Constraining a class of B − L extended models fromvacuum stability and perturbativity

Joydeep Chakrabortty,1 Partha Konar,2,* and Tanmoy Mondal2,†1Department of Physics, Indian Institute of Technology, Kanpur 208016, India

2Physical Research Laboratory, Ahmedabad 380009, India(Received 24 September 2013; published 25 March 2014)

The precise knowledge of the Standard Model (SM) Higgs boson and top-quark masses and couplingsare crucial to understand the physics beyond it. An SM-like Higgs boson having a mass in the range of123–127 GeV squeezes the parameters for physics beyond the Standard Model. In recent the LHC eramany TeV-scale neutrino mass models have earned much attention as they pose many interestingphenomenological aspects. We have contemplated B − L extended models which are theoretically wellmotivated and phenomenologically interesting, and they successfully explain neutrino mass generation. Inthis article we analyze the detailed structures of the scalar potentials for such models. We compute thecriteria which guarantee that the vacuum is bounded from below in all directions. In addition, perturbativity(triviality) bounds are also necessitated. Incorporating all such effects, we constrain the parameters of suchmodels by performing their renormalization-group evolutions.

DOI: 10.1103/PhysRevD.89.056014 PACS numbers: 11.10.Hi, 14.60.Pq, 14.60.St, 14.80.Ec

I. INTRODUCTION

The recent announcements from both ATLAS [1] andCMS [2] have revealed the existence of a new boson havinga mass in the range 123–127 GeV. The data so far indicatesa close resemblance to one having some of the measuredproperties of the Standard Model (SM) Higgs. However, ithas yet to be confirmed whether this boson is the SM Higgsor a beyond the Standard Model artifact. This long awaitedquest will only be examined more vigorously in the nearfuture with the help of more data.If the newly discovered particle is indeed the SM Higgs

boson then its mass can carry a signature of new physicswhich embeds SM at low energy. The Higgs mass can berecast solely in terms of the Higgs quartic coupling, λh. Thestability of the electroweak (EW) vacuum demands apositive λh. Now if the SM is the only existing theory innature then this condition, λh > 0,1 must be maintained ateach scale of its evolution up to the Planck scale (MPl). Theevolution of λh with the renormalization (mass) scale limitstwo boundary values—one at the EW scale for which wehave λhðMPlÞ ¼ π, and one at the Planck scale for which wehave 0—from the demands of perturbativity of the coupling(triviality) and the stability of the vacuum (vacuum stabil-ity), respectively. It has been noted in Refs. [3–5] that theSM electroweak vacuum is not stable up to the Planck scalefor most of the SM parameters (top-quark mass, Higgsmass and strong coupling αs). Thus it indicates that somenew physics might be there before the SM vacuum attains

instability. Thus the physics beyond Standard Model isexpected to take care of the stability of the vacuum of thefull scalar potential along with the electroweak ones. Inbrief, the present range of the SM-like Higgs mass enter-tains the presence of new physics solely from the vacuumstability point of view.Apart from this, we already have hints of new physics

beyond the Standard Model from the neutrino sector. Manyexperimental observations, like neutrino oscillations, con-firm that neutrinos have tiny nonzero masses which cannotbe accommodated naturally within the SM. Thus we musthave physics beyond the Standard Model to explain thisfeature. Among the neutrino mass generation proceduresthe seesaw mechanism [6–15] is very popular. In usual(natural) seesaw models light neutrino masses are ∼m2

D=Mwhere the Dirac-type mass mD ∼ 100 GeV and M is theMajorana mass of a heavy fermion which gets integratedout during the process. The mass of this heavy fermion,M,determines the scale of the seesaw models which needs tobe very high (∼1011 GeV) to avoid any fine-tuning in mD.As the natural scale of the seesaw is very high these modelssuffer from a lack of testability. But it is also possible toconstruct low-scale (∼TeV) models by either importingsome new fields [16] or incorporating higher-dimensionaloperators [17–21]. These models not only generate thecorrect order of neutrino masses and mixing, but are alsophenomenologically interesting as the scale of these the-ories are well within the reach of present experiments likethe LHC. These models are extended by some extra gaugesymmetry and/or new particles. The presence of these newfields might affect the evolution of the SM couplings—likethe gauge, Higgs quartic, and top Yukawa couplings—ifthey couple to the SM particles. Hence it is necessary toexamine the status of the SM vacuum once these new

*konar@prl.res.in†tanmoym@prl.res.in1This is the necessary—but not sufficient—condition to

confirm the sole existence of the SM up to the Planck scale.

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physics models come into play. Thus by using knowledgeof the SM parameters and from the demand of vacuumstability2 the new parameters involved in the theory mightbe severely constrained. In the literature the stability of thevacua was discussed in several scenarios consideringbeyond Standard Models. These models are extended bythe extra gauge symmetry and/or the addition of newparticles. Quantum corrections of the quartic couplingsdepend on the spin of the particles belonging to a particularmodel. The fermion-loop contributions contain a relativeminus sign compared to the bosonic fields. Thus theYukawa couplings tend to spoil the stability unlike thegauge and other scalar self-couplings. Vacuum stability indifferent variants of seesaw models has been adjudged inRefs. [22–28] which has a richer particle spectrum com-pared to the SM. In the context of gauge extensions, thevacuum stability for the alternative left-right symmetricmodel has been discussed in Ref. [29].In a theory involving multiple scalar fields the structure

of the potential is complicated. The vacuum stabilitycriteria depend on some combinations of the scalar quarticcouplings. Moreover, the perturbativity (triviality) boundsalso play crucial roles in finding a consistent parameterspace compatible with the choice of new physics scales.Nontachyonic scalar masses are guaranteed with theseconstraints. It has been noted that some of the quarticcouplings can be recast in terms of the heavy scalar massesand thus can be constrained from a phenomenological pointof view. On the contrary, few of them do not have that muchimpact on scalar masses; rather, they determine the splittingamong the narrowly spaced massive scalar modes. Ourpresent collider experiments are still not sensitive enough toaddress these fine splittings, and thus the quartic couplingsare beyond the reach of any experimental verification. Butthese couplings can be constrained through vacuum sta-bility and perturbativity (triviality) depending on the choiceof the scale of new physics.In this paper we concentrate on the Uð1ÞB−L extended

models which are classified into two categories: SM ⊗Uð1ÞB−L or the left-right (LR) symmetry. We have adoptedtwo variants of the LR-symmetric models containing(i) two SU(2) triplet scalars ΔLðRÞ, and (ii) two SU(2)doublet scalars, HLðRÞ. In Sec. II we introduce the basicstructures of these models. Then we include the renorm-alization group evolutions of all the necessary couplingsand show how the vacuum stability and perturbativity(triviality) bounds constrain the parameter space of eachmodel in Sec. III. We have analyze the structure of the

potentials in detail and compute the criteria for vacuumstability using the formalism shown in Ref. [30]. Allvacuum stability conditions corresponding to differentmodels are listed in Appendix B.

II. MODELS

The Standard Model symmetry group is expressed asSUð3ÞC ⊗ SUð2ÞL ⊗ Uð1ÞY . It has been noted in Ref. [31]that an extra U(1) gauge symmetry along with the SM canprovide solutions to some of the unaddressed issues in theStandard Model. These extra Abelian symmetry groupscan, in general, originate from different high-scale grandunified theories (GUTs), like SO(10) and E(6). These largergroups contain Uð1ÞB−L as a part of the intermediate gaugesymmetries. In nonsupersymmetric GUT models theUð1ÞB−L breaking scale can be lowered to a few TeV3

[32], which is consistent with unification pictures. In ourpresent study we concentrate on TeV-scale Uð1ÞB−Lextended models where neutrino mass generation can beexplained. However, any high-scale roots of these modelsare not considered and are kept for future work.

A. Uð1ÞB−LThe gauge group under consideration is SUð3ÞC ⊗

SUð2ÞL ⊗ Uð1ÞY ⊗ Uð1ÞB−L. This minimal model con-tains an extra complex singlet scalar field S and this extraB − L symmetry is broken once it acquires a vacuumexpectation value (VEV) [33–35]. Thus the VEV deter-mines the symmetry-breaking scale of this symmetry andalso the mass of the extra neutral gauge boson ZB−L. Forthe purpose of our study we will focus only on the relevantpart of the Lagrangian, namely the scalar kinetic andpotential terms and the lepton Yukawa couplings. Thescalar kinetic term is

Ls ¼ ðDμΦÞ†ðDμΦÞ þ ðDμSÞ†ðDμSÞ − VðΦ; SÞ: (1)

Here the potential VðΦ; SÞ is given as

VðΦ; SÞ ¼ m2Φ†Φþ μ2jSj2 þ λ1ðΦ†ΦÞ2 þ λ2jSj4þ λ3Φ†ΦjSj2; (2)

where Φ and S are the complex scalar doublet and singletfields, respectively. After gauging away the extra modesand acquiring the VEVs these fields are redefined as

Φ≡�

01ffiffi2

p ðvþ ϕÞ�; S≡ 1ffiffiffi

2p ðvB−L þ sÞ; (3)

where the EW-symmetry-breaking VEV v and the B − L-breaking VEV vB−L are real and positive.

2In this paper we are considering stability up to the Planckscale. We are not considering the metastability which does notrequire the vacuum to be bounded from below. If the decaylifetime of a vacuum is larger than the lifetime of the Universethen that vacuum is metastable. But as our procedure concernsonly the boundedness of the scalar potential it fails to pin downthe existence of the metastable vacuum.

3This is also true for supersymmetric GUT models; seeRef. [32].

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We also find the scalar mass matrix in the followingform:

M ¼

λ1v2λ3vB−Lv

2

λ3vB−Lv2

λ2v2B−L

!¼�M11 M12

M21 M22

�: (4)

After diagonalizing this mass matrix we construct twophysical scalar states: a light h and a heavy H, havingmasses Mh and MH, respectively,

M2H;h ¼

1

2

hM11 þM22 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðM11 −M22Þ2 þ 4M2

12

q i:

(5)

The scalar mixing angle α can be expressed as

tanð2αÞ ¼ 2M12

M11 −M22

¼ λ3vvB−Lλ1v2 − λ2v2B−L

: (6)

Using Eqs. (5) and (6) the quartic coupling constants λ1, λ2,and λ3 can be recast in the following forms:

λ1 ¼1

4v2fðM2

H þM2hÞ − cos 2αðM2

H −M2hÞg;

λ2 ¼1

4v2B−LfðM2

H þM2hÞ þ cos 2αðM2

H −M2hÞg;

λ3 ¼1

2vvB−Lfsin 2αðM2

H −M2hÞg: (7)

It can be noted from the last equation in Eq. (7) that wewould get a duplicate set of solutions with inverted signsfor both α and λ3. Hence one choice of positive α suffices aspresented at Sec. III A.Due to the presence of an extra Uð1ÞB−L gauge theory

the SM gauge kinetic terms are modified by

LKEB−L ¼ − 1

4F0μνF0

μν; (8)

where,

F0μν ¼ ∂μB0

ν − ∂νB0μ: (9)

The covariant derivative for SUð2ÞL ⊗ Uð1ÞY ⊗Uð1ÞB−L sector in this model is modified as

Dμ ≡ ∂μ þ ig2TaWaμ þ ig1YBμ þ ið~gY þ gB−LYB−LÞB0

μ:

(10)

The SM gauge bosons Bμ and W3μ will mix with the new

gauge boson B0μ to create two massive physical fields Z and

ZB−L and one massless photon field A. Assuming there isno kinetic mixing at tree level, i.e., ~g ¼ 0 at the EW scale,the physical gauge-boson masses are given as

M2Z ¼ 1

4ðg21 þ g22Þv2; (11)

M2ZB−L ¼ 4g2B−Lv2B−L: (12)

Along with the Standard Model particles, three right-handed neutrinos (νR) are introduced.

4 The relevant term ofthe Lagrangian of the Yukawa interactions can be written as

−LY ¼ ylijliL ~Φ νjR þ yhijðνRÞci νjRSþ H:c:; (13)

where ~Φ ¼ iσ2Φ� with σ2 being the Pauli matrix. Thesecond term of the above equation is the Majoranamass term. Note from Eq. (13) that the conservation ofB − L charge requires that the singlet scalar field, S,must have QB−L ¼ −2. When the SM Higgs and singletscalar S acquire VEVs the neutrino mass matrix takes theform

Mν ¼�

0 mD

mTD mR

�; (14)

where mD ¼ yl vffiffi2

p and mR ¼ ffiffiffi2

pyhvB−L. The light (mνl )

and heavy (mνh ) neutrino masses are

mνl ¼ −mTDm

−1R mD; (15)

mνh ¼ mR: (16)

In this model the heavy neutrino mass mR is alsogenerated through the Yukawa terms unlike the gauge-invariant Majorana mass term in type-I seesaw models. Itcan be noted that with mR ∼O ðTeVÞ, yl needs to be verysmall to generate light neutrino masses ∼O ðeVÞ. But yhcan be large ∼Oð1Þ as vB−L is around the TeV scale. Thussuccessful light neutrino mass generation does not con-strain yh. But as the heavy neutrino is also coupled to theSM-like Higgs, yh affects the vacuum stability of the scalarpotential in this model and gets constrained. The gaugecoupling gB−L and the VEVof the B − L-breaking scale arealso free parameters. In the following section we show howthese parameters are constrained from the vacuum stabilityof the scalar potential and also from the perturbativity(triviality) of the couplings.

B. Left-right symmetry

The full LR-symmetric gauge group is written asSUð3ÞC ⊗ SUð2ÞL ⊗ SUð2ÞR ⊗ Uð1ÞB−L. The SUð2ÞR ⊗Uð1ÞB−L is broken to Uð1ÞY at a scale higher than theEW-symmetry-breaking one. Thus the hypercharge gen-erator is a linear combination of the SUð2ÞR and Uð1ÞB−Lgenerators. In this model, the hypercharge, Y, can be

4One right-handed neutrino (QB−L ¼ −1) for each generationis required for the sake of gauge anomaly cancellation.

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reconstructed from the SUð2ÞR and Uð1ÞB−L quantumnumbers as

Y ¼ T3R þ ðB − LÞ=2; (17)

with T3R being the third component of SUð2ÞR isospin.Here we briefly present two variants of minimal left-right

symmetric models:(i) The scalar sector consists of a bidoublet (Φ), one

left-handed triplet (ΔL), and one right-handed triplet(ΔR) [36–39].

(ii) The scalar sector consists of a bidoublet (Φ), oneleft-handed doublet (HL), and one right-handeddoublet (HR) [40–42].

1. LR model with triplet scalars

The most generic scalar potential of this model withbidoublet and triplet scalars (Φ;ΔL;R) is given inAppendix 2. The explicit structures of the scalars can bepresented in the following form:

Φ¼ ϕ01 ϕþ

1

ϕ−2 ϕ0

2

!; ΔL;R¼

δþL;R=

ffiffiffi2

pδþþL;R

δ0L;R −δþL;R=ffiffiffi2

p!:

These fields transform under SUð2ÞL ⊗ SUð2ÞR ⊗Uð1ÞB−L gauge groups in the following manner:

Φ≡ ð2;2;0Þ; ΔR≡ ð1;3;2Þ; ΔL≡ ð3;1;2Þ: (18)

Once neutral components of these scalars acquire vacuumexpectation values, they can be written in the followingform:

hΦi ¼�v1 0

0 v2eiθ

�; hΔLi ¼

�0 0

vL 0

�;

hΔRi ¼�

0 0

vR 0

�; (19)

where, for simplicity we have chosen v2 ¼ 0without loss ofgenerality. With these structures for the vacuum expectationvalues, symmetry breaking occurs in two stages. Thesymmetry group SUð2ÞL ⊗ SUð2ÞR ⊗ Uð1ÞB−L breaksdown to SUð2ÞL ⊗ Uð1ÞY by vR at the high scale.Consequently, the vacuum expectation value v1 of thebidoublet breaks SUð2ÞL ⊗ Uð1ÞY to Uð1ÞEM. So the totalnumber of Goldstone bosons will be six. Now the Higgssector has 20 degrees of freedom (eight real fields for thebidoublet and six each for the triplet fields). Hence, theremaining 14 fields will be massive scalars and they are asfollows:(1) Two doubly charged scalars ðH��

1 ; H��2 Þ.

(2) Two singly charged scalars ðH�1 ; H

�2 Þ.

(3) Four neutral CP-even scalars ðH00; H

01; H

02; H

03Þ.

(4) Two neutral (CP-odd) pseudoscalars ðA00; A

01Þ.

Since we already mentioned that the scale vR is muchhigher than the VEVof electroweak breaking v1, the scalarmasses can be expressed in leading-order terms5 [43,44],

M2H0

0

≃ 2λ1v21;

M2H0

1

≃ 1

2λ12v2R;

M2H0

2

≃M2A01

≃M2H�

2

≃ 2λ5v2R;

M2H0

3

≃M2A02

≃M2H�

1

≃M2H��

1

≃ 1

2ðλ7 − 2λ5Þv2R;

M2H��

2

≃ 2λ6v2R: (20)

MH00is the Standard Model Higgs boson and is denoted as

Mh from here onwards. For simplicity and to reduce thenumber of free parameters, we consider degenerate heavyscalars at the vR scale, i.e., MH0

1¼ MH0

2¼ MH0

3¼

MH��2

¼ MH. It is important to note that the remainingquartic couplings only contribute in the scalar massesas subleading terms and they are proportional to thev21 at the electroweak symmetry-breaking (EWSB) scale.Hence, λ2, λ3, λ4, λ8, λ9, λ10, and λ11 induce only therelative mass splittings among these heavy scalars whichare almost phenomenologically unaccessible at presentexperiments.The kinetic term of the scalar part can be written as

Lkin ¼ Tr½ðDμΦÞ†ðDμΦÞ� þ Tr½ðDμΔLÞ†ðDμΔLÞ�þ Tr½ðDμΔRÞ†ðDμΔRÞ�; (21)

where,

DμΦ ¼ ∂μΦ − ig2LTaWaLμΦþ ig2RΦTaWa

Rμ;

DμΔðL=RÞ ¼ ∂μΔðL=RÞ − igð2L=2RÞ½TaWaðL=RÞμ;ΔðL=RÞ�

− igB−LBμΔðL=RÞ: (22)

We choose the gauge couplings g2L and g2R for theSUð2ÞL and SUð2ÞR gauge groups, respectively, to be samefor the sake of minimality of the model in terms of thenumber of parameters. After spontaneous breaking of theLR and EW symmetries, two chargedW�

L=R and two neutralZL=R gauge bosons become massive, while the photon Aremains massless,

5These leading-order terms match exactly with the masses ofthe heavy scalars at the scale vR, i.e., before the EWSB. Afterthe EWSB, some correction terms are generated which areproportional to v21. But as vR ≫ v1, the splitting among themasses of these heavy scalars are negligible compared to theirrelative masses. It is important to note that this “≃” will bereplaced by “¼” in Eq. (20) when these masses are given at thevR scale.

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M2W�

L¼ 1

4g22v

21; M2

W�R¼ 1

4g22ðv21 þ 2v2RÞ;

M2ZL;R

¼ 1

4½ðg22v21 þ 2v2Rðg22 þ g2B−LÞÞ∓

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifg22v21 þ 2v2Rðg22 þ g2B−LÞg2 − 4g2ðg22 þ 2g2B−LÞv21v2R

q�: (23)

Under the gauge group SUð2ÞL ⊗ SUð2ÞR ⊗ Uð1ÞB−Lquarks and leptons are doublets,

LiðL=RÞ ¼�νili

�ðL=RÞ

; QiðL=RÞ ¼�uidi

�ðL=RÞ

: (24)

The most general lepton Yukawa Lagrangian can be writtenas,

−LY ¼ ½L̄LðylΦþ ~yl ~ΦÞLR þ h:c� þ yhLLcR~ΔLLL

þ yhRL̄cL~ΔRLR; (25)

where, ~Φ ¼ iσ2Φ� and ~ΔL=R ¼ iσ2ΔL=R. Here we haveconsidered that the Yukawa matrices are diagonal.6 Theneutral fermion masses are generated once the Φ and Δacquire VEVs. The neutral fermion mass matrix is given as

Mν ¼�mII

ν mD

mTD mR

�; mD ¼ 1ffiffiffi

2p ylv1;

mR ¼ffiffiffi2

pyhvR; mII

ν ¼ffiffiffi2

pyhvL; (26)

where, yhL ¼ yhR ¼ yh because of left-right symmetry. Thusthe light neutrino mass

mνl ¼ mIIν −mT

Dm−1R mD; (27)

is generated through type-II (first term) and type-I (secondterm) seesaw mechanisms.As the VEV of the left-handed triplet scalar is con-

strained by ρ parameter of the SM it cannot be larger than∼Oðfew GeVÞ. Thus it is indeed possible to generate lightneutrino masses ∼eV with vL ∼ eV while the neutrinoYukawa coupling can be ∼Oð1Þ. In our further analysis weconsider vL ¼ 0, and thus the type-II seesaw mechanism isabsent here. The heavy neutrino mass mR is also generatedthrough the Yukawa terms and is proportional to vR. It canbe noted that withmR ∼O ðTeVÞ, the Dirac termmD needsto be very small to generate light neutrino masses∼O ðeVÞ. But yh can be as large as ∼Oð1Þ even whenvR is around the TeV scale. Thus successful light neutrinomass generation is still possible while keeping yh as large as∼Oð1Þ. But yh affects the vacuum stability of the scalar

potential in this model as the heavy neutrino is also coupledto the SM-like Higgs. In the following section we showhow these parameters are constrained due to vacuumstability and perturbativity (triviality).It has been noted that the minimal left-right symmetric

model is constrained by flavor-changing neutral currents(FCNCs) [46–49]. The model we have worked withcontains the bidoublet whose one of the VEV is zero.Thus there is no FCNC problem in this model. There arealso constraints from neutral kaon mixing, i.e., the kaonmass difference. Our choice of the vR scale and themasses for the heavy neutral scalars takes care of thosebounds. As the VEVs and the Yukawa couplings in ourscenario are real there is neither a source of norspontaneous or explicit CP violation. But since we haveconsidered the Yukawa matrices to be diagonal we willboil down to the trivial, i.e., identity Cabibbo-Kobayashi-Maskawa and Pontecorvo-Maki-Nakagawa-Sakata matri-ces. To fit all the masses and mixings we need to go forthe nonminimal extension of this model and that certainlymodify the set of renormalization group evolutions(RGEs) that we have used here.

2. LR model with doublet scalars

In this case the scalar sector consists of a bidoublet (Φ),one left-handed doublet (HL), and one right-handed dou-blet (HR). The scalar potential is depicted in Appendix A 3.In terms of the SUð2ÞL ⊗ SUð2ÞR ⊗ Uð1ÞB−L gauge groupthese fields can be written as

Φ≡ ð2; 2; 0Þ; HL ≡ ð2; 1; 1Þ; and HR ≡ ð1; 2; 1Þ:(28)

The structure of HL=R is written as

HL=R ¼�hþL=Rh0L=R

�: (29)

The neutral components of Φ andHL=R acquire the vacuumexpectation values

hΦi ¼�v1 0

0 v2eiθ

�; hHLi ¼

�0

vL

�;

hHRi ¼�

0

vR

�: (30)

6There exist two different discrete symmetries which can relateleft- and right-handed fields [45]. Yukawa matrices are diagonalas we have considered the parity operation as defined in Ref. [43]to relate L and R fields.

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As before, we put v2 ¼ 0. The scalar sector consists of 16real scalar fields out of which six will be Goldstone bosons.Finally we will have four CP-even scalars: two CP-oddscalars and two charged scalars. Among the CP-evenscalars one is the Standard Model Higgs boson with massMh and the other three are taken as degenerate heavyscalars having mass MH. The parameters in the Higgspotential can be recast in terms of the masses of the neutraland charged scalars. The details about the scalar sector havebeen discussed in Ref. [50]. The gauge sector is similar tothe previous case, i.e., the LR model with triplet scalars.In the limit vR ≫ v1 and assuming all the heavy scalars

are degenerate, we have

f1 ¼ ðMH=vRÞ2 ¼ κ1 ¼ −κ2; (31)

whereas the minimization of the potential requires

v21v2R

¼ f1 − 2β14λ1

:

The structure of the covariant derivative in thismodel is very similar to that for the triplet scenario [seeEq. (22)],

DμΦ ¼ ∂μ − ig2LTaWaLμΦþ ig2RΦTaWa

Rμ;

DμHðL=RÞ ¼ ∂μHðL=RÞ − igð2L=2RÞTaWaðL=RÞμHðL=RÞ

− igB−LBμHðL=RÞ: (32)

Following the previous convention we also set g2L ¼ g2R ¼g2. After spontaneous breaking of the SUð2ÞL ⊗SUð2ÞR ⊗ Uð1ÞB−L symmetry, two charged W�

L=R andtwo neutral ZL=R gauge bosons become massive, whilephoton the A remains massless,

M2W�

L¼ 1

4g22v

21; M2

W�R¼ 1

4g22ðv21þv2RÞ;

M2ZL;R

¼ 1

8½ð2g22v21þv2Rðg22þg2B−LÞÞ

∓ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi4g42v

41þðg22þg2B−LÞv4R−4g22g

2B−Lv21v2R

q�: (33)

In the left-right symmetric model with a doublet scalarthe leptonic part of the Yukawa interaction can bewritten as

−L ¼ L̄Lðy1Φþ y2 ~ΦÞLR þ H:c:; (34)

where the SUð2ÞL ⊗ SUð2ÞR quantum numbers of LLand LR are (2,1) and (1,2), respectively. So from thisLagrangian the Dirac mass term for the neutrinos can bewritten as

mD ¼ y1v1: (35)

Here, it is not possible to write the renormalizableMajorana mass term for the light and heavy neutrinos.But we can add nonrenormalizable effective terms as

Leff ¼ηLM

LLLLHLHL þ ηRM

LRLRHRHR; (36)

where, M is some very high scale and the η’s aredimensionless parameters that denote the strength of thesenonrenormalizable couplings. OnceHR acquires a VEV theright-handed neutrino mass is generated as

mR ≃ ηRv2RM

:

Here we consider that hHLi ¼ vL ¼ 0, and thus thiseffective term does not contribute to the light neutrinomass. The neutrino mass matrix in the (νl, νh) basis reads as

Mν ¼�

0 mD

mTD mR

�; (37)

and the light neutrino mass can be written as

mνl ¼ −mTDm

−1R mD; (38)

which is a variant of the type-I seesaw mechanism.In the left-right symmetric model associated with two

doublet scalars, neutrino masses cannot be generatedthrough a type-II seesaw mechanism due to the lack of aleft-handed triplet scalar.7 Thus the type-I seesaw mecha-nism is the natural choice in this case. But the right-handedneutrino masses are generated through an effective operatorsuppressed by a heavy scale. This may provide a possibleexplanation how the right-handed neutrinos can be loweredto the TeV scale. Here, the correct order of light neutrinomasses are generated if the Dirac-type neutrino Yukawacoupling needs to be very small, unless one considers thespecial textures for the Dirac Yukawa couplings. Thenvacuum stability is automatically satisfied as these DiracYukawa couplings are much smaller. Thus here only thequartic couplings get constrained through the vacuumstability and perturbativity (triviality) of the couplings.Within a framework very similar to this it is indeed possibleto generate light neutrinomasses of the correct orderwithoutlowering the Yukawa coupling as the light neutrino massesare independent of vR but are suppressed by some high scale[51,52]. In that case the vacuum stability constraints cannotbe avoided and play the most crucial role in constraining theYukawa couplings and other parameters.

7Although, through an effective operator the Majorana massterm for light neutrinos can be generated; see Eq. (36). But thiscontribution is absent here as we have set vL ¼ 0.

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III. VACUUM STABILITY

The presence of new physics introduces exotic non-SMparticles in the theory and if they couple to the SM fieldsthen the RGEs of the Higgs quartic coupling (λh) will bemodified. Moreover, additional quartic interactions of extrascalar fields should also be introduced. Extended gaugeinteractions from the larger gauge groups as well asYukawa interactions would contribute to these evolutionequations. Now the question arises of whether or not thevacuum is stable in the presence of the new physics. Inparticular, when we have narrowed down a preferred rangeof the Higgs mass between 123–127 GeV, the new physicscould be constrained by the vacuum stability criteria. Toadjudge the stability of these models we have consideredthe one-loop RGEs of all the required parameters. Inpassing we would like to mention that the allowedparameter space in our analysis is the minimal set whichwill be extended once one includes the higher-orderrenormalization group (RG) effects. The RGEs for theSM and each of the B − L models which are used in ourcalculation are given in an Appendix. Since we are dealingwith the TeV-scale models, all the SM RGEs will bemodified once the new physics effects are switched on.Thus from the EW scale to the TeV scale (specific valuesare dictated in plots) the RGEs will be SM-like and fromthe TeV scale to the Planck scale they will be the modifiedones, and during the process proper matching conditionsare incorporated at the TeV scale.

A. Uð1ÞB−L model

It is clear from the structure of the potential as shown inEq. (2) for the Uð1ÞB−L model, that the vacuum stabilityconditions are different from those for the SM due to thepresence of an extra singlet scalar. If all the quarticcouplings are positive, the potential will be triviallybounded from below, i.e., the vacuum is stable and thesestability conditions read simply as λ1;2;3 > 0. But it isindeed possible to allow λ3 to be negative and still have thevacuum be stable. Thus vacuum stability conditions beyondthe trivial ones allow for a larger parameter space and needto be accommodated in these conditions. We find thenontrivial vacuum stability criteria using the proposaldictated in Ref. [30], which are shown in Appendix B 1,

4λ1λ2 − λ23 > 0; λ1 > 0; λ2 > 0. (39)

Together with these we have also incorporated perturba-tivity constraints on quartic couplings by demanding anupper limit, i.e., jλij < 1ði ¼ 1; 2; 3Þ.Noting from Eqs. (5) and (6) that the physical Higgs field

is an admixture of two scalar fields ϕ and s, in our study thescalar mixing angle α is considered to be a free parameterinstead of the quartic couplings λiði ¼ 1; 2; 3Þ. This modelconsists of two different scales in the theory: the EW scale

and the B − L symmetry-breaking scale. Thus two RGEsare invoked for the analysis. As we have two Abeliancouplings in this model, there might be mixing betweenthem [53,54]. To simplify the situation, and of coursewithout hampering any other conclusions, we impose nomixing between the ZB−L and Z gauge bosons at the treelevel. This follows from the condition ~gðQEWÞ ¼ 0 asalready discussed above [Eq. (11)]. As a consequencethe B − L-breaking VEV vB−L relates to the new ZB−Lboson mass given in Eq. (12). For demonstration, we havepicked the perturbative value of this additional gaugecoupling at the breaking scale as, gB−L ¼ 0.1. For sim-plicity we further assume that heavy neutrinos are degen-erate and fixed at m1;2;3

νh ≡mνh ≃ 200 GeV, which arewithin the allowed values. We have used the central valueof the light Higgs mass (Mh) at 125 GeV, the top-quarkmass at 173.2 GeV and the strong coupling constant αs at0.1184. Thus the remaining free parameters in our study areMH, α and vB−L. We have explored the correlatedconstraints on these parameters from vacuum stability.The set of RGEs of different couplings that we have used

in our analysis are encoded in Appendix C 2 [35]. Theparameter space consistent with vacuum stability in the

200

400

600

800

1000

1200

1400

-1.5 -1 -0.5 0 0.5 1 1.5

MH

(G

eV)

α (radian)

vB-L = 7.5 TeVgB-L = 0.1

Input Allowed107 GeV

1010 GeV1019 GeV

λ 3 < 0

λ 3 > 0

FIG. 1 (color online). The allowed parameter space in the heavyHiggs mass (MH) and scalar mixing angle ðαÞ plane consistentwith vacuum stability and perturbativity bounds are shown. Thegray region is the domain of allowed input parameters. The red(dark gray), green (light gray), and black sub-parameter spacesshow the domain of MH and α for which this B − L theory isvalid up to 107, 1010 and 1019 GeV, respectively and expectedlythe region in both sides squeeze for demanding higher scale ofvalidity of the theory. The Majorana neutrino mass is fixed at200 GeV and the B − L-breaking VEV (vB−L) is set at 7.5 TeV.The Uð1ÞB−L gauge coupling is taken to be 0.1 which impliesMZB−L ¼ 1.5 TeV. The shaded region satisfies λ3 < 0 [as well asα < 0 from Eq. (7)]. Thus the nontrivial vacuum stabilityconditions are satisfied in this region. These conditions are morestringent than the trivial one applied in the positive-α region.Although the pattern of the allowed parameter space is verysimilar for both the positive and negative α regions, the α > 0region covers a larger parameter space.

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heavyHiggsmass (MH) and scalarmixing angle ðαÞ plane isdepicted in Fig. 1. All the couplings are perturbativethroughout their evolutions. The gray region is the domainof allowed input parameters. The red, green, and black sub-parameter spaces show the domain of MH and α for whichthis B − L theory is valid up to 107, 1010 and 1019 GeV,respectively. In this figure, for a particular heavy scalar masseach part of this allowed domain is restricted at someminimum (maximum) value of α due to the vacuum stability(perturbativity) of the quartic couplings. The Majorananeutrino mass is fixed at 200 GeV and the B − L-breakingVEV (vB−L) is set at 7.5 TeV. The Uð1ÞB−L gauge couplingis taken to be 0.1which impliesMZB−L ¼ 1.5 TeVconsistentwith present experimental bounds [55]. The yellow shadedregion contains the set of allowed parameters for λ3 < 0[as well as α < 0 from Eq. (7)]. Though the pattern of theallowed parameter space in the positive-λ3 region is verysimilar, it is not exactly symmetric. The outer boundariesabove each color in Fig. 1 match exactly for both thepositive- and negative-α region. This is not surprisingbecause the outer boundary is determined by the perturba-tivity of the couplings and thus is not affected by the

vacuum stability conditions which are different for differentsigns of λ3. However, the lower boundaries are the outcomeof the need to satisfy the criteria of vacuum stability.Allowed parameters in the yellow shaded region (whichrepresents λ3 < 0) in Fig. 1 are reflected by the nontrivialvacuum stability condition in Eq. (39), which sequentiallyplays a role in determining the lower boundaries in theallowed parameters. Thus expectedly in the positive-αregion the allowed parameter space is larger than that fornegative α. Also, note that α ¼ 0 leads to the decouplinglimit when the heavy scalar will not affect the vacuumstability. The parameter space has also shrunk as the validityof the model must be closer to the Planck scale as can beinferred from Fig. 1.To study the dependence of different parameters as shown

in Fig. 1, we plot the allowed parameter space in theMH − αplane which remains consistent with vacuum stability andwhere all the couplings are perturbative up to the PlanckScale. In Fig. 2(a) the Majorana neutrino Yukawa couplingyh is varied while keeping vB−L and gB−L fixed. As yh

increases, the allowed parameter space is shrunk since theYukawa coupling affects the quartic couplings negatively in

MH

(G

eV)

Scalar Mixing Angle α (Radian)

vB-L = 7.5 TeVgB-L = 0.1

(a)yh

0.040.20

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1400

1600

1800

2000

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9

MH

(G

eV)

Scalar Mixing Angle α (Radian)

gB-L = 0.1yh = 0.04

(b)vB-L

15 TeV7.5TeV

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1400

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0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9

MH

(G

eV)

Scalar Mixing Angle α (Radian)

vB-L = 7.5 TeVyh = 0.04

gB-L0.30.1

200

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800

1000

1200

1400

1600

1800

2000

0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9

(c)

FIG. 2 (color online). Allowed parameter space in the MH − α plane, with α varying between ½0;−π=2�, consistent with vacuumstability and perturbativity (triviality) bounds up to the Planck scale. (a): The Majorana neutrino Yukawa coupling yh is varied keepingvB−L and gB−L fixed. (b): Two different sets of the B − L-breaking VEV, vB−L are chosen keeping gB−L and yh fixed. (c): In this plotgB−L varies while vB−L and yh are kept constant. In our analysis any value of gB−L for vB−L ¼ 7.5 TeV that is greater than 0.34 isdisallowed as the coupling becomes nonperturbative before the Planck scale. Corresponding regions for positive α are not shown here, asthey remain unaffected and are the same as those given in the blue strip in Fig. 1 owing to the trivial conditions.

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their RG evolutions. Thus larger Yukawa couplings spoil thevacuum stability. Figure 2(b) shows the dependence on theB − L-breaking VEV for fixed gB−L and yh. vB−L deter-mines the scale of new physics beyond the Standard Model,i.e., from where the RGEs are being modified due to thepresence of new particles. The larger vB−L implies that newset of RGEs come into play later. In the B − L extendedmodel λ3 is inversely proportional to vB−L at the EW scale[see Eq. (7)]. Thus for the same set of values ofMH and α, λ3is smaller for larger vB−L at 15 TeV. The RGE of λ3 is suchthat for our choice of parameters it grows with the massscale. Thus there is a possibility of generating a large λ3 suchthat vacuum stability and perturbativity conditions are notvalidated at some higher scale. This plot therefore shows thatit is possible to have a larger allowed parameter space forlarger values of vB−L. Finally in Fig. 2(c), gB−L varies whilevB−L and Yν are kept constant. As the larger values of thegauge couplings affect the RGEs of the quartic couplingspositively, the vacuum stability is improved. Thus with alarger value of gauge coupling a larger parameter space isallowed. But the U(1) couplings increases with the massscale. Hence the couplings with much larger values at thelow scale might be nonperturbative at the high scale. Inour analysis, when vB−L is at 7.5 TeV, any value of gB−Lgreater than 0.34 is disallowed as the coupling becomesnonperturbative before the Planck scale.

B. Left-right symmetry

1. LR model with triplet scalars

In this model the scalar potential for the left-rightsymmetric model with a triplet scalar as shown inAppendix A 2 contains many quartic couplings. To findthe condition of vacuum stability we have considered alltwo-field, three-field and four-field directions and foundtheir stability criteria. Detailed field directions correspond-ing to the potential together with calculated stabilityconditions are listed in Appendix B 2. Finally, the effectivenontrivial vacuum stability conditions which are necessaryand sufficient are

λ1 > 0; λ5 > 0; λ5 þ λ6 > 0;

λ5 þ 2λ6 > 0; λ12 − 2ffiffiffiffiffiffiffiffiffiλ1λ5

p< 0. (40)

Along with the above conditions, we find an additionalcondition λ12 > 0 from Eq. (20).The renormalization group evolutions that we have

considered in our analysis are depicted in Appendix C 3[46]. In Fig. 3 we show the constraints on the universalquartic coupling λuð≡λ2; λ3; λ4; λ8; λ9; λ10; λ11Þ for the LRmodel with triplet scalars in the low-vR region. The yellowshaded region is disallowed from low-energy data(MWR

> 3.5 TeV) [56–59] and the green shaded regionis excluded from direct searches at the LHC(MWR

> 2.5 TeV) [60–63]. These limits can be extracted

using Eq. (33). In our analysis we also set the MajoranaYukawa, yh at 0.25. We note that, for any particular heavyscalar mass (MH), the universal quartic coupling λu isdisallowed above the corresponding line shown in thefigure. For example, as seen from the plot, the maximumallowed value of the universal quartic coupling is 0.024 ifone considers an LR-breaking scale at 10 TeV and a heavyscalar mass at 1 TeV. The allowed maximum quarticcoupling is lowered for a heavier scalar which can beunderstood from vacuum stability and perturbativity.In Fig. 4 we check the compatibility for the stable

vacuum in the left-right symmetric breaking scale vR andheavy scalar MH allowed region in the LR model with atriplet scalar. Each color represents a particular set of lightHiggs masses (Mh) and top masses (Mt) in their respectiveplots. In Fig. 4(a) the Higgs mass is fixed at 125 GeV andthe top-quark mass varies from 170 GeV to 175 GeVwhereas, in Fig. 4(b) the top-quark mass is fixed at173.2 GeV and the Higgs mass varies from 122 GeV to127 GeV. The upper-left region (shaded with light blue)above the line MH ¼ vR is disallowed since quarticcouplings are nonperturbative in this domain. The blank(white) strip is also ruled out as the values of the couplingsin this region are such that they become nonperturbativebefore reaching the Planck scale. In the lower-right region(shaded with light pink) the quartic coupling related to theheavy Higgs mass becomes extremely small [≤ Oð10−7Þ].We choose the universal quartic coupling λ2, λ3, λ4, λ8, λ9,λ10, λ11 ¼ λu to be fixed at 0.03. This choice of λu allowsonly vR ≥ 100 TeV which can be inferred from Fig. 3. Theinsets in both panels show the higher vR scale where colorpatches terminate, representing the very scale where in factthe Standard Model breaks down for a particular Higgsmass or top-quark mass at one loop.

0.005

0.01

0.015

0.02

0.025

0.03

0.035

1 5 10 50 100

Qua

rtic

Cou

plin

gs (

λ u)

vR (TeV)

Mh = 125 GeVMt = 173.2 GeV

MH1 TeV 2 TeV 3 TeV 4 TeV 5 TeV

LHC Exclusion

Excludedfrom Low Energy Data

λumax(vR=10 TeV) = 0.024

FIG. 3 (color online). Constraints on the universal quarticcoupling λuð≡λ2; λ3; λ4; λ8; λ9; λ10; λ11Þ for the LR model withtriplet scalars in the low-vR region. The yellow shaded region isdisallowed from low-energy data (MWR

> 3.5 TeV) and thegreen shaded region is excluded from direct searches at theLHC (MWR

> 2.5 TeV).

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2. LR model with doublet scalars

Using a similar technique as that used in the previoussection we depict all the multiple-field directions of thepotential and the corresponding stability criteria inAppendix B 3. We find the nontrivial vacuum stabilityconditions which read as

λ1 > 0; 2β1 þ f1 > 0; 2β1 − f1 > 0: (41)

We have also noted the required RGEs for our analysis inAppendix 4 [64]. In Fig. 5 we constrain the universalquartic coupling λuð≡λ2;−λ3Þ for the LR model with

doublet scalars in the low-vR region for different sets ofheavy scalar masses MH. Similar to the previous case, theyellow shaded region in the plot is disallowed from low-energy data (MWR

> 3.5 TeV) and the green shadedregion is excluded from direct searches at theLHC (MWR

> 2.5 TeV).As we noticed in Fig. 5, for any particular heavy scalar

mass (MH), the universal quartic coupling λu is disallowedabove the corresponding line. For example, as seen fromthe plot, the maximum allowed value of the universalquartic coupling is 0.033 if one considers an LR-breakingscale at 10 TeVand a heavy scalar mass at 1 TeV. As before,the allowed maximum quartic coupling is lowered for aheavier scalar.In Fig. 6 we check the compatibility for the stable

vacuum in the vR and heavy scalar MH allowed region inthe LR model with doublet scalars. Each color representsa particular set of light Higgs masses (Mh) and topmasses (Mt) in their respective plots. In Fig. 6(a) theHiggs mass is fixed at 125 GeV and the top-quark massvaries from 170 GeV to 175 GeV whereas, in Fig. 6(b)the top-quark mass is fixed at 173.2 GeV and the Higgsmass varies from 122 GeV to 127 GeV. The upper-leftregion (shaded with light blue) above the line MH ¼ vRis disallowed since the quartic couplings are nonpertur-bative at the low scale itself in this domain. The blank(white) strip is also ruled out as the values of thecouplings in this region are such that they becomenonperturbative before reaching the Planck scale. Inthe lower-right region (shaded with light pink) the quarticcoupling related to a heavy Higgs mass becomesextremely small [≤ Oð10−7Þ]. We choose the universalquartic coupling λ1 ¼ −λ2 ¼ λu to be fixed at 0.04. Here,the choice of λu allows only vR ≥ 100 TeV. The insets in

0.01

0.015

0.02

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0.03

0.035

0.04

0.045

1 5 10 50 100

Qua

rtic

Cou

plin

gs (

λ u)

vR (TeV)

Mh = 125 GeVMt = 173.2 GeV

MH1 TeV

2 TeV 3 TeV 4 TeV 5 TeV

λumax(vR=10 TeV) = 0.033

LHC Exclusion

Excludedfrom Low Energy Data

FIG. 5 (color online). Constraints on the universal quarticcoupling λuð≡λ2;−λ3Þ for the LR model with doublet scalarsin the low-vR region for different sets of heavy scalar massesMH.The yellow shaded region is disallowed from low-energy data(MWR

> 3.5 TeV) and the green shaded region is excluded fromdirect searches at the LHC (MWR

> 2.5 TeV).

MH

(G

eV)

vR (GeV)

Mt (GeV)175174173172171170

103

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1010

105 106 107 108 109 1010

(a)λu = 0.03

M H = v R

Mh = 125 GeV

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MH

(G

eV)

vR (GeV)

Mh (GeV)122123124125126127

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109

1010

105 106 107 108 109 1010

(b)λu = 0.03

M H = v R

Mt = 173.2 GeV

104

106

108

1010

108 109 1010

FIG. 4 (color online). Compatibility for the stable vacuum in the vR and heavy scalarMH allowed region in the LR model with a tripletscalar. Each color (or shade) represents a particular set of light Higgs masses (Mh) and top masses (Mt) in their respective plots. In (a) theHiggs mass is fixed at 125 GeV for different top-quark masses whereas, in (b) the top-quark mass is fixed at 173.2 GeV and the Higgsmass is allowed to vary. The upper-left region (shaded with light blue/gray) above the line MH ¼ vR is disallowed since quarticcouplings are nonperturbative in this domain. In the lower-right region (shaded with light pink/gray) the quartic coupling related to theheavy scalar mass becomes extremely small [≤ Oð10−7Þ]. We fix the universal quartic coupling λu at 0.03. The insets in both panelsshow the higher vR scale where color/shade patches terminate, representing the very scale where in fact the Standard Model breaks downfor a particular Higgs mass or top-quark mass at one loop. As expected, from left to right the top quark mass decreases in (a), whereas theHiggs mass increses in (b).

JOYDEEP CHAKRABORTTY, PARTHA KONAR, AND TANMOY MONDAL PHYSICAL REVIEW D 89, 056014 (2014)

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both panels show the higher vR scale where color patchesterminate, representing the very scale where in fact theStandard Model breaks down for a particular Higgs massor top-quark mass at one loop.

IV. CONCLUSIONS

We have noted that one needs to study the scalarpotential to understand the structure of the vacuum andits compatibility with successful spontaneous symmetrybreaking. In addition, the perturbativity (triviality) ofthe couplings also plays a crucial role. We haveanalyzed the structure of the scalar potentials of B −L extended models—namely, SM ⊗ Uð1ÞB−L and theleft-right symmetry—with different scalar representa-tions. We have computed the criteria for the potentialto be bounded from below, i.e., the conditions forvacuum stability. We also performed the renormalizationgroup evolutions of the parameters (couplings) of thesemodels at the one-loop level with proper matchingconditions. We have shown how the phenomenologi-cally inaccessible couplings can be constrained fordifferent choices of scales of new physics. They inturn also affect the RGEs of the other couplings. Wehave noted that the new physics effects must beswitched on before the SM vacuum faces the instability.This helps the vacuum stability of the full scalarpotential and achieves a consistent spontaneous sym-metry breaking. We have analyzed these aspects byvarying the Higgs and top-quark mass over theirallowed ranges. In summary, it is meaningful to mentionthat more precise knowledge of the SM parameters—like the Higgs mass, the top-quark mass and the strongcoupling—will constrain the parameters (couplings,masses, scales) of new physics and might direct us

towards the correct theory for beyond the StandardModel physics. In principle one can study the left-rightsymmetric models including the radiative correction inthe scalar potential and use the Coleman-Weinbergmechanism, e.g., Ref. [50] has considered this scenarioand calculated the flat directions using the one-loopeffective potential. This will certainly change the corre-lations among the parameters of the scalar potentialleading to the stable vacuum. While submitting ourpaper Ref. [65] appeared, where the vacuum stability forSM ⊗ Uð1ÞB−L was discussed. The viewpoints of ouranalysis are quite different from that work.

ACKNOWLEDGMENTS

The work of J. C. is supported by Department of Science& Technology, Government of INDIA under the GrantAgreement number IFA12-PH-34 (INSPIRE FacultyAward). The authors want to thank Srubabati Goswamiand Namit Mahajan for useful discussions.

APPENDIX A: SCALAR POTENTIAL FORDIFFERENT MODELS

1. Uð1ÞB−L model

VðΦ; SÞ ¼ m2Φ†Φþ μ2∣S∣2 þ λ1ðΦ†ΦÞ2 þ λ2∣S∣4þ λ3Φ†Φ∣S∣2: (A1)

2. LR model with triplet scalars

The most general form of the scalar potential can bewritten as in Ref. [64],

MH

(G

eV)

vR (GeV)

Mt (GeV)175174173172171170

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104

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106

107

108

109

1010

105 106 107 108 109 1010

(a)λu = 0.04

M H = vR

Mh = 125 GeV

104

106

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1010

108 109 1010

MH

(G

eV)

vR (GeV)

Mh (GeV)122123124125126127

103

104

105

106

107

108

109

1010

105 106 107 108 109 1010

(b)λu = 0.04

M H = vR

Mt = 173.2 GeV

104

106

108

1010

108 109 1010

FIG. 6 (color online). Compatibility for stable vacuum in vR and heavy Higgs MH allowed region in LR model with doublet scalars.Each color (or shade) represents a particular set of light Higgs mass (Mh) and top mass (Mt) in respective plot. In figure (a) Higgs mass isfixed at 125 GeVand top quark mass is varying, where as, in figure (b) top quark mass is fixed at 173.2 GeVand Higgs mass is varying.Upper-left region (shaded with light blue/gray) above the lineMH ¼ vR is disallowed since quartic couplings are non-perturbative at thelow scale itself in this domain. Lower-right region (shaded with light pink/gray) quartic coupling related with heavy Higgs mass becomesextremely small (≤ Oð10−7Þ).We choose universal quartic coupling λu fixed at 0.04. Inset to both figures shows the higher vR scalewherecolor/shade patches terminate, representing the very scale where in fact Standard Model breaks down for a particular Higgs mass or topquark mass at one loop. As expected, from left to right the top quark mass decreases in (a), whereas the Higgs mass increases in (b).

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VLRTðΦ;ΔL;ΔRÞ¼ −μ21fTr½Φ†Φ�g − μ22fTr½ ~ΦΦ†� þ Tr½ ~Φ†Φ�g − μ23fTr½Δ†

LΔL� þ Tr½Δ†RΔR�g þ λ1fðTr½Φ†Φ�Þ2g

þ λ2fðTr½ ~ΦΦ†�Þ2 þ ðTr½ ~Φ†Φ�Þ2g þ λ3fTr½ ~ΦΦ†�Tr½ ~Φ†Φ�g þ λ4fTr½Φ†Φ�ðTr½ ~ΦΦ†� þ Tr½ ~Φ†Φ�Þgþ λ5fðTr½ΔLΔ

†L�Þ2þðΔRΔ

†RÞ2gþ λ6fTr½ΔLΔL�Tr½Δ†

LΔ†L�þTr½ΔRΔR�Tr½Δ†

RΔ†R�gþλ7fTr½ΔLΔ

†L�Tr½ΔRΔ

†R�g

þ λ8½ΔLΔ†L�fTr½ΔLΔ

†L�Tr½ΔRΔ

†R�g þ λ9fTr½Φ†Φ�ðTr½ΔLΔ

†L�þTr½ΔRΔ

†R�Þgþðλ10 þ iλ11ÞfTr½Φ ~Φ†�Tr½ΔRΔ

†R�

þ Tr½Φ† ~Φ�Tr½ΔLΔ†L�g þ ðλ10 − iλ11ÞfTr½Φ† ~Φ�Tr½ΔRΔ†

R� þ Tr½ ~Φ†Φ�Tr½ΔLΔ†L�g þ λ12fTr½ΦΦ†ΔLΔ†

L�þ Tr½Φ†ΦΔRΔ

†R�g þ λ13fTr½ΦΔRΦ†Δ†

L� þ Tr½Φ†ΔLΦΔ†R�g þ λ14fTr½ ~ΦΔRΦ†Δ†

L� þ Tr½ ~Φ†ΔLΦΔ†R�g

þ λ15fTr½ΦΔR~Φ†Δ†

L� þ Tr½Φ†ΔL~ΦΔ†

R�g;

where all the coupling constants are real.

3. LR model with doublet scalars

The scalar potential for the LR model with doublet scalars can be written as

VLRDðΦ; HL;HRÞ ¼ 4λ1ðTr½Φ†Φ�Þ2 þ 4λ2ðTr½Φ† ~Φ� þ Tr½Φ ~Φ†�Þ2 þ 4λ3ðTr½Φ† ~Φ� − Tr½Φ ~Φ†�Þ2

þ κ12ðH†

LHL þH†RHRÞ2 þ

κ22ðH†

LHL −H†RHRÞ2 þ β1ðTr½Φ† ~Φ� þ Tr½Φ ~Φ†�ÞðH†

LHL þH†RHRÞ

þ f1ðH†Lð ~Φ ~Φ† − ΦΦ†ÞHL −H†

RðΦ†Φ − ~Φ† ~ΦÞHRÞ:

APPENDIX B: CALCULATION OF NONTRIVIALVACUUM STABILITY CONDITIONS

Herewe have gathered the structure of the scalar potentialin the two-, three-, and four-field directions. We have calcu-lated vacuum stability conditions from these field directionswhile keeping inmind that the conditions should covermostof the parameter space spanned by the quartic couplings.

1. Uð1ÞB−L model

For the Uð1ÞB−L model the potential has a simplestructure and the stability conditions can be calculatedeasily. The quartic potential has the form

λ1jΦj4 þ λ2jSj4 þ λ3jΦj2jSj2;and we can easily write this potential as� ffiffiffiffiffi

λ1p

jΦj2 þ λ32ffiffiffiffiffiλ1

p jSj2�

2

þ�λ2 − λ23

4λ1

�jSj4:

Clearly the above equation is positive definite if

λ1 > 0; λ2 > 0; 4λ1λ2 − λ23 > 0:

These are the nontrivial vacuum stability conditions withλ3 < 0. The trivial boundary conditions are when allthe λ1;2;3 > 0.

2. LR model with triplet scalars

The absence of any tachyonic pseudoscalar modesimposes the condition λ12 > 0.

a. Two-field directions and stability conditions

2FV1ðϕ01;ϕ

þ1 Þ ¼ λ1ðϕ0

12 þ ϕþ

12Þ2;

2FV2ðϕ01; δ

0Þ ¼ λ5δ04 þ λ1ϕ

014;

2FV3ðϕ01; δ

þÞ ¼ ðλ5 þ λ6Þδþ4 þ λ1ϕ014

þ 1

2ðλ12 þ 2λ9Þδþ2ϕ0

12;

2FV4ðϕ01; δ

þþÞ ¼ λ5δþþ4 þ λ1ϕ

014 þ λ12δ

þþ2ϕ012;

2FV5ðϕþ1 ; δ

0Þ ¼ λ5δ04 þ λ1ϕ

þ14 þ λ12δ

02ϕþ12;

2FV6ðϕþ1 ; δ

þÞ ¼ ðλ5 þ λ6Þδþ4 þ λ1ϕþ14

þ 1

2ðλ12 þ 2λ9Þδþ2ϕþ

12;

2FV7ðϕþ1 ; δ

þþÞ ¼ λ5δþþ4 þ λ1ϕ

014;

2FV8ðδ0; δþÞ ¼ λ5ðδ02 þ δþ2Þ2 þ λ6δþ4;

2FV9ðδ0; δþþÞ ¼ λ5ðδ02 þ δþþ2Þ2 þ 4λ6δþ2δ02;

2FV10ðδþ; δþþÞ ¼ λ5ðδþ2 þ δþþ2Þ2 þ λ6δþ4:

Stability conditions

2FV1 → λ1 > 0;2FV2; 2FV4; 2FV5; 2FV7 → λ1 > 0; λ5 > 0;

2FV3; 2FV6 → λ1 > 0; λ5 þ λ6 > 0;2FV8; 2FV9; 2FV10 → λ5 > 0; λ5 þ λ6 > 0;

JOYDEEP CHAKRABORTTY, PARTHA KONAR, AND TANMOY MONDAL PHYSICAL REVIEW D 89, 056014 (2014)

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b. Three-field directions and stability conditions

3FV1ðϕ01;ϕ

þ1 ; δ

0Þ ¼ λ1ðϕ012 þ ϕþ

12Þ2 þ λ5δ

02 þ λ12δ02ϕ0

12;

3FV2ðϕ01;ϕ

þ1 ; δ

þÞ ¼ λ1ðϕ012 þ ϕþ

12Þ2 þ ðλ5 þ λ6Þδþ4 þ 1

2ðλ12 þ 2λ9Þðϕ0

12 þ ϕþ

12Þδþ2;

3FV3ðϕ01;ϕ

þ1 ; δ

þþÞ ¼ λ1ðϕ012 þ ϕþ

12Þ2 þ λ5δ

þþ4 þ λ12ϕ012δþþ2;

3FV4ðϕ01; δ

0; δþÞ ¼ λ1ϕ014 þ λ5ðδ02 þ δþ2Þ2 þ λ6δ

þ4 þ 1

2ðλ12 þ 2λ9Þϕ0

12δþ2;

3FV5ðϕ01; δ

0; δþþÞ ¼ λ5ðδ02 þ δþþ2Þ2 þ λ1ϕ014 þ 4λ6δ0

2δþþ2 þ λ12δþþ2ϕ0

12 þ 2λ9δ

0δþþϕ012;

3FV6ðϕ01; δ

þ; δþþÞ ¼ λ1ϕ014 þ λ5ðδþþ2 þ δþ2Þ2 þ λ6δ

þ4 þ 1

2λ12ϕ

012ð2δþþ2 þ δþ2Þ þ λ9δ

þ2ϕ012;

3FV7ðϕþ1 ; δ

0; δþÞ ¼ λ1ϕþ14 þ λ5ðδ02 þ δþ2Þ2 þ λ6δ

þ4 þ 1

2λ12ϕ

þ12ð2δ02 þ δþ2Þ þ λ9δ

þ2ϕþ12;

3FV8ðϕþ1 ; δ

0; δþþÞ ¼ λ5ðδ02 þ δþþ2Þ2 þ λ1ϕþ14 þ 4λ6δ0

2δþþ2 þ λ12δ02ϕþ

12 þ 2λ9δ

0δþþϕþ12;

3FV9ðϕþ1 ; δ

þ; δþþÞ ¼ λ1ϕþ14 þ λ5ðδþ2 þ δþþ2Þ2 þ λ6δ

þ4 þ 1

2ðλ12 þ 2λ9Þϕþ

12δþ2;

3FV10ðδ0; δþ; δþþÞ ¼ λ5ðδ02 þ δþ2 þ δþþ2Þ2 þ λ6ðδþ2 þ 2δ0δþþÞ2:

Stability conditions

3FV1; 3FV3 → λ1 > 0; λ5 > 0;3FV2 → λ1 > 0; λ5 þ λ6 > 0;

3FV4; 3FV6; 3FV7; 3FV9 → λ1 > 0; λ5 > 0; λ5 þ λ6 > 0;3FV5; 3FV8 → λ1 > 0; λ5 > 0; λ5 þ 2λ6 > 0;

3FV10 → λ5 > 0; λ5 þ λ6 > 0; λ5 þ 2λ6 > 0.

c. Four-field directions and stability conditions

4FV1ðϕ01;ϕ

þ1 ; δ

0; δþÞ ¼ λ5ðδ02 þ δþ2Þ2 þ λ6δþ4 þ λ1ðϕ0

12 þ ϕþ

12Þ2

þ 1

2λ12ð2δ02ϕþ

12 þ 2

ffiffiffi2

pϕ01ϕ

þ1 δ

0δþ þ δþ2ðϕ012 þ ϕþ

12ÞÞ þ λ9δ

þ2ðϕ012 þ ϕþ

12Þ;

4FV2ðϕ01;ϕ

þ1 ; δ

0; δþþÞ ¼ λ5ðδ02 þ δþþ2Þ2 þ 4λ6δ02δþþ2 þ λ1ðϕ0

12 þ ϕþ

12Þ2 þ λ12ðδþþ2ϕ0

12 þ δ02ϕþ

12Þ

þ 2λ9δ0δþþðϕ0

12 þ ϕþ

12Þ;

4FV3ðϕ01;ϕ

þ1 ; δ

þ; δþþÞ ¼ λ5ðδþ2 þ δþþ2Þ2 þ λ6δþ4 þ λ1ðϕ0

12 þ ϕþ

12Þ2 þ 1

2λ12ð2δþþ2ϕ0

12 − 2

ffiffiffi2

pϕ01ϕ

þ1 δ

þδþþ

þ δþ2ðϕ012 þ ϕþ

12ÞÞ þ λ9δ

þ2ðϕ012 þ ϕþ

12Þ;

4FV4ðϕ01; δ

0; δþ; δþþÞ ¼ λ5ðδ02 þ δþ2 þ δþþ2Þ2 þ λ6ðδþ2 þ 2δ0δþþÞ2 þ λ1ϕ014 þ 1

2λ12ϕ

012ð2δ02 þ δþ2Þ

þ λ9ϕ012ðδþ2 þ 2δ0δþþÞ;

4FV5ðϕþ1 ; δ

0; δþ; δþþÞ ¼ λ5ðδ02 þ δþ2 þ δþþ2Þ2 þ λ6ðδþ2 þ 2δ0δþþÞ2 þ λ1ϕþ14

þ 1

2λ12ϕ

þ12ð2δ02 þ δþ2Þ þ λ9ϕ

þ12ðδþ2 þ 2δ0δþþÞ:

CONSTRAINING A CLASS OF B − L EXTENDED MODELS … PHYSICAL REVIEW D 89, 056014 (2014)

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Stability conditions

4FV1 → λ1 > 0; λ5 > 0; λ5 þ 2λ6 > 0;4FV2 → λ1 > 0; λ5 > 0; λ5 þ λ6 > 0;

4FV3 → λ1 > 0; λ5 > 0; λ5 þ λ6 > 0; λ12 − 2ffiffiffiffiffiffiffiffiffiffiffi2λ1λ5

p< 0;

4FV4; 4FV5 → λ1 > 0; λ5 > 0; λ5 þ λ6 > 0; λ5 þ 2λ6 > 0;

3. LR model with doublet scalars

a. Two-field directions and stability conditions

2FV1ðϕ01;ϕ

þ1 Þ ¼ λ1ðϕ0

12 þ ϕþ

12Þ2;

2FV2ðϕþ1 ; h

þR Þ ¼ λ1ϕ

þ14 þ 2β1 þ f1

2hþR

2ϕþ12;

2FV3ðϕ01; h

þR Þ ¼ λ1ϕ

014 þ 2β1 − f1

2hþR

2ϕ012;

2FV4ðϕþ1 ; h

0RÞ ¼ λ1ϕ

þ14 þ 2β1 − f1

2h0R

2ϕþ12;

2FV5ðϕ01; h

0RÞ ¼ λ1ϕ

014 þ 2β1 þ f1

2h0R

2ϕ012:

Stability conditions

2FV1 → λ1 > 0; 2FV2; 2FV5 → λ1 > 0; 2β1 þ f1 > 0; 2FV3; 2FV4 → λ1 > 0; 2β1 − f1 > 0;

b. Three-field directions and stability conditions

3FV1ðϕ01;ϕ

þ1 ; h

0RÞ ¼ λ1ðϕ0

12 þ ϕþ

12Þ2 þ h0R

2

�β1ðϕ0

12 þ ϕþ

12Þ þ 1

2f1ðϕ0

12 − ϕþ

12Þ�;

3FV2ðϕ01;ϕ

þ1 ; h

þR Þ ¼ λ1ðϕ0

12 þ ϕþ

12Þ2 þ hþR

2

�β1ðϕ0

12 þ ϕþ

12Þ þ 1

2f1ðϕþ

12 − ϕ0

12Þ�;

3FV3ðϕ01; h

0R; h

þR Þ ¼

1

2ϕ012ðf1ðh0R2 − hþR

2Þ þ 2β1ðh0R2 þ hþR2Þ þ 2λ1ϕ

012Þ;

3FV3ðϕþ1 ; h

0R; h

þR Þ ¼

1

2ϕþ12ðf1ðhþR 2 − h0R

2Þ þ 2β1ðh0R2 þ hþR2Þ þ 2λ1ϕ

þ12Þ:

Stability conditions

3FV1; 3FV2; 3FV3; 3FV4 → λ1 > 0; 2β1 þ f1 > 0; 2β1 − f1 > 0.

c. Four-field directions and stability conditions

4FV1ðϕ01;ϕ

þ1 ; h

0R; h

þR Þ ¼

1

2ðf1ðhþR ðϕ0

1 − ϕþ1 Þ þ h0Rðϕ0

1 þ ϕþ1 ÞÞðh0Rðϕ0

1 − ϕþ1 Þ − hþR ðϕ0

1 þ ϕþ1 ÞÞ

þ 2ðϕ012 þ ϕþ

12Þððh0R2 þ hþR

2Þβ1 þ λ1ðϕ012 þ ϕþ

12ÞÞÞ:

Stability conditions

4FV1 → λ1 > 0; 2β1 þ f1 > 0; 2β1 − f1 > 0:

JOYDEEP CHAKRABORTTY, PARTHA KONAR, AND TANMOY MONDAL PHYSICAL REVIEW D 89, 056014 (2014)

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APPENDIX C: RENORMALIZATION GROUPEVOLUTION EQUATIONS

1. Standard Model RGEs

For the Standard Model we have used the renormaliza-tion group evolution equations from Ref. [66] with match-ing conditions for the top Yukawa and Higgs quarticcouplings at their pole masses.

2. Uð1ÞB−L model

a. Gauge RG equations

The renormalization group equations for SUð3ÞC andSUð2ÞL gauge couplings g3 and g2 are

16π2ddt

g3 ¼ g33

�−1þ 4

3ng

�¼ g33

16π2½−7�;

16π2ddt

g2 ¼ g32

�− 22

3þ 4

3ng þ

1

6

�¼ g32

16π2

�− 19

6

�;

where ng is the number of generations.The renormalization group equations for the Abelian

gauge couplings g1; gB−L and ~g are

16π2ddt

g1 ¼�41

6g31

�;

16π2ddt

gB−L ¼�12g3B−L þ 32

3gB−L ~gþ

41

6gB−L ~g2

�;

16π2ddt

~g ¼�41

6~gð~g2 þ 2g21Þ þ

32

3gB−Lð~g2 þ g21Þ

þ 12g2B−L ~g�;

b. Fermion RG equations

The RG evolution equation for the top-quark Yukawacoupling Yt is

16π2ddt

Yt ¼ Yt

�9

2Y2t − 8g23 − 9

4g22 − 17

12g21 − 17

12~g2

− 2

3g2B−L − 5

3~ggB−L

�:

In the case of right-handed neutrinos the RGEs we areconsidering degenerate the right-handed neutrino Yukawacoupling and we are in a basis where these couplings arediagonal; then, we have

16π2ddt

yhi ¼ yhi ½4ðyhi Þ2 þ 2Tr½ðyhÞ2� − 6g2B−L�:

c. Scalar RG equations

The RGEs for the scalar couplings λ1; λ2 and λ3 are

16π2ddt

λ1 ¼�24λ21 þ λ23 − 6Y4

t þ9

8g42 þ

3

8g41 þ

3

4g22g

21 þ

3

4g22 ~g

2 þ 3

4g21 ~g

2 þ 3

8~g4 þ 12λ1Y2

t − 9λ1g22 − 3λ1g21 − 3λ1 ~g2�;

8π2ddt

λ2 ¼�10λ22 þ λ23 − 1

2Tr½ðyhÞ4� þ 48g4B−L þ 4λ2Tr½ðyhÞ2� − 24λ2g2B−L

�;

8π2ddt

λ3 ¼ λ3

�6λ1 þ 4λ2 þ 2λ3 þ 3Y2

t − 3

4ð3g22 − g21 − ~g2Þ þ 2Tr½ðyhÞ2� − 12g2B−L

�þ 6~g2g2B−L:

3. LR model with triplet scalars

a. Gauge RG equations

16π2ddt

g3 ¼ g33ð−7Þ; 16π2 ddt

g2 ¼ g32

�− 15

6

�; 16π2

ddt

gB−L ¼ g3B−L�28

9

�:

Note that in our case g2L ¼ g2R ¼ g2.b. Fermion RG equations

16π2ddt

Yt ¼�8Y3

t − Yt

�2

3g21 − 9

2g22 − 8g23

��;

16π2ddt

YMi ¼

�2YM

i

�− 3

4g21 − 9

4g22

�þ 2YM

i Tr½ðYMÞ2� þ 6ðYMi Þ3�:

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056014-15

c. Scalar RG equations

To write down the scalar RG equations, we classified 15 scalar couplings into three categories depending on how theycoupled with scalar fields.

(i) Coefficients with Φ4

16π2ddt

λ1 ¼ 32λ21 þ5

3λ212 þ

1

2λ213 þ 2λ214 þ 64λ22 þ 16λ1λ3 þ 16λ23 þ 48λ24 þ 6λ12λ9 þ 6λ29 þ 12λ1Y2

t − 6Y4t

− 18λ1g22 þ 3g42;

16π2ddt

λ2 ¼ 6ðλ210 − λ112Þ þ 3

2λ14λ15 þ 24λ1λ2 þ 48λ2λ3 þ 12λ24 þ 12λ2Y2

t − 18λ2g22;

16π2ddt

λ3 ¼ 12ðλ210 þ λ211Þ − ðλ212 − λ213Þ − 1

2ðλ214 þ λ215Þ þ 128λ22 þ 24λ1λ3 þ 16λ23 þ 24λ24 þ 12λ3Y2

t þ 3Y4t

− 18λ3g22 þ3

2g22;

16π2ddt

λ4 ¼ 48λ4ðλ1 þ 2λ2 þ λ3Þ þ 6λ10ð2λ9 þ λ12Þ þ3

2λ13ðλ14 þ λ15Þ þ 12λ4Y2

t − 18λ4g22:

(ii) Coefficients with Δ4

16π2ddt

λ5 ¼ 28λ25 þ 16λ6ðλ5 þ λ6Þ þ 16ðλ210 þ λ211Þ þ 2λ212 þ 3λ27 þ 4λ9ðλ9 þ λ12Þ þ 2λ5Y2t − 16Y4

t − 12λ5g2B−L

þ 6g4B−L þ 12g2B−Lg22 − 24λ5g22 þ 9g42;

16π2ddt

λ6 ¼ 12λ6ðλ6 þ 2λ5 − g2B−L − 2g22Þ þ 12λ28 − λ212 þ 8Y4t þ 8λ6Y2

t − 12g2B−Lg22 þ 3g42;

16π2ddt

λ7 ¼ 4λ27 þ 16λ7ð2λ5 þ λ6Þ þ 32ðλ210 − λ211Þ þ 2ðλ212 þ λ213Þ þ 4ðλ214 þ λ215Þ þ 32λ28 þ 8λ12λ9 þ λ29

þ 8λ7Y2t − 12λ7ðg2B−L þ g22Þ þ 12g4B−L;

16π2ddt

λ8 ¼ λ213 þ 4λ14λ15 þ 8λ8ðλ5 þ 5λ6 þ λ7 þ Y2t Þ − 12λ8ð2g2B−L þ g22Þ:

(iii) Coefficients with Φ2Δ2

16π2ddtλ9¼ λ9ð20λ1þ8λ3þ16λ5þ8λ6þ6λ7þ4λ9þ6Y2

t þ4Tr½ðYMÞ2�−6g2B−L−21g22Þþ6g42

þ16ðλ210þλ211Þþλ12ð8λ1þλ12Þþ3λ213þ12λ214þ8λ12λ3þ48λ10λ4þλ12ð6λ5þ8λ6þ3λ7Þ;

16π2ddtλ10¼ λ10ð4λ1þ4λ12þ48λ2þ16λ3þ16λ4þ16λ5þ8λ6þ6λ7þ8λ9

þ6Y2t þ4Tr½ðYMÞ2�−6g2B−L−21g22Þ−3λ13ðλ14þλ15Þþ12λ4λ9;

16π2ddtλ11¼ λ11ð4λ1þ4λ12−48λ2þ16λ3þ16λ5þ8λ6−6λ7þ8λ9þ6Y2

t þ4Tr½ðYMÞ2�−6g2B−L−21g22Þ;

16π2ddtλ12¼ λ12ð4λ1þ4λ12−8λ3þ4λ5−8λ6þ8λ9þ6Y2

t 4Tr½ðYMÞ2�−6g2B−L−21g22Þ−12ðλ214− λ215Þ;

16π2ddtλ13¼ λ13ð4λ1þ4λ12þ8λ3þ2λ7þ8λ8þ8λ9þ3Y2

t þTr½ðYMÞ2�−6g2B−L−21g22Þþð8λ4þ16λ10Þðλ14þλ15Þ;

16π2ddtλ14¼ λ14ð4λ1−4λ12þ2λ7þ8λ9þ6Y2

t þ4Tr½ðYMÞ2�−6g2B−L−21g22Þþ4λ13ðλ4þ2λ10Þþ8λ15ð2λ2þλ8Þ;

16π2ddtλ15¼ λ15ð4λ1þ12λ12þ2λ7þ8λ9þ6Y2

t þ4Tr½ðYMÞ2�−6g2B−L−21g22Þþ4λ13ðλ4þ4λ10Þþ8λ14ð2λ2þλ8Þ:

JOYDEEP CHAKRABORTTY, PARTHA KONAR, AND TANMOY MONDAL PHYSICAL REVIEW D 89, 056014 (2014)

056014-16

4. LR model with doublet scalars

a. Gauge RG equations

16π2ddt

g3 ¼ g33ð−7Þ; 16π2ddt

g2 ¼ g32

�− 17

6

�; 16π2

ddt

g2B−L ¼ g3B−Lð3Þ:

Note that in our case g2L ¼ g2R ¼ g2.

b. Fermion RG equations

64π2ddt

Yt ¼�− 2

9g2B−L − 9g22 − 32g23

�Yt þ 7Y3

t :

c. Scalar RG equations

(i) Coefficients with Φ4

128π2ddt

λ1 ¼ λ1ð−72g22 þ 256ðλ1 þ λ2 − λ3Þ þ 24Y2t Þ þ 1024ðλ21 þ λ22Þ þ 32β21 þ 8f21 þ 9g42 − 12Y − Y4

t ;

512π2ddt

λ2 ¼ λ2ð−288g22 þ 768λ1 þ 3072λ2 þ 1024λ3 þ 96Y2t Þ − 8f21 þ 3g42 − 3Y4

t ;

256π2ddt

λ3 ¼ λ3ð−144g22 − 384λ1 − 512λ2 − 1536λ3 þ 48Y2t Þ þ 4f21 − 3g42 − 3Y4

t :

(ii) Coefficients with H4L=R

512π2ddt

κ1 ¼ κ1ð−96g2B−L − 144g22 þ 576κ1 þ 384κ2Þ þ 192κ22 þ 256β21 þ 128f21 þ 24g4B−l þ 12g2B−Lg22 þ 9g42;

512π2ddt

κ2 ¼ κ2ð−96g2B−L − 144g22 þ 512κ1 þ 384κ2Þ þ 128f21 þ 12g2B−Lg22 þ 9g42:

(iii) Coefficeients with Φ2H2L=R

256π2ddt

β1 ¼ −4β1½−8β1 þ 6g2B−L þ 27g22 − 2ð20κ1 þ 4κ2 þ 40λ1 þ 32λ2 − 32λ3 þ 3Y2t Þ� þ 24f21 þ 9g42;

256π2ddt

f1 ¼ f1ð16β1 − 6g2B−L − 27g22 þ 8ðκ1 þ κ2Þ þ 16ðλ1 − 4λ2Þ þ 64λ3 þ 6Y2t Þ:

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