conditions

120575120573120572

sim 120582120583

120582 ≪ 1

120575 lt 1

(25)

so that ⟨Ψ⟩ ≪ ⟨Φ⟩ and ⟨119879⟩ gt 0

Near vacuum fields 119884119885Ψ and 119879 obtain large effectivemasses above inflation scale while 119883Φ are much lighter Atthe inflation scale the heavy fields should be integrated outThe 119865-term flatness conditions for fields 119884 and 119885 are

119865119884 = 119890minus120573119879

minus 120583Ψ = 0

119865119885 = ΨΦ minus 120582 = 0

(26)

in which the supergravity corrections 1198701198841198851198821 are neglectedas both119884 and119885 get close to zero during inflation IntegratingoutΨ and 119879 from (26) we obtain the effective superpotential1198821015840 from1198821

1198821015840= 120590120582

119883

Φ((

120583120582

Φ)

120572120573

minus 120575) (27)

Given 120572 ≪ 120573 the effective superpotential contains a termwith fractional power Inflation driven by complex potentialwith fractional power was considered in [49] to get sufficientlarge axion decay constant Here the supersymmetric fieldmonodromy naturally leads to the superpotential with frac-tional power which arises from the small hierarchy of axiondecay constants in two nonperturbative terms

Helical phase inflation is described by the effective super-potential 1198821015840 the role of phase-axion alignment is not clearfrom119882

1015840 since it is hidden in the procedure of integrating outheavy fields

The 119865-term flatness conditions (26) fix four degrees offreedom for the extra degree of freedom they correspond tothe transformations free from constraints (26)

Ψ 997888rarr Ψ119890minus119906minus119894V

Φ 997888rarr Φ119890119906+119894V

119879 997888rarr 119879 +119906

120573+

119894V120573

(28)

Parameter 119906 corresponds to the norm variation of complexfield Φ which is fixed by the supergravity correction on thescalar potential 119890119870(ΦΦ) Parameter V relates to 119880(1) transfor-mation which leads to the phase monodromy from the firstterm of1198821The scalar potential including the inflation termdepends on the superpositions among phases ofΨΦ and theaxion Im(119879) which are constrained as in (28) Among thesefields the phase ofΦ has the lightestmass after canonical fieldnormalization Similar physical picture appears in the axion-axion alignment where inflation is triggered by the axionsuperposition along the flat direction

After integrating out the heavy fields the Kahler potentialis

119870 = ΦΦ +1205822

ΦΦ+ sdot sdot sdot (29)

As 120582 ≪ 1 the Kahler potential is dominated by ΦΦ asdiscussed before the extra terms like 1205822ΦΦ only give small

Advances in High Energy Physics 7

corrections to the field stabilization Helical phase inflationcan be simply described by the supergravity

119870 = ΦΦ + 119883119883 +

119882 = 119886119883

Φ(Φminus119887

minus 119888)

(30)

where 119887 = 120572120573 ≪ 1 and 119888 asymp 1 The scalar potential given bythe above Kahler potential and superpotential is

119881 = 1198901199032 1198862

1199032(119903minus2119887

+ 1198882minus 2119888119903minus119887 cos (119887120579)) (31)

in which we have usedΦ equiv 119903119890119894120579 As usual the norm 119903 couples

with the phase in the scalar potential To generate singlefield inflation the norm component needs to be stabilizedwith heavy mass above the Hubble scale while the phasecomponent remains light Due to the phase-norm couplingthe phase and norm are likely to obtain comparable effectivemasses during inflation so it is difficult to realize singlefield inflation However it is a bit different in helical phaseinflation The above scalar potential can be rewritten asfollows

119881 = 1198901199032 1198862

1199032(119903minus119887

minus 119888)2+ 1198901199032 41198862119888

1199032+119887(sin 119887

2120579)

2

(32)

So its vacuum locates at ⟨119903⟩ = 1199030 = 119888minus1119887 and 120579 = 0 Besides

the coefficient of the phase term 1198901199032(41198881199032+119887

) reaches itsminimal value at 1199031 = radic1 + 1198872 For 119888 asymp 1 we have theapproximation 1199030 asymp 1199031 asymp 1 The extra terms in 119870 givesmall corrections to 1199030 and 1199031 but the approximation is stillvalid So with the parameters in (25) the norm 119903 in thetwo terms of scalar potential 119881 can be stabilized at the closeregion 119903 asymp 1 separately If the parameters are tuned so that1199030 = 1199031 then the norm of complex field is strictly stabilizedat the vacuum value during inflation Without such tuninga small difference between 1199030 and 1199031 is expected but theshift of 119903 during inflation is rather small and inflation is stillapproximate to the single field inflation driven by the phaseterm prop (sin(1198872)120579)2 The helicoid structure of the potential(31) is shown in Figure 2

It is known that to realize aligned axion mechanism insupergravity [50] it is very difficult to stabilize the moduli asthey couple with the axions In [40] the moduli stabilizationis fulfilled with gauged anomalous 119880(1) symmetries since119880(1) 119863-terms only depend on the norm |Φ| or Re(119879) andthen directly separate the norms and phases of matter fieldsIn helical phase inflation the modulus and matter fieldsexcept the phase are stabilized at higher scale or by thesupergravity scalar potential Only the phase can be aninflaton candidate because of the protection from the global119880(1) symmetry in Kahler potential and approximate 119880(1)

symmetry in superpotential

5 Helical Phase Inflation and the KYY Model

120578 problem in supergravity inflation can be solved both inhelical phase inflation and in the shift symmetry in KYY

1

1

00

minus1

minus1

10

8

6

4

V(a

2)

120601x120601y

Figure 2 The helicoid structure of potential (31) scaled by 1198862 The

parameters with 119888 = 096 and 119887 = 01 are adopted in the graph Itis shown that the local valley locates around 119903 asymp 1 Note that thepotential gets flatter at the top of the graph

model The physics in these two solutions are obviously dif-ferent For helical phase inflation the solution employs 119880(1)

symmetry in Kahler potential of the minimal supergravityand the superpotential admits a phase monodromy arisingfrom the global 119880(1) transformation

119870 = ΦΦ + 119883119883 + sdot sdot sdot

119882 = 119886119883

ΦlnΦ

(33)

The inflation is driven by the phase of complex field Φ

and several special virtues appear in the model For theKYY model with shift symmetry [6] the Kahler potential isadjusted so that it admits a shift symmetry along the directionof Im(119879)

119870 =1

2(119879 + 119879)

2+ 119883119883 + sdot sdot sdot

119882 = 119886119883119879

(34)

and inflation is driven by Im(119879) The shift symmetry isendowedwith axions so thismechanism is attractive for axioninflation

Here we will show that although the physical pictures aremuch different in helical phase inflation andKYY typemodeljust considering the lower order terms in the Kahler potentialof redefined complex field helical phase inflation can reduceto the KYY model

Because the phase of Φ rotates under the global 119880(1)

transformation to connect helical phase inflation with theKYY model a natural guess is to take the following fieldredefinitionΦ = 119890

119879 then helical phase inflation (33) becomes

119870 = 119890119879+119879

+ sdot sdot sdot = 1 + 119879 + 119879 +1

2(119879 + 119879)

2+ sdot sdot sdot

119882 = 119886119883119879119890minus119879

(35)

8 Advances in High Energy Physics

TheKahler manifold of complex field119879 is invariant under theholomorphic Kahler transformation

119870(119879 119879) 997888rarr 119870(119879 119879) + 119865 (119879) + 119865 (119879) (36)

in which 119865(119879) is a holomorphic function of 119879 To keepthe whole Lagrangian also invariant under the Kahler trans-formation the superpotential transforms under the Kahlertransformation

119882 997888rarr 119890minus119865(119879)

119882 (37)

For the supergravity model in (35) taking the Kahler trans-formation with 119865(119879) = minus(12) minus 119879 the Kahler potential andsuperpotential become

119870 =1

2(119879 + 119879)

2+ sdot sdot sdot

119882 = 119886radic119890119883119879

(38)

which is just the KYY model (34) with higher order cor-rections in the Kahler potential These higher order termsvanish after field stabilization and have no effect on inflationprocess The field relation Φ = 119890

119879 also gives a map betweenthe simplest helical phase inflation and the KYY model suchas the inflaton arg(Φ) rarr Im(119879) and field stabilization|Φ| = 1 rarr Re(119879) = 0

Nevertheless helical phase inflation is not equivalent tothe KYY model There are higher order corrections to theKahler potential in the map from helical phase inflation toKYY model which have no effect on inflation after fieldstabilization but indicate different physics in two models Bydropping these terms certain information is lost so the mapis irreversible Specifically the inverse function 119879 = lnΦ ofthe field redefinition Φ = 119890

119879 cannot reproduce helical phaseinflation from the KYY model unless one introduces higherdimensional operators in the Kahler potential exactly thesame as given in the expansion (35) Such strict constraint oneach higher dimensional operator is not guaranteed by anyknown rule and so is unphysical As there is no pole or singu-larity in the KYY model it is unlikely to introduce pole andsingularity at origin with phase monodromy through well-defined field redefinition Actually the singularity with phasemonodromy in the superpotential indicates rich physics inthe scale above inflation

6 Helical Phase Inflation inNo-Scale Supergravity

The no-scale supergravity is an attractive frame for GUTscale phenomenology it is interesting to realize helicalphase inflation in no-scale supergravity Generally theKahler manifold of the no-scale supergravity is equippedwith 119878119880(119873 1)119878119880(119873) times 119880(1) symmetry For the no-scalesupergravity with exact 119878119880(1 1)119880(1) symmetry withoutextra fields no inflation can be realized so the case with119878119880(2 1)119878119880(2) times 119880(1) symmetry is the simplest one thatadmits inflation

The Kahler potential with 119878119880(2 1)119878119880(2)times119880(1) symme-try is

119870 = minus3 ln(119879 + 119879 minusΦΦ

3) (39)

In the symmetry of the Kahler manifold there is 119880(1)

subsector the phase rotation of complex fieldΦ which can beemployed for helical phase inflation Besides the modulus 119879should be stabilized during inflation which can be fulfilled byintroducing extra terms on119879 As a simple example of no-scalehelical phase inflation here we follow the simplification in [9]that the modulus 119879 has already been stabilized at ⟨119879⟩ = 119888Different from the minimal supergravity the kinetic termgiven by the no-scale Kahler potential is noncanonical

119871119870 = 119870ΦΦ120597120583Φ120597120583Φ

=2119888

(2119888 minus 11990323)2(120597120583119903120597

120583119903 + 1199032120597120583120579120597120583120579)

(40)

in which Φ equiv 119903119890119894120579 is used The 119865-term scalar potential is

119881 = 119890minus(23)119870 1003816100381610038161003816119882Φ

1003816100381610038161003816

2=

1003816100381610038161003816119882Φ1003816100381610038161003816

2

(119879 + 119879 minus ΦΦ3)2 (41)

where the superpotential 119882 is a holomorphic function ofsuperfieldΦ and119879 = ⟨119879⟩ = 119888 It requires a phasemonodromyin superpotential119882 for phase inflation the simple choice is

119882 =119886

Φln Φ

Λ (42)

The scalar potential given by this superpotential is

119881 =91198862

(6119888 minus 1199032)21199034

((ln 119903

119890Λ)

2

+ 1205792) (43)

As in the minimal supergravity the norm and phase ofcomplex field Φ are separated in the scalar potential For the119903-dependent coefficient factor 1(6119888 minus 119903

2)21199034 its minimum

locates at 1199030 = radic3119888 and another term (ln(119903119890Λ))2 reaches its

minimum at 1199031 = 119890Λ Given that the parameter Λ is tuned sothat 1199030 = 1199031 in the radial direction the potential has a globalminimum at 119903 = 1199030 Similar to helical phase inflation in theminimal supergravity the potential (43) also shows helicoidstructure as presented in Figure 3

The physical mass of 119903 in the region near the vacuum is

1198982119903 =

(2119888 minus 11990323)2

4119888

1205972119881

1205971199032

100381610038161003816100381610038161003816100381610038161003816100381610038161003816119903=radic3119888

= (4 +1

21205792)1198672 (44)

where119867 is theHubble constant during inflation So the norm119903 is strongly stabilized at 1199030 If 1199030 and 1199031 are not equal butclose to each other 119903 will slightly shift during inflation butits mass remains above the Hubble scale and inflation is stillapproximately the single field inflation

Advances in High Energy Physics 9

30

20

10

0

minus10

minus10

minus05

minus05

0000

05

05

10

10

120601x (radic

3c) 120601 y(radic3

c)

V(a

2 c4 )

Figure 3The helicoid structure of potential (43) scaled by 11988621198884 Inthe graph the parameter Λ is tuned so that 1199030 = 1199031 The complexfieldΦ has been rescaled byradic3119888 and the scale of field norm at localvalley is determined by the parameter 119888 instead of the Planck mass

Instead of stabilizing 119879 independently with the inflationprocess we can realize helical phase inflation from the no-scale supergravity with dynamical119879There is a natural reasonfor such consideration The phase monodromy requiresmatter fields transforming as rotations under 119880(1) and alsoa modulus 119879 as a shift under 119880(1) The Kahler potential of 119879is shift invariant Interestingly for the no-scale supergravitywith 119878119880(2 1)119878119880(2) times 119880(1) symmetry as shown in (39) theKahler potential 119870 is automatically endowed with the shiftsymmetry and global 119880(1) symmetry

119879 997888rarr 119879 + 119894119862

Φ 997888rarr Φ119890119894120579

(45)

Therefore the no-scale Kahler potential fits with the phasemonodromy in (16) and (22) initiatively

The Kahler potentials of superfields 119911 isin 119883 119884 119885 areof the minimal type 119911119911 while for Ψ its Kahler potentialcan be the minimal type ΨΨ or the no-scale type 119870 =

minus3 ln(119879+119879minus (ΦΦ+ΨΨ)3) which extends the symmetry ofKahler manifold to 119878119880(3 1)119878119880(3) times 119880(1) In this scenariothe process to integrate out heavy fields is the same asbefore Besides the potential of phase proportional to |119882119883|

2

is insensitive to the formula of Kahler potential due to119880(1) symmetry The major difference appears in the fieldstabilization The scalar potential from phase monodromy in(16) is

119881 = 119890119870 1003816100381610038161003816119882119883

1003816100381610038161003816

2=

1

(ln 1199032120572 + 119888 minus 11990323 minus 120582211990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792) asymp

1

(119888 minus 11990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792)

(46)

with 120572 ≫ 1 and 120582 ≪ 1 The field norm 119903 is stabilized byminimizing the coefficient 11199032(119888 minus 119903

23)3 and the parameter

119888 from ⟨119879⟩ determines the scale of ⟨119903⟩ = radic31198882 This isdifferent from helical phase inflation in the minimal super-gravity which minimizes the norm based on the coefficient1198901199032(11199032) and then the scale of ⟨119903⟩ is close to119872119875 the unique

energy scale of supergravity corrections Similarly combiningthe no-scale Kahlermanifold with phasemonodromy in (22)we can obtain natural inflation

7 UV-Sensitivity and Helical Phase Inflation

For the slow-roll inflation the Lyth bound [21] provides arelationship between the tensor-to-scalar ratio r and the fieldexcursion Δ120601 Roughly it requires

Δ120601

119872119875

⩾ (r

001)

12

(47)

To get large tensor-to-scalar ratio r ⩾ 001 such as in chaoticinflation or natural inflation the field excursion should bemuch larger than the Planck mass The super-Planckianfield excursion makes the description based on the effectivefield theory questionable In the Wilsonian sense the lowenergy field theory is an effective theory with higher ordercorrections introduced by the physics at the cut-off scalelike quantum gravity and these terms are irrelevant in theeffective field theory since they are suppressed by the cut-offenergy scale However for the inflation process the inflatonhas super-Planckian field excursion which is much largerthan the cut-off scaleThus the higher order terms cannot besuppressed by the Planckmass andmay affect the inflationaryobservations significantly And then they may significantlyaffect inflation or even destroy the inflation process Forexample considering the corrections

Δ119881 = 119888119894119881(120601

119872119875

)

119894

+ sdot sdot sdot (48)

to the original inflaton potential 119881 as long as 119888119894 are of theorder 10minus119894 in the initial stage of inflation 120601 sim 119874(10)119872119875 thehigher order terms can be as large as the original potential119881 So for large field inflation it is sensitive to the physics atthe cut-off scale and the predictions of inflation just based oneffective field theory are questionable

In consideration of the UV sensitivity of large fieldinflation a possible choice is to realize inflation in UV-completed theory like string theory (for a review see [22])To realize inflation in string theory there are a lot of problemsto solve besides inflation such as the moduli stabilizationMinkowskide Sitter vacua and effects of 1205721015840- and string loop-corrections Alternatively in the bottom-up approach onemay avoid the higher order corrections by introducing anextra shift symmetry in the theory The shift symmetry istechnically natural and safe under quantum loop-correctionsHowever the global symmetry can be broken by the quantumgravity effect So it is still questionable whether the shiftsymmetry can safely evade the higher corrections like in (48)

10 Advances in High Energy Physics

The UV-completion problem is dodged in helical phaseinflation Since the super-Planckian field excursion is thephase of a complex field and the phase component is notdirectly involved in the gravity interaction there are nodangerous high-order corrections like in (48) for the phasepotentialThe inflaton evolves along the helical trajectory anddoes not relate to the physics in the region above the Planckscale For helical phase inflation in the minimal supergravitythe norm of field is stabilized at the marginal point of thePlanck scale where the supergravity correction on the scalarpotential gets important based onwhich the normof complexfield can be strongly stabilized The extra corrections arelikely to appear in the Kahler potential however they canonly slightly affect the field stabilization while the phaseinflation is protected by the global 119880(1) symmetry in theKahler potential in consequence helical phase inflation isnot sensitive to these corrections at all For the helical phaseinflation in no-scale supergravity the normof complex field isstabilized at the scale of themodulus ⟨119879⟩ instead of the Planckscale one can simply adjust the scale of ⟨119879⟩ to keep themodelaway from super-Planckian region

Helical phase inflation is free from the UV-sensitivityproblem and it is just a typical physical process at theGUT scale with special superpotential that admits phasemonodromy So it provides an inflationary model that can bereliably studied just in supersymmetric field theory

8 Discussions and Conclusion

In this work we have studied the details of helical phase infla-tion from several aspects Helical phase inflation is realizedin supergravity with global 119880(1) symmetry 119880(1) symmetryis built in the Kahler potential so that helical phase inflationcan be realized by the ordinary Kahler potentials such as theminimal or no-scale types Helical phase inflation directlyleads to the phasemonodromy in the superpotential which issingular and an effective field theory arising from integratingout heavy fieldsThe phasemonodromy originates from119880(1)

rotation of the superpotential at higher scale Generically thesuperpotential can be separated into two parts119882 = 119882119868 +119882119878where119882119878 admits the global119880(1) symmetrywhile119882119868 breaks itexplicitly at scale much lower than 119882119878 Under 119880(1) rotationthe inflation term 119882119868 is slightly changed which realizes thephase monodromy in the effective theory and introducesa flat potential along the direction of phase rotation Bybreaking the global119880(1) symmetry in different ways we mayget different kinds of inflation such as quadratic inflationnatural inflation or the other types of inflation that are notpresented in this work

An amazing fact of helical phase inflation is that itdeeply relates to several interesting points of inflation andnaturally combines them in a rather simple potential withhelicoid structure The features of helical phase inflation canbe summarized as follows

(i) The global 119880(1) symmetry is built in the Kahlerpotential so helical phase inflation provides a naturalsolution to 120578 problem

(ii) The phase excursion requires phase monodromy inthe superpotential So helical phase inflation providesin the mathematical sense a new type of monodromyin supersymmetric field theory

(iii) The singularity in the superpotential together withthe supergravity scalar potential provides strong fieldstabilization which is consistent with phase inflation

(iv) The super-Planckian field excursion is realized bythe phase of a complex field instead of any otherldquophysicalrdquo fields that directly couple with gravity Sothere are no polynomial higher order corrections forthe phase and thus inflation is not sensitive to thequantum gravity corrections

To summarize helical phase inflation introduces a newtype of inflation that can be effectively described by super-symmetric field theory at the GUT scale Generically thesuper-Planckian field excursion makes the inflationary pre-dictions based on effective field theory questionable sincethe higher order corrections from quantum gravity arelikely to affect the inflation process significantly One of thesolutions is to realize inflation in a UV-completed theory likestring theory nevertheless there are many difficult issues instring theory to resolve before realizing inflation completelyHelical phase inflation is another simple solution to the UV-sensitivity problem It is based on the supersymmetric fieldtheory and the physics are clear and much easier to controlFurthermore helical phase inflation makes the unification ofinflation theory with GUT more natural since both of themare triggered at the scale of 1016 GeV and can be effectivelystudied based on supersymmetric field theory

Besides we have shown that helical phase inflation alsorelates to several interesting developments in inflation theoryIt can be easily modified for natural inflation and realizethe phase-axion alignment indirectly which is similar tothe axion-axion alignment mechanism for super-Planckianaxion decay constant [34] The phase-axion alignment isnot shown in the final supergravity model which exhibitsexplicit phase monodromy only However the phase-axionalignment is hidden in the process when the heavy fields areintegrating out For 120578 problem in the supergravity inflationthere is another well-known solution the KYY model withshift symmetry We showed that through field redefinitionhelical phase inflation given by (4) and (5) reduces to theKYYmodel where the higher order corrections in the Kahlerpotential have no effect on inflation process However thereis no inverse transformation from the KYY model to helicalphase inflation since no well-defined field redefinition canintroduce the pole and phase singularity needed for phasemonodromy Helical phase inflation can be realized in no-scale supergravity The no-scale Kahler potential automati-cally provides the symmetry needed by phase monodromyIn the no-scale supergravity the norm of complex field isstabilized at the scale of the modulus

Our inflation models are constructed within the super-gravity theory with global 119880(1) symmetry broken explicitlyby the subleading order superpotential term So it is justtypical GUT scale physics and indicates that a UV-completed

Advances in High Energy Physics 11

framework seems to be not prerequisite to effectively describesuch inflation process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work of DVN was supported in part by the DOEGrant DE-FG03-95-ER-40917 The work of Tianjun Li issupported in part by theNatural Science Foundation of ChinaunderGrants nos 10821504 11075194 11135003 11275246 and11475238 and by the National Basic Research Program ofChina (973 Program) under Grant no 2010CB833000

References

[1] A H Guth ldquoInflationary universe a possible solution to thehorizon and flatness problemsrdquo Physical Review D vol 23 no2 pp 347ndash356 1981

[2] A D Linde ldquoColeman-Weinberg theory and the new inflation-ary universe scenariordquo Physics Letters B vol 114 no 6 pp 431ndash435 1982

[3] A Albrecht and P J Steinhardt ldquoCosmology for grand unifiedtheories with radiatively induced symmetry breakingrdquo PhysicalReview Letters vol 48 no 17 pp 1220ndash1223 1982

[4] P A R Ade N Aghanim C Armitage-Caplan et al ldquoPlanck2013 results XXII Constraints on inflationrdquo Astronomy ampAstrophysics vol 571 article A22 42 pages 2014

[5] P A R Ade R W Aikin D Barkats et al ldquoDetection of B-mode polarization at degree angular scales by BICEP2rdquo PhysicalReview Letters vol 112 no 24 Article ID 241101 2014

[6] M Kawasaki M Yamaguchi and T Yanagida ldquoNatural chaoticinflation in supergravityrdquo Physical Review Letters vol 85 no 17pp 3572ndash3575 2000

[7] T Li Z Li and D V Nanopoulos ldquoSupergravity inflation withbroken shift symmetry and large tensor-to-scalar ratiordquo Journalof Cosmology and Astroparticle Physics vol 2014 no 2 article028 2014

[8] K Harigaya and T T Yanagida ldquoDiscovery of large scale tensormode and chaotic inflation in supergravityrdquo Physics Letters Bvol 734 pp 13ndash16 2014

[9] J Ellis D V Nanopoulos and K A Olive ldquoNo-scale supergrav-ity realization of the starobinsky model of inflationrdquo PhysicalReview Letters vol 111 no 11 Article ID 111301 2013

[10] T Li Z Li and D V Nanopoulos ldquoNo-scale ripple inflationrevisitedrdquo Journal of Cosmology and Astroparticle Physics vol04 article 018 2014

[11] C Pallis ldquoInduced-gravity inflation in no-scale supergravityand beyondrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 8 article 057 2014

[12] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky modelrdquo Fortschritte der Physik vol 62 no 7 pp 573ndash5832014

[13] J Ellis M A G Garca D V Nanopoulos and K A OliveldquoResurrecting quadratic inflation in no-scale supergravity inlight of BICEP2rdquo Journal of Cosmology andAstroparticle Physicsvol 2014 no 5 article 037 2014

[14] T Li Z Li and D V Nanopoulos ldquoChaotic inflation in no-scale supergravitywith string inspiredmoduli stabilizationrdquoTheEuropean Physical Journal C vol 75 no 2 article 55 2015

[15] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky model and higher curvature correctionsrdquo Fortschritte derPhysik vol 63 no 1 pp 2ndash11 2015

[16] S V Ketov andT Terada ldquoInflation in supergravity with a singlechiral superfieldrdquo Physics Letters B vol 736 pp 272ndash277 2014

[17] C Kounnas D Lust and N Toumbas ldquoR2 inflation fromscale invariant supergravity and anomaly free superstrings withfluxesrdquo Fortschritte der Physik vol 63 no 1 pp 12ndash35 2015

[18] G A Diamandis B C Georgalas K Kaskavelis PKouroumalou A B Lahanas and G Pavlopoulos ldquoInflationin R2 supergravity with non-minimal superpotentialsrdquo PhysicsLetters B vol 744 pp 74ndash81 2015

[19] C Pallis and Q Shafi ldquoGravity waves from non-minimalquadratic inflationrdquo Journal of Cosmology and AstroparticlePhysics vol 2015 no 3 article 023 2015

[20] T Li Z Li and D V Nanopoulos ldquoHelical phase inflationrdquoPhysical Review D vol 91 no 6 Article ID 061303 2015

[21] D H Lyth ldquoWhat would we learn by detecting a gravitationalwave signal in the cosmic microwave background anisotropyrdquoPhysical Review Letters vol 78 no 10 article 1861 1997

[22] D Baumann and L McAllister ldquoInflation and stringtheoryrdquohttparxivorgabs14042601

[23] F Marchesano G Shiu and A M Uranga ldquoF-term axionmonodromy inflationrdquo Journal of High Energy Physics vol 2014no 9 article 184 2014

[24] C Long L McAllister and P McGuirk ldquoAligned naturalinflation in string theoryrdquo Physical Review D vol 90 no 2Article ID 023501 2014

[25] X Gao T Li and P Shukla ldquoCombining universal and odd RRaxions for aligned natural inflationrdquo Journal of Cosmology andAstroparticle Physics vol 2014 no 10 article 048 2014

[26] I Ben-Dayan F G Pedro and A Westphal ldquoTowards naturalinflation in string theoryrdquo Physical Review D vol 92 no 2Article ID 023515 2015

[27] R Blumenhagen D Herschmann and E Plauschinn ldquoThechallenge of realizing F-term axion monodromy inflation instring theoryrdquo Journal of High Energy Physics vol 2015 article7 2015

[28] H Abe T Kobayashi and H Otsuka ldquoTowards natural infla-tion from weakly coupled heterotic string theoryrdquo Progress ofTheoretical and Experimental Physics Article ID 063E02 2015

[29] T Ali S S Haque and V Jejjala ldquoNatural inflation from nearalignment in heterotic string theoryrdquo Physical Review D vol 91no 8 Article ID 083516 9 pages 2015

[30] R Flauger LMcAllister E Silverstein and AWestphal ldquoDrift-ing oscillations in axionmonodromyrdquo httparxivorgabs14121814

[31] K Freese J A Frieman andAVOlinto ldquoNatural inflationwithpseudo nambu-goldstone bosonsrdquo Physical Review Letters vol65 no 26 pp 3233ndash3236 1990

[32] F C Adams J R Bond K Freese J A Frieman and AV Olinto ldquoNatural inflation particle physics models power-law spectra for large-scale structure and constraints from thecosmic background explorerrdquo Physical Review D vol 47 no 2pp 426ndash455 1993

[33] G German A Mazumdar and A Perez-Lorenzana ldquoAngularinflation from supergravityrdquo Modern Physics Letters A vol 17no 25 pp 1627ndash1634 2002

12 Advances in High Energy Physics

[34] J E Kim H P Nilles and M Peloso ldquoCompleting naturalinflationrdquo Journal of Cosmology and Astroparticle Physics vol0501 article 005 2005

[35] D Baumann and D Green ldquoInflating with baryonsrdquo Journal ofHigh Energy Physics vol 2011 no 4 article 071 2011

[36] JMcDonald ldquoSub-Planckian two-field inflation consistent withthe Lyth boundrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 9 article 027 2014

[37] J McDonald ldquoA minimal sub-Planckian axion inflation modelwith large tensor-to-scalar ratiordquo Journal of Cosmology andAstroparticle Physics vol 2015 no 1 article 018 2015

[38] C D Carone J Erlich A Sensharma and Z Wang ldquoTwo-fieldaxion-monodromy hybrid inflation model Dantersquos WaterfallrdquoPhysical ReviewD vol 91 no 4 Article ID 043512 8 pages 2015

[39] G Barenboim and W-I Park ldquoSpiral Inflationrdquo Physics LettersB vol 741 pp 252ndash255 2015

[40] T Li Z Li andDVNanopoulos ldquoAligned natural inflation andmoduli stabilization from anomalous U(1) gauge symmetriesrdquoJournal of High Energy Physics vol 2014 no 11 article 012 2014

[41] T Li Z Li and D V Nanopoulos ldquoNatural inflation withnatural trans-planckian axion decay constant from anomalousU(1)119883rdquo Journal of High Energy Physics vol 2014 no 7 article052 2014

[42] E Silverstein and A Westphal ldquoMonodromy in the CMBgravity waves and string inflationrdquo Physical Review D vol 78no 10 Article ID 106003 21 pages 2008

[43] L McAllister E Silverstein and A Westphal ldquoGravity wavesand linear inflation from axion monodromyrdquo Physical ReviewD vol 82 no 4 Article ID 046003 2010

[44] K Choi H Kim and S Yun ldquoNatural inflation with multiplesub-Planckian axionsrdquo Physical Review D vol 90 no 2 ArticleID 023545 2014

[45] S-H H Tye and S S C Wong ldquoHelical inflation and cosmic-stringsrdquo httparxivorgabs14046988

[46] R Kappl S Krippendorf and H P Nilles ldquoAligned naturalinflation monodromies of two axionsrdquo Physics Letters B vol737 pp 124ndash128 2014

[47] G rsquotHooft ldquoNaturalness chiral symmetry and spontaneouschiral symmetry breakingrdquo in Recent Developments in GaugeTheories G rsquotHooft C Itzykson A Jaffe et al Eds vol 59of NATO Advanced Study Institutes Series pp 135ndash157 PlenumPress New York NY USA 1980

[48] R Blumenhagen M Cvetic S Kachru and T Weigand ldquoD-brane instantons in type II string theoryrdquo Annual Review ofNuclear and Particle Science vol 59 pp 269ndash296 2009

[49] K Harigaya andM Ibe ldquoSimple realization of inflaton potentialon a Riemann surfacerdquo Physics Letters B vol 738 pp 301ndash3042014

[50] R Kallosh A Linde and B Vercnocke ldquoNatural inflation insupergravity and beyondrdquo Physical Review D vol 90 Article ID041303(R) 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

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Biophysics

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ThermodynamicsJournal of

6 Advances in High Energy Physics

as shown in the last section Furthermore the quantumloop effects during integrating out heavy fields can introducecorrections to the Kahler potential as well However becauseof 119880(1) symmetry built in the Kahler potential these termstogether with (21) can only mildly affect the field stabiliza-tion and phase inflation is not sensitive to the correctionsin Kahler potential As to the superpotential it is protectedby the nonrenormalized theorem and free from radiativecorrections

4 Natural Inflation in Helical Phase Inflation

In the superpotential 1198820 the inflation term is perturbativecoupling of complex field119879which shifts under global119880(1) aninterestingmodification is to consider inflation given by non-perturbative coupling of 119879 Such term gives a modified 119880(1)

phase monodromy in the superpotential And it leads to nat-ural inflation as a special type of helical phase inflation withphase-axion alignment which is similar to the axion-axionalignment mechanism proposed in [34] for natural inflationwith super-Planckian axion decay constant Specifically itcan be shown that the phase monodromy realized in super-symmetric field theory has similar physical picture to themodified axion alignment mechanism provided in [44 45]

To realize natural inflation the superpotential 1198820 in (16)just needs to be slightly modified

1198821 = 120590119883Ψ(119890minus120572119879

minus 120575) + 119884 (119890minus120573119879

minus 120583Ψ)

+ 119885 (ΨΦ minus 120582)

(22)

in which 1 ≪ 120572 ≪ 120573 Again there is a global 119880(1)

symmetry in the last two terms of 1198821 and the fields transferunder119880(1) like in (17) The first term which is hierarchicallysmaller breaks the 119880(1) symmetry explicitly besides ashift symmetry of 119879 is needed in the Kahler potential Themonodromy of1198821 under a circular 119880(1) rotation is

Ψ 997888rarr Ψ119890minus1198942120587

1198821 997888rarr 1198821 + 120590119883Ψ119890minus120572119879

(119890minus2120587119894(120572120573)

minus 1)

(23)

The supersymmetric field theory given by (22) admits thefollowing supersymmetric Minkowski vacuum

⟨119883⟩ = ⟨119884⟩ = ⟨119885⟩ = 0

⟨119879⟩ = minus1

120572ln 120575

⟨Ψ⟩ =1

120583120575120573120572

⟨Φ⟩ = 120582120583120575minus120573120572

(24)

The parameters are set to satisfy conditions

120575120573120572

sim 120582120583

120582 ≪ 1

120575 lt 1

(25)

so that ⟨Ψ⟩ ≪ ⟨Φ⟩ and ⟨119879⟩ gt 0

Near vacuum fields 119884119885Ψ and 119879 obtain large effectivemasses above inflation scale while 119883Φ are much lighter Atthe inflation scale the heavy fields should be integrated outThe 119865-term flatness conditions for fields 119884 and 119885 are

119865119884 = 119890minus120573119879

minus 120583Ψ = 0

119865119885 = ΨΦ minus 120582 = 0

(26)

in which the supergravity corrections 1198701198841198851198821 are neglectedas both119884 and119885 get close to zero during inflation IntegratingoutΨ and 119879 from (26) we obtain the effective superpotential1198821015840 from1198821

1198821015840= 120590120582

119883

Φ((

120583120582

Φ)

120572120573

minus 120575) (27)

Given 120572 ≪ 120573 the effective superpotential contains a termwith fractional power Inflation driven by complex potentialwith fractional power was considered in [49] to get sufficientlarge axion decay constant Here the supersymmetric fieldmonodromy naturally leads to the superpotential with frac-tional power which arises from the small hierarchy of axiondecay constants in two nonperturbative terms

Helical phase inflation is described by the effective super-potential 1198821015840 the role of phase-axion alignment is not clearfrom119882

1015840 since it is hidden in the procedure of integrating outheavy fields

The 119865-term flatness conditions (26) fix four degrees offreedom for the extra degree of freedom they correspond tothe transformations free from constraints (26)

Ψ 997888rarr Ψ119890minus119906minus119894V

Φ 997888rarr Φ119890119906+119894V

119879 997888rarr 119879 +119906

120573+

119894V120573

(28)

Parameter 119906 corresponds to the norm variation of complexfield Φ which is fixed by the supergravity correction on thescalar potential 119890119870(ΦΦ) Parameter V relates to 119880(1) transfor-mation which leads to the phase monodromy from the firstterm of1198821The scalar potential including the inflation termdepends on the superpositions among phases ofΨΦ and theaxion Im(119879) which are constrained as in (28) Among thesefields the phase ofΦ has the lightestmass after canonical fieldnormalization Similar physical picture appears in the axion-axion alignment where inflation is triggered by the axionsuperposition along the flat direction

After integrating out the heavy fields the Kahler potentialis

119870 = ΦΦ +1205822

ΦΦ+ sdot sdot sdot (29)

As 120582 ≪ 1 the Kahler potential is dominated by ΦΦ asdiscussed before the extra terms like 1205822ΦΦ only give small

Advances in High Energy Physics 7

corrections to the field stabilization Helical phase inflationcan be simply described by the supergravity

119870 = ΦΦ + 119883119883 +

119882 = 119886119883

Φ(Φminus119887

minus 119888)

(30)

where 119887 = 120572120573 ≪ 1 and 119888 asymp 1 The scalar potential given bythe above Kahler potential and superpotential is

119881 = 1198901199032 1198862

1199032(119903minus2119887

+ 1198882minus 2119888119903minus119887 cos (119887120579)) (31)

in which we have usedΦ equiv 119903119890119894120579 As usual the norm 119903 couples

with the phase in the scalar potential To generate singlefield inflation the norm component needs to be stabilizedwith heavy mass above the Hubble scale while the phasecomponent remains light Due to the phase-norm couplingthe phase and norm are likely to obtain comparable effectivemasses during inflation so it is difficult to realize singlefield inflation However it is a bit different in helical phaseinflation The above scalar potential can be rewritten asfollows

119881 = 1198901199032 1198862

1199032(119903minus119887

minus 119888)2+ 1198901199032 41198862119888

1199032+119887(sin 119887

2120579)

2

(32)

So its vacuum locates at ⟨119903⟩ = 1199030 = 119888minus1119887 and 120579 = 0 Besides

the coefficient of the phase term 1198901199032(41198881199032+119887

) reaches itsminimal value at 1199031 = radic1 + 1198872 For 119888 asymp 1 we have theapproximation 1199030 asymp 1199031 asymp 1 The extra terms in 119870 givesmall corrections to 1199030 and 1199031 but the approximation is stillvalid So with the parameters in (25) the norm 119903 in thetwo terms of scalar potential 119881 can be stabilized at the closeregion 119903 asymp 1 separately If the parameters are tuned so that1199030 = 1199031 then the norm of complex field is strictly stabilizedat the vacuum value during inflation Without such tuninga small difference between 1199030 and 1199031 is expected but theshift of 119903 during inflation is rather small and inflation is stillapproximate to the single field inflation driven by the phaseterm prop (sin(1198872)120579)2 The helicoid structure of the potential(31) is shown in Figure 2

It is known that to realize aligned axion mechanism insupergravity [50] it is very difficult to stabilize the moduli asthey couple with the axions In [40] the moduli stabilizationis fulfilled with gauged anomalous 119880(1) symmetries since119880(1) 119863-terms only depend on the norm |Φ| or Re(119879) andthen directly separate the norms and phases of matter fieldsIn helical phase inflation the modulus and matter fieldsexcept the phase are stabilized at higher scale or by thesupergravity scalar potential Only the phase can be aninflaton candidate because of the protection from the global119880(1) symmetry in Kahler potential and approximate 119880(1)

symmetry in superpotential

5 Helical Phase Inflation and the KYY Model

120578 problem in supergravity inflation can be solved both inhelical phase inflation and in the shift symmetry in KYY

1

1

00

minus1

minus1

10

8

6

4

V(a

2)

120601x120601y

Figure 2 The helicoid structure of potential (31) scaled by 1198862 The

parameters with 119888 = 096 and 119887 = 01 are adopted in the graph Itis shown that the local valley locates around 119903 asymp 1 Note that thepotential gets flatter at the top of the graph

model The physics in these two solutions are obviously dif-ferent For helical phase inflation the solution employs 119880(1)

symmetry in Kahler potential of the minimal supergravityand the superpotential admits a phase monodromy arisingfrom the global 119880(1) transformation

119870 = ΦΦ + 119883119883 + sdot sdot sdot

119882 = 119886119883

ΦlnΦ

(33)

The inflation is driven by the phase of complex field Φ

and several special virtues appear in the model For theKYY model with shift symmetry [6] the Kahler potential isadjusted so that it admits a shift symmetry along the directionof Im(119879)

119870 =1

2(119879 + 119879)

2+ 119883119883 + sdot sdot sdot

119882 = 119886119883119879

(34)

and inflation is driven by Im(119879) The shift symmetry isendowedwith axions so thismechanism is attractive for axioninflation

Here we will show that although the physical pictures aremuch different in helical phase inflation andKYY typemodeljust considering the lower order terms in the Kahler potentialof redefined complex field helical phase inflation can reduceto the KYY model

Because the phase of Φ rotates under the global 119880(1)

transformation to connect helical phase inflation with theKYY model a natural guess is to take the following fieldredefinitionΦ = 119890

119879 then helical phase inflation (33) becomes

119870 = 119890119879+119879

+ sdot sdot sdot = 1 + 119879 + 119879 +1

2(119879 + 119879)

2+ sdot sdot sdot

119882 = 119886119883119879119890minus119879

(35)

8 Advances in High Energy Physics

TheKahler manifold of complex field119879 is invariant under theholomorphic Kahler transformation

119870(119879 119879) 997888rarr 119870(119879 119879) + 119865 (119879) + 119865 (119879) (36)

in which 119865(119879) is a holomorphic function of 119879 To keepthe whole Lagrangian also invariant under the Kahler trans-formation the superpotential transforms under the Kahlertransformation

119882 997888rarr 119890minus119865(119879)

119882 (37)

For the supergravity model in (35) taking the Kahler trans-formation with 119865(119879) = minus(12) minus 119879 the Kahler potential andsuperpotential become

119870 =1

2(119879 + 119879)

2+ sdot sdot sdot

119882 = 119886radic119890119883119879

(38)

which is just the KYY model (34) with higher order cor-rections in the Kahler potential These higher order termsvanish after field stabilization and have no effect on inflationprocess The field relation Φ = 119890

119879 also gives a map betweenthe simplest helical phase inflation and the KYY model suchas the inflaton arg(Φ) rarr Im(119879) and field stabilization|Φ| = 1 rarr Re(119879) = 0

Nevertheless helical phase inflation is not equivalent tothe KYY model There are higher order corrections to theKahler potential in the map from helical phase inflation toKYY model which have no effect on inflation after fieldstabilization but indicate different physics in two models Bydropping these terms certain information is lost so the mapis irreversible Specifically the inverse function 119879 = lnΦ ofthe field redefinition Φ = 119890

119879 cannot reproduce helical phaseinflation from the KYY model unless one introduces higherdimensional operators in the Kahler potential exactly thesame as given in the expansion (35) Such strict constraint oneach higher dimensional operator is not guaranteed by anyknown rule and so is unphysical As there is no pole or singu-larity in the KYY model it is unlikely to introduce pole andsingularity at origin with phase monodromy through well-defined field redefinition Actually the singularity with phasemonodromy in the superpotential indicates rich physics inthe scale above inflation

6 Helical Phase Inflation inNo-Scale Supergravity

The no-scale supergravity is an attractive frame for GUTscale phenomenology it is interesting to realize helicalphase inflation in no-scale supergravity Generally theKahler manifold of the no-scale supergravity is equippedwith 119878119880(119873 1)119878119880(119873) times 119880(1) symmetry For the no-scalesupergravity with exact 119878119880(1 1)119880(1) symmetry withoutextra fields no inflation can be realized so the case with119878119880(2 1)119878119880(2) times 119880(1) symmetry is the simplest one thatadmits inflation

The Kahler potential with 119878119880(2 1)119878119880(2)times119880(1) symme-try is

119870 = minus3 ln(119879 + 119879 minusΦΦ

3) (39)

In the symmetry of the Kahler manifold there is 119880(1)

subsector the phase rotation of complex fieldΦ which can beemployed for helical phase inflation Besides the modulus 119879should be stabilized during inflation which can be fulfilled byintroducing extra terms on119879 As a simple example of no-scalehelical phase inflation here we follow the simplification in [9]that the modulus 119879 has already been stabilized at ⟨119879⟩ = 119888Different from the minimal supergravity the kinetic termgiven by the no-scale Kahler potential is noncanonical

119871119870 = 119870ΦΦ120597120583Φ120597120583Φ

=2119888

(2119888 minus 11990323)2(120597120583119903120597

120583119903 + 1199032120597120583120579120597120583120579)

(40)

in which Φ equiv 119903119890119894120579 is used The 119865-term scalar potential is

119881 = 119890minus(23)119870 1003816100381610038161003816119882Φ

1003816100381610038161003816

2=

1003816100381610038161003816119882Φ1003816100381610038161003816

2

(119879 + 119879 minus ΦΦ3)2 (41)

where the superpotential 119882 is a holomorphic function ofsuperfieldΦ and119879 = ⟨119879⟩ = 119888 It requires a phasemonodromyin superpotential119882 for phase inflation the simple choice is

119882 =119886

Φln Φ

Λ (42)

The scalar potential given by this superpotential is

119881 =91198862

(6119888 minus 1199032)21199034

((ln 119903

119890Λ)

2

+ 1205792) (43)

As in the minimal supergravity the norm and phase ofcomplex field Φ are separated in the scalar potential For the119903-dependent coefficient factor 1(6119888 minus 119903

2)21199034 its minimum

locates at 1199030 = radic3119888 and another term (ln(119903119890Λ))2 reaches its

minimum at 1199031 = 119890Λ Given that the parameter Λ is tuned sothat 1199030 = 1199031 in the radial direction the potential has a globalminimum at 119903 = 1199030 Similar to helical phase inflation in theminimal supergravity the potential (43) also shows helicoidstructure as presented in Figure 3

The physical mass of 119903 in the region near the vacuum is

1198982119903 =

(2119888 minus 11990323)2

4119888

1205972119881

1205971199032

100381610038161003816100381610038161003816100381610038161003816100381610038161003816119903=radic3119888

= (4 +1

21205792)1198672 (44)

where119867 is theHubble constant during inflation So the norm119903 is strongly stabilized at 1199030 If 1199030 and 1199031 are not equal butclose to each other 119903 will slightly shift during inflation butits mass remains above the Hubble scale and inflation is stillapproximately the single field inflation

Advances in High Energy Physics 9

30

20

10

0

minus10

minus10

minus05

minus05

0000

05

05

10

10

120601x (radic

3c) 120601 y(radic3

c)

V(a

2 c4 )

Figure 3The helicoid structure of potential (43) scaled by 11988621198884 Inthe graph the parameter Λ is tuned so that 1199030 = 1199031 The complexfieldΦ has been rescaled byradic3119888 and the scale of field norm at localvalley is determined by the parameter 119888 instead of the Planck mass

Instead of stabilizing 119879 independently with the inflationprocess we can realize helical phase inflation from the no-scale supergravity with dynamical119879There is a natural reasonfor such consideration The phase monodromy requiresmatter fields transforming as rotations under 119880(1) and alsoa modulus 119879 as a shift under 119880(1) The Kahler potential of 119879is shift invariant Interestingly for the no-scale supergravitywith 119878119880(2 1)119878119880(2) times 119880(1) symmetry as shown in (39) theKahler potential 119870 is automatically endowed with the shiftsymmetry and global 119880(1) symmetry

119879 997888rarr 119879 + 119894119862

Φ 997888rarr Φ119890119894120579

(45)

Therefore the no-scale Kahler potential fits with the phasemonodromy in (16) and (22) initiatively

The Kahler potentials of superfields 119911 isin 119883 119884 119885 areof the minimal type 119911119911 while for Ψ its Kahler potentialcan be the minimal type ΨΨ or the no-scale type 119870 =

minus3 ln(119879+119879minus (ΦΦ+ΨΨ)3) which extends the symmetry ofKahler manifold to 119878119880(3 1)119878119880(3) times 119880(1) In this scenariothe process to integrate out heavy fields is the same asbefore Besides the potential of phase proportional to |119882119883|

2

is insensitive to the formula of Kahler potential due to119880(1) symmetry The major difference appears in the fieldstabilization The scalar potential from phase monodromy in(16) is

119881 = 119890119870 1003816100381610038161003816119882119883

1003816100381610038161003816

2=

1

(ln 1199032120572 + 119888 minus 11990323 minus 120582211990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792) asymp

1

(119888 minus 11990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792)

(46)

with 120572 ≫ 1 and 120582 ≪ 1 The field norm 119903 is stabilized byminimizing the coefficient 11199032(119888 minus 119903

23)3 and the parameter

119888 from ⟨119879⟩ determines the scale of ⟨119903⟩ = radic31198882 This isdifferent from helical phase inflation in the minimal super-gravity which minimizes the norm based on the coefficient1198901199032(11199032) and then the scale of ⟨119903⟩ is close to119872119875 the unique

energy scale of supergravity corrections Similarly combiningthe no-scale Kahlermanifold with phasemonodromy in (22)we can obtain natural inflation

7 UV-Sensitivity and Helical Phase Inflation

For the slow-roll inflation the Lyth bound [21] provides arelationship between the tensor-to-scalar ratio r and the fieldexcursion Δ120601 Roughly it requires

Δ120601

119872119875

⩾ (r

001)

12

(47)

To get large tensor-to-scalar ratio r ⩾ 001 such as in chaoticinflation or natural inflation the field excursion should bemuch larger than the Planck mass The super-Planckianfield excursion makes the description based on the effectivefield theory questionable In the Wilsonian sense the lowenergy field theory is an effective theory with higher ordercorrections introduced by the physics at the cut-off scalelike quantum gravity and these terms are irrelevant in theeffective field theory since they are suppressed by the cut-offenergy scale However for the inflation process the inflatonhas super-Planckian field excursion which is much largerthan the cut-off scaleThus the higher order terms cannot besuppressed by the Planckmass andmay affect the inflationaryobservations significantly And then they may significantlyaffect inflation or even destroy the inflation process Forexample considering the corrections

Δ119881 = 119888119894119881(120601

119872119875

)

119894

+ sdot sdot sdot (48)

to the original inflaton potential 119881 as long as 119888119894 are of theorder 10minus119894 in the initial stage of inflation 120601 sim 119874(10)119872119875 thehigher order terms can be as large as the original potential119881 So for large field inflation it is sensitive to the physics atthe cut-off scale and the predictions of inflation just based oneffective field theory are questionable

In consideration of the UV sensitivity of large fieldinflation a possible choice is to realize inflation in UV-completed theory like string theory (for a review see [22])To realize inflation in string theory there are a lot of problemsto solve besides inflation such as the moduli stabilizationMinkowskide Sitter vacua and effects of 1205721015840- and string loop-corrections Alternatively in the bottom-up approach onemay avoid the higher order corrections by introducing anextra shift symmetry in the theory The shift symmetry istechnically natural and safe under quantum loop-correctionsHowever the global symmetry can be broken by the quantumgravity effect So it is still questionable whether the shiftsymmetry can safely evade the higher corrections like in (48)

10 Advances in High Energy Physics

The UV-completion problem is dodged in helical phaseinflation Since the super-Planckian field excursion is thephase of a complex field and the phase component is notdirectly involved in the gravity interaction there are nodangerous high-order corrections like in (48) for the phasepotentialThe inflaton evolves along the helical trajectory anddoes not relate to the physics in the region above the Planckscale For helical phase inflation in the minimal supergravitythe norm of field is stabilized at the marginal point of thePlanck scale where the supergravity correction on the scalarpotential gets important based onwhich the normof complexfield can be strongly stabilized The extra corrections arelikely to appear in the Kahler potential however they canonly slightly affect the field stabilization while the phaseinflation is protected by the global 119880(1) symmetry in theKahler potential in consequence helical phase inflation isnot sensitive to these corrections at all For the helical phaseinflation in no-scale supergravity the normof complex field isstabilized at the scale of themodulus ⟨119879⟩ instead of the Planckscale one can simply adjust the scale of ⟨119879⟩ to keep themodelaway from super-Planckian region

Helical phase inflation is free from the UV-sensitivityproblem and it is just a typical physical process at theGUT scale with special superpotential that admits phasemonodromy So it provides an inflationary model that can bereliably studied just in supersymmetric field theory

8 Discussions and Conclusion

In this work we have studied the details of helical phase infla-tion from several aspects Helical phase inflation is realizedin supergravity with global 119880(1) symmetry 119880(1) symmetryis built in the Kahler potential so that helical phase inflationcan be realized by the ordinary Kahler potentials such as theminimal or no-scale types Helical phase inflation directlyleads to the phasemonodromy in the superpotential which issingular and an effective field theory arising from integratingout heavy fieldsThe phasemonodromy originates from119880(1)

rotation of the superpotential at higher scale Generically thesuperpotential can be separated into two parts119882 = 119882119868 +119882119878where119882119878 admits the global119880(1) symmetrywhile119882119868 breaks itexplicitly at scale much lower than 119882119878 Under 119880(1) rotationthe inflation term 119882119868 is slightly changed which realizes thephase monodromy in the effective theory and introducesa flat potential along the direction of phase rotation Bybreaking the global119880(1) symmetry in different ways we mayget different kinds of inflation such as quadratic inflationnatural inflation or the other types of inflation that are notpresented in this work

An amazing fact of helical phase inflation is that itdeeply relates to several interesting points of inflation andnaturally combines them in a rather simple potential withhelicoid structure The features of helical phase inflation canbe summarized as follows

(i) The global 119880(1) symmetry is built in the Kahlerpotential so helical phase inflation provides a naturalsolution to 120578 problem

(ii) The phase excursion requires phase monodromy inthe superpotential So helical phase inflation providesin the mathematical sense a new type of monodromyin supersymmetric field theory

(iii) The singularity in the superpotential together withthe supergravity scalar potential provides strong fieldstabilization which is consistent with phase inflation

(iv) The super-Planckian field excursion is realized bythe phase of a complex field instead of any otherldquophysicalrdquo fields that directly couple with gravity Sothere are no polynomial higher order corrections forthe phase and thus inflation is not sensitive to thequantum gravity corrections

To summarize helical phase inflation introduces a newtype of inflation that can be effectively described by super-symmetric field theory at the GUT scale Generically thesuper-Planckian field excursion makes the inflationary pre-dictions based on effective field theory questionable sincethe higher order corrections from quantum gravity arelikely to affect the inflation process significantly One of thesolutions is to realize inflation in a UV-completed theory likestring theory nevertheless there are many difficult issues instring theory to resolve before realizing inflation completelyHelical phase inflation is another simple solution to the UV-sensitivity problem It is based on the supersymmetric fieldtheory and the physics are clear and much easier to controlFurthermore helical phase inflation makes the unification ofinflation theory with GUT more natural since both of themare triggered at the scale of 1016 GeV and can be effectivelystudied based on supersymmetric field theory

Besides we have shown that helical phase inflation alsorelates to several interesting developments in inflation theoryIt can be easily modified for natural inflation and realizethe phase-axion alignment indirectly which is similar tothe axion-axion alignment mechanism for super-Planckianaxion decay constant [34] The phase-axion alignment isnot shown in the final supergravity model which exhibitsexplicit phase monodromy only However the phase-axionalignment is hidden in the process when the heavy fields areintegrating out For 120578 problem in the supergravity inflationthere is another well-known solution the KYY model withshift symmetry We showed that through field redefinitionhelical phase inflation given by (4) and (5) reduces to theKYYmodel where the higher order corrections in the Kahlerpotential have no effect on inflation process However thereis no inverse transformation from the KYY model to helicalphase inflation since no well-defined field redefinition canintroduce the pole and phase singularity needed for phasemonodromy Helical phase inflation can be realized in no-scale supergravity The no-scale Kahler potential automati-cally provides the symmetry needed by phase monodromyIn the no-scale supergravity the norm of complex field isstabilized at the scale of the modulus

Our inflation models are constructed within the super-gravity theory with global 119880(1) symmetry broken explicitlyby the subleading order superpotential term So it is justtypical GUT scale physics and indicates that a UV-completed

Advances in High Energy Physics 11

framework seems to be not prerequisite to effectively describesuch inflation process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work of DVN was supported in part by the DOEGrant DE-FG03-95-ER-40917 The work of Tianjun Li issupported in part by theNatural Science Foundation of ChinaunderGrants nos 10821504 11075194 11135003 11275246 and11475238 and by the National Basic Research Program ofChina (973 Program) under Grant no 2010CB833000

References

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[2] A D Linde ldquoColeman-Weinberg theory and the new inflation-ary universe scenariordquo Physics Letters B vol 114 no 6 pp 431ndash435 1982

[3] A Albrecht and P J Steinhardt ldquoCosmology for grand unifiedtheories with radiatively induced symmetry breakingrdquo PhysicalReview Letters vol 48 no 17 pp 1220ndash1223 1982

[4] P A R Ade N Aghanim C Armitage-Caplan et al ldquoPlanck2013 results XXII Constraints on inflationrdquo Astronomy ampAstrophysics vol 571 article A22 42 pages 2014

[5] P A R Ade R W Aikin D Barkats et al ldquoDetection of B-mode polarization at degree angular scales by BICEP2rdquo PhysicalReview Letters vol 112 no 24 Article ID 241101 2014

[6] M Kawasaki M Yamaguchi and T Yanagida ldquoNatural chaoticinflation in supergravityrdquo Physical Review Letters vol 85 no 17pp 3572ndash3575 2000

[7] T Li Z Li and D V Nanopoulos ldquoSupergravity inflation withbroken shift symmetry and large tensor-to-scalar ratiordquo Journalof Cosmology and Astroparticle Physics vol 2014 no 2 article028 2014

[8] K Harigaya and T T Yanagida ldquoDiscovery of large scale tensormode and chaotic inflation in supergravityrdquo Physics Letters Bvol 734 pp 13ndash16 2014

[9] J Ellis D V Nanopoulos and K A Olive ldquoNo-scale supergrav-ity realization of the starobinsky model of inflationrdquo PhysicalReview Letters vol 111 no 11 Article ID 111301 2013

[10] T Li Z Li and D V Nanopoulos ldquoNo-scale ripple inflationrevisitedrdquo Journal of Cosmology and Astroparticle Physics vol04 article 018 2014

[11] C Pallis ldquoInduced-gravity inflation in no-scale supergravityand beyondrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 8 article 057 2014

[12] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky modelrdquo Fortschritte der Physik vol 62 no 7 pp 573ndash5832014

[13] J Ellis M A G Garca D V Nanopoulos and K A OliveldquoResurrecting quadratic inflation in no-scale supergravity inlight of BICEP2rdquo Journal of Cosmology andAstroparticle Physicsvol 2014 no 5 article 037 2014

[14] T Li Z Li and D V Nanopoulos ldquoChaotic inflation in no-scale supergravitywith string inspiredmoduli stabilizationrdquoTheEuropean Physical Journal C vol 75 no 2 article 55 2015

[15] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky model and higher curvature correctionsrdquo Fortschritte derPhysik vol 63 no 1 pp 2ndash11 2015

[16] S V Ketov andT Terada ldquoInflation in supergravity with a singlechiral superfieldrdquo Physics Letters B vol 736 pp 272ndash277 2014

[17] C Kounnas D Lust and N Toumbas ldquoR2 inflation fromscale invariant supergravity and anomaly free superstrings withfluxesrdquo Fortschritte der Physik vol 63 no 1 pp 12ndash35 2015

[18] G A Diamandis B C Georgalas K Kaskavelis PKouroumalou A B Lahanas and G Pavlopoulos ldquoInflationin R2 supergravity with non-minimal superpotentialsrdquo PhysicsLetters B vol 744 pp 74ndash81 2015

[19] C Pallis and Q Shafi ldquoGravity waves from non-minimalquadratic inflationrdquo Journal of Cosmology and AstroparticlePhysics vol 2015 no 3 article 023 2015

[20] T Li Z Li and D V Nanopoulos ldquoHelical phase inflationrdquoPhysical Review D vol 91 no 6 Article ID 061303 2015

[21] D H Lyth ldquoWhat would we learn by detecting a gravitationalwave signal in the cosmic microwave background anisotropyrdquoPhysical Review Letters vol 78 no 10 article 1861 1997

[22] D Baumann and L McAllister ldquoInflation and stringtheoryrdquohttparxivorgabs14042601

[23] F Marchesano G Shiu and A M Uranga ldquoF-term axionmonodromy inflationrdquo Journal of High Energy Physics vol 2014no 9 article 184 2014

[24] C Long L McAllister and P McGuirk ldquoAligned naturalinflation in string theoryrdquo Physical Review D vol 90 no 2Article ID 023501 2014

[25] X Gao T Li and P Shukla ldquoCombining universal and odd RRaxions for aligned natural inflationrdquo Journal of Cosmology andAstroparticle Physics vol 2014 no 10 article 048 2014

[26] I Ben-Dayan F G Pedro and A Westphal ldquoTowards naturalinflation in string theoryrdquo Physical Review D vol 92 no 2Article ID 023515 2015

[27] R Blumenhagen D Herschmann and E Plauschinn ldquoThechallenge of realizing F-term axion monodromy inflation instring theoryrdquo Journal of High Energy Physics vol 2015 article7 2015

[28] H Abe T Kobayashi and H Otsuka ldquoTowards natural infla-tion from weakly coupled heterotic string theoryrdquo Progress ofTheoretical and Experimental Physics Article ID 063E02 2015

[29] T Ali S S Haque and V Jejjala ldquoNatural inflation from nearalignment in heterotic string theoryrdquo Physical Review D vol 91no 8 Article ID 083516 9 pages 2015

[30] R Flauger LMcAllister E Silverstein and AWestphal ldquoDrift-ing oscillations in axionmonodromyrdquo httparxivorgabs14121814

[31] K Freese J A Frieman andAVOlinto ldquoNatural inflationwithpseudo nambu-goldstone bosonsrdquo Physical Review Letters vol65 no 26 pp 3233ndash3236 1990

[32] F C Adams J R Bond K Freese J A Frieman and AV Olinto ldquoNatural inflation particle physics models power-law spectra for large-scale structure and constraints from thecosmic background explorerrdquo Physical Review D vol 47 no 2pp 426ndash455 1993

[33] G German A Mazumdar and A Perez-Lorenzana ldquoAngularinflation from supergravityrdquo Modern Physics Letters A vol 17no 25 pp 1627ndash1634 2002

12 Advances in High Energy Physics

[34] J E Kim H P Nilles and M Peloso ldquoCompleting naturalinflationrdquo Journal of Cosmology and Astroparticle Physics vol0501 article 005 2005

[35] D Baumann and D Green ldquoInflating with baryonsrdquo Journal ofHigh Energy Physics vol 2011 no 4 article 071 2011

[36] JMcDonald ldquoSub-Planckian two-field inflation consistent withthe Lyth boundrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 9 article 027 2014

[37] J McDonald ldquoA minimal sub-Planckian axion inflation modelwith large tensor-to-scalar ratiordquo Journal of Cosmology andAstroparticle Physics vol 2015 no 1 article 018 2015

[38] C D Carone J Erlich A Sensharma and Z Wang ldquoTwo-fieldaxion-monodromy hybrid inflation model Dantersquos WaterfallrdquoPhysical ReviewD vol 91 no 4 Article ID 043512 8 pages 2015

[39] G Barenboim and W-I Park ldquoSpiral Inflationrdquo Physics LettersB vol 741 pp 252ndash255 2015

[40] T Li Z Li andDVNanopoulos ldquoAligned natural inflation andmoduli stabilization from anomalous U(1) gauge symmetriesrdquoJournal of High Energy Physics vol 2014 no 11 article 012 2014

[41] T Li Z Li and D V Nanopoulos ldquoNatural inflation withnatural trans-planckian axion decay constant from anomalousU(1)119883rdquo Journal of High Energy Physics vol 2014 no 7 article052 2014

[42] E Silverstein and A Westphal ldquoMonodromy in the CMBgravity waves and string inflationrdquo Physical Review D vol 78no 10 Article ID 106003 21 pages 2008

[43] L McAllister E Silverstein and A Westphal ldquoGravity wavesand linear inflation from axion monodromyrdquo Physical ReviewD vol 82 no 4 Article ID 046003 2010

[44] K Choi H Kim and S Yun ldquoNatural inflation with multiplesub-Planckian axionsrdquo Physical Review D vol 90 no 2 ArticleID 023545 2014

[45] S-H H Tye and S S C Wong ldquoHelical inflation and cosmic-stringsrdquo httparxivorgabs14046988

[46] R Kappl S Krippendorf and H P Nilles ldquoAligned naturalinflation monodromies of two axionsrdquo Physics Letters B vol737 pp 124ndash128 2014

[47] G rsquotHooft ldquoNaturalness chiral symmetry and spontaneouschiral symmetry breakingrdquo in Recent Developments in GaugeTheories G rsquotHooft C Itzykson A Jaffe et al Eds vol 59of NATO Advanced Study Institutes Series pp 135ndash157 PlenumPress New York NY USA 1980

[48] R Blumenhagen M Cvetic S Kachru and T Weigand ldquoD-brane instantons in type II string theoryrdquo Annual Review ofNuclear and Particle Science vol 59 pp 269ndash296 2009

[49] K Harigaya andM Ibe ldquoSimple realization of inflaton potentialon a Riemann surfacerdquo Physics Letters B vol 738 pp 301ndash3042014

[50] R Kallosh A Linde and B Vercnocke ldquoNatural inflation insupergravity and beyondrdquo Physical Review D vol 90 Article ID041303(R) 2014

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Advances in High Energy Physics 7

corrections to the field stabilization Helical phase inflationcan be simply described by the supergravity

119870 = ΦΦ + 119883119883 +

119882 = 119886119883

Φ(Φminus119887

minus 119888)

(30)

where 119887 = 120572120573 ≪ 1 and 119888 asymp 1 The scalar potential given bythe above Kahler potential and superpotential is

119881 = 1198901199032 1198862

1199032(119903minus2119887

+ 1198882minus 2119888119903minus119887 cos (119887120579)) (31)

in which we have usedΦ equiv 119903119890119894120579 As usual the norm 119903 couples

with the phase in the scalar potential To generate singlefield inflation the norm component needs to be stabilizedwith heavy mass above the Hubble scale while the phasecomponent remains light Due to the phase-norm couplingthe phase and norm are likely to obtain comparable effectivemasses during inflation so it is difficult to realize singlefield inflation However it is a bit different in helical phaseinflation The above scalar potential can be rewritten asfollows

119881 = 1198901199032 1198862

1199032(119903minus119887

minus 119888)2+ 1198901199032 41198862119888

1199032+119887(sin 119887

2120579)

2

(32)

So its vacuum locates at ⟨119903⟩ = 1199030 = 119888minus1119887 and 120579 = 0 Besides

the coefficient of the phase term 1198901199032(41198881199032+119887

) reaches itsminimal value at 1199031 = radic1 + 1198872 For 119888 asymp 1 we have theapproximation 1199030 asymp 1199031 asymp 1 The extra terms in 119870 givesmall corrections to 1199030 and 1199031 but the approximation is stillvalid So with the parameters in (25) the norm 119903 in thetwo terms of scalar potential 119881 can be stabilized at the closeregion 119903 asymp 1 separately If the parameters are tuned so that1199030 = 1199031 then the norm of complex field is strictly stabilizedat the vacuum value during inflation Without such tuninga small difference between 1199030 and 1199031 is expected but theshift of 119903 during inflation is rather small and inflation is stillapproximate to the single field inflation driven by the phaseterm prop (sin(1198872)120579)2 The helicoid structure of the potential(31) is shown in Figure 2

It is known that to realize aligned axion mechanism insupergravity [50] it is very difficult to stabilize the moduli asthey couple with the axions In [40] the moduli stabilizationis fulfilled with gauged anomalous 119880(1) symmetries since119880(1) 119863-terms only depend on the norm |Φ| or Re(119879) andthen directly separate the norms and phases of matter fieldsIn helical phase inflation the modulus and matter fieldsexcept the phase are stabilized at higher scale or by thesupergravity scalar potential Only the phase can be aninflaton candidate because of the protection from the global119880(1) symmetry in Kahler potential and approximate 119880(1)

symmetry in superpotential

5 Helical Phase Inflation and the KYY Model

120578 problem in supergravity inflation can be solved both inhelical phase inflation and in the shift symmetry in KYY

1

1

00

minus1

minus1

10

8

6

4

V(a

2)

120601x120601y

Figure 2 The helicoid structure of potential (31) scaled by 1198862 The

parameters with 119888 = 096 and 119887 = 01 are adopted in the graph Itis shown that the local valley locates around 119903 asymp 1 Note that thepotential gets flatter at the top of the graph

model The physics in these two solutions are obviously dif-ferent For helical phase inflation the solution employs 119880(1)

symmetry in Kahler potential of the minimal supergravityand the superpotential admits a phase monodromy arisingfrom the global 119880(1) transformation

119870 = ΦΦ + 119883119883 + sdot sdot sdot

119882 = 119886119883

ΦlnΦ

(33)

The inflation is driven by the phase of complex field Φ

and several special virtues appear in the model For theKYY model with shift symmetry [6] the Kahler potential isadjusted so that it admits a shift symmetry along the directionof Im(119879)

119870 =1

2(119879 + 119879)

2+ 119883119883 + sdot sdot sdot

119882 = 119886119883119879

(34)

and inflation is driven by Im(119879) The shift symmetry isendowedwith axions so thismechanism is attractive for axioninflation

Here we will show that although the physical pictures aremuch different in helical phase inflation andKYY typemodeljust considering the lower order terms in the Kahler potentialof redefined complex field helical phase inflation can reduceto the KYY model

Because the phase of Φ rotates under the global 119880(1)

transformation to connect helical phase inflation with theKYY model a natural guess is to take the following fieldredefinitionΦ = 119890

119879 then helical phase inflation (33) becomes

119870 = 119890119879+119879

+ sdot sdot sdot = 1 + 119879 + 119879 +1

2(119879 + 119879)

2+ sdot sdot sdot

119882 = 119886119883119879119890minus119879

(35)

8 Advances in High Energy Physics

TheKahler manifold of complex field119879 is invariant under theholomorphic Kahler transformation

119870(119879 119879) 997888rarr 119870(119879 119879) + 119865 (119879) + 119865 (119879) (36)

in which 119865(119879) is a holomorphic function of 119879 To keepthe whole Lagrangian also invariant under the Kahler trans-formation the superpotential transforms under the Kahlertransformation

119882 997888rarr 119890minus119865(119879)

119882 (37)

For the supergravity model in (35) taking the Kahler trans-formation with 119865(119879) = minus(12) minus 119879 the Kahler potential andsuperpotential become

119870 =1

2(119879 + 119879)

2+ sdot sdot sdot

119882 = 119886radic119890119883119879

(38)

which is just the KYY model (34) with higher order cor-rections in the Kahler potential These higher order termsvanish after field stabilization and have no effect on inflationprocess The field relation Φ = 119890

119879 also gives a map betweenthe simplest helical phase inflation and the KYY model suchas the inflaton arg(Φ) rarr Im(119879) and field stabilization|Φ| = 1 rarr Re(119879) = 0

Nevertheless helical phase inflation is not equivalent tothe KYY model There are higher order corrections to theKahler potential in the map from helical phase inflation toKYY model which have no effect on inflation after fieldstabilization but indicate different physics in two models Bydropping these terms certain information is lost so the mapis irreversible Specifically the inverse function 119879 = lnΦ ofthe field redefinition Φ = 119890

119879 cannot reproduce helical phaseinflation from the KYY model unless one introduces higherdimensional operators in the Kahler potential exactly thesame as given in the expansion (35) Such strict constraint oneach higher dimensional operator is not guaranteed by anyknown rule and so is unphysical As there is no pole or singu-larity in the KYY model it is unlikely to introduce pole andsingularity at origin with phase monodromy through well-defined field redefinition Actually the singularity with phasemonodromy in the superpotential indicates rich physics inthe scale above inflation

6 Helical Phase Inflation inNo-Scale Supergravity

The no-scale supergravity is an attractive frame for GUTscale phenomenology it is interesting to realize helicalphase inflation in no-scale supergravity Generally theKahler manifold of the no-scale supergravity is equippedwith 119878119880(119873 1)119878119880(119873) times 119880(1) symmetry For the no-scalesupergravity with exact 119878119880(1 1)119880(1) symmetry withoutextra fields no inflation can be realized so the case with119878119880(2 1)119878119880(2) times 119880(1) symmetry is the simplest one thatadmits inflation

The Kahler potential with 119878119880(2 1)119878119880(2)times119880(1) symme-try is

119870 = minus3 ln(119879 + 119879 minusΦΦ

3) (39)

In the symmetry of the Kahler manifold there is 119880(1)

subsector the phase rotation of complex fieldΦ which can beemployed for helical phase inflation Besides the modulus 119879should be stabilized during inflation which can be fulfilled byintroducing extra terms on119879 As a simple example of no-scalehelical phase inflation here we follow the simplification in [9]that the modulus 119879 has already been stabilized at ⟨119879⟩ = 119888Different from the minimal supergravity the kinetic termgiven by the no-scale Kahler potential is noncanonical

119871119870 = 119870ΦΦ120597120583Φ120597120583Φ

=2119888

(2119888 minus 11990323)2(120597120583119903120597

120583119903 + 1199032120597120583120579120597120583120579)

(40)

in which Φ equiv 119903119890119894120579 is used The 119865-term scalar potential is

119881 = 119890minus(23)119870 1003816100381610038161003816119882Φ

1003816100381610038161003816

2=

1003816100381610038161003816119882Φ1003816100381610038161003816

2

(119879 + 119879 minus ΦΦ3)2 (41)

where the superpotential 119882 is a holomorphic function ofsuperfieldΦ and119879 = ⟨119879⟩ = 119888 It requires a phasemonodromyin superpotential119882 for phase inflation the simple choice is

119882 =119886

Φln Φ

Λ (42)

The scalar potential given by this superpotential is

119881 =91198862

(6119888 minus 1199032)21199034

((ln 119903

119890Λ)

2

+ 1205792) (43)

As in the minimal supergravity the norm and phase ofcomplex field Φ are separated in the scalar potential For the119903-dependent coefficient factor 1(6119888 minus 119903

2)21199034 its minimum

locates at 1199030 = radic3119888 and another term (ln(119903119890Λ))2 reaches its

minimum at 1199031 = 119890Λ Given that the parameter Λ is tuned sothat 1199030 = 1199031 in the radial direction the potential has a globalminimum at 119903 = 1199030 Similar to helical phase inflation in theminimal supergravity the potential (43) also shows helicoidstructure as presented in Figure 3

The physical mass of 119903 in the region near the vacuum is

1198982119903 =

(2119888 minus 11990323)2

4119888

1205972119881

1205971199032

100381610038161003816100381610038161003816100381610038161003816100381610038161003816119903=radic3119888

= (4 +1

21205792)1198672 (44)

where119867 is theHubble constant during inflation So the norm119903 is strongly stabilized at 1199030 If 1199030 and 1199031 are not equal butclose to each other 119903 will slightly shift during inflation butits mass remains above the Hubble scale and inflation is stillapproximately the single field inflation

Advances in High Energy Physics 9

30

20

10

0

minus10

minus10

minus05

minus05

0000

05

05

10

10

120601x (radic

3c) 120601 y(radic3

c)

V(a

2 c4 )

Figure 3The helicoid structure of potential (43) scaled by 11988621198884 Inthe graph the parameter Λ is tuned so that 1199030 = 1199031 The complexfieldΦ has been rescaled byradic3119888 and the scale of field norm at localvalley is determined by the parameter 119888 instead of the Planck mass

Instead of stabilizing 119879 independently with the inflationprocess we can realize helical phase inflation from the no-scale supergravity with dynamical119879There is a natural reasonfor such consideration The phase monodromy requiresmatter fields transforming as rotations under 119880(1) and alsoa modulus 119879 as a shift under 119880(1) The Kahler potential of 119879is shift invariant Interestingly for the no-scale supergravitywith 119878119880(2 1)119878119880(2) times 119880(1) symmetry as shown in (39) theKahler potential 119870 is automatically endowed with the shiftsymmetry and global 119880(1) symmetry

119879 997888rarr 119879 + 119894119862

Φ 997888rarr Φ119890119894120579

(45)

Therefore the no-scale Kahler potential fits with the phasemonodromy in (16) and (22) initiatively

The Kahler potentials of superfields 119911 isin 119883 119884 119885 areof the minimal type 119911119911 while for Ψ its Kahler potentialcan be the minimal type ΨΨ or the no-scale type 119870 =

minus3 ln(119879+119879minus (ΦΦ+ΨΨ)3) which extends the symmetry ofKahler manifold to 119878119880(3 1)119878119880(3) times 119880(1) In this scenariothe process to integrate out heavy fields is the same asbefore Besides the potential of phase proportional to |119882119883|

2

is insensitive to the formula of Kahler potential due to119880(1) symmetry The major difference appears in the fieldstabilization The scalar potential from phase monodromy in(16) is

119881 = 119890119870 1003816100381610038161003816119882119883

1003816100381610038161003816

2=

1

(ln 1199032120572 + 119888 minus 11990323 minus 120582211990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792) asymp

1

(119888 minus 11990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792)

(46)

with 120572 ≫ 1 and 120582 ≪ 1 The field norm 119903 is stabilized byminimizing the coefficient 11199032(119888 minus 119903

23)3 and the parameter

119888 from ⟨119879⟩ determines the scale of ⟨119903⟩ = radic31198882 This isdifferent from helical phase inflation in the minimal super-gravity which minimizes the norm based on the coefficient1198901199032(11199032) and then the scale of ⟨119903⟩ is close to119872119875 the unique

energy scale of supergravity corrections Similarly combiningthe no-scale Kahlermanifold with phasemonodromy in (22)we can obtain natural inflation

7 UV-Sensitivity and Helical Phase Inflation

For the slow-roll inflation the Lyth bound [21] provides arelationship between the tensor-to-scalar ratio r and the fieldexcursion Δ120601 Roughly it requires

Δ120601

119872119875

⩾ (r

001)

12

(47)

To get large tensor-to-scalar ratio r ⩾ 001 such as in chaoticinflation or natural inflation the field excursion should bemuch larger than the Planck mass The super-Planckianfield excursion makes the description based on the effectivefield theory questionable In the Wilsonian sense the lowenergy field theory is an effective theory with higher ordercorrections introduced by the physics at the cut-off scalelike quantum gravity and these terms are irrelevant in theeffective field theory since they are suppressed by the cut-offenergy scale However for the inflation process the inflatonhas super-Planckian field excursion which is much largerthan the cut-off scaleThus the higher order terms cannot besuppressed by the Planckmass andmay affect the inflationaryobservations significantly And then they may significantlyaffect inflation or even destroy the inflation process Forexample considering the corrections

Δ119881 = 119888119894119881(120601

119872119875

)

119894

+ sdot sdot sdot (48)

to the original inflaton potential 119881 as long as 119888119894 are of theorder 10minus119894 in the initial stage of inflation 120601 sim 119874(10)119872119875 thehigher order terms can be as large as the original potential119881 So for large field inflation it is sensitive to the physics atthe cut-off scale and the predictions of inflation just based oneffective field theory are questionable

In consideration of the UV sensitivity of large fieldinflation a possible choice is to realize inflation in UV-completed theory like string theory (for a review see [22])To realize inflation in string theory there are a lot of problemsto solve besides inflation such as the moduli stabilizationMinkowskide Sitter vacua and effects of 1205721015840- and string loop-corrections Alternatively in the bottom-up approach onemay avoid the higher order corrections by introducing anextra shift symmetry in the theory The shift symmetry istechnically natural and safe under quantum loop-correctionsHowever the global symmetry can be broken by the quantumgravity effect So it is still questionable whether the shiftsymmetry can safely evade the higher corrections like in (48)

10 Advances in High Energy Physics

The UV-completion problem is dodged in helical phaseinflation Since the super-Planckian field excursion is thephase of a complex field and the phase component is notdirectly involved in the gravity interaction there are nodangerous high-order corrections like in (48) for the phasepotentialThe inflaton evolves along the helical trajectory anddoes not relate to the physics in the region above the Planckscale For helical phase inflation in the minimal supergravitythe norm of field is stabilized at the marginal point of thePlanck scale where the supergravity correction on the scalarpotential gets important based onwhich the normof complexfield can be strongly stabilized The extra corrections arelikely to appear in the Kahler potential however they canonly slightly affect the field stabilization while the phaseinflation is protected by the global 119880(1) symmetry in theKahler potential in consequence helical phase inflation isnot sensitive to these corrections at all For the helical phaseinflation in no-scale supergravity the normof complex field isstabilized at the scale of themodulus ⟨119879⟩ instead of the Planckscale one can simply adjust the scale of ⟨119879⟩ to keep themodelaway from super-Planckian region

Helical phase inflation is free from the UV-sensitivityproblem and it is just a typical physical process at theGUT scale with special superpotential that admits phasemonodromy So it provides an inflationary model that can bereliably studied just in supersymmetric field theory

8 Discussions and Conclusion

In this work we have studied the details of helical phase infla-tion from several aspects Helical phase inflation is realizedin supergravity with global 119880(1) symmetry 119880(1) symmetryis built in the Kahler potential so that helical phase inflationcan be realized by the ordinary Kahler potentials such as theminimal or no-scale types Helical phase inflation directlyleads to the phasemonodromy in the superpotential which issingular and an effective field theory arising from integratingout heavy fieldsThe phasemonodromy originates from119880(1)

rotation of the superpotential at higher scale Generically thesuperpotential can be separated into two parts119882 = 119882119868 +119882119878where119882119878 admits the global119880(1) symmetrywhile119882119868 breaks itexplicitly at scale much lower than 119882119878 Under 119880(1) rotationthe inflation term 119882119868 is slightly changed which realizes thephase monodromy in the effective theory and introducesa flat potential along the direction of phase rotation Bybreaking the global119880(1) symmetry in different ways we mayget different kinds of inflation such as quadratic inflationnatural inflation or the other types of inflation that are notpresented in this work

An amazing fact of helical phase inflation is that itdeeply relates to several interesting points of inflation andnaturally combines them in a rather simple potential withhelicoid structure The features of helical phase inflation canbe summarized as follows

(i) The global 119880(1) symmetry is built in the Kahlerpotential so helical phase inflation provides a naturalsolution to 120578 problem

(ii) The phase excursion requires phase monodromy inthe superpotential So helical phase inflation providesin the mathematical sense a new type of monodromyin supersymmetric field theory

(iii) The singularity in the superpotential together withthe supergravity scalar potential provides strong fieldstabilization which is consistent with phase inflation

(iv) The super-Planckian field excursion is realized bythe phase of a complex field instead of any otherldquophysicalrdquo fields that directly couple with gravity Sothere are no polynomial higher order corrections forthe phase and thus inflation is not sensitive to thequantum gravity corrections

To summarize helical phase inflation introduces a newtype of inflation that can be effectively described by super-symmetric field theory at the GUT scale Generically thesuper-Planckian field excursion makes the inflationary pre-dictions based on effective field theory questionable sincethe higher order corrections from quantum gravity arelikely to affect the inflation process significantly One of thesolutions is to realize inflation in a UV-completed theory likestring theory nevertheless there are many difficult issues instring theory to resolve before realizing inflation completelyHelical phase inflation is another simple solution to the UV-sensitivity problem It is based on the supersymmetric fieldtheory and the physics are clear and much easier to controlFurthermore helical phase inflation makes the unification ofinflation theory with GUT more natural since both of themare triggered at the scale of 1016 GeV and can be effectivelystudied based on supersymmetric field theory

Besides we have shown that helical phase inflation alsorelates to several interesting developments in inflation theoryIt can be easily modified for natural inflation and realizethe phase-axion alignment indirectly which is similar tothe axion-axion alignment mechanism for super-Planckianaxion decay constant [34] The phase-axion alignment isnot shown in the final supergravity model which exhibitsexplicit phase monodromy only However the phase-axionalignment is hidden in the process when the heavy fields areintegrating out For 120578 problem in the supergravity inflationthere is another well-known solution the KYY model withshift symmetry We showed that through field redefinitionhelical phase inflation given by (4) and (5) reduces to theKYYmodel where the higher order corrections in the Kahlerpotential have no effect on inflation process However thereis no inverse transformation from the KYY model to helicalphase inflation since no well-defined field redefinition canintroduce the pole and phase singularity needed for phasemonodromy Helical phase inflation can be realized in no-scale supergravity The no-scale Kahler potential automati-cally provides the symmetry needed by phase monodromyIn the no-scale supergravity the norm of complex field isstabilized at the scale of the modulus

Our inflation models are constructed within the super-gravity theory with global 119880(1) symmetry broken explicitlyby the subleading order superpotential term So it is justtypical GUT scale physics and indicates that a UV-completed

Advances in High Energy Physics 11

framework seems to be not prerequisite to effectively describesuch inflation process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work of DVN was supported in part by the DOEGrant DE-FG03-95-ER-40917 The work of Tianjun Li issupported in part by theNatural Science Foundation of ChinaunderGrants nos 10821504 11075194 11135003 11275246 and11475238 and by the National Basic Research Program ofChina (973 Program) under Grant no 2010CB833000

References

[1] A H Guth ldquoInflationary universe a possible solution to thehorizon and flatness problemsrdquo Physical Review D vol 23 no2 pp 347ndash356 1981

[2] A D Linde ldquoColeman-Weinberg theory and the new inflation-ary universe scenariordquo Physics Letters B vol 114 no 6 pp 431ndash435 1982

[3] A Albrecht and P J Steinhardt ldquoCosmology for grand unifiedtheories with radiatively induced symmetry breakingrdquo PhysicalReview Letters vol 48 no 17 pp 1220ndash1223 1982

[4] P A R Ade N Aghanim C Armitage-Caplan et al ldquoPlanck2013 results XXII Constraints on inflationrdquo Astronomy ampAstrophysics vol 571 article A22 42 pages 2014

[5] P A R Ade R W Aikin D Barkats et al ldquoDetection of B-mode polarization at degree angular scales by BICEP2rdquo PhysicalReview Letters vol 112 no 24 Article ID 241101 2014

[6] M Kawasaki M Yamaguchi and T Yanagida ldquoNatural chaoticinflation in supergravityrdquo Physical Review Letters vol 85 no 17pp 3572ndash3575 2000

[7] T Li Z Li and D V Nanopoulos ldquoSupergravity inflation withbroken shift symmetry and large tensor-to-scalar ratiordquo Journalof Cosmology and Astroparticle Physics vol 2014 no 2 article028 2014

[8] K Harigaya and T T Yanagida ldquoDiscovery of large scale tensormode and chaotic inflation in supergravityrdquo Physics Letters Bvol 734 pp 13ndash16 2014

[9] J Ellis D V Nanopoulos and K A Olive ldquoNo-scale supergrav-ity realization of the starobinsky model of inflationrdquo PhysicalReview Letters vol 111 no 11 Article ID 111301 2013

[10] T Li Z Li and D V Nanopoulos ldquoNo-scale ripple inflationrevisitedrdquo Journal of Cosmology and Astroparticle Physics vol04 article 018 2014

[11] C Pallis ldquoInduced-gravity inflation in no-scale supergravityand beyondrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 8 article 057 2014

[12] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky modelrdquo Fortschritte der Physik vol 62 no 7 pp 573ndash5832014

[13] J Ellis M A G Garca D V Nanopoulos and K A OliveldquoResurrecting quadratic inflation in no-scale supergravity inlight of BICEP2rdquo Journal of Cosmology andAstroparticle Physicsvol 2014 no 5 article 037 2014

[14] T Li Z Li and D V Nanopoulos ldquoChaotic inflation in no-scale supergravitywith string inspiredmoduli stabilizationrdquoTheEuropean Physical Journal C vol 75 no 2 article 55 2015

[15] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky model and higher curvature correctionsrdquo Fortschritte derPhysik vol 63 no 1 pp 2ndash11 2015

[16] S V Ketov andT Terada ldquoInflation in supergravity with a singlechiral superfieldrdquo Physics Letters B vol 736 pp 272ndash277 2014

[17] C Kounnas D Lust and N Toumbas ldquoR2 inflation fromscale invariant supergravity and anomaly free superstrings withfluxesrdquo Fortschritte der Physik vol 63 no 1 pp 12ndash35 2015

[18] G A Diamandis B C Georgalas K Kaskavelis PKouroumalou A B Lahanas and G Pavlopoulos ldquoInflationin R2 supergravity with non-minimal superpotentialsrdquo PhysicsLetters B vol 744 pp 74ndash81 2015

[19] C Pallis and Q Shafi ldquoGravity waves from non-minimalquadratic inflationrdquo Journal of Cosmology and AstroparticlePhysics vol 2015 no 3 article 023 2015

[20] T Li Z Li and D V Nanopoulos ldquoHelical phase inflationrdquoPhysical Review D vol 91 no 6 Article ID 061303 2015

[21] D H Lyth ldquoWhat would we learn by detecting a gravitationalwave signal in the cosmic microwave background anisotropyrdquoPhysical Review Letters vol 78 no 10 article 1861 1997

[22] D Baumann and L McAllister ldquoInflation and stringtheoryrdquohttparxivorgabs14042601

[23] F Marchesano G Shiu and A M Uranga ldquoF-term axionmonodromy inflationrdquo Journal of High Energy Physics vol 2014no 9 article 184 2014

[24] C Long L McAllister and P McGuirk ldquoAligned naturalinflation in string theoryrdquo Physical Review D vol 90 no 2Article ID 023501 2014

[25] X Gao T Li and P Shukla ldquoCombining universal and odd RRaxions for aligned natural inflationrdquo Journal of Cosmology andAstroparticle Physics vol 2014 no 10 article 048 2014

[26] I Ben-Dayan F G Pedro and A Westphal ldquoTowards naturalinflation in string theoryrdquo Physical Review D vol 92 no 2Article ID 023515 2015

[27] R Blumenhagen D Herschmann and E Plauschinn ldquoThechallenge of realizing F-term axion monodromy inflation instring theoryrdquo Journal of High Energy Physics vol 2015 article7 2015

[28] H Abe T Kobayashi and H Otsuka ldquoTowards natural infla-tion from weakly coupled heterotic string theoryrdquo Progress ofTheoretical and Experimental Physics Article ID 063E02 2015

[29] T Ali S S Haque and V Jejjala ldquoNatural inflation from nearalignment in heterotic string theoryrdquo Physical Review D vol 91no 8 Article ID 083516 9 pages 2015

[30] R Flauger LMcAllister E Silverstein and AWestphal ldquoDrift-ing oscillations in axionmonodromyrdquo httparxivorgabs14121814

[31] K Freese J A Frieman andAVOlinto ldquoNatural inflationwithpseudo nambu-goldstone bosonsrdquo Physical Review Letters vol65 no 26 pp 3233ndash3236 1990

[32] F C Adams J R Bond K Freese J A Frieman and AV Olinto ldquoNatural inflation particle physics models power-law spectra for large-scale structure and constraints from thecosmic background explorerrdquo Physical Review D vol 47 no 2pp 426ndash455 1993

[33] G German A Mazumdar and A Perez-Lorenzana ldquoAngularinflation from supergravityrdquo Modern Physics Letters A vol 17no 25 pp 1627ndash1634 2002

12 Advances in High Energy Physics

[34] J E Kim H P Nilles and M Peloso ldquoCompleting naturalinflationrdquo Journal of Cosmology and Astroparticle Physics vol0501 article 005 2005

[35] D Baumann and D Green ldquoInflating with baryonsrdquo Journal ofHigh Energy Physics vol 2011 no 4 article 071 2011

[36] JMcDonald ldquoSub-Planckian two-field inflation consistent withthe Lyth boundrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 9 article 027 2014

[37] J McDonald ldquoA minimal sub-Planckian axion inflation modelwith large tensor-to-scalar ratiordquo Journal of Cosmology andAstroparticle Physics vol 2015 no 1 article 018 2015

[38] C D Carone J Erlich A Sensharma and Z Wang ldquoTwo-fieldaxion-monodromy hybrid inflation model Dantersquos WaterfallrdquoPhysical ReviewD vol 91 no 4 Article ID 043512 8 pages 2015

[39] G Barenboim and W-I Park ldquoSpiral Inflationrdquo Physics LettersB vol 741 pp 252ndash255 2015

[40] T Li Z Li andDVNanopoulos ldquoAligned natural inflation andmoduli stabilization from anomalous U(1) gauge symmetriesrdquoJournal of High Energy Physics vol 2014 no 11 article 012 2014

[41] T Li Z Li and D V Nanopoulos ldquoNatural inflation withnatural trans-planckian axion decay constant from anomalousU(1)119883rdquo Journal of High Energy Physics vol 2014 no 7 article052 2014

[42] E Silverstein and A Westphal ldquoMonodromy in the CMBgravity waves and string inflationrdquo Physical Review D vol 78no 10 Article ID 106003 21 pages 2008

[43] L McAllister E Silverstein and A Westphal ldquoGravity wavesand linear inflation from axion monodromyrdquo Physical ReviewD vol 82 no 4 Article ID 046003 2010

[44] K Choi H Kim and S Yun ldquoNatural inflation with multiplesub-Planckian axionsrdquo Physical Review D vol 90 no 2 ArticleID 023545 2014

[45] S-H H Tye and S S C Wong ldquoHelical inflation and cosmic-stringsrdquo httparxivorgabs14046988

[46] R Kappl S Krippendorf and H P Nilles ldquoAligned naturalinflation monodromies of two axionsrdquo Physics Letters B vol737 pp 124ndash128 2014

[47] G rsquotHooft ldquoNaturalness chiral symmetry and spontaneouschiral symmetry breakingrdquo in Recent Developments in GaugeTheories G rsquotHooft C Itzykson A Jaffe et al Eds vol 59of NATO Advanced Study Institutes Series pp 135ndash157 PlenumPress New York NY USA 1980

[48] R Blumenhagen M Cvetic S Kachru and T Weigand ldquoD-brane instantons in type II string theoryrdquo Annual Review ofNuclear and Particle Science vol 59 pp 269ndash296 2009

[49] K Harigaya andM Ibe ldquoSimple realization of inflaton potentialon a Riemann surfacerdquo Physics Letters B vol 738 pp 301ndash3042014

[50] R Kallosh A Linde and B Vercnocke ldquoNatural inflation insupergravity and beyondrdquo Physical Review D vol 90 Article ID041303(R) 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Advances in Condensed Matter Physics

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

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Statistical MechanicsInternational Journal of

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GravityJournal of

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AstrophysicsJournal of

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Solid State PhysicsJournal of

 Computational  Methods in Physics

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Soft MatterJournal of

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AerodynamicsJournal of

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PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Advances in High Energy Physics

TheKahler manifold of complex field119879 is invariant under theholomorphic Kahler transformation

119870(119879 119879) 997888rarr 119870(119879 119879) + 119865 (119879) + 119865 (119879) (36)

in which 119865(119879) is a holomorphic function of 119879 To keepthe whole Lagrangian also invariant under the Kahler trans-formation the superpotential transforms under the Kahlertransformation

119882 997888rarr 119890minus119865(119879)

119882 (37)

For the supergravity model in (35) taking the Kahler trans-formation with 119865(119879) = minus(12) minus 119879 the Kahler potential andsuperpotential become

119870 =1

2(119879 + 119879)

2+ sdot sdot sdot

119882 = 119886radic119890119883119879

(38)

which is just the KYY model (34) with higher order cor-rections in the Kahler potential These higher order termsvanish after field stabilization and have no effect on inflationprocess The field relation Φ = 119890

119879 also gives a map betweenthe simplest helical phase inflation and the KYY model suchas the inflaton arg(Φ) rarr Im(119879) and field stabilization|Φ| = 1 rarr Re(119879) = 0

Nevertheless helical phase inflation is not equivalent tothe KYY model There are higher order corrections to theKahler potential in the map from helical phase inflation toKYY model which have no effect on inflation after fieldstabilization but indicate different physics in two models Bydropping these terms certain information is lost so the mapis irreversible Specifically the inverse function 119879 = lnΦ ofthe field redefinition Φ = 119890

119879 cannot reproduce helical phaseinflation from the KYY model unless one introduces higherdimensional operators in the Kahler potential exactly thesame as given in the expansion (35) Such strict constraint oneach higher dimensional operator is not guaranteed by anyknown rule and so is unphysical As there is no pole or singu-larity in the KYY model it is unlikely to introduce pole andsingularity at origin with phase monodromy through well-defined field redefinition Actually the singularity with phasemonodromy in the superpotential indicates rich physics inthe scale above inflation

6 Helical Phase Inflation inNo-Scale Supergravity

The no-scale supergravity is an attractive frame for GUTscale phenomenology it is interesting to realize helicalphase inflation in no-scale supergravity Generally theKahler manifold of the no-scale supergravity is equippedwith 119878119880(119873 1)119878119880(119873) times 119880(1) symmetry For the no-scalesupergravity with exact 119878119880(1 1)119880(1) symmetry withoutextra fields no inflation can be realized so the case with119878119880(2 1)119878119880(2) times 119880(1) symmetry is the simplest one thatadmits inflation

The Kahler potential with 119878119880(2 1)119878119880(2)times119880(1) symme-try is

119870 = minus3 ln(119879 + 119879 minusΦΦ

3) (39)

In the symmetry of the Kahler manifold there is 119880(1)

subsector the phase rotation of complex fieldΦ which can beemployed for helical phase inflation Besides the modulus 119879should be stabilized during inflation which can be fulfilled byintroducing extra terms on119879 As a simple example of no-scalehelical phase inflation here we follow the simplification in [9]that the modulus 119879 has already been stabilized at ⟨119879⟩ = 119888Different from the minimal supergravity the kinetic termgiven by the no-scale Kahler potential is noncanonical

119871119870 = 119870ΦΦ120597120583Φ120597120583Φ

=2119888

(2119888 minus 11990323)2(120597120583119903120597

120583119903 + 1199032120597120583120579120597120583120579)

(40)

in which Φ equiv 119903119890119894120579 is used The 119865-term scalar potential is

119881 = 119890minus(23)119870 1003816100381610038161003816119882Φ

1003816100381610038161003816

2=

1003816100381610038161003816119882Φ1003816100381610038161003816

2

(119879 + 119879 minus ΦΦ3)2 (41)

where the superpotential 119882 is a holomorphic function ofsuperfieldΦ and119879 = ⟨119879⟩ = 119888 It requires a phasemonodromyin superpotential119882 for phase inflation the simple choice is

119882 =119886

Φln Φ

Λ (42)

The scalar potential given by this superpotential is

119881 =91198862

(6119888 minus 1199032)21199034

((ln 119903

119890Λ)

2

+ 1205792) (43)

As in the minimal supergravity the norm and phase ofcomplex field Φ are separated in the scalar potential For the119903-dependent coefficient factor 1(6119888 minus 119903

2)21199034 its minimum

locates at 1199030 = radic3119888 and another term (ln(119903119890Λ))2 reaches its

minimum at 1199031 = 119890Λ Given that the parameter Λ is tuned sothat 1199030 = 1199031 in the radial direction the potential has a globalminimum at 119903 = 1199030 Similar to helical phase inflation in theminimal supergravity the potential (43) also shows helicoidstructure as presented in Figure 3

The physical mass of 119903 in the region near the vacuum is

1198982119903 =

(2119888 minus 11990323)2

4119888

1205972119881

1205971199032

100381610038161003816100381610038161003816100381610038161003816100381610038161003816119903=radic3119888

= (4 +1

21205792)1198672 (44)

where119867 is theHubble constant during inflation So the norm119903 is strongly stabilized at 1199030 If 1199030 and 1199031 are not equal butclose to each other 119903 will slightly shift during inflation butits mass remains above the Hubble scale and inflation is stillapproximately the single field inflation

Advances in High Energy Physics 9

30

20

10

0

minus10

minus10

minus05

minus05

0000

05

05

10

10

120601x (radic

3c) 120601 y(radic3

c)

V(a

2 c4 )

Figure 3The helicoid structure of potential (43) scaled by 11988621198884 Inthe graph the parameter Λ is tuned so that 1199030 = 1199031 The complexfieldΦ has been rescaled byradic3119888 and the scale of field norm at localvalley is determined by the parameter 119888 instead of the Planck mass

Instead of stabilizing 119879 independently with the inflationprocess we can realize helical phase inflation from the no-scale supergravity with dynamical119879There is a natural reasonfor such consideration The phase monodromy requiresmatter fields transforming as rotations under 119880(1) and alsoa modulus 119879 as a shift under 119880(1) The Kahler potential of 119879is shift invariant Interestingly for the no-scale supergravitywith 119878119880(2 1)119878119880(2) times 119880(1) symmetry as shown in (39) theKahler potential 119870 is automatically endowed with the shiftsymmetry and global 119880(1) symmetry

119879 997888rarr 119879 + 119894119862

Φ 997888rarr Φ119890119894120579

(45)

Therefore the no-scale Kahler potential fits with the phasemonodromy in (16) and (22) initiatively

The Kahler potentials of superfields 119911 isin 119883 119884 119885 areof the minimal type 119911119911 while for Ψ its Kahler potentialcan be the minimal type ΨΨ or the no-scale type 119870 =

minus3 ln(119879+119879minus (ΦΦ+ΨΨ)3) which extends the symmetry ofKahler manifold to 119878119880(3 1)119878119880(3) times 119880(1) In this scenariothe process to integrate out heavy fields is the same asbefore Besides the potential of phase proportional to |119882119883|

2

is insensitive to the formula of Kahler potential due to119880(1) symmetry The major difference appears in the fieldstabilization The scalar potential from phase monodromy in(16) is

119881 = 119890119870 1003816100381610038161003816119882119883

1003816100381610038161003816

2=

1

(ln 1199032120572 + 119888 minus 11990323 minus 120582211990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792) asymp

1

(119888 minus 11990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792)

(46)

with 120572 ≫ 1 and 120582 ≪ 1 The field norm 119903 is stabilized byminimizing the coefficient 11199032(119888 minus 119903

23)3 and the parameter

119888 from ⟨119879⟩ determines the scale of ⟨119903⟩ = radic31198882 This isdifferent from helical phase inflation in the minimal super-gravity which minimizes the norm based on the coefficient1198901199032(11199032) and then the scale of ⟨119903⟩ is close to119872119875 the unique

energy scale of supergravity corrections Similarly combiningthe no-scale Kahlermanifold with phasemonodromy in (22)we can obtain natural inflation

7 UV-Sensitivity and Helical Phase Inflation

For the slow-roll inflation the Lyth bound [21] provides arelationship between the tensor-to-scalar ratio r and the fieldexcursion Δ120601 Roughly it requires

Δ120601

119872119875

⩾ (r

001)

12

(47)

To get large tensor-to-scalar ratio r ⩾ 001 such as in chaoticinflation or natural inflation the field excursion should bemuch larger than the Planck mass The super-Planckianfield excursion makes the description based on the effectivefield theory questionable In the Wilsonian sense the lowenergy field theory is an effective theory with higher ordercorrections introduced by the physics at the cut-off scalelike quantum gravity and these terms are irrelevant in theeffective field theory since they are suppressed by the cut-offenergy scale However for the inflation process the inflatonhas super-Planckian field excursion which is much largerthan the cut-off scaleThus the higher order terms cannot besuppressed by the Planckmass andmay affect the inflationaryobservations significantly And then they may significantlyaffect inflation or even destroy the inflation process Forexample considering the corrections

Δ119881 = 119888119894119881(120601

119872119875

)

119894

+ sdot sdot sdot (48)

to the original inflaton potential 119881 as long as 119888119894 are of theorder 10minus119894 in the initial stage of inflation 120601 sim 119874(10)119872119875 thehigher order terms can be as large as the original potential119881 So for large field inflation it is sensitive to the physics atthe cut-off scale and the predictions of inflation just based oneffective field theory are questionable

In consideration of the UV sensitivity of large fieldinflation a possible choice is to realize inflation in UV-completed theory like string theory (for a review see [22])To realize inflation in string theory there are a lot of problemsto solve besides inflation such as the moduli stabilizationMinkowskide Sitter vacua and effects of 1205721015840- and string loop-corrections Alternatively in the bottom-up approach onemay avoid the higher order corrections by introducing anextra shift symmetry in the theory The shift symmetry istechnically natural and safe under quantum loop-correctionsHowever the global symmetry can be broken by the quantumgravity effect So it is still questionable whether the shiftsymmetry can safely evade the higher corrections like in (48)

10 Advances in High Energy Physics

The UV-completion problem is dodged in helical phaseinflation Since the super-Planckian field excursion is thephase of a complex field and the phase component is notdirectly involved in the gravity interaction there are nodangerous high-order corrections like in (48) for the phasepotentialThe inflaton evolves along the helical trajectory anddoes not relate to the physics in the region above the Planckscale For helical phase inflation in the minimal supergravitythe norm of field is stabilized at the marginal point of thePlanck scale where the supergravity correction on the scalarpotential gets important based onwhich the normof complexfield can be strongly stabilized The extra corrections arelikely to appear in the Kahler potential however they canonly slightly affect the field stabilization while the phaseinflation is protected by the global 119880(1) symmetry in theKahler potential in consequence helical phase inflation isnot sensitive to these corrections at all For the helical phaseinflation in no-scale supergravity the normof complex field isstabilized at the scale of themodulus ⟨119879⟩ instead of the Planckscale one can simply adjust the scale of ⟨119879⟩ to keep themodelaway from super-Planckian region

Helical phase inflation is free from the UV-sensitivityproblem and it is just a typical physical process at theGUT scale with special superpotential that admits phasemonodromy So it provides an inflationary model that can bereliably studied just in supersymmetric field theory

8 Discussions and Conclusion

In this work we have studied the details of helical phase infla-tion from several aspects Helical phase inflation is realizedin supergravity with global 119880(1) symmetry 119880(1) symmetryis built in the Kahler potential so that helical phase inflationcan be realized by the ordinary Kahler potentials such as theminimal or no-scale types Helical phase inflation directlyleads to the phasemonodromy in the superpotential which issingular and an effective field theory arising from integratingout heavy fieldsThe phasemonodromy originates from119880(1)

rotation of the superpotential at higher scale Generically thesuperpotential can be separated into two parts119882 = 119882119868 +119882119878where119882119878 admits the global119880(1) symmetrywhile119882119868 breaks itexplicitly at scale much lower than 119882119878 Under 119880(1) rotationthe inflation term 119882119868 is slightly changed which realizes thephase monodromy in the effective theory and introducesa flat potential along the direction of phase rotation Bybreaking the global119880(1) symmetry in different ways we mayget different kinds of inflation such as quadratic inflationnatural inflation or the other types of inflation that are notpresented in this work

An amazing fact of helical phase inflation is that itdeeply relates to several interesting points of inflation andnaturally combines them in a rather simple potential withhelicoid structure The features of helical phase inflation canbe summarized as follows

(i) The global 119880(1) symmetry is built in the Kahlerpotential so helical phase inflation provides a naturalsolution to 120578 problem

(ii) The phase excursion requires phase monodromy inthe superpotential So helical phase inflation providesin the mathematical sense a new type of monodromyin supersymmetric field theory

(iii) The singularity in the superpotential together withthe supergravity scalar potential provides strong fieldstabilization which is consistent with phase inflation

(iv) The super-Planckian field excursion is realized bythe phase of a complex field instead of any otherldquophysicalrdquo fields that directly couple with gravity Sothere are no polynomial higher order corrections forthe phase and thus inflation is not sensitive to thequantum gravity corrections

To summarize helical phase inflation introduces a newtype of inflation that can be effectively described by super-symmetric field theory at the GUT scale Generically thesuper-Planckian field excursion makes the inflationary pre-dictions based on effective field theory questionable sincethe higher order corrections from quantum gravity arelikely to affect the inflation process significantly One of thesolutions is to realize inflation in a UV-completed theory likestring theory nevertheless there are many difficult issues instring theory to resolve before realizing inflation completelyHelical phase inflation is another simple solution to the UV-sensitivity problem It is based on the supersymmetric fieldtheory and the physics are clear and much easier to controlFurthermore helical phase inflation makes the unification ofinflation theory with GUT more natural since both of themare triggered at the scale of 1016 GeV and can be effectivelystudied based on supersymmetric field theory

Besides we have shown that helical phase inflation alsorelates to several interesting developments in inflation theoryIt can be easily modified for natural inflation and realizethe phase-axion alignment indirectly which is similar tothe axion-axion alignment mechanism for super-Planckianaxion decay constant [34] The phase-axion alignment isnot shown in the final supergravity model which exhibitsexplicit phase monodromy only However the phase-axionalignment is hidden in the process when the heavy fields areintegrating out For 120578 problem in the supergravity inflationthere is another well-known solution the KYY model withshift symmetry We showed that through field redefinitionhelical phase inflation given by (4) and (5) reduces to theKYYmodel where the higher order corrections in the Kahlerpotential have no effect on inflation process However thereis no inverse transformation from the KYY model to helicalphase inflation since no well-defined field redefinition canintroduce the pole and phase singularity needed for phasemonodromy Helical phase inflation can be realized in no-scale supergravity The no-scale Kahler potential automati-cally provides the symmetry needed by phase monodromyIn the no-scale supergravity the norm of complex field isstabilized at the scale of the modulus

Our inflation models are constructed within the super-gravity theory with global 119880(1) symmetry broken explicitlyby the subleading order superpotential term So it is justtypical GUT scale physics and indicates that a UV-completed

Advances in High Energy Physics 11

framework seems to be not prerequisite to effectively describesuch inflation process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work of DVN was supported in part by the DOEGrant DE-FG03-95-ER-40917 The work of Tianjun Li issupported in part by theNatural Science Foundation of ChinaunderGrants nos 10821504 11075194 11135003 11275246 and11475238 and by the National Basic Research Program ofChina (973 Program) under Grant no 2010CB833000

References

[1] A H Guth ldquoInflationary universe a possible solution to thehorizon and flatness problemsrdquo Physical Review D vol 23 no2 pp 347ndash356 1981

[2] A D Linde ldquoColeman-Weinberg theory and the new inflation-ary universe scenariordquo Physics Letters B vol 114 no 6 pp 431ndash435 1982

[3] A Albrecht and P J Steinhardt ldquoCosmology for grand unifiedtheories with radiatively induced symmetry breakingrdquo PhysicalReview Letters vol 48 no 17 pp 1220ndash1223 1982

[4] P A R Ade N Aghanim C Armitage-Caplan et al ldquoPlanck2013 results XXII Constraints on inflationrdquo Astronomy ampAstrophysics vol 571 article A22 42 pages 2014

[5] P A R Ade R W Aikin D Barkats et al ldquoDetection of B-mode polarization at degree angular scales by BICEP2rdquo PhysicalReview Letters vol 112 no 24 Article ID 241101 2014

[6] M Kawasaki M Yamaguchi and T Yanagida ldquoNatural chaoticinflation in supergravityrdquo Physical Review Letters vol 85 no 17pp 3572ndash3575 2000

[7] T Li Z Li and D V Nanopoulos ldquoSupergravity inflation withbroken shift symmetry and large tensor-to-scalar ratiordquo Journalof Cosmology and Astroparticle Physics vol 2014 no 2 article028 2014

[8] K Harigaya and T T Yanagida ldquoDiscovery of large scale tensormode and chaotic inflation in supergravityrdquo Physics Letters Bvol 734 pp 13ndash16 2014

[9] J Ellis D V Nanopoulos and K A Olive ldquoNo-scale supergrav-ity realization of the starobinsky model of inflationrdquo PhysicalReview Letters vol 111 no 11 Article ID 111301 2013

[10] T Li Z Li and D V Nanopoulos ldquoNo-scale ripple inflationrevisitedrdquo Journal of Cosmology and Astroparticle Physics vol04 article 018 2014

[11] C Pallis ldquoInduced-gravity inflation in no-scale supergravityand beyondrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 8 article 057 2014

[12] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky modelrdquo Fortschritte der Physik vol 62 no 7 pp 573ndash5832014

[13] J Ellis M A G Garca D V Nanopoulos and K A OliveldquoResurrecting quadratic inflation in no-scale supergravity inlight of BICEP2rdquo Journal of Cosmology andAstroparticle Physicsvol 2014 no 5 article 037 2014

[14] T Li Z Li and D V Nanopoulos ldquoChaotic inflation in no-scale supergravitywith string inspiredmoduli stabilizationrdquoTheEuropean Physical Journal C vol 75 no 2 article 55 2015

[15] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky model and higher curvature correctionsrdquo Fortschritte derPhysik vol 63 no 1 pp 2ndash11 2015

[16] S V Ketov andT Terada ldquoInflation in supergravity with a singlechiral superfieldrdquo Physics Letters B vol 736 pp 272ndash277 2014

[17] C Kounnas D Lust and N Toumbas ldquoR2 inflation fromscale invariant supergravity and anomaly free superstrings withfluxesrdquo Fortschritte der Physik vol 63 no 1 pp 12ndash35 2015

[18] G A Diamandis B C Georgalas K Kaskavelis PKouroumalou A B Lahanas and G Pavlopoulos ldquoInflationin R2 supergravity with non-minimal superpotentialsrdquo PhysicsLetters B vol 744 pp 74ndash81 2015

[19] C Pallis and Q Shafi ldquoGravity waves from non-minimalquadratic inflationrdquo Journal of Cosmology and AstroparticlePhysics vol 2015 no 3 article 023 2015

[20] T Li Z Li and D V Nanopoulos ldquoHelical phase inflationrdquoPhysical Review D vol 91 no 6 Article ID 061303 2015

[21] D H Lyth ldquoWhat would we learn by detecting a gravitationalwave signal in the cosmic microwave background anisotropyrdquoPhysical Review Letters vol 78 no 10 article 1861 1997

[22] D Baumann and L McAllister ldquoInflation and stringtheoryrdquohttparxivorgabs14042601

[23] F Marchesano G Shiu and A M Uranga ldquoF-term axionmonodromy inflationrdquo Journal of High Energy Physics vol 2014no 9 article 184 2014

[24] C Long L McAllister and P McGuirk ldquoAligned naturalinflation in string theoryrdquo Physical Review D vol 90 no 2Article ID 023501 2014

[25] X Gao T Li and P Shukla ldquoCombining universal and odd RRaxions for aligned natural inflationrdquo Journal of Cosmology andAstroparticle Physics vol 2014 no 10 article 048 2014

[26] I Ben-Dayan F G Pedro and A Westphal ldquoTowards naturalinflation in string theoryrdquo Physical Review D vol 92 no 2Article ID 023515 2015

[27] R Blumenhagen D Herschmann and E Plauschinn ldquoThechallenge of realizing F-term axion monodromy inflation instring theoryrdquo Journal of High Energy Physics vol 2015 article7 2015

[28] H Abe T Kobayashi and H Otsuka ldquoTowards natural infla-tion from weakly coupled heterotic string theoryrdquo Progress ofTheoretical and Experimental Physics Article ID 063E02 2015

[29] T Ali S S Haque and V Jejjala ldquoNatural inflation from nearalignment in heterotic string theoryrdquo Physical Review D vol 91no 8 Article ID 083516 9 pages 2015

[30] R Flauger LMcAllister E Silverstein and AWestphal ldquoDrift-ing oscillations in axionmonodromyrdquo httparxivorgabs14121814

[31] K Freese J A Frieman andAVOlinto ldquoNatural inflationwithpseudo nambu-goldstone bosonsrdquo Physical Review Letters vol65 no 26 pp 3233ndash3236 1990

[32] F C Adams J R Bond K Freese J A Frieman and AV Olinto ldquoNatural inflation particle physics models power-law spectra for large-scale structure and constraints from thecosmic background explorerrdquo Physical Review D vol 47 no 2pp 426ndash455 1993

[33] G German A Mazumdar and A Perez-Lorenzana ldquoAngularinflation from supergravityrdquo Modern Physics Letters A vol 17no 25 pp 1627ndash1634 2002

12 Advances in High Energy Physics

[34] J E Kim H P Nilles and M Peloso ldquoCompleting naturalinflationrdquo Journal of Cosmology and Astroparticle Physics vol0501 article 005 2005

[35] D Baumann and D Green ldquoInflating with baryonsrdquo Journal ofHigh Energy Physics vol 2011 no 4 article 071 2011

[36] JMcDonald ldquoSub-Planckian two-field inflation consistent withthe Lyth boundrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 9 article 027 2014

[37] J McDonald ldquoA minimal sub-Planckian axion inflation modelwith large tensor-to-scalar ratiordquo Journal of Cosmology andAstroparticle Physics vol 2015 no 1 article 018 2015

[38] C D Carone J Erlich A Sensharma and Z Wang ldquoTwo-fieldaxion-monodromy hybrid inflation model Dantersquos WaterfallrdquoPhysical ReviewD vol 91 no 4 Article ID 043512 8 pages 2015

[39] G Barenboim and W-I Park ldquoSpiral Inflationrdquo Physics LettersB vol 741 pp 252ndash255 2015

[40] T Li Z Li andDVNanopoulos ldquoAligned natural inflation andmoduli stabilization from anomalous U(1) gauge symmetriesrdquoJournal of High Energy Physics vol 2014 no 11 article 012 2014

[41] T Li Z Li and D V Nanopoulos ldquoNatural inflation withnatural trans-planckian axion decay constant from anomalousU(1)119883rdquo Journal of High Energy Physics vol 2014 no 7 article052 2014

[42] E Silverstein and A Westphal ldquoMonodromy in the CMBgravity waves and string inflationrdquo Physical Review D vol 78no 10 Article ID 106003 21 pages 2008

[43] L McAllister E Silverstein and A Westphal ldquoGravity wavesand linear inflation from axion monodromyrdquo Physical ReviewD vol 82 no 4 Article ID 046003 2010

[44] K Choi H Kim and S Yun ldquoNatural inflation with multiplesub-Planckian axionsrdquo Physical Review D vol 90 no 2 ArticleID 023545 2014

[45] S-H H Tye and S S C Wong ldquoHelical inflation and cosmic-stringsrdquo httparxivorgabs14046988

[46] R Kappl S Krippendorf and H P Nilles ldquoAligned naturalinflation monodromies of two axionsrdquo Physics Letters B vol737 pp 124ndash128 2014

[47] G rsquotHooft ldquoNaturalness chiral symmetry and spontaneouschiral symmetry breakingrdquo in Recent Developments in GaugeTheories G rsquotHooft C Itzykson A Jaffe et al Eds vol 59of NATO Advanced Study Institutes Series pp 135ndash157 PlenumPress New York NY USA 1980

[48] R Blumenhagen M Cvetic S Kachru and T Weigand ldquoD-brane instantons in type II string theoryrdquo Annual Review ofNuclear and Particle Science vol 59 pp 269ndash296 2009

[49] K Harigaya andM Ibe ldquoSimple realization of inflaton potentialon a Riemann surfacerdquo Physics Letters B vol 738 pp 301ndash3042014

[50] R Kallosh A Linde and B Vercnocke ldquoNatural inflation insupergravity and beyondrdquo Physical Review D vol 90 Article ID041303(R) 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 9

30

20

10

0

minus10

minus10

minus05

minus05

0000

05

05

10

10

120601x (radic

3c) 120601 y(radic3

c)

V(a

2 c4 )

Figure 3The helicoid structure of potential (43) scaled by 11988621198884 Inthe graph the parameter Λ is tuned so that 1199030 = 1199031 The complexfieldΦ has been rescaled byradic3119888 and the scale of field norm at localvalley is determined by the parameter 119888 instead of the Planck mass

Instead of stabilizing 119879 independently with the inflationprocess we can realize helical phase inflation from the no-scale supergravity with dynamical119879There is a natural reasonfor such consideration The phase monodromy requiresmatter fields transforming as rotations under 119880(1) and alsoa modulus 119879 as a shift under 119880(1) The Kahler potential of 119879is shift invariant Interestingly for the no-scale supergravitywith 119878119880(2 1)119878119880(2) times 119880(1) symmetry as shown in (39) theKahler potential 119870 is automatically endowed with the shiftsymmetry and global 119880(1) symmetry

119879 997888rarr 119879 + 119894119862

Φ 997888rarr Φ119890119894120579

(45)

Therefore the no-scale Kahler potential fits with the phasemonodromy in (16) and (22) initiatively

The Kahler potentials of superfields 119911 isin 119883 119884 119885 areof the minimal type 119911119911 while for Ψ its Kahler potentialcan be the minimal type ΨΨ or the no-scale type 119870 =

minus3 ln(119879+119879minus (ΦΦ+ΨΨ)3) which extends the symmetry ofKahler manifold to 119878119880(3 1)119878119880(3) times 119880(1) In this scenariothe process to integrate out heavy fields is the same asbefore Besides the potential of phase proportional to |119882119883|

2

is insensitive to the formula of Kahler potential due to119880(1) symmetry The major difference appears in the fieldstabilization The scalar potential from phase monodromy in(16) is

119881 = 119890119870 1003816100381610038161003816119882119883

1003816100381610038161003816

2=

1

(ln 1199032120572 + 119888 minus 11990323 minus 120582211990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792) asymp

1

(119888 minus 11990323)3

sdot1

1199032((ln 119903 minus lnΛ)

2+ 1205792)

(46)

with 120572 ≫ 1 and 120582 ≪ 1 The field norm 119903 is stabilized byminimizing the coefficient 11199032(119888 minus 119903

23)3 and the parameter

119888 from ⟨119879⟩ determines the scale of ⟨119903⟩ = radic31198882 This isdifferent from helical phase inflation in the minimal super-gravity which minimizes the norm based on the coefficient1198901199032(11199032) and then the scale of ⟨119903⟩ is close to119872119875 the unique

energy scale of supergravity corrections Similarly combiningthe no-scale Kahlermanifold with phasemonodromy in (22)we can obtain natural inflation

7 UV-Sensitivity and Helical Phase Inflation

For the slow-roll inflation the Lyth bound [21] provides arelationship between the tensor-to-scalar ratio r and the fieldexcursion Δ120601 Roughly it requires

Δ120601

119872119875

⩾ (r

001)

12

(47)

To get large tensor-to-scalar ratio r ⩾ 001 such as in chaoticinflation or natural inflation the field excursion should bemuch larger than the Planck mass The super-Planckianfield excursion makes the description based on the effectivefield theory questionable In the Wilsonian sense the lowenergy field theory is an effective theory with higher ordercorrections introduced by the physics at the cut-off scalelike quantum gravity and these terms are irrelevant in theeffective field theory since they are suppressed by the cut-offenergy scale However for the inflation process the inflatonhas super-Planckian field excursion which is much largerthan the cut-off scaleThus the higher order terms cannot besuppressed by the Planckmass andmay affect the inflationaryobservations significantly And then they may significantlyaffect inflation or even destroy the inflation process Forexample considering the corrections

Δ119881 = 119888119894119881(120601

119872119875

)

119894

+ sdot sdot sdot (48)

to the original inflaton potential 119881 as long as 119888119894 are of theorder 10minus119894 in the initial stage of inflation 120601 sim 119874(10)119872119875 thehigher order terms can be as large as the original potential119881 So for large field inflation it is sensitive to the physics atthe cut-off scale and the predictions of inflation just based oneffective field theory are questionable

In consideration of the UV sensitivity of large fieldinflation a possible choice is to realize inflation in UV-completed theory like string theory (for a review see [22])To realize inflation in string theory there are a lot of problemsto solve besides inflation such as the moduli stabilizationMinkowskide Sitter vacua and effects of 1205721015840- and string loop-corrections Alternatively in the bottom-up approach onemay avoid the higher order corrections by introducing anextra shift symmetry in the theory The shift symmetry istechnically natural and safe under quantum loop-correctionsHowever the global symmetry can be broken by the quantumgravity effect So it is still questionable whether the shiftsymmetry can safely evade the higher corrections like in (48)

10 Advances in High Energy Physics

The UV-completion problem is dodged in helical phaseinflation Since the super-Planckian field excursion is thephase of a complex field and the phase component is notdirectly involved in the gravity interaction there are nodangerous high-order corrections like in (48) for the phasepotentialThe inflaton evolves along the helical trajectory anddoes not relate to the physics in the region above the Planckscale For helical phase inflation in the minimal supergravitythe norm of field is stabilized at the marginal point of thePlanck scale where the supergravity correction on the scalarpotential gets important based onwhich the normof complexfield can be strongly stabilized The extra corrections arelikely to appear in the Kahler potential however they canonly slightly affect the field stabilization while the phaseinflation is protected by the global 119880(1) symmetry in theKahler potential in consequence helical phase inflation isnot sensitive to these corrections at all For the helical phaseinflation in no-scale supergravity the normof complex field isstabilized at the scale of themodulus ⟨119879⟩ instead of the Planckscale one can simply adjust the scale of ⟨119879⟩ to keep themodelaway from super-Planckian region

Helical phase inflation is free from the UV-sensitivityproblem and it is just a typical physical process at theGUT scale with special superpotential that admits phasemonodromy So it provides an inflationary model that can bereliably studied just in supersymmetric field theory

8 Discussions and Conclusion

In this work we have studied the details of helical phase infla-tion from several aspects Helical phase inflation is realizedin supergravity with global 119880(1) symmetry 119880(1) symmetryis built in the Kahler potential so that helical phase inflationcan be realized by the ordinary Kahler potentials such as theminimal or no-scale types Helical phase inflation directlyleads to the phasemonodromy in the superpotential which issingular and an effective field theory arising from integratingout heavy fieldsThe phasemonodromy originates from119880(1)

rotation of the superpotential at higher scale Generically thesuperpotential can be separated into two parts119882 = 119882119868 +119882119878where119882119878 admits the global119880(1) symmetrywhile119882119868 breaks itexplicitly at scale much lower than 119882119878 Under 119880(1) rotationthe inflation term 119882119868 is slightly changed which realizes thephase monodromy in the effective theory and introducesa flat potential along the direction of phase rotation Bybreaking the global119880(1) symmetry in different ways we mayget different kinds of inflation such as quadratic inflationnatural inflation or the other types of inflation that are notpresented in this work

An amazing fact of helical phase inflation is that itdeeply relates to several interesting points of inflation andnaturally combines them in a rather simple potential withhelicoid structure The features of helical phase inflation canbe summarized as follows

(i) The global 119880(1) symmetry is built in the Kahlerpotential so helical phase inflation provides a naturalsolution to 120578 problem

(ii) The phase excursion requires phase monodromy inthe superpotential So helical phase inflation providesin the mathematical sense a new type of monodromyin supersymmetric field theory

(iii) The singularity in the superpotential together withthe supergravity scalar potential provides strong fieldstabilization which is consistent with phase inflation

(iv) The super-Planckian field excursion is realized bythe phase of a complex field instead of any otherldquophysicalrdquo fields that directly couple with gravity Sothere are no polynomial higher order corrections forthe phase and thus inflation is not sensitive to thequantum gravity corrections

To summarize helical phase inflation introduces a newtype of inflation that can be effectively described by super-symmetric field theory at the GUT scale Generically thesuper-Planckian field excursion makes the inflationary pre-dictions based on effective field theory questionable sincethe higher order corrections from quantum gravity arelikely to affect the inflation process significantly One of thesolutions is to realize inflation in a UV-completed theory likestring theory nevertheless there are many difficult issues instring theory to resolve before realizing inflation completelyHelical phase inflation is another simple solution to the UV-sensitivity problem It is based on the supersymmetric fieldtheory and the physics are clear and much easier to controlFurthermore helical phase inflation makes the unification ofinflation theory with GUT more natural since both of themare triggered at the scale of 1016 GeV and can be effectivelystudied based on supersymmetric field theory

Besides we have shown that helical phase inflation alsorelates to several interesting developments in inflation theoryIt can be easily modified for natural inflation and realizethe phase-axion alignment indirectly which is similar tothe axion-axion alignment mechanism for super-Planckianaxion decay constant [34] The phase-axion alignment isnot shown in the final supergravity model which exhibitsexplicit phase monodromy only However the phase-axionalignment is hidden in the process when the heavy fields areintegrating out For 120578 problem in the supergravity inflationthere is another well-known solution the KYY model withshift symmetry We showed that through field redefinitionhelical phase inflation given by (4) and (5) reduces to theKYYmodel where the higher order corrections in the Kahlerpotential have no effect on inflation process However thereis no inverse transformation from the KYY model to helicalphase inflation since no well-defined field redefinition canintroduce the pole and phase singularity needed for phasemonodromy Helical phase inflation can be realized in no-scale supergravity The no-scale Kahler potential automati-cally provides the symmetry needed by phase monodromyIn the no-scale supergravity the norm of complex field isstabilized at the scale of the modulus

Our inflation models are constructed within the super-gravity theory with global 119880(1) symmetry broken explicitlyby the subleading order superpotential term So it is justtypical GUT scale physics and indicates that a UV-completed

Advances in High Energy Physics 11

framework seems to be not prerequisite to effectively describesuch inflation process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work of DVN was supported in part by the DOEGrant DE-FG03-95-ER-40917 The work of Tianjun Li issupported in part by theNatural Science Foundation of ChinaunderGrants nos 10821504 11075194 11135003 11275246 and11475238 and by the National Basic Research Program ofChina (973 Program) under Grant no 2010CB833000

References

[1] A H Guth ldquoInflationary universe a possible solution to thehorizon and flatness problemsrdquo Physical Review D vol 23 no2 pp 347ndash356 1981

[2] A D Linde ldquoColeman-Weinberg theory and the new inflation-ary universe scenariordquo Physics Letters B vol 114 no 6 pp 431ndash435 1982

[3] A Albrecht and P J Steinhardt ldquoCosmology for grand unifiedtheories with radiatively induced symmetry breakingrdquo PhysicalReview Letters vol 48 no 17 pp 1220ndash1223 1982

[4] P A R Ade N Aghanim C Armitage-Caplan et al ldquoPlanck2013 results XXII Constraints on inflationrdquo Astronomy ampAstrophysics vol 571 article A22 42 pages 2014

[5] P A R Ade R W Aikin D Barkats et al ldquoDetection of B-mode polarization at degree angular scales by BICEP2rdquo PhysicalReview Letters vol 112 no 24 Article ID 241101 2014

[6] M Kawasaki M Yamaguchi and T Yanagida ldquoNatural chaoticinflation in supergravityrdquo Physical Review Letters vol 85 no 17pp 3572ndash3575 2000

[7] T Li Z Li and D V Nanopoulos ldquoSupergravity inflation withbroken shift symmetry and large tensor-to-scalar ratiordquo Journalof Cosmology and Astroparticle Physics vol 2014 no 2 article028 2014

[8] K Harigaya and T T Yanagida ldquoDiscovery of large scale tensormode and chaotic inflation in supergravityrdquo Physics Letters Bvol 734 pp 13ndash16 2014

[9] J Ellis D V Nanopoulos and K A Olive ldquoNo-scale supergrav-ity realization of the starobinsky model of inflationrdquo PhysicalReview Letters vol 111 no 11 Article ID 111301 2013

[10] T Li Z Li and D V Nanopoulos ldquoNo-scale ripple inflationrevisitedrdquo Journal of Cosmology and Astroparticle Physics vol04 article 018 2014

[11] C Pallis ldquoInduced-gravity inflation in no-scale supergravityand beyondrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 8 article 057 2014

[12] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky modelrdquo Fortschritte der Physik vol 62 no 7 pp 573ndash5832014

[13] J Ellis M A G Garca D V Nanopoulos and K A OliveldquoResurrecting quadratic inflation in no-scale supergravity inlight of BICEP2rdquo Journal of Cosmology andAstroparticle Physicsvol 2014 no 5 article 037 2014

[14] T Li Z Li and D V Nanopoulos ldquoChaotic inflation in no-scale supergravitywith string inspiredmoduli stabilizationrdquoTheEuropean Physical Journal C vol 75 no 2 article 55 2015

[15] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky model and higher curvature correctionsrdquo Fortschritte derPhysik vol 63 no 1 pp 2ndash11 2015

[16] S V Ketov andT Terada ldquoInflation in supergravity with a singlechiral superfieldrdquo Physics Letters B vol 736 pp 272ndash277 2014

[17] C Kounnas D Lust and N Toumbas ldquoR2 inflation fromscale invariant supergravity and anomaly free superstrings withfluxesrdquo Fortschritte der Physik vol 63 no 1 pp 12ndash35 2015

[18] G A Diamandis B C Georgalas K Kaskavelis PKouroumalou A B Lahanas and G Pavlopoulos ldquoInflationin R2 supergravity with non-minimal superpotentialsrdquo PhysicsLetters B vol 744 pp 74ndash81 2015

[19] C Pallis and Q Shafi ldquoGravity waves from non-minimalquadratic inflationrdquo Journal of Cosmology and AstroparticlePhysics vol 2015 no 3 article 023 2015

[20] T Li Z Li and D V Nanopoulos ldquoHelical phase inflationrdquoPhysical Review D vol 91 no 6 Article ID 061303 2015

[21] D H Lyth ldquoWhat would we learn by detecting a gravitationalwave signal in the cosmic microwave background anisotropyrdquoPhysical Review Letters vol 78 no 10 article 1861 1997

[22] D Baumann and L McAllister ldquoInflation and stringtheoryrdquohttparxivorgabs14042601

[23] F Marchesano G Shiu and A M Uranga ldquoF-term axionmonodromy inflationrdquo Journal of High Energy Physics vol 2014no 9 article 184 2014

[24] C Long L McAllister and P McGuirk ldquoAligned naturalinflation in string theoryrdquo Physical Review D vol 90 no 2Article ID 023501 2014

[25] X Gao T Li and P Shukla ldquoCombining universal and odd RRaxions for aligned natural inflationrdquo Journal of Cosmology andAstroparticle Physics vol 2014 no 10 article 048 2014

[26] I Ben-Dayan F G Pedro and A Westphal ldquoTowards naturalinflation in string theoryrdquo Physical Review D vol 92 no 2Article ID 023515 2015

[27] R Blumenhagen D Herschmann and E Plauschinn ldquoThechallenge of realizing F-term axion monodromy inflation instring theoryrdquo Journal of High Energy Physics vol 2015 article7 2015

[28] H Abe T Kobayashi and H Otsuka ldquoTowards natural infla-tion from weakly coupled heterotic string theoryrdquo Progress ofTheoretical and Experimental Physics Article ID 063E02 2015

[29] T Ali S S Haque and V Jejjala ldquoNatural inflation from nearalignment in heterotic string theoryrdquo Physical Review D vol 91no 8 Article ID 083516 9 pages 2015

[30] R Flauger LMcAllister E Silverstein and AWestphal ldquoDrift-ing oscillations in axionmonodromyrdquo httparxivorgabs14121814

[31] K Freese J A Frieman andAVOlinto ldquoNatural inflationwithpseudo nambu-goldstone bosonsrdquo Physical Review Letters vol65 no 26 pp 3233ndash3236 1990

[32] F C Adams J R Bond K Freese J A Frieman and AV Olinto ldquoNatural inflation particle physics models power-law spectra for large-scale structure and constraints from thecosmic background explorerrdquo Physical Review D vol 47 no 2pp 426ndash455 1993

[33] G German A Mazumdar and A Perez-Lorenzana ldquoAngularinflation from supergravityrdquo Modern Physics Letters A vol 17no 25 pp 1627ndash1634 2002

12 Advances in High Energy Physics

[34] J E Kim H P Nilles and M Peloso ldquoCompleting naturalinflationrdquo Journal of Cosmology and Astroparticle Physics vol0501 article 005 2005

[35] D Baumann and D Green ldquoInflating with baryonsrdquo Journal ofHigh Energy Physics vol 2011 no 4 article 071 2011

[36] JMcDonald ldquoSub-Planckian two-field inflation consistent withthe Lyth boundrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 9 article 027 2014

[37] J McDonald ldquoA minimal sub-Planckian axion inflation modelwith large tensor-to-scalar ratiordquo Journal of Cosmology andAstroparticle Physics vol 2015 no 1 article 018 2015

[38] C D Carone J Erlich A Sensharma and Z Wang ldquoTwo-fieldaxion-monodromy hybrid inflation model Dantersquos WaterfallrdquoPhysical ReviewD vol 91 no 4 Article ID 043512 8 pages 2015

[39] G Barenboim and W-I Park ldquoSpiral Inflationrdquo Physics LettersB vol 741 pp 252ndash255 2015

[40] T Li Z Li andDVNanopoulos ldquoAligned natural inflation andmoduli stabilization from anomalous U(1) gauge symmetriesrdquoJournal of High Energy Physics vol 2014 no 11 article 012 2014

[41] T Li Z Li and D V Nanopoulos ldquoNatural inflation withnatural trans-planckian axion decay constant from anomalousU(1)119883rdquo Journal of High Energy Physics vol 2014 no 7 article052 2014

[42] E Silverstein and A Westphal ldquoMonodromy in the CMBgravity waves and string inflationrdquo Physical Review D vol 78no 10 Article ID 106003 21 pages 2008

[43] L McAllister E Silverstein and A Westphal ldquoGravity wavesand linear inflation from axion monodromyrdquo Physical ReviewD vol 82 no 4 Article ID 046003 2010

[44] K Choi H Kim and S Yun ldquoNatural inflation with multiplesub-Planckian axionsrdquo Physical Review D vol 90 no 2 ArticleID 023545 2014

[45] S-H H Tye and S S C Wong ldquoHelical inflation and cosmic-stringsrdquo httparxivorgabs14046988

[46] R Kappl S Krippendorf and H P Nilles ldquoAligned naturalinflation monodromies of two axionsrdquo Physics Letters B vol737 pp 124ndash128 2014

[47] G rsquotHooft ldquoNaturalness chiral symmetry and spontaneouschiral symmetry breakingrdquo in Recent Developments in GaugeTheories G rsquotHooft C Itzykson A Jaffe et al Eds vol 59of NATO Advanced Study Institutes Series pp 135ndash157 PlenumPress New York NY USA 1980

[48] R Blumenhagen M Cvetic S Kachru and T Weigand ldquoD-brane instantons in type II string theoryrdquo Annual Review ofNuclear and Particle Science vol 59 pp 269ndash296 2009

[49] K Harigaya andM Ibe ldquoSimple realization of inflaton potentialon a Riemann surfacerdquo Physics Letters B vol 738 pp 301ndash3042014

[50] R Kallosh A Linde and B Vercnocke ldquoNatural inflation insupergravity and beyondrdquo Physical Review D vol 90 Article ID041303(R) 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

10 Advances in High Energy Physics

The UV-completion problem is dodged in helical phaseinflation Since the super-Planckian field excursion is thephase of a complex field and the phase component is notdirectly involved in the gravity interaction there are nodangerous high-order corrections like in (48) for the phasepotentialThe inflaton evolves along the helical trajectory anddoes not relate to the physics in the region above the Planckscale For helical phase inflation in the minimal supergravitythe norm of field is stabilized at the marginal point of thePlanck scale where the supergravity correction on the scalarpotential gets important based onwhich the normof complexfield can be strongly stabilized The extra corrections arelikely to appear in the Kahler potential however they canonly slightly affect the field stabilization while the phaseinflation is protected by the global 119880(1) symmetry in theKahler potential in consequence helical phase inflation isnot sensitive to these corrections at all For the helical phaseinflation in no-scale supergravity the normof complex field isstabilized at the scale of themodulus ⟨119879⟩ instead of the Planckscale one can simply adjust the scale of ⟨119879⟩ to keep themodelaway from super-Planckian region

Helical phase inflation is free from the UV-sensitivityproblem and it is just a typical physical process at theGUT scale with special superpotential that admits phasemonodromy So it provides an inflationary model that can bereliably studied just in supersymmetric field theory

8 Discussions and Conclusion

In this work we have studied the details of helical phase infla-tion from several aspects Helical phase inflation is realizedin supergravity with global 119880(1) symmetry 119880(1) symmetryis built in the Kahler potential so that helical phase inflationcan be realized by the ordinary Kahler potentials such as theminimal or no-scale types Helical phase inflation directlyleads to the phasemonodromy in the superpotential which issingular and an effective field theory arising from integratingout heavy fieldsThe phasemonodromy originates from119880(1)

rotation of the superpotential at higher scale Generically thesuperpotential can be separated into two parts119882 = 119882119868 +119882119878where119882119878 admits the global119880(1) symmetrywhile119882119868 breaks itexplicitly at scale much lower than 119882119878 Under 119880(1) rotationthe inflation term 119882119868 is slightly changed which realizes thephase monodromy in the effective theory and introducesa flat potential along the direction of phase rotation Bybreaking the global119880(1) symmetry in different ways we mayget different kinds of inflation such as quadratic inflationnatural inflation or the other types of inflation that are notpresented in this work

An amazing fact of helical phase inflation is that itdeeply relates to several interesting points of inflation andnaturally combines them in a rather simple potential withhelicoid structure The features of helical phase inflation canbe summarized as follows

(i) The global 119880(1) symmetry is built in the Kahlerpotential so helical phase inflation provides a naturalsolution to 120578 problem

(ii) The phase excursion requires phase monodromy inthe superpotential So helical phase inflation providesin the mathematical sense a new type of monodromyin supersymmetric field theory

(iii) The singularity in the superpotential together withthe supergravity scalar potential provides strong fieldstabilization which is consistent with phase inflation

(iv) The super-Planckian field excursion is realized bythe phase of a complex field instead of any otherldquophysicalrdquo fields that directly couple with gravity Sothere are no polynomial higher order corrections forthe phase and thus inflation is not sensitive to thequantum gravity corrections

To summarize helical phase inflation introduces a newtype of inflation that can be effectively described by super-symmetric field theory at the GUT scale Generically thesuper-Planckian field excursion makes the inflationary pre-dictions based on effective field theory questionable sincethe higher order corrections from quantum gravity arelikely to affect the inflation process significantly One of thesolutions is to realize inflation in a UV-completed theory likestring theory nevertheless there are many difficult issues instring theory to resolve before realizing inflation completelyHelical phase inflation is another simple solution to the UV-sensitivity problem It is based on the supersymmetric fieldtheory and the physics are clear and much easier to controlFurthermore helical phase inflation makes the unification ofinflation theory with GUT more natural since both of themare triggered at the scale of 1016 GeV and can be effectivelystudied based on supersymmetric field theory

Besides we have shown that helical phase inflation alsorelates to several interesting developments in inflation theoryIt can be easily modified for natural inflation and realizethe phase-axion alignment indirectly which is similar tothe axion-axion alignment mechanism for super-Planckianaxion decay constant [34] The phase-axion alignment isnot shown in the final supergravity model which exhibitsexplicit phase monodromy only However the phase-axionalignment is hidden in the process when the heavy fields areintegrating out For 120578 problem in the supergravity inflationthere is another well-known solution the KYY model withshift symmetry We showed that through field redefinitionhelical phase inflation given by (4) and (5) reduces to theKYYmodel where the higher order corrections in the Kahlerpotential have no effect on inflation process However thereis no inverse transformation from the KYY model to helicalphase inflation since no well-defined field redefinition canintroduce the pole and phase singularity needed for phasemonodromy Helical phase inflation can be realized in no-scale supergravity The no-scale Kahler potential automati-cally provides the symmetry needed by phase monodromyIn the no-scale supergravity the norm of complex field isstabilized at the scale of the modulus

Our inflation models are constructed within the super-gravity theory with global 119880(1) symmetry broken explicitlyby the subleading order superpotential term So it is justtypical GUT scale physics and indicates that a UV-completed

Advances in High Energy Physics 11

framework seems to be not prerequisite to effectively describesuch inflation process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work of DVN was supported in part by the DOEGrant DE-FG03-95-ER-40917 The work of Tianjun Li issupported in part by theNatural Science Foundation of ChinaunderGrants nos 10821504 11075194 11135003 11275246 and11475238 and by the National Basic Research Program ofChina (973 Program) under Grant no 2010CB833000

References

[1] A H Guth ldquoInflationary universe a possible solution to thehorizon and flatness problemsrdquo Physical Review D vol 23 no2 pp 347ndash356 1981

[2] A D Linde ldquoColeman-Weinberg theory and the new inflation-ary universe scenariordquo Physics Letters B vol 114 no 6 pp 431ndash435 1982

[3] A Albrecht and P J Steinhardt ldquoCosmology for grand unifiedtheories with radiatively induced symmetry breakingrdquo PhysicalReview Letters vol 48 no 17 pp 1220ndash1223 1982

[4] P A R Ade N Aghanim C Armitage-Caplan et al ldquoPlanck2013 results XXII Constraints on inflationrdquo Astronomy ampAstrophysics vol 571 article A22 42 pages 2014

[5] P A R Ade R W Aikin D Barkats et al ldquoDetection of B-mode polarization at degree angular scales by BICEP2rdquo PhysicalReview Letters vol 112 no 24 Article ID 241101 2014

[6] M Kawasaki M Yamaguchi and T Yanagida ldquoNatural chaoticinflation in supergravityrdquo Physical Review Letters vol 85 no 17pp 3572ndash3575 2000

[7] T Li Z Li and D V Nanopoulos ldquoSupergravity inflation withbroken shift symmetry and large tensor-to-scalar ratiordquo Journalof Cosmology and Astroparticle Physics vol 2014 no 2 article028 2014

[8] K Harigaya and T T Yanagida ldquoDiscovery of large scale tensormode and chaotic inflation in supergravityrdquo Physics Letters Bvol 734 pp 13ndash16 2014

[9] J Ellis D V Nanopoulos and K A Olive ldquoNo-scale supergrav-ity realization of the starobinsky model of inflationrdquo PhysicalReview Letters vol 111 no 11 Article ID 111301 2013

[10] T Li Z Li and D V Nanopoulos ldquoNo-scale ripple inflationrevisitedrdquo Journal of Cosmology and Astroparticle Physics vol04 article 018 2014

[11] C Pallis ldquoInduced-gravity inflation in no-scale supergravityand beyondrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 8 article 057 2014

[12] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky modelrdquo Fortschritte der Physik vol 62 no 7 pp 573ndash5832014

[13] J Ellis M A G Garca D V Nanopoulos and K A OliveldquoResurrecting quadratic inflation in no-scale supergravity inlight of BICEP2rdquo Journal of Cosmology andAstroparticle Physicsvol 2014 no 5 article 037 2014

[14] T Li Z Li and D V Nanopoulos ldquoChaotic inflation in no-scale supergravitywith string inspiredmoduli stabilizationrdquoTheEuropean Physical Journal C vol 75 no 2 article 55 2015

[15] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky model and higher curvature correctionsrdquo Fortschritte derPhysik vol 63 no 1 pp 2ndash11 2015

[16] S V Ketov andT Terada ldquoInflation in supergravity with a singlechiral superfieldrdquo Physics Letters B vol 736 pp 272ndash277 2014

[17] C Kounnas D Lust and N Toumbas ldquoR2 inflation fromscale invariant supergravity and anomaly free superstrings withfluxesrdquo Fortschritte der Physik vol 63 no 1 pp 12ndash35 2015

[18] G A Diamandis B C Georgalas K Kaskavelis PKouroumalou A B Lahanas and G Pavlopoulos ldquoInflationin R2 supergravity with non-minimal superpotentialsrdquo PhysicsLetters B vol 744 pp 74ndash81 2015

[19] C Pallis and Q Shafi ldquoGravity waves from non-minimalquadratic inflationrdquo Journal of Cosmology and AstroparticlePhysics vol 2015 no 3 article 023 2015

[20] T Li Z Li and D V Nanopoulos ldquoHelical phase inflationrdquoPhysical Review D vol 91 no 6 Article ID 061303 2015

[21] D H Lyth ldquoWhat would we learn by detecting a gravitationalwave signal in the cosmic microwave background anisotropyrdquoPhysical Review Letters vol 78 no 10 article 1861 1997

[22] D Baumann and L McAllister ldquoInflation and stringtheoryrdquohttparxivorgabs14042601

[23] F Marchesano G Shiu and A M Uranga ldquoF-term axionmonodromy inflationrdquo Journal of High Energy Physics vol 2014no 9 article 184 2014

[24] C Long L McAllister and P McGuirk ldquoAligned naturalinflation in string theoryrdquo Physical Review D vol 90 no 2Article ID 023501 2014

[25] X Gao T Li and P Shukla ldquoCombining universal and odd RRaxions for aligned natural inflationrdquo Journal of Cosmology andAstroparticle Physics vol 2014 no 10 article 048 2014

[26] I Ben-Dayan F G Pedro and A Westphal ldquoTowards naturalinflation in string theoryrdquo Physical Review D vol 92 no 2Article ID 023515 2015

[27] R Blumenhagen D Herschmann and E Plauschinn ldquoThechallenge of realizing F-term axion monodromy inflation instring theoryrdquo Journal of High Energy Physics vol 2015 article7 2015

[28] H Abe T Kobayashi and H Otsuka ldquoTowards natural infla-tion from weakly coupled heterotic string theoryrdquo Progress ofTheoretical and Experimental Physics Article ID 063E02 2015

[29] T Ali S S Haque and V Jejjala ldquoNatural inflation from nearalignment in heterotic string theoryrdquo Physical Review D vol 91no 8 Article ID 083516 9 pages 2015

[30] R Flauger LMcAllister E Silverstein and AWestphal ldquoDrift-ing oscillations in axionmonodromyrdquo httparxivorgabs14121814

[31] K Freese J A Frieman andAVOlinto ldquoNatural inflationwithpseudo nambu-goldstone bosonsrdquo Physical Review Letters vol65 no 26 pp 3233ndash3236 1990

[32] F C Adams J R Bond K Freese J A Frieman and AV Olinto ldquoNatural inflation particle physics models power-law spectra for large-scale structure and constraints from thecosmic background explorerrdquo Physical Review D vol 47 no 2pp 426ndash455 1993

[33] G German A Mazumdar and A Perez-Lorenzana ldquoAngularinflation from supergravityrdquo Modern Physics Letters A vol 17no 25 pp 1627ndash1634 2002

12 Advances in High Energy Physics

[34] J E Kim H P Nilles and M Peloso ldquoCompleting naturalinflationrdquo Journal of Cosmology and Astroparticle Physics vol0501 article 005 2005

[35] D Baumann and D Green ldquoInflating with baryonsrdquo Journal ofHigh Energy Physics vol 2011 no 4 article 071 2011

[36] JMcDonald ldquoSub-Planckian two-field inflation consistent withthe Lyth boundrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 9 article 027 2014

[37] J McDonald ldquoA minimal sub-Planckian axion inflation modelwith large tensor-to-scalar ratiordquo Journal of Cosmology andAstroparticle Physics vol 2015 no 1 article 018 2015

[38] C D Carone J Erlich A Sensharma and Z Wang ldquoTwo-fieldaxion-monodromy hybrid inflation model Dantersquos WaterfallrdquoPhysical ReviewD vol 91 no 4 Article ID 043512 8 pages 2015

[39] G Barenboim and W-I Park ldquoSpiral Inflationrdquo Physics LettersB vol 741 pp 252ndash255 2015

[40] T Li Z Li andDVNanopoulos ldquoAligned natural inflation andmoduli stabilization from anomalous U(1) gauge symmetriesrdquoJournal of High Energy Physics vol 2014 no 11 article 012 2014

[41] T Li Z Li and D V Nanopoulos ldquoNatural inflation withnatural trans-planckian axion decay constant from anomalousU(1)119883rdquo Journal of High Energy Physics vol 2014 no 7 article052 2014

[42] E Silverstein and A Westphal ldquoMonodromy in the CMBgravity waves and string inflationrdquo Physical Review D vol 78no 10 Article ID 106003 21 pages 2008

[43] L McAllister E Silverstein and A Westphal ldquoGravity wavesand linear inflation from axion monodromyrdquo Physical ReviewD vol 82 no 4 Article ID 046003 2010

[44] K Choi H Kim and S Yun ldquoNatural inflation with multiplesub-Planckian axionsrdquo Physical Review D vol 90 no 2 ArticleID 023545 2014

[45] S-H H Tye and S S C Wong ldquoHelical inflation and cosmic-stringsrdquo httparxivorgabs14046988

[46] R Kappl S Krippendorf and H P Nilles ldquoAligned naturalinflation monodromies of two axionsrdquo Physics Letters B vol737 pp 124ndash128 2014

[47] G rsquotHooft ldquoNaturalness chiral symmetry and spontaneouschiral symmetry breakingrdquo in Recent Developments in GaugeTheories G rsquotHooft C Itzykson A Jaffe et al Eds vol 59of NATO Advanced Study Institutes Series pp 135ndash157 PlenumPress New York NY USA 1980

[48] R Blumenhagen M Cvetic S Kachru and T Weigand ldquoD-brane instantons in type II string theoryrdquo Annual Review ofNuclear and Particle Science vol 59 pp 269ndash296 2009

[49] K Harigaya andM Ibe ldquoSimple realization of inflaton potentialon a Riemann surfacerdquo Physics Letters B vol 738 pp 301ndash3042014

[50] R Kallosh A Linde and B Vercnocke ldquoNatural inflation insupergravity and beyondrdquo Physical Review D vol 90 Article ID041303(R) 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 11

framework seems to be not prerequisite to effectively describesuch inflation process

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The work of DVN was supported in part by the DOEGrant DE-FG03-95-ER-40917 The work of Tianjun Li issupported in part by theNatural Science Foundation of ChinaunderGrants nos 10821504 11075194 11135003 11275246 and11475238 and by the National Basic Research Program ofChina (973 Program) under Grant no 2010CB833000

References

[1] A H Guth ldquoInflationary universe a possible solution to thehorizon and flatness problemsrdquo Physical Review D vol 23 no2 pp 347ndash356 1981

[2] A D Linde ldquoColeman-Weinberg theory and the new inflation-ary universe scenariordquo Physics Letters B vol 114 no 6 pp 431ndash435 1982

[3] A Albrecht and P J Steinhardt ldquoCosmology for grand unifiedtheories with radiatively induced symmetry breakingrdquo PhysicalReview Letters vol 48 no 17 pp 1220ndash1223 1982

[4] P A R Ade N Aghanim C Armitage-Caplan et al ldquoPlanck2013 results XXII Constraints on inflationrdquo Astronomy ampAstrophysics vol 571 article A22 42 pages 2014

[5] P A R Ade R W Aikin D Barkats et al ldquoDetection of B-mode polarization at degree angular scales by BICEP2rdquo PhysicalReview Letters vol 112 no 24 Article ID 241101 2014

[6] M Kawasaki M Yamaguchi and T Yanagida ldquoNatural chaoticinflation in supergravityrdquo Physical Review Letters vol 85 no 17pp 3572ndash3575 2000

[7] T Li Z Li and D V Nanopoulos ldquoSupergravity inflation withbroken shift symmetry and large tensor-to-scalar ratiordquo Journalof Cosmology and Astroparticle Physics vol 2014 no 2 article028 2014

[8] K Harigaya and T T Yanagida ldquoDiscovery of large scale tensormode and chaotic inflation in supergravityrdquo Physics Letters Bvol 734 pp 13ndash16 2014

[9] J Ellis D V Nanopoulos and K A Olive ldquoNo-scale supergrav-ity realization of the starobinsky model of inflationrdquo PhysicalReview Letters vol 111 no 11 Article ID 111301 2013

[10] T Li Z Li and D V Nanopoulos ldquoNo-scale ripple inflationrevisitedrdquo Journal of Cosmology and Astroparticle Physics vol04 article 018 2014

[11] C Pallis ldquoInduced-gravity inflation in no-scale supergravityand beyondrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 8 article 057 2014

[12] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky modelrdquo Fortschritte der Physik vol 62 no 7 pp 573ndash5832014

[13] J Ellis M A G Garca D V Nanopoulos and K A OliveldquoResurrecting quadratic inflation in no-scale supergravity inlight of BICEP2rdquo Journal of Cosmology andAstroparticle Physicsvol 2014 no 5 article 037 2014

[14] T Li Z Li and D V Nanopoulos ldquoChaotic inflation in no-scale supergravitywith string inspiredmoduli stabilizationrdquoTheEuropean Physical Journal C vol 75 no 2 article 55 2015

[15] S Ferrara A Kehagias andA Riotto ldquoThe imaginary Starobin-sky model and higher curvature correctionsrdquo Fortschritte derPhysik vol 63 no 1 pp 2ndash11 2015

[16] S V Ketov andT Terada ldquoInflation in supergravity with a singlechiral superfieldrdquo Physics Letters B vol 736 pp 272ndash277 2014

[17] C Kounnas D Lust and N Toumbas ldquoR2 inflation fromscale invariant supergravity and anomaly free superstrings withfluxesrdquo Fortschritte der Physik vol 63 no 1 pp 12ndash35 2015

[18] G A Diamandis B C Georgalas K Kaskavelis PKouroumalou A B Lahanas and G Pavlopoulos ldquoInflationin R2 supergravity with non-minimal superpotentialsrdquo PhysicsLetters B vol 744 pp 74ndash81 2015

[19] C Pallis and Q Shafi ldquoGravity waves from non-minimalquadratic inflationrdquo Journal of Cosmology and AstroparticlePhysics vol 2015 no 3 article 023 2015

[20] T Li Z Li and D V Nanopoulos ldquoHelical phase inflationrdquoPhysical Review D vol 91 no 6 Article ID 061303 2015

[21] D H Lyth ldquoWhat would we learn by detecting a gravitationalwave signal in the cosmic microwave background anisotropyrdquoPhysical Review Letters vol 78 no 10 article 1861 1997

[22] D Baumann and L McAllister ldquoInflation and stringtheoryrdquohttparxivorgabs14042601

[23] F Marchesano G Shiu and A M Uranga ldquoF-term axionmonodromy inflationrdquo Journal of High Energy Physics vol 2014no 9 article 184 2014

[24] C Long L McAllister and P McGuirk ldquoAligned naturalinflation in string theoryrdquo Physical Review D vol 90 no 2Article ID 023501 2014

[25] X Gao T Li and P Shukla ldquoCombining universal and odd RRaxions for aligned natural inflationrdquo Journal of Cosmology andAstroparticle Physics vol 2014 no 10 article 048 2014

[26] I Ben-Dayan F G Pedro and A Westphal ldquoTowards naturalinflation in string theoryrdquo Physical Review D vol 92 no 2Article ID 023515 2015

[27] R Blumenhagen D Herschmann and E Plauschinn ldquoThechallenge of realizing F-term axion monodromy inflation instring theoryrdquo Journal of High Energy Physics vol 2015 article7 2015

[28] H Abe T Kobayashi and H Otsuka ldquoTowards natural infla-tion from weakly coupled heterotic string theoryrdquo Progress ofTheoretical and Experimental Physics Article ID 063E02 2015

[29] T Ali S S Haque and V Jejjala ldquoNatural inflation from nearalignment in heterotic string theoryrdquo Physical Review D vol 91no 8 Article ID 083516 9 pages 2015

[30] R Flauger LMcAllister E Silverstein and AWestphal ldquoDrift-ing oscillations in axionmonodromyrdquo httparxivorgabs14121814

[31] K Freese J A Frieman andAVOlinto ldquoNatural inflationwithpseudo nambu-goldstone bosonsrdquo Physical Review Letters vol65 no 26 pp 3233ndash3236 1990

[32] F C Adams J R Bond K Freese J A Frieman and AV Olinto ldquoNatural inflation particle physics models power-law spectra for large-scale structure and constraints from thecosmic background explorerrdquo Physical Review D vol 47 no 2pp 426ndash455 1993

[33] G German A Mazumdar and A Perez-Lorenzana ldquoAngularinflation from supergravityrdquo Modern Physics Letters A vol 17no 25 pp 1627ndash1634 2002

12 Advances in High Energy Physics

[34] J E Kim H P Nilles and M Peloso ldquoCompleting naturalinflationrdquo Journal of Cosmology and Astroparticle Physics vol0501 article 005 2005

[35] D Baumann and D Green ldquoInflating with baryonsrdquo Journal ofHigh Energy Physics vol 2011 no 4 article 071 2011

[36] JMcDonald ldquoSub-Planckian two-field inflation consistent withthe Lyth boundrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 9 article 027 2014

[37] J McDonald ldquoA minimal sub-Planckian axion inflation modelwith large tensor-to-scalar ratiordquo Journal of Cosmology andAstroparticle Physics vol 2015 no 1 article 018 2015

[38] C D Carone J Erlich A Sensharma and Z Wang ldquoTwo-fieldaxion-monodromy hybrid inflation model Dantersquos WaterfallrdquoPhysical ReviewD vol 91 no 4 Article ID 043512 8 pages 2015

[39] G Barenboim and W-I Park ldquoSpiral Inflationrdquo Physics LettersB vol 741 pp 252ndash255 2015

[40] T Li Z Li andDVNanopoulos ldquoAligned natural inflation andmoduli stabilization from anomalous U(1) gauge symmetriesrdquoJournal of High Energy Physics vol 2014 no 11 article 012 2014

[41] T Li Z Li and D V Nanopoulos ldquoNatural inflation withnatural trans-planckian axion decay constant from anomalousU(1)119883rdquo Journal of High Energy Physics vol 2014 no 7 article052 2014

[42] E Silverstein and A Westphal ldquoMonodromy in the CMBgravity waves and string inflationrdquo Physical Review D vol 78no 10 Article ID 106003 21 pages 2008

[43] L McAllister E Silverstein and A Westphal ldquoGravity wavesand linear inflation from axion monodromyrdquo Physical ReviewD vol 82 no 4 Article ID 046003 2010

[44] K Choi H Kim and S Yun ldquoNatural inflation with multiplesub-Planckian axionsrdquo Physical Review D vol 90 no 2 ArticleID 023545 2014

[45] S-H H Tye and S S C Wong ldquoHelical inflation and cosmic-stringsrdquo httparxivorgabs14046988

[46] R Kappl S Krippendorf and H P Nilles ldquoAligned naturalinflation monodromies of two axionsrdquo Physics Letters B vol737 pp 124ndash128 2014

[47] G rsquotHooft ldquoNaturalness chiral symmetry and spontaneouschiral symmetry breakingrdquo in Recent Developments in GaugeTheories G rsquotHooft C Itzykson A Jaffe et al Eds vol 59of NATO Advanced Study Institutes Series pp 135ndash157 PlenumPress New York NY USA 1980

[48] R Blumenhagen M Cvetic S Kachru and T Weigand ldquoD-brane instantons in type II string theoryrdquo Annual Review ofNuclear and Particle Science vol 59 pp 269ndash296 2009

[49] K Harigaya andM Ibe ldquoSimple realization of inflaton potentialon a Riemann surfacerdquo Physics Letters B vol 738 pp 301ndash3042014

[50] R Kallosh A Linde and B Vercnocke ldquoNatural inflation insupergravity and beyondrdquo Physical Review D vol 90 Article ID041303(R) 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

12 Advances in High Energy Physics

[34] J E Kim H P Nilles and M Peloso ldquoCompleting naturalinflationrdquo Journal of Cosmology and Astroparticle Physics vol0501 article 005 2005

[35] D Baumann and D Green ldquoInflating with baryonsrdquo Journal ofHigh Energy Physics vol 2011 no 4 article 071 2011

[36] JMcDonald ldquoSub-Planckian two-field inflation consistent withthe Lyth boundrdquo Journal of Cosmology and Astroparticle Physicsvol 2014 no 9 article 027 2014

[37] J McDonald ldquoA minimal sub-Planckian axion inflation modelwith large tensor-to-scalar ratiordquo Journal of Cosmology andAstroparticle Physics vol 2015 no 1 article 018 2015

[38] C D Carone J Erlich A Sensharma and Z Wang ldquoTwo-fieldaxion-monodromy hybrid inflation model Dantersquos WaterfallrdquoPhysical ReviewD vol 91 no 4 Article ID 043512 8 pages 2015

[39] G Barenboim and W-I Park ldquoSpiral Inflationrdquo Physics LettersB vol 741 pp 252ndash255 2015

[40] T Li Z Li andDVNanopoulos ldquoAligned natural inflation andmoduli stabilization from anomalous U(1) gauge symmetriesrdquoJournal of High Energy Physics vol 2014 no 11 article 012 2014

[41] T Li Z Li and D V Nanopoulos ldquoNatural inflation withnatural trans-planckian axion decay constant from anomalousU(1)119883rdquo Journal of High Energy Physics vol 2014 no 7 article052 2014

[42] E Silverstein and A Westphal ldquoMonodromy in the CMBgravity waves and string inflationrdquo Physical Review D vol 78no 10 Article ID 106003 21 pages 2008

[43] L McAllister E Silverstein and A Westphal ldquoGravity wavesand linear inflation from axion monodromyrdquo Physical ReviewD vol 82 no 4 Article ID 046003 2010

[44] K Choi H Kim and S Yun ldquoNatural inflation with multiplesub-Planckian axionsrdquo Physical Review D vol 90 no 2 ArticleID 023545 2014

[45] S-H H Tye and S S C Wong ldquoHelical inflation and cosmic-stringsrdquo httparxivorgabs14046988

[46] R Kappl S Krippendorf and H P Nilles ldquoAligned naturalinflation monodromies of two axionsrdquo Physics Letters B vol737 pp 124ndash128 2014

[47] G rsquotHooft ldquoNaturalness chiral symmetry and spontaneouschiral symmetry breakingrdquo in Recent Developments in GaugeTheories G rsquotHooft C Itzykson A Jaffe et al Eds vol 59of NATO Advanced Study Institutes Series pp 135ndash157 PlenumPress New York NY USA 1980

[48] R Blumenhagen M Cvetic S Kachru and T Weigand ldquoD-brane instantons in type II string theoryrdquo Annual Review ofNuclear and Particle Science vol 59 pp 269ndash296 2009

[49] K Harigaya andM Ibe ldquoSimple realization of inflaton potentialon a Riemann surfacerdquo Physics Letters B vol 738 pp 301ndash3042014

[50] R Kallosh A Linde and B Vercnocke ldquoNatural inflation insupergravity and beyondrdquo Physical Review D vol 90 Article ID041303(R) 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of