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Nonperturbative scales in AdSCFT

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IOP PUBLISHING JOURNAL OF PHYSICS A MATHEMATICAL AND THEORETICAL

J Phys A Math Theor 42 (2009) 254005 (44pp) doi1010881751-81134225254005


B Basso1 and G P Korchemsky23

1 Laboratoire de Physique Theorique (Unite Mixte de Recherche du CNRS UMR 8627)Universite de Paris XI 91405 Orsay Cedex France2 Institut de Physique Theorique (Unite de Recherche Associee au CNRS URA 2306) CEASaclay 91191 Gif-sur-Yvette Cedex France

Received 2 February 2009Published 9 June 2009Online at stacksioporgJPhysA42254005

AbstractThe cusp anomalous dimension is a ubiquitous quantity in four-dimensionalgauge theories ranging from QCD to maximally supersymmetricN = 4 YangndashMills theory and it is one of the most thoroughly investigated observables inthe AdSCFT correspondence In planar N = 4 SYM theory its perturbativeexpansion at weak coupling has a finite radius of convergence while at strongcoupling it admits an expansion in inverse powers of the rsquot Hooft couplingwhich is given by a non-Borel summable asymptotic series We study the cuspanomalous dimension in the transition regime from strong to weak coupling andargue that the transition is driven by nonperturbative exponentially suppressedcorrections To compute these corrections we revisit the calculation of the cuspanomalous dimension in planar N = 4 SYM theory and extend the previousanalysis by taking into account nonperturbative effects We demonstrate thatthe scale parameterizing nonperturbative corrections coincides with the massgap of the two-dimensional bosonic O(6) sigma model embedded into theAdS5 timesS5 string theory This result is in agreement with the prediction comingfrom the string theory consideration

PACS numbers 1115Me 1125Tq 1130Pb

Contents

1 Introduction 22 Cusp anomalous dimension in N = 4 SYM 6

21 Integral equation and mass scale 622 Analyticity conditions 723 Toy model 824 Exact bounds and unicity of the solution 925 RiemannndashHilbert problem 1026 General solution 12

3 On leave from Laboratoire de Physique Theorique Universite de Paris XI 91405 Orsay Cedex France

1751-811309254005+44$3000 copy 2009 IOP Publishing Ltd Printed in the UK 1

J Phys A Math Theor 42 (2009) 254005 B Basso and G P Korchemsky

27 Quantization conditions 1328 Cusp anomalous dimension in the toy model 14

3 Solving the quantization conditions 1931 Quantization conditions 1932 Numerical solution 2033 Strong coupling solution 21

4 Mass scale 2441 General expression 2442 Strong coupling expansion 2543 Nonperturbative corrections to the cusp anomalous dimension 2744 Relation between the cusp anomalous dimension and mass gap 29

5 Conclusions 31Acknowledgments 32Appendix A Weak coupling expansion 33Appendix B Constructing general solution 34Appendix C Wronskian-like relations 36Appendix D Relation to the Whittaker functions 38

D1 Integral representations 38D2 Asymptotic expansions 39

Appendix E Expression for the mass gap 41References 42

1 Introduction

The AdSCFT correspondence provides a powerful framework for studying maximallysupersymmetric N = 4 YangndashMills theory (SYM) at strong coupling [1] At present one ofthe best-studied examples of the conjectured gaugestring duality is the relationship betweenthe anomalous dimensions of Wilson operators in planar N = 4 theory in the so-called SL(2)

sector and energy spectrum of folded strings spinning on AdS5 times S5 [2 3] The Wilsonoperators in this sector are given by single trace operators built from L copies of the samecomplex scalar field and N light-cone components of the covariant derivatives These quantumnumbers define correspondingly the twist and the Lorentz spin of the Wilson operators inN = 4 SYM theory (for a review see [4]) In a dual string theory description [2 3] they areidentified as angular momenta of the string spinning on S5 and AdS5 parts of the background

In general anomalous dimensions in planar N = 4 theory in the SL(2) sector arenontrivial functions of the rsquot Hooft coupling g2 = g2

YMNc(4π)2 and quantum numbers ofWilson operatorsmdashtwist L and Lorentz spin N Significant simplification occurs in the limit[5] when the Lorentz spin grows exponentially with the twist L sim ln N with N rarr infin Inthis limit the anomalous dimensions scale logarithmically with N for arbitrary coupling andthe minimal anomalous dimension has the following scaling behavior [5ndash9]

γNL(g) = [2cusp(g) + ε(g j)] ln N + middot middot middot (11)

where j = L ln N is an appropriate scaling variable and the ellipses denote terms suppressedby powers of 1L Here the coefficient in front of ln N is split into the sum of two functionsin such a way that ε(g j) carries the dependence on the twist and it vanishes for j = 0 Thefirst term inside the square brackets in (11) has a universal twist independent form [10 11]It involves the function of the coupling constant known as the cusp anomalous dimensionThis anomalous dimension was introduced in [10] to describe specific (cusp) ultraviolet (UV)divergences of Wilson loops [12 13] with a light-like cusp on the integration contour [14]

2


The cusp anomalous dimension plays a distinguished role in N = 4 theory and in general infour-dimensional YangndashMills theories since aside from logarithmic scaling of the anomalousdimension (11) it also controls infrared divergences of scattering amplitudes [15] Sudakovasymptotics of elastic form factors [16] gluon Regge trajectories [17] etc

According to (11) the asymptotic behavior of the minimal anomalous dimension isdetermined by two independent functions cusp(g) and ε(g j) At weak coupling thesefunctions are given by series in powers of g2 and the first few terms of the expansion can becomputed in perturbation theory At strong coupling the AdSCFT correspondence allowsus to obtain the expansion of cusp(g) and ε(g j) in powers of 1g from the semiclassicalexpansion of the energy of the folded spinning string Being combined together the weak andstrong coupling expansions define asymptotic behavior of these functions at the boundaries of(semi-infinite) interval 0 g lt infin The following questions arise what are the correspondinginterpolating functions for arbitrary g How does the transition from the weak to strongcoupling regimes occur These are the questions that we address in this paper

At weak coupling the functions cusp(g) and ε(g j) can be found in a generic(supersymmetric) YangndashMills theory in the planar limit by making use of the remarkableproperty of integrability The Bethe ansatz approach to computing these functions at weakcoupling was developed in [5 11 18] It was extended in [7 19] to all loops in N = 4 SYMtheory leading to integral BESFRS equations for cusp(g) and ε(g j) valid in the planarlimit for arbitrary values of the scaling parameter j and the coupling constant g For the cuspanomalous dimension the solution to the BES equation at weak coupling is in agreement withthe most advanced explicit four-loop perturbative calculation [20] and it yields a perturbativeseries for cusp(g) which has a finite radius of convergence [19] The BES equation was alsoanalyzed at strong coupling [21ndash24] but constructing its solution for cusp(g) turned out to bea nontrivial task

The problem was solved in [25 26] where the cusp anomalous dimension was found inthe form of an asymptotic series in 1g It turned out that the coefficients of this expansion havethe same sign and grow factorially at higher orders As a result the asymptotic 1g expansionof cusp(g) is given by a non-Borel summable series which suffers from ambiguities that areexponentially small for g rarr infin This suggests that the cusp anomalous dimension receivesnonperturbative corrections at strong coupling [25]

cusp(g) =infinsum

k=minus1

ckgk minus σ

4radic

2m2

cusp + o(m2

cusp

) (12)

Here the dependence of the nonperturbative scale m2cusp on the coupling constant mcusp sim

g14 eminusπg follows through a standard analysis [27 28] from the large order behavior of theexpansion coefficients ck sim

(k + 1

2

)for k rarr infin The value of the coefficient σ in (12)

depends on the regularization of Borel singularities in the perturbative 1g expansion and thenumerical prefactor was introduced for later convenience

Note that the expression for the nonperturbative scale m2cusp looks similar to that for the

mass gap in an asymptotically free field theory with the coupling constant sim 1g An importantdifference is however that m2

cusp is a dimensionless function of the rsquot Hooft coupling Thisis perfectly consistent with the fact that the N = 4 model is a conformal field theory andtherefore it does not involve any dimensionfull scale Nevertheless as we will show in thispaper the nonperturbative scale m2

cusp is indeed related to the mass gap in the two-dimensionalbosonic O(6) sigma model

Relation (12) sheds light on the properties of cusp(g) in the transition region g sim 1Going from g 1 to g = 1 we find that m2

cusp increases and as a consequence

3


nonperturbative O(m2

cusp

)corrections to cusp(g) become comparable with perturbative

O(1g) corrections We will argue in this paper that the nonperturbative corrections playa crucial role in the transition from the strong to weak coupling regime To describe thetransition we present a simplified model for the cusp anomalous dimension This modelcorrectly captures the properties of cusp(g) at strong coupling and most importantly it allowsus to obtain a closed expression for the cusp anomalous dimension which turns out to beremarkably close to the exact value of cusp(g) throughout the entire range of the couplingconstant

In the AdSCFT correspondence relation (12) should follow from the semiclassicalexpansion of the energy of the quantized folded spinning string [2 3] On the right-handside of (12) the coefficient cminus1 corresponds to the classical energy and ck describes (k + 1)thloop correction Indeed the explicit two-loop stringy calculation [29] yields the expressionsfor cminus1 c0 and c1 which are in a perfect agreement with (12)4 However the semiclassicalapproach does not allow us to calculate nonperturbative corrections to cusp(g) and verificationof (12) remains a challenge for the string theory

Recently Alday and Maldacena [6] put forward an interesting proposal that the scalingfunction ε(g j) entering (11) can be found exactly at strong coupling in terms of a nonlinearO(6) bosonic sigma model embedded into the AdS5 times S5 model More precisely using thedual description of Wilson operators as folded strings spinning on AdS5 times S5 and taking intoaccount the one-loop stringy corrections to these states [8] they conjectured that the scalingfunction ε(g j) should be related at strong coupling to the energy density εO(6) in the groundstate of the O(6) model corresponding to the particle density ρO(6) = j2

εO(6) = ε(g j) + j

2 mO(6) = kg14 eminusπg [1 + O(1g)] (13)

This relation should hold at strong coupling and jmO(6) = fixed Here the scale mO(6) isidentified as the dynamically generated mass gap in the O(6) model with k = 234π14

(54

)being the normalization factor

The O(6) sigma model is an exactly solvable theory [31ndash34] and the dependence of εO(6)

on the mass scale mO(6) and the density of particles ρO(6) can be found exactly with the helpof thermodynamical Bethe ansatz equations Together with (13) this allows us to determinethe scaling function ε(g j) at strong coupling In particular for jmO(6) 1 the asymptoticbehavior of ε(g j) follows from the known expression for the energy density of the O(6)model in the (nonperturbative) regime of small density of particles [6 34ndash36]

ε(j g) + j = m2

[j

m+

π2

24

(j

m

)3

+ O(j 4m4)

] (14)

with m equiv mO(6) For jmO(6) 1 the scaling function ε(g j) admits a perturbative expansionin inverse powers of g with the coefficients enhanced by powers of ln (with = j(4g) 1[6 8]

ε(g j) + j = 22

[g +

1

π

(3

4minus ln

)+

1

4π2g

(q02

2minus 3 ln + 4(ln )2

)+O(1g2)

]+ O(4)

This expansion was derived both in string theory [37] and in gauge theory [30 38 39]yielding however different results for the constant q02 The reason for the disagreementremains unclear The first two terms of this expansion were also computed in string theory[37] and they were found to be in a disagreement with gauge theory calculation [30 38 39]

4 The same result was obtained using a different approach from the quantum string Bethe ansatz in [9 30]

4


Remarkably enough relation (13) was established in planar N = 4 SYM theory at strongcoupling [35] using the conjectured integrability of the dilatation operator [7] The mass scalemO(6) was computed both numerically [36] and analytically [35 38] and it was found to be ina perfect agreement with (13) This result is extremely nontrivial given the fact that the scalemO(6) has a different origin in gauge and in string theory sides of AdSCFT In string theoryit is generated by the dimensional transmutation mechanism in two-dimensional effectivetheory describing dynamics of massless modes in the AdS5 times S5 sigma model In gaugetheory the same scale parameterizes nonperturbative corrections to anomalous dimensions infour-dimensional YangndashMills theory at strong coupling It is interesting to note that a similarphenomenon when two different quantities computed in four-dimensional gauge theory and indual two-dimensional sigma model coincide has already been observed in the BPS spectrumin N = 2 supersymmetric YangndashMills theory [40 41] We would like the mention that theprecise matching of the leading coefficients in perturbative expansion of spinning string energyand anomalous dimensions on the gauge side was previously found in [42 43 44] Relation(13) implies that for the anomalous dimensions (11) the gaugestring correspondence holdsat the level of nonperturbative corrections

As we just explained the functions cusp(g) and ε(g j) entering (11) receivenonperturbative contributions at strong coupling described by the scales mcusp and mO(6)respectively In N = 4 SYM theory these functions satisfy two different integral equations[7 19] and there is no a priori reason why the scales mcusp and mO(6) should be related to eachother Nevertheless examining their leading order expressions equations (12) and (13) wenote that they have the same dependence on the coupling constant One may wonder whethersubleading O(1g) corrections are also related to each other In this paper we show that thetwo scales coincide at strong coupling to any order of 1g expansion

mcusp = mO(6) (15)

thus proving that nonperturbative corrections to the cusp anomalous dimension (12) and tothe scaling function (14) are parameterized by the same scale

Relations (12) and (15) also have an interpretation in string theory The cusp anomalousdimension has the meaning of the energy density of a folded string spinning on AdS3 [2 6]As such it receives quantum corrections from both massive and massless excitations of thisstring in the AdS5 times S5 sigma model The O(6) model emerges in this context as the effectivetheory describing the dynamics of massless modes In distinction with the scaling functionε(g j) for which the massive modes decouple in the limit jmO(6) = fixed and g rarr infinthe cusp anomalous dimension is not described entirely by the O(6) model Neverthelessit is expected that the leading nonperturbative corrections to cusp(g) should originate fromnontrivial infrared dynamics of the massless excitations and therefore they should be related tononperturbative corrections to the vacuum energy density in the O(6) model As a consequencecusp(g) should receive exponentially suppressed corrections proportional to the square of theO(6) mass gap sim m2

O(6) We show in this paper by explicit calculation that this is indeed thecase

The paper is organized as follows In section 2 we revisit the calculation of the cuspanomalous dimension in planar N = 4 SYM theory and construct the exact solution forcusp(g) In section 3 we analyze the expressions obtained at strong coupling and identifynonperturbative corrections to cusp(g) In section 4 we compute subleading corrections tothe nonperturbative scales mcusp and mO(6) and show that they are the same for the two scalesThen we extend our analysis to higher orders in 1g and demonstrate that the two scalescoincide Section 5 contains concluding remarks Some technical details of our calculationsare presented in appendices

5


2 Cusp anomalous dimension in N = 4 SYM

The cusp anomalous dimension can be found in planar N = 4 SYM theory for arbitrarycoupling as a solution to the BES equation [19] At strong coupling cusp(g) was constructedin [25 26] in the form of perturbative expansion in 1g The coefficients of this series growfactorially at higher orders thus indicating that cusp(g) receives nonperturbative correctionswhich are exponentially small at strong coupling (equation (12)) To identity such correctionswe revisit in this section the calculation of the cusp anomalous dimension and construct theexact solution to the BES equation for arbitrary coupling

21 Integral equation and mass scale

In the Bethe ansatz approach the cusp anomalous dimension is determined by the behavioraround the origin of the auxiliary function γ (t) related to the density of Bethe roots

cusp(g) = minus8ig2 limtrarr0

γ (t)t (21)

The function γ (t) depends on the rsquot Hooft coupling and has the form

γ (t) = γ+(t) + iγminus(t) (22)

where γplusmn(t) are real functions of t with a definite parity γplusmn(plusmnt) = plusmnγplusmn(t) For arbitrarycoupling the functions γplusmn(t) satisfy the (infinite-dimensional) system of integral equationsint infin

0

dt

tJ2nminus1(t)

[γminus(t)

1 minus eminust(2g)+

γ+(t)

et(2g) minus 1

]= 1

2δn1

(23)int infin

0

dt

tJ2n(t)

[γ+(t)

1 minus eminust(2g)minus γminus(t)

et(2g) minus 1

]= 0

with n 1 and Jn(t) being the Bessel functions These relations are equivalent to the BESequation [19] provided that γplusmn(t) verify certain analyticity conditions specified in section 22

As was shown in [25 35] equations (23) can be significantly simplified with the help ofthe transformation γ (t) rarr (t)5

(t) =(

1 + i cotht

4g

)γ (t) equiv +(t) + iminus(t) (24)

We find from (21) and (24) the following representation for the cusp anomalous dimension

cusp(g) = minus2g(0) (25)

It follows from (22) and (23) that plusmn(t) are real functions with a definite parity plusmn(minust) =plusmnplusmn(t) satisfying the system of integral equationsint infin

0dt cos(ut)[minus(t) minus +(t)] = 2

(26)int infin

0dt sin(ut)[minus(t) + +(t)] = 0

with u being an arbitrary real parameter such that minus1 u 1 Since plusmn(t) take real valueswe can rewrite these relations in a compact formint infin

0dt[eiutminus(t) minus eminusiut+(t)] = 2 (27)

5 With a slight abuse of notations we use here the same notation as for the Euler gamma function

6


To recover (23) we apply (24) replace in (26) the trigonometric functions by their Besselseries expansions

cos(ut) = 2sumn1

(2n minus 1)cos((2n minus 1)ϕ)

cos ϕ

J2nminus1(t)

t

(28)sin(ut) = 2

sumn1

(2n)sin(2nϕ)

cos ϕ

J2n(t)

t

with u = sin ϕ and finally compare coefficients in front of cos((2n minus 1)ϕ)cos ϕ andsin(2nϕ)cos ϕ on both sides of (26) It is important to stress that doing this calculationwe interchanged the sum over n with the integral over t This is only justified for ϕ real andtherefore relation (26) only holds for minus1 u 1

Comparing (27) and (23) we observe that the transformation γplusmn rarr plusmn eliminates thedependence of the integral kernel on the left-hand side of (27) on the coupling constantOne may then wonder where does the dependence of the functions plusmn(t) on the couplingconstant come from We will show in the following subsection that it is dictated by additionalconditions imposed on analytical properties of solutions to (27)

Relations (25) and (26) were used in [25] to derive an asymptotic (perturbative) expansionof cusp(g) in powers of 1g This series however suffers from Borel singularities and weexpect that the cusp anomalous dimension should receive nonperturbative corrections sim eminus2πg

exponentially small at strong coupling As was already mentioned in section 1 similarcorrections are also present in the scaling function ε(g j) which controls the asymptoticbehavior of the anomalous dimensions (11) in the limit when the Lorentz spin of Wilsonoperators grows exponentially with their twist According to (13) for jmO(6) = fixed andg rarr infin the scaling function coincides with the energy density of the O(6) model embeddedinto AdS5 times S5 The mass gap of this model defines a new nonperturbative scale mO(6) inAdSCFT Its dependence on the coupling g follows univocally from the FRS equation and ithas the following form [35 38]

mO(6) = 8radic

2

π2eminusπg minus 8g

πeminusπgRe

[int infin

0

dt ei(tminusπ4)

t + iπg(+(t) + iminus(t))

] (29)

where plusmn(t) are solutions to (27) To compute the mass gap (29) we have to solve theintegral equation (27) and then substitute the resulting expression for plusmn(t) into (29) Notethat the same functions also determine the cusp anomalous dimension (25)

Later in the paper we will construct a solution to the integral equation (27) and thenapply (25) to compute nonperturbative corrections to cusp(g) at strong coupling

22 Analyticity conditions

The integral equations (27) and (23) determine plusmn(t) and γplusmn(t) or equivalently the functions(t) and γ (t) up to a contribution of zero modes The latter satisfy the same integral equations(27) and (23) but without an inhomogeneous term on the right-hand side

To fix the zero modes we have to impose additional conditions on solutions to (27)and (23) These conditions follow unambiguously from the BES equation [23 25] and theycan be formulated as a requirement that γplusmn(t) should be entire functions of t which admit arepresentation in the form of Neumann series over the Bessel functions

γminus(t) = 2sumn1

(2n minus 1)J2nminus1(t)γ2nminus1 γ+(t) = 2sumn1

(2n)J2n(t)γ2n (210)

7


with the expansion coefficients γ2nminus1 and γ2n depending on the coupling constant This impliesin particular that the series on the right-hand side of (210) are convergent on the real axisUsing orthogonality conditions for the Bessel functions we obtain from (210)

γ2nminus1 =int infin

0

dt

tJ2nminus1(t)γminus(t) γ2n =

int infin

0

dt

tJ2n(t)γ+(t) (211)

Here we assumed that the sum over n on the right-hand side of (210) can be interchangedwith the integral over t We will show below that relations (210) and (211) determine aunique solution to the system (23)

The coefficient γ1 plays a special role in our analysis since it determines the cuspanomalous dimension (21)

cusp(g) = 8g2γ1(g) (212)

Here we applied (22) and (210) and took into account small-t behavior of the Bessel functionsJn(t) sim tn as t rarr 0

Let us now translate (210) and (211) into properties of the functions plusmn(t) orequivalently (t) It is convenient to rewrite relation (24) as

(it) = γ (it)sin

(t

4g+ π

4

)sin

(t

4g

)sin

(π4

) = γ (it)radic

2infinprod

k=minusinfin

t minus 4πg(k minus 1

4

)t minus 4πgk

(213)

Since γ (it) is an entire function in the complex t-plane we conclude from (213) that (it)has an infinite number of zeros (itzeros) = 0 and poles (it) sim 1(t minus tpoles) on a realt-axis located at

tzeros = 4πg( minus 1

4

) tpoles = 4πgprime (214)

where prime isin Z and prime = 0 so that (it) is regular at the origin (see equation (21)) Note that(it) has an additional (infinite) set of zeros coming from the function γ (it) but in distinctionwith (214) their position is not fixed Later in the paper we will construct a solution to theintegral equation (26) which satisfies relations (214)

23 Toy model

To understand the relationship between analytical properties of (it) and properties of thecusp anomalous dimension it is instructive to slightly simplify the problem and consider alsquotoyrsquo model in which the function (it) is replaced with (toy)(it)

We require that (toy)(it) satisfies the same integral equation (26) and define following(25) the cusp anomalous dimension in the toy model as

(toy)cusp (g) = minus2g(toy)(0) (215)

The only difference compared to (it) is that (toy)(it) has different analytical propertiesdictated by the relation

(toy)(it) = γ (toy)(it)t + πg

t (216)

while γ (toy)(it) has the same analytical properties as the function γ (it)6 This relation canbe considered as a simplified version of (213) Indeed it can be obtained from (213) if we

6 Note that the function γ (toy)(t) does no longer satisfy the integral equation (23) Substitution of (216) into (27)yields an integral equation for γ (toy)(t) which can be obtained from (23) by replacing 1(1 minus eminust(2g)) rarr πg

2t+ 1

2and 1(et(2g) minus 1) rarr πg

2tminus 1

2 in the kernel on the left-hand side of (23)

8


retained in the product only one term with k = 0 As compared with (214) the function(toy)(it) does not have poles and it vanishes for t = minusπg

The main advantage of the toy model is that as we will show in section 28 the expressionfor

(toy)cusp (g) can be found in a closed form for an arbitrary value of the coupling constant (see

equation (250)) We will then compare it with the exact expression for cusp(g) and identifythe difference between the two functions

24 Exact bounds and unicity of the solution

Before we turn to finding a solution to (26) let us demonstrate that this integral equationsupplemented with the additional conditions (210) and (211) on its solutions leads tonontrivial constraints for the cusp anomalous dimension valid for arbitrary coupling g

Let us multiply both sides of the two relations in (23) by 2(2n minus 1)γ2nminus1 and 2(2n)γ2nrespectively and perform summation over n 1 Then we convert the sums into functionsγplusmn(t) using (210) and add the second relation to the first one to obtain7

γ1 =int infin

0

dt

t

(γ+(t))2 + (γminus(t))2

1 minus eminust(2g) (217)

Since γplusmn(t) are real functions of t and the denominator is positively definite for 0 t lt infinthis relation leads to the following inequality

γ1 int infin

0

dt

t(γminus(t))2 2γ 2

1 0 (218)

Here we replaced the function γminus(t) by its Bessel series (210) and made use of theorthogonality condition for the Bessel functions with odd indices We deduce from (218) that

0 γ1 12 (219)

and then apply (212) to translate this inequality into the following relation for the cuspanomalous dimension

0 cusp(g) 4g2 (220)

We would like to stress that this relation should hold in planar N = 4 SYM theory for arbitrarycoupling g

Note that the lower bound on the cusp anomalous dimension cusp(g) 0 holds inany gauge theory [11] It is the upper bound cusp(g) 4g2 that is a distinguished featureof N = 4 theory Let us verify the validity of (220) At weak coupling cusp(g) admitsperturbative expansion in powers of g2 [20]

cusp(g) = 4g2

[1 minus 1

3π2g2 +

11

45π4g4 minus 2

(73

630π6 + 4ζ 2

3

)g6 + middot middot middot

] (221)

while at strong coupling it has the form [25 26 29]

cusp(g) = 2g

[1 minus 3 ln 2

4πgminus1 minus K

16π2gminus2 minus

(3K ln 2

64π3+

27ζ3

2048π3

)gminus3 + O(gminus4)

] (222)

with K being the Catalan constant It is easy to see that relations (221) and (222) are in anagreement with (220)

For arbitrary g we can verify relation (220) by using the results for the cusp anomalousdimension obtained from a numerical solution of the BES equation [25 45] The comparisonis shown in figure 1 We observe that the upper bound condition cusp(g)(2g) 2g is indeedsatisfied for arbitrary g gt 0

7 Our analysis here goes along the same lines as in appendix A of [35]

9


0

02

04

06

08

1

05 1 15 2 25 3

Figure 1 Dependence of the cusp anomalous dimension cusp(g)(2g) on the coupling constantThe dashed line denotes the upper bound 2g

We are ready to show that the analyticity conditions formulated in section 22 specify aunique solution to (23) As was already mentioned solutions to (23) are defined modulocontribution of zero modes γ (t) rarr γ (t)+γ (0)(t) with γ (0)(t) being a solution to homogenousequations Going through the same steps that led us to (217) we obtain

0 =int infin

0

dt

t

(γ(0)+ (t))2 + (γ

(0)minus (t))2


where the zero on the left-hand side is due to absence of the inhomogeneous term Sincethe integrand is a positively definite function we immediately deduce that γ (0)(t) = 0 andtherefore the solution for γ (t) is unique

25 RiemannndashHilbert problem

Let us now construct the exact solution to the integral equations (27) and (23) To this endit is convenient to Fourier transform functions (22) and (24)

(k) =int infin

minusinfin

dt

2πeikt(t) γ (k) =

int infin

minusinfin

dt

2πeiktγ (t) (224)

According to (22) and (210) the function γ (t) is given by the Neumann series over theBessel functions Then we perform the Fourier transform on both sides of (210) and use thewell-known fact that the Fourier transform of the Bessel function Jn(t) vanishes for k2 gt 1 todeduce that the same is true for γ (t) leading to

γ (k) = 0 for k2 gt 1 (225)

10


This implies that the Fourier integral for γ (t) only involves modes with minus1 k 1 andtherefore the function γ (t) behaves at large (complex) t as

γ (t) sim e|t | for |t | rarr infin (226)

Let us now examine the function (k) We find from (224) and (213) that (k) admits thefollowing representation

(k) =int infin

minusinfin

dt

2πeikt

sinh(

t4g

+ iπ4

)sinh

(t

4g

)sin

(π4

)γ (t) (227)

Here the integrand has poles along the imaginary axis at t = 4π ign (with n = plusmn1plusmn2 )8It is suggestive to evaluate the integral (227) by deforming the integration contour to

infinity and by picking up residues at the poles However taking into account relation (226)we find that the contribution to (227) at infinity can be neglected for k2 gt 1 only In this caseclosing the integration contour into the upper (or lower) half-plane for k gt 1 (or k lt minus1) wefind

(k)k2gt1= θ(k minus 1)

sumn1

c+(n g) eminus4πng(kminus1) + θ(minusk minus 1)sumn1

cminus(n g) eminus4πng(minuskminus1) (228)

Here the notation was introduced for k-independent expansion coefficients

cplusmn(n g) = ∓4gγ (plusmn4π ign) eminus4πng (229)

where the factor eminus4πng is inserted to compensate the exponential growth of γ (plusmn4π ign) sime4πng at large n (see equation (226)) For k2 1 we are not allowed to neglect the contributionto (227) at infinity and relation (228) does no longer hold As we will see in a moment fork2 1 the function (k) can be found from (27)

Comparing relations (225) and (228) we conclude that in distinction with γ (k) thefunction (k) does not vanish for k2 gt 1 Moreover each term on the right-hand sideof (228) is exponentially small at strong coupling and the function scales at large k as(k) sim eminus4πg(|k|minus1) This implies that nonzero values of (k) for k2 gt 1 are of nonperturbativeorigin Indeed in a perturbative approach of [25] the function (t) is given by the Besselfunction series analogous to (210) and similar to (225) the function (k) vanishes for k2 gt 1to any order in 1g expansion

We note that the sum on the right-hand side of (228) runs over poles of the function (it)specified in (214) We recall that in the toy model (216) (toy)(it) and γ (toy)(it) are entirefunctions of t At large t they have the same asymptotic behavior as the Bessel functions(toy)(it) sim γ (toy)(it) sim eplusmnit Performing their Fourier transformation (224) we find

γ (toy)(k) = (toy)(k) = 0 for k2 gt 1 (230)

in a close analogy with (225) Comparison with (228) shows that the coefficients (229)vanish in the toy model for arbitrary n and g

c(toy)+ (n g) = c

(toy)minus (n g) = 0 (231)

Relation (228) defines the function (k) for k2 gt 1 but it involves the coefficientscplusmn(n g) that need to be determined In addition we have to construct the same function fork2 1 To achieve both goals let us return to the integral equations (26) and replace plusmn(t)

by Fourier integrals (see equations (224) and (24))

+(t) =int infin

minusinfindk cos(kt)(k)

(232)

minus(t) = minusint infin

minusinfindk sin(kt)(k)

8 We recall that γ (t) = O(t) and therefore the integrand is regular at t = 0

11


In this way we obtain from (26) the following remarkably simple integral equation for (k)

minusint infin

minusinfin

dk(k)

k minus u+ π(u) = minus2 (minus1 u 1) (233)

where the integral is defined using the principal value prescription This relation is equivalentto the functional equation obtained in [24] (see equation (55) there) Let us split the integralin (233) into k2 1 and k2 gt 1 and rewrite (233) in the form of a singular integral equationfor the function (k) on the interval minus1 k 1

(u) +1

πminusint 1

minus1

dk(k)

k minus u= φ(u) (minus1 u 1) (234)

where the inhomogeneous term is given by

φ(u) = minus 1

π

(2 +

int minus1

minusinfin

dk(k)

k minus u+

int infin

1

dk(k)

k minus u

) (235)

Since integration in (235) goes over k2 gt 1 the function (k) can be replaced on theright-hand side of (235) by its expression (228) in terms of the coefficients cplusmn(n g)

The integral equation (234) can be solved by standard methods [46] A general solutionfor (k) reads as (for minus1 k 1)

(k) = 1

2φ(k) minus 1

2π

(1 + k

1 minus k

)14

minusint 1

minus1

duφ(u)

u minus k

(1 minus u

1 + u

)14

minusradic

2

π

(1 + k

1 minus k

)14c

1 + k (236)

where the last term describes the zero mode contribution with c being an arbitrary function ofthe coupling We replace φ(u) by its expression (235) interchange the order of integrationand find after some algebra

(k)k21= minus

radic2

π

(1 + k

1 minus k

)14[

1 +c

1 + k+

1

2

int infin

minusinfin

dp(p)

p minus k

(p minus 1

p + 1

)14

θ(p2 minus 1)

] (237)

Note that the integral on the right-hand side of (237) goes along the real axis except theinterval [minus1 1] and therefore (p) can be replaced by its expression (228)

Being combined together relations (228) and (237) define the function (k) forminusinfin lt k lt infin in terms of a (infinite) set of yet unknown coefficients cplusmn(n g) and c(g)To fix these coefficients we will first perform the Fourier transform of (k) to obtain thefunction (t) and then require that (t) should have correct analytical properties (214)

26 General solution

We are now ready to write down a general expression for the function (t) According to(224) it is related to (k) through the inverse Fourier transformation

(t) =int 1

minus1dk eminusikt (k) +

int minus1

minusinfindk eminusikt (k) +

int infin

1dk eminusikt (k) (238)

where we split the integral into three terms since (k) has a different form for k lt minus1minus1 k 1 and k gt 1 Then we use the expressions obtained for (k) equations (228) and (237)to find after some algebra the following remarkable relation (see appendix B for details)

(it) = f0(t)V0(t) + f1(t)V1(t) (239)

12


Here the notation was introduced for

f0(t) = minus1 +sumn1

t

[c+(n g)

U+1 (4πng)

4πng minus t+ cminus(n g)

Uminus1 (4πng)

4πng + t

]

(240)

f1(t) = minusc(g) +sumn1

4πng

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

4πng + t

]

Also Vn and Uplusmnn (with n = 0 1) stand for integrals

Vn(x) =radic

2

π

int 1

minus1du(1 + u)14minusn(1 minus u)minus14 eux

(241)

Uplusmnn (x) = 1

2

int infin

1du(u plusmn 1)minus14(u ∓ 1)14minusn eminus(uminus1)x

which can be expressed in terms of the Whittaker functions of first and second kinds [47] (seeappendix D) We would like to emphasize that solution (239) is exact for arbitrary couplingg gt 0 and that the only undetermined ingredients in (239) are the expansion coefficientscplusmn(n g) and c(g)

In the special case of the toy model equation (231) the expansion coefficients vanishc(toy)plusmn (n g) = 0 and relation (240) takes a simple form

f(toy)

0 (t) = minus1 f(toy)

1 (t) = minusc(toy)(g) (242)

Substituting these expressions into (239) we obtain a general solution to the integral equation(27) in the toy model

(toy)(it) = minusV0(t) minus c(toy)(g)V1(t) (243)

It involves an arbitrary g-dependent constant c(toy) which will be determined in section 28

27 Quantization conditions

Relation (239) defines a general solution to the integral equation (27) It still depends on thecoefficients cplusmn(n g) and c(g) that need to be determined We recall that (it) should havepoles and zeros specified in (214)

Let us first examine poles on the right-hand side of (239) It follows from (241) thatV0(t) and V1(t) are entire functions of t and therefore poles can only come from the functionsf0(t) and f1(t) Indeed the sums entering (240) produce an infinite sequence of poles locatedat t = plusmn4πn (with n 1) and as a result solution (239) has a correct pole structure (214)Let us now require that (it) should vanish for t = tzero specified in (214) This leads to aninfinite set of relations

(4π ig

( minus 1

4

)) = 0 isin Z (244)

Replacing (it) by its expression (239) we rewrite these relations in an equivalent form

f0(t)V0(t) + f1(t)V1(t) = 0 t = 4πg( minus 1

4

) (245)

Relations (244) and (245) provide the quantization conditions for the coefficients c(g) andcplusmn(n g) respectively that we will analyze in section 3

Let us substitute (239) into expression (25) for the cusp anomalous dimension Theresult involves the functions Vn(t) and fn(t) (with n = 1 2) evaluated at t = 0 It is easyto see from (241) that V0(0) = 1 and V1(0) = 2 In addition we obtain from (240) thatf0(0) = minus1 for arbitrary coupling leading to

cusp(g) = 2g[1 minus 2f1(0)] (246)

13


Replacing f1(0) by its expression (240) we find the following relation for the cusp anomalousdimension in terms of the coefficients c and cplusmn

cusp(g) = 2g

⎧⎨⎩1 + 2c(g) minus 2sumn1

[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

]⎫⎬⎭ (247)

We would like to stress that relations (246) and (247) are exact and hold for arbitrary couplingg This implies that at weak coupling it should reproduce the known expansion of cusp(g)

in positive integer powers of g2 [20] Similarly at strong coupling it should reproduce theknown 1g expansion [25 26] and most importantly describe nonperturbative exponentiallysuppressed corrections to cusp(g)

28 Cusp anomalous dimension in the toy model

As before the situation simplifies for the toy model (243) In this case we have only onequantization condition (toy)(minusπ ig) = 0 which follows from (216) Together with (243) itallows us to fix the coefficient c(toy)(g) as

c(toy)(g) = minusV0(minusπg)

V1(minusπg) (248)

Then we substitute relations (248) and (231) into (247) and obtain

(toy)cusp (g) = 2g[1 + 2c(toy)(g)] = 2g

[1 minus 2

V0(minusπg)

V1(minusπg)

] (249)

Replacing V0(minusπg) and V1(minusπg) by their expressions in terms of the Whittaker function ofthe first kind (see equation (D2)) we find the following remarkable relation

(toy)cusp (g) = 2g

[1 minus (2πg)minus12 M1412(2πg)

Mminus140(2πg)

] (250)

which defines the cusp anomalous dimension in the toy model for arbitrary coupling g gt 0Using (250) it is straightforward to compute

(toy)cusp (g) for arbitrary positive g By

construction (toy)cusp (g) should be different from cusp(g) Nevertheless evaluating (250) for

0 g 3 we found that the numerical values of (toy)cusp (g) are very close to the exact values

of the cusp anomalous dimension shown by the solid line in figure 1 Also as we will showin a moment the two functions have similar properties at strong coupling To compare thesefunctions it is instructive to examine the asymptotic behavior of

(toy)cusp (g) at weak and strong

couplings

281 Weak coupling At weak coupling we find from (250)

(toy)cusp (g) = 3

2πg2 minus 1

2π2g3 minus 1

64π3g4 +

5

64π4g5 minus 11

512π5g6 minus 3

512π6g7 + O(g8) (251)

Comparison with (221) shows that this expansion is quite different from the weak couplingexpansion of the cusp anomalous dimension In distinction with cusp(g) the expansion in(251) runs in both even and odd powers of the coupling In addition the coefficient in frontof gn on the right-hand side of (251) has transcendentality (nminus 1) while for cusp(g) it equals(n minus 2) (with n taking even values only)

Despite this and similar to the weak coupling expansion of the cusp anomalous dimension[19] the series (251) has a finite radius of convergence |g0| = 0796 It is determined by theposition of the zero of the Whittaker function closest to the origin Mminus140(2πg0) = 0 for

14


g0 = minus0297 plusmn i0739 Moreover numerical analysis indicates that (toy)cusp (g) has an infinite

number of poles in the complex g-plane The poles are located on the left-half side of thecomplex plane Reg lt 0 symmetrically with respect to the real axis and they approachprogressively the imaginary axis as one goes away from the origin

282 Strong coupling At strong coupling we can replace the Whittaker functions in (250)by their asymptotic expansion for g 1 It is convenient however to apply (249) andreplace the functions V0(minusπg) and V1(minusπg) by their expressions given in (D14) and (D16)respectively In particular we have (see equation (D14))

V0(minusπg) = e1(2α) α54

(

34

)[F

(1

4

5

4|α + i0

)+ 2F

(minus1

4

3

4| minus α

)] α = 1(2πg)

(252)

where the parameter 2 is defined as

2 = σαminus12 eminus1α

(34

)

(54

) σ = eminus3iπ4 (253)

Here F(a b| minus α) is expressed in terms of the confluent hypergeometric function of thesecond kind (see equations (D12) and (D7) in appendix D and equation (256)) [47]

F

(1

4

5

4| minus α

)= αminus54U+

0 (1(2α))

(5

4

)

(254)

F

(minus1

4

3

4| minus α

)= αminus34Uminus

0 (1(2α))

(3

4

)

The function F(a b| minus α) defined in this way is an analytical function of α with a cut alongthe negative semi-axis

For positive α = 1(2πg) the function F(minus 1

4 34 | minus α

)entering (252) is defined away

from the cut and its large g expansion is given by the Borel-summable asymptotic series (fora = minus 1

4 and b = 34 )

F(a b| minus α) =sumk0

(minusα)k

k

(a + k)(b + k)

(a)(b)= 1 minus αab + O(α2) (255)

with the expansion coefficients growing factorially to higher orders in α This series canbe immediately resummed by means of the Borel resummation method Namely replacing(a + k) by its integral representation and performing the sum over k we find for Reα gt 0

F(a b| minus α) = αminusa

(a)

int infin

0dssaminus1(1 + s)minusb eminussα (256)

in agreement with (254) and (241)Relation (255) holds in fact for arbitrary complex α and the functions F(a b|α plusmn i0)

defined for α gt 0 above and below the cut respectively are given by the same asymptoticexpansion (255) with α replaced by minusα The important difference is that now the series(255) is no longer Borel summable Indeed if one attempted to resum this series using theBorel summation method one would immediately find a branch point singularity along theintegration contour at s = 1

F(a b|α plusmn i0) = αminusa

(a)

int infin

0dssaminus1(1 minus s ∓ i0)minusb eminussα (257)

15


The ambiguity related to the choice of the prescription to integrate over the singularity is knownas Borel ambiguity In particular deforming the s-integration contour above or below the cutone obtains two different functions F(a b|α plusmn i0) They define analytical continuation of thesame function F(a b| minus α) from Re α gt 0 to the upper and lower edges of the cut runningalong the negative semi-axis Its discontinuity across the cut F(a b|α + i0)minusF(a b| α minus i0)is exponentially suppressed at small α gt 0 and is proportional to the nonperturbative scale 2

(see equation (D17)) This property is perfectly consistent with the fact that function (252)is an entire function of α Indeed it takes the same form if one used α minus i0 prescription in thefirst term in the right-hand side of (252) and replaced σ in (253) by its complex conjugatedvalue

We can now elucidate the reason for decomposing the entire V0-function in (252) intothe sum of two F-functions In spite of the fact that the analytical properties of the formerfunction are simpler compared to the latter functions its asymptotic behavior at large g ismore complicated Indeed the F-functions admit asymptotic expansions in the whole complexg-plane and they can be unambiguously defined through the Borel resummation once theiranalytical properties are specified (we recall that the function F(a b|α) has a cut along thepositive semi-axis) In distinction with this the entire function V0(minusπg) admits differentasymptotic behavior for positive and negative values of g in virtue of the Stokes phenomenonNot only does it restrict the domain of validity of each asymptotic expansion but it also forcesus to keep track of both perturbative and nonperturbative contributions in the transition regionfrom positive to negative g including the transition from the strong to weak coupling

We are now in position to discuss the strong coupling expansion of the cusp anomalousdimension in the toy model including into our consideration both perturbative andnonperturbative contributions Substituting (252) and a similar relation for V1(minusπg) (seeequation (D16)) into (249) we find (for α+ equiv α + i0 and α = 1(2πg))

(toy)cusp (g)(2g) = 1 minus α

F(

14 5

4 |α+)

+ 2F(minus 1

4 34 | minus α

)F

(14 1

4 |α+)

+ 142αF

(34 3

4 | minus α) (258)

Since the parameter 2 is exponentially suppressed at strong coupling equation (253) and atthe same time the F-functions are all of the same order it makes sense to expand the right-handside of (258) in powers of 2 and then study separately each coefficient function In thisway we identify the leading 2 independent term as perturbative contribution to

(toy)cusp (g)

and the O(2) term as the leading nonperturbative correction More precisely expanding theright-hand side of (258) in powers of 2 we obtain

(toy)cusp (g)(2g) = C0(α) minus α2C2(α) + 1

4α24C4(α) + O(6) (259)

Here the expansion runs in even powers of and the coefficient functions Ck(α) are givenby algebraic combinations of F-functions

C0 = 1 minus αF

(14 5

4 |α+)

F(

14 1

4 |α+) C2 = 1[

F(

14 1

4 |α+)]2 C4 = F

(34 3

4 | minus α)[

F(

14 1

4 |α+)]3 (260)

where we applied (D9) and (D12) to simplify the last two relations Since the coefficientfunctions are expressed in terms of the functions F(a b|α+) and F(a b| minus α) having the cutalong the positive and negative semi-axes respectively Ck(α) are analytical functions of α inthe upper-half plane

Let us now examine the strong coupling expansion of the coefficient functions (260)Replacing F-functions in (260) by their asymptotic series representation (255) we get

16


C0 = 1 minus α minus 1

4α2 minus 3

8α3 minus 61

64α4 minus 433

128α5 + O(α6)

C2 = 1 minus 1

8α minus 11

128α2 minus 151

1024α3 minus 13085

32768α4 + O(α5) (261)

C4 = 1 minus 3

4α minus 27

32α2 minus 317

128α3 + O(α4)

Not surprisingly these expressions inherit the properties of the F-functionsmdashthe series(261) are asymptotic and non-Borel summable If one simply substituted relations(261) into the right-hand side of (259) one would then worry about the meaning ofnonperturbative O(2) corrections to (259) given the fact that the strong coupling expansionof perturbative contribution C0(α) suffers from Borel ambiguity We recall that the appearanceof exponentially suppressed corrections to

(toy)cusp (g) is ultimately related to the Stokes

phenomenon for the function V0(minusπg) (equation (252)) As was already mentioned this doesnot happen for the F-function and as a consequence its asymptotic expansion supplementedwith the additional analyticity conditions allows us to reconstruct the F-function through theBorel transformation (equations (256) and (257)) Since the coefficient functions (260) areexpressed in terms of the F-functions we may expect that the same should be true for theC-functions Indeed it follows from the unicity condition of asymptotic expansion [27] thatthe functions C0(α) C2(α) C4(α) are uniquely determined by their series representations(261) as soon as the latter are understood as asymptotic expansions for the functions analyticalin the upper-half plane Im α 0 This implies that the exact expressions for functions(260) can be unambiguously constructed by means of the Borel resummation but the explicitconstruction remains beyond the scope of the present study

Since expression (258) is exact for arbitrary coupling g we may now address the questionformulated in section 1 how does the transition from the strong to the weak coupling regimeoccur We recall that in the toy model

(toy)cusp (g)(2g) is given for g 1 and g 1

by relations (251) and (259) respectively Let us choose some sufficiently small valueof the coupling constant say g = 14 and compute

(toy)cusp (g)(2g) using three different

representations First we substitute g = 025 into (258) and find the exact value as 04424(3)Then we use the weak coupling expansion (251) and obtain a close value 04420(2) Finallywe use the strong coupling expansion (259) and evaluate the first few terms on the right-handside of (259) for g = 025 to get

Equation(259) = (02902 minus 01434i) + (01517 + 01345i)

+ (00008 + 00086i) minus (00002 minus 00003i) + middot middot middot = 04425 + middot middot middot (262)

Here the four expressions inside the round brackets correspond to contributions proportionalto 024 and 6 respectively with 2(g = 025) = 03522 times eminus3iπ4 being thenonperturbative scale (253)

We observe that each term in (262) takes complex values and their sum is remarkablyclose to the exact value In addition the leading O(2) nonperturbative correction (the secondterm) is comparable with the perturbative correction (the first term) Moreover the formerterm starts to dominate over the latter one as we go to smaller values of the coupling constantThus the transition from the strong to weak coupling regime is driven by nonperturbativecorrections parameterized by the scale 2 Moreover the numerical analysis indicates thatthe expansion of

(toy)cusp (g) in powers of 2 is convergent for Reg gt 0

283 From toy model to the exact solution Relation (259) is remarkably similar tothe expected strong coupling expansion of the cusp anomalous dimension (12) with the

17


function C0(α) providing perturbative contribution and 2 defining the leading nonperturbativecontribution Let us compare C0(α) with the known perturbative expansion (222) of cusp(g)In terms of the coupling α = 1(2πg) the first few terms of this expansion look as

cusp(g)(2g) = 1 minus 3 ln 2

2α minus K

4α2 minus

(3K ln 2

8+

27ζ3

256

)α3 + (263)

where the ellipses denote both higher order corrections in α and nonperturbative corrections in2 Comparing (263) and the first term C0(α) on the right-hand side of (259) we observethat both expressions approach the same value 1 as α rarr 0

As was already mentioned the expansion coefficients of the two series have differenttranscendentalitymdashthey are rational for the toy model (equation (261)) and have maximaltranscendentality for the cusp anomalous dimension (equation (263)) Note that the two serieswould coincide if one formally replaced the transcendental numbers in (263) by appropriaterational constants In particular replacing

3 ln 2

2rarr 1

K

2rarr 1

2

9ζ3

32rarr 1

3 (264)

one obtains from (263) the first few terms of perturbative expansion (261) of the function C0

in the toy model This rule can be generalized to all loops as follows Introducing an auxiliaryparameter τ we define the generating function for the transcendental numbers in (264) andrewrite (264) as

exp

[3 ln 2

2τ minus K

2τ 2 +

9ζ3

32τ 3 +

]rarr exp

[τ minus τ 2

2+

τ 3

3+

] (265)

Going to higher loops we have to add higher order terms in τ to both exponents On theright-hand side these terms are resummed into exp(ln(1 + τ)) = 1 + τ while on the left-handside they produce the ratio of Euler gamma functions leading to

(

14

)

(1 + τ

4

)

(34 minus τ

4

)

(34

)

(1 minus τ

4

)

(14 + τ

4

) rarr (1 + τ) (266)

Taking logarithms on both sides of this relation and subsequently expanding them in powersof τ we obtain the substitution rules which generalize (264) to the complete family oftranscendental numbers entering into the strong coupling expansion (263) At this pointrelation (266) can be thought of as an empirical rule which allows us to map the strongcoupling expansion of the cusp anomalous dimension (263) into that in the toy model (equation(261)) We will clarify its origin in section 42

In spite of the fact that the numbers entering both sides of (264) have differenttranscendentality we may compare their numerical values Taking into account that3 ln 22 = 10397(2) K2 = 04579(8) and 9ζ332 = 03380(7) we observe that relation(264) defines a meaningful approximation to the transcendental numbers Moreoverexamining the coefficients in front of τn on both sides of (265) at large n we find thatthe accuracy of approximation increases as n rarr infin This is in agreement with the observationmade in the beginning of section 28 the cusp anomalous dimension in the toy model (toy)

cusp (g)

is close numerically to the exact expression cusp(g) In addition the same property suggests

that the coefficients in the strong coupling expansion of (toy)cusp (g) and cusp(g) should have the

same large order behavior It was found in [25] that the expansion coefficients on the right-hand side of (263) grow at higher orders as cusp(g) sim sum

k (k + 1

2

)αk It is straightforward

to verify using (255) and (260) that the expansion coefficients of C0(α) in the toy model havethe same behavior This suggests that nonperturbative corrections to cusp(g) and

(toy)cusp (g)

18


are parameterized by the same scale 2 defined in (253) Indeed we will show this in thefollowing section by explicit calculation

We demonstrated in this section that nonperturbative corrections in the toy model followunambiguously from the exact solution (250) In the following section we will extendanalysis to the cusp anomalous dimension and work out the strong coupling expansion ofcusp(g)(2g) analogous to (259)

3 Solving the quantization conditions

Let us now solve the quantization conditions (245) for the cusp anomalous dimensionRelation (245) involves two sets of functions The functions V0(t) and V1(t) are given by theWhittaker function of the first kind (see equation (D2)) At the same time the functions f0(t)

and f1(t) are defined in (240) and they depend on the (infinite) set of expansion coefficientsc(g) and cplusmn(n g) Having determined these coefficients from the quantization conditions(245) we can then compute the cusp anomalous dimension for arbitrary coupling with thehelp of (247)

We expect that at strong coupling the resulting expression for cusp(g) will have form(12) Examining (247) we observe that the dependence on the coupling resides bothin the expansion coefficients and in the functions Uplusmn

0 (4πg) The latter are given by theWhittaker functions of the second kind (see equation (D7)) and as such they are given byBorel-summable sign-alternating asymptotic series in 1g Therefore nonperturbativecorrections to the cusp anomalous dimension (247) could only come from the coefficientscplusmn(n g) and c(g)


Let us replace f0(t) and f1(t) in (245) by their explicit expressions (240) and rewrite thequantization conditions (245) as

V0(4πgx) + c(g)V1(4πgx) =sumn1

[c+(n g)A+(n x) + cminus(n g)Aminus(n x)] (31)

where x = minus 14 (with = 0plusmn1plusmn2 ) and the notation was introduced for

Aplusmn(n x) = nV1(4πgx)Uplusmn0 (4πng) + xV0(4πgx)U

plusmn1 (4πng)

n ∓ x

(32)

Relation (31) provides an infinite system of linear equations for cplusmn(g n) and c(g) Thecoefficients in this system depend on V01(4πgx) and Uplusmn

01(4πng) which are known functionsdefined in appendix D We would like to stress that relation (31) holds for arbitrary g gt 0

Let us show that the quantization conditions (31) lead to c(g) = 0 for arbitrary couplingTo this end we examine (31) for |x| 1 In this limit for g = fixed we are allowed toreplace the functions V0(4πgx) and V1(4πgx) on both sides of (31) by their asymptoticbehavior at infinity Making use of (D10) and (D12) we find for |x| 1

r(x) equiv V1(4πgx)

V0(4πgx)=

minus16πgx + middot middot middot (x lt 0)

12 + middot middot middot (x gt 0)

(33)

where the ellipses denote terms suppressed by powers of 1(gx) and eminus8πg|x| We divideboth sides of (31) by V1(4πgx) and observe that for x rarr minusinfin the first term on the left-hand side of (31) is subleading and can be safely neglected In a similar manner one has

19


Table 1 Comparison of the numerical value of cusp(g)(2g) found from (31) and (247) fornmax = 40 with the exact one [25 45] for different values of the coupling constant g

g 01 02 04 06 08 10 12 14 16 18

Numer 01976 03616 05843 07096 07825 08276 08576 08787 08944 09065Exact 01939 03584 05821 07080 07813 08267 08568 08781 08938 09059

Table 2 Dependence of cusp(g)(2g) on the truncation parameter nmax for g = 1 and c(g) = 0The last column describes the exact result

nmax 10 20 30 40 50 60 70 infinNumer 08305 08286 08279 08276 08274 08273 08272 08267

Aplusmn(n x)V1(4πgx) = O(1x) for fixed n on the right-hand side of (31) Therefore goingto the limit x rarr minusinfin on both sides of (31) we get

c(g) = 0 (34)

for arbitrary g We verify in appendix A by explicit calculation that this relation indeed holdsat weak coupling

Arriving at (34) we tacitly assumed that the sum over n in (31) remains finite in the limitx rarr minusinfin Taking into account large n behavior of the functions Uplusmn

0 (4πng) and Uplusmn1 (4πng)

(see equation (D12)) we obtain that this condition translates into the following condition forasymptotic behavior of the coefficients at large n

c+(n g) = o(n14) cminus(n g) = o(nminus14) (35)

These relations also ensure that the sum in expression (247) for the cusp anomalous dimensionis convergent

32 Numerical solution

To begin with let us solve the infinite system of linear equations (31) numerically In orderto verify (34) we decided to do it in two steps we first solve (31) for cplusmn(n g) assumingc(g) = 0 and then repeat the same analysis by relaxing condition (34) and treating c(g) asunknown

For c(g) = 0 we truncate the infinite sums on the right-hand side of (31) at some largenmax and then use (31) for = 1 minus nmax nmax to find numerical values of cplusmn(n g) with1 n nmax for given coupling g Substituting the resulting expressions for cplusmn(n g) into(247) we compute the cusp anomalous dimension Taking the limit nmax rarr infin we expect torecover the exact result Results of our analysis are summarized in two tables Table 1 showsthe dependence of the cusp anomalous dimension on the coupling constant Table 2 showsthe dependence of the cusp anomalous dimension on the truncation parameter nmax for fixedcoupling

For c(g) arbitrary we use (31) for = minusnmax nmax to find numerical values ofc(g) and cplusmn(n g) with 1 n nmax for given coupling g In this manner we computecusp(g)(2g) and c(g) and then compare them with the exact expressions correspondingto nmax rarr infin For the cusp anomalous dimension our results for cusp(g)(2g) are inremarkable agreement with the exact expression Namely for nmax = 40 their differenceequals 5480times10minus6 for g = 1 and it decreases down to 8028times10minus7 for g = 18 The reason

20


Table 3 Dependence of c(g) on the truncation parameter nmax for g = 1 derived from thequantization condition (31)

nmax 10 20 30 40 50 60 70 infinminusc(g) 00421 00357 00323 00301 00285 00272 00262 0

why the agreement is better compared to the c(g) = 0 case (see table 1) is that c(g) takeseffectively into account a reminder of the sum on the right-hand side of (31) correspondingto n gt nmax The dependence of the expression obtained for c(g) on the truncation parameternmax is shown in table 3 We observe that in agreement with (34) c(g) vanishes as nmax rarr infin

Our numerical analysis shows that the cusp anomalous dimension (247) can be determinedfrom the quantization conditions (31) and (34) for arbitrary coupling g In distinction withthe toy model (250) the resulting expression for cusp(g) does not admit a closed formrepresentation Still as we will show in the following subsection the quantization conditions(31) can be solved analytically for g 1 leading to asymptotic expansion for the cuspanomalous dimension at strong coupling

33 Strong coupling solution

Let us divide both sides of (31) by V0(4πgx) and use (34) to get (for x = minus 14 and isin Z)

1 =sumn1

c+(n g)

[nU+

0 (4πng)r(x) + U+1 (4πng)x

n minus x

]

+sumn1

cminus(n g)

[nUminus

0 (4πng)r(x) + Uminus1 (4πng)x

n + x

] (36)

where the function r(x) was defined in (33)Let us now examine the large g asymptotics of the coefficient functions accompanying

cplusmn(n g) on the right-hand side of (36) The functions Uplusmn0 (4πng) and Uplusmn

1 (4πng) admitasymptotic expansion in 1g given by (D12) For the function r(x) the situation isdifferent As follows from its definition equations (33) and (D10) large g expansionof r(x) runs in two parameters perturbative 1g and nonperturbative exponentially smallparameter 2 sim g12 eminus2πg which we already encountered in the toy model (equation (253))Moreover we deduce from (33) and (D10) that the leading nonperturbative correction tor(x) scales as

δr(x) = O(|8minus2|)(x = minus 1

4 isin Z) (37)

so that the power of grows with We observe that O(2) corrections are only present inr(x) for = 0 Therefore as far as the leading O(2) correction to the solutions to (36) areconcerned we are allowed to neglect nonperturbative (2-dependent) corrections to r(x) onthe right-hand side of (36) for = 0 and retain them for = 0 only

Since the coefficient functions in the linear equations (36) admit a double series expansionin powers of 1g and 2 we expect that the same should be true for their solutions cplusmn(n g)Let us determine the first few terms of this expansion using the following ansatz

cplusmn(n g) = (8πgn)plusmn14

[aplusmn(n) +

bplusmn(n)

4πg+ middot middot middot

]+ 2

[αplusmn(n) +

βplusmn(n)


]+ O(4)

(38)

21


where 2 is a nonperturbative parameter defined in (253)

2 = σ(2πg)12 eminus2πg

(34

)

(54

) (39)

and the ellipses denote terms suppressed by powers of 1g Here the functionsaplusmn(n) bplusmn(n) are assumed to be g-independent We recall that the functions cplusmn(n g)

have to verify relation (35) This implies that the functions aplusmn(n) bplusmn(n) should vanishas n rarr infin To determine them we substitute (38) into (36) and compare the coefficients infront of powers of 1g and 2 on both sides of (36)

331 Perturbative corrections Let us start with the lsquoperturbativersquo 2-independent part of(38) and compute the functions aplusmn(n) and bplusmn(n)

To determine aplusmn(n) we substitute (38) into (36) replace the functions Uplusmn01(4πgn) and

r(x) by their large g asymptotic expansion equations (D12) and (33) respectively neglectcorrections in 2 and compare the leading O(g0) terms on both sides of (36) In this waywe obtain from (36) the following relations for aplusmn(n) (with x = minus 1

4 )

2x

(5

4

) sumn1

a+(n)

n minus x

= 1 ( 1)

(310)

minus2x

(3

4

) sumn1

aminus(n)

n + x

= 1 ( 0)

One can verify that the solutions to this system satisfying aplusmn(n) rarr 0 for n rarr infin have theform

a+(n) = 2(n + 1

4

)(n + 1)2

(14

)

(311)

aminus(n) = (n + 3

4

)2(n + 1)2

(34

)

In a similar manner we compare the subleading O(1g) terms on both sides of (36) and findthat the functions bplusmn(n) satisfy the following relations (with x = minus 1

4 )

2x

(5

4

) sumn1

b+(n)

n minus x

= minus 3

32x

minus 3π

64minus 15

32ln 2 ( 1)

(312)

minus 2x

(3

4

) sumn1

bminus(n)

n + x

= minus 5

32x

minus 5π

64+

9

32ln 2 ( 0)

where on the right-hand side we made use of (311) The solutions to these relations are

b+(n) = minusa+(n)

(3 ln 2

4+

3

32n

) bminus(n) = aminus(n)

(3 ln 2

4+

5

32n

) (313)

It is straightforward to extend the analysis to subleading perturbative corrections to cplusmn(n g)Let us substitute (38) into expression (247) for the cusp anomalous dimension Taking

into account the identities (D12) we find the lsquoperturbativersquo contribution to cusp(g) as

cusp(g) = 2g minussumn1

(2πn)minus1

[

(5

4

) (a+(n) +

b+(n)


)(1 minus 5

128πgn+

)

+

(3

4

) (aminus(n) +

bminus(n)


)(1 +

3

128πgn+ middot middot middot

)]+ O(2) (314)

22


Replacing aplusmn(n) and bplusmn(n) by their expressions (311) and (313) we find after some algebra

cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K

16π2g2+ O(1g3)

]+ O(2) (315)

where K is the Catalan number This relation is in agreement with the known result obtainedboth in N = 4 SYM theory [25 26] and in string theory [29]

332 Nonperturbative corrections Let us now compute the leading O(2) nonperturbativecorrection to the coefficients cplusmn(n g) According to (38) it is described by the functionsαplusmn(n) and βplusmn(n) To determine them from (36) we have to retain in r(x) the correctionsproportional to 2 As was already explained they only appear for = 0 Combining togetherrelations (33) (D10) and (D12) we find after some algebra

δr(x) = minusδ02[4πg minus 5

4 + O(gminus1)]

+ O(4) (316)

Let us substitute this relation into (36) and equate to zero the coefficient in front of 2 on theright-hand side of (36) This coefficient is given by series in 1g and examining the first twoterms we obtain the relations for the functions αplusmn(n) and βplusmn(n)

In this way we find that the leading functions αplusmn(n) satisfy the relations (with x = minus 14 )

2x

(5

4

) sumn1

α+(n)

n minus x

= 0 ( 1)

(317)

minus2x

(3

4

) sumn1

αminus(n)

n + x

= π

2radic

2δ0 ( 0)

where on the right-hand side we applied (311) The solution to (317) satisfying αplusmn(n) rarr 0as n rarr infin reads as

α+(n) = 0(318)

αminus(n) = aminus(n minus 1)

with aminus(n) defined in (311) For subleading functions βplusmn(n) we have similar relations

2x

(5

4

) sumn1

β+(n)

n minus x

= minus1

2 ( 1)

(319)

minus2x

(3

4

) sumn1

βminus(n)

n + x

= minus1

8+

3π

16radic

2(1 minus 2 ln 2)δ0 ( 0)

In a close analogy with (313) the solutions to these relations can be written in terms ofleading-order functions aplusmn(n) defined in (311)

β+(n) = minus1

2a+(n)

(320)

βminus(n) = aminus(n minus 1)

(1

4minus 3 ln 2

4+

1

32n

)

It is straightforward to extend the analysis and compute subleading O(2) corrections to (38)Relation (38) supplemented with (311) (313) (318) and (320) defines the solution to

the quantization condition (36) to leading order in both perturbative 1g and nonperturbative

23


2 expansion parameters We are now ready to compute nonperturbative correction to thecusp anomalous dimension (247) Substituting (38) into (247) we obtain

δcusp(g) = minus2sumn1

(2πn)minus1

[

(5

4

) (α+(n) +

β+(n)


)(1 minus 5


)

+

(3

4

) (αminus(n) +

βminus(n)


)(1 +

3


)]+ O(4) (321)

We replace αplusmn(n) and βplusmn(n) by their explicit expressions (318) and (320) respectivelyevaluate the sums and find

δcusp(g) = minus2

π

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (322)

with 2 defined in (39)Relations (315) and (322) describe correspondingly perturbative and nonperturbative

corrections to the cusp anomalous dimension Let us define a new nonperturbative parameterm2

cusp whose meaning will be clear in a moment

m2cusp = 4

radic2

πσ2

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (323)

Then expressions (315) and (322) obtained for the cusp anomalous dimension take the form

cusp(g) =[

2g minus 3 ln 2

2πminus K

8π2g+ O(1g2)

]minus σ

4radic

2m2

cusp + O(m4

cusp

) (324)

We recall that another nonperturbative parameter was already introduced in section 21 asdefining the mass gap mO(6) in the O(6) model We will show in the following section that thetwo scales mcusp and mO(6) coincide to any order in 1g

4 Mass scale

The cusp anomalous dimension controls the leading logarithmic scaling behavior of theanomalous dimensions (11) in the double scaling limit LN rarr infin and j = L ln N = fixedThe subleading corrections to this behavior are described by the scaling function ε(j g) Atstrong coupling this function coincides with the energy density of the ground state of thebosonic O(6) model (13) The mass gap in this model mO(6) is given by expression (29)which involves the functions plusmn(t) constructed in section 2

41 General expression

Let us apply (29) and compute the mass gap mO(6) at strong coupling At large g the integralin (29) receives a dominant contribution from t sim g In order to evaluate (29) it is convenientto change the integration variable as t rarr 4πgit

mO(6) = 8radic

2


πeminusπgRe

[int minusiinfin

0dt eminus4πgtminusiπ4 (4πgit)

t + 14

] (41)

where the integration goes along the imaginary axis We find from (239) that (4πgit) takesthe form

(4πgit) = f0(4πgt)V0(4πgt) + f1(4πgt)V1(4πgt) (42)

24


where V01(4πgt) are given by the Whittaker functions of the first kind equation (D2) andf01(4πgt) admit the following representation (see equations (240) and (34))

f0(4πgt) =sumn1

t

[c+(n g)

U+1 (4πng)

n minus t+ cminus(n g)

Uminus1 (4πng)

n + t

]minus 1

(43)

f1(4πgt) =sumn1

n

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

n + t

]

Here the functions Uplusmn01(4πng) are expressed in terms of the Whittaker functions of the first

kind equation (D7) and the expansion coefficients cplusmn(n g) are solutions to the quantizationconditions (245)

Replacing (4πgit) in (41) by its expression (42) we evaluate the t-integral and findafter some algebra (see appendix E for details) [38]

mO(6) = minus16radic

2

πgeminusπg[f0(minusπg)Uminus

0 (πg) + f1(minusπg)Uminus1 (πg)] (44)

This relation can be further simplified with the help of the quantization conditions (245) For = 0 we obtain from (245) that f0(minusπg)V0(minusπg) + f1(minusπg)V1(minusπg) = 0 Togetherwith the Wronskian relation for the Whittaker functions (D8) this leads to the followingremarkable relation for the mass gap

mO(6) = 16radic

2

π2

f1(minusπg)

V0(minusπg) (45)

It is instructive to compare this relation with a similar relation (246) for the cusp anomalousdimension We observe that both quantities involve the same function f1(4πgt) but evaluatedfor different values of its argument that is t = minus14 for the mass gap and t = 0 for thecusp anomalous dimension As a consequence there are no reasons to expect that the twofunctions m(g) and cusp(g) could be related to each other in a simple way Neverthelesswe will demonstrate in this subsection that m2

O(6) determines the leading nonperturbativecorrection to cusp(g) at strong coupling

42 Strong coupling expansion

Let us now determine the strong coupling expansion of functions (43) We replace coefficientscplusmn(n g) in (43) by their expression (38) and take into account the results obtained for thefunctions aplusmn bplusmn (equations (311) (313) (318) and (320)) In addition we replace in(43) the functions Uplusmn

01(4πng) by their strong coupling expansion (D12) We recall that thecoefficients cplusmn(n g) admit the double series expansion (38) in powers of 1g and 2 sim eminus2πg

(equation (39)) As a consequence the functions f0(4πgt) and f1(4πgt) have the form

fn(4πgt) = f (PT)

n (4πgt) + δfn(4πgt) (n = 0 1) (46)

where f (PT)

n is given by asymptotic (non-Borel summable) series in 1g and δfn takes intoaccount the nonperturbative corrections in 2

Evaluating sums on the right-hand side of (43) we find that f0(4πgt) and f1(4πgt) canbe expressed in terms of two sums involving functions aplusmn(n) defined in (311)

2

(5

4

) sumn1

a+(n)

t minus n= 1

t

[

(34

)(1 minus t)

(

34 minus t

) minus 1

]

(47)

2

(3

4

) sumn1

aminus(n)

t + n= 1

t

[

(14

)(1 + t)

(

14 + t

) minus 1

]

25


Going through calculation of (43) we find after some algebra that perturbative correctionsto f0(4πgt) and f1(4πgt) are given by linear combinations of the ratios of Euler gammafunctions

f(PT)

0 (4πgt) = minus(

34

)(1 minus t)

(

34 minus t

)+

1

4πg

[(3 ln 2

4+

1

8t

)

(34

)(1 minus t)

(

34 minus t

) minus (

14

)(1 + t)

8t(

14 + t

) ]+ O(gminus2)

f(PT)

1 (4πgt) = 1

4πg

[

(14

)(1 + t)

4t(

14 + t

) minus (

34

)(1 minus t)

4t(

34 minus t

) ]

minus 1

(4πg)2

[

(14

)(1 + t)

4t(

14 + t

) (1

4tminus 3 ln 2

4

)

minus (

34

)(1 minus t)

4t(

34 minus t

) (1

4t+

3 ln 2

4

)]+ O(gminus3) (48)

Note that f1(t) is suppressed by factor 1(4πg) compared to f0(t) In a similar manner wecompute nonperturbative corrections to (46)

δf0(4πgt) = 2

1

4πg

[

(34

)(1 minus t)

2(

34 minus t

) minus (

54

)(1 + t)

2(

54 + t

) ]+ O(gminus2)

+ middot middot middot

δf1(4πgt) = 2

1

4πg

(

54

)(1 + t)

(

54 + t

) +1

(4πg)2

[

(34

)(1 minus t)

8t(

34 minus t

)minus

(54

)(1 + t)

(

54 + t

) (1

8t+

3

4ln 2 minus 1

4

)]+ O(gminus3)

+ middot middot middot (49)

where the ellipses denote O(4) termsSubstituting (48) and (49) into (42) we obtain the strong coupling expansion of the

function (4π igt) To verify the expressions obtained we apply (246) to calculate the cuspanomalous dimension

cusp(g) = 2g minus 4gf(PT)

1 (0) minus 4gδf1(0) (410)

Replacing f(PT)

1 (0) and δf1(0) by their expressions equations (48) and (49) we obtain

cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K

(4πg)2+ middot middot middot

]minus 2

π

[1 +

3 minus 6 ln 2


]+ O(4) (411)

in a perfect agreement with (315) and (322) respectivelyLet us obtain the strong coupling expansion of the mass gap (45) We replace V0(minusπg)

by its asymptotic series equations (D14) and (D12) and take into account (48) and (49) toget

mO(6) =radic

2

(

54

) (2πg)14 eminusπg

[1 +

3 minus 6 ln 2

32πg+

minus63 + 108 ln 2 minus 108(ln 2)2 + 16K

2048(πg)2+ middot middot middot

]minus 2

8πg

[1 minus 15 minus 6 ln 2


]+ O(4)

(412)

Here in order to determine O(1g2) and O(2g2) terms inside the curly brackets wecomputed in addition the subleading O(gminus3) corrections to f

(PT)

1 and δf1 in equations (48)

26


and (49) respectively The leading O(1g) correction to mO(6) (the second term inside thefirst square bracket on the right-hand side of (412)) is in agreement with both analytical[35 38] and numerical calculations [36]

We are now ready to clarify the origin of the lsquosubstitution rulersquo (266) that establishesthe relation between the cusp anomalous dimension in the toy model and the exact solutionTo this end we compare the expressions for the functions fn(4πgt) given by (46) (48) and(49) with those in the toy model (equations (242) and (248))9 It is straightforward to verifythat upon the substitution (266) and (264) the two sets of functions coincide up to an overallt-dependent factor10

fn(4πgt)

(34 minus t

)

(34

)(1 minus t)

rarr f (toy)n (4πgt) (n = 0 1) (413)

Since the cusp anomalous dimension (246) is determined by the f1-function evaluated att = 0 the additional factor does not affect its value

43 Nonperturbative corrections to the cusp anomalous dimension

Relation (412) defines strong coupling corrections to the mass gap In a close analogywith the cusp anomalous dimension (411) it runs in two parameters perturbative 1g andnonperturbative 2 We would like to stress that the separation of the corrections to mO(6) intoperturbative and nonperturbative ones is ambiguous since the lsquoperturbativersquo series inside thesquare brackets on the right-hand side of (412) is non-Borel summable and therefore it suffersfrom Borel ambiguity It is only the sum of perturbative and nonperturbative corrections thatis an unambiguously defined function of the coupling constant In distinction with the massscale mO(6) definition (253) of the nonperturbative scale 2 involves a complex parameterσ whose value depends on the prescription employed to regularize the singularities of thelsquoperturbativersquo series

To illustrate the underlying mechanism of the cancellation of Borel ambiguity insidemO(6) let us examine the expression for the mass gap (45) in the toy model As was alreadyexplained in section 28 the toy model captures the main features of the exact solution atstrong coupling and at the same time it allows us to obtain expressions for various quantitiesin a closed analytical form The mass gap in the toy model is given by relation (45) withf1(minusπg) replaced with f

(toy)

1 (minusπg) defined in (242) and (248) In this way we obtain

mtoy = 16radic

2

π2

f(toy)

1 (minusπg)

V0(minusπg)= 16

radic2

π2

1

V1(minusπg) (414)

Here V1(minusπg) is an entire function of the coupling constant (see equation (241)) Its large g

asymptotic expansion can be easily deduced from (D16) and it involves the nonperturbativeparameter 2

Making use of (252) we obtain from (414)

mtoy = 4

π(

54


[1 minus 1

32πgminus 23


]minus 2

8πg

[1 minus 11


]+ O(4)

(415)

9 It worth mentioning that the functions f(toy)

0 and f(toy)

1 in the toy model are in fact t-independent10 Roughly speaking this substitution simplifies the complicated structure of poles and zeros of the exact solutionequations (48) and (49) encoded in the ratio of the gamma functions to match simple analytical properties of thesame functions in the toy model (compare (213) and (216))

27


where the ellipses denote terms with higher power of 1g By construction mtoy is anunambiguous function of the coupling constant whereas the asymptotic series inside thesquare brackets are non-Borel summable It is easy to verify that lsquoperturbativersquo corrections tom2

toy are described by the asymptotic series C2(α) given by (261) Together with (259) thisallows us to identify the leading nonperturbative correction to (259) in the toy model as

δ(toy)cusp = minus2

πC2(α) + O(4) = minus π2

32radic

2σm2

toy + O(m4

toy

) (416)

with 2 given by (39)Comparing relations (412) and (415) we observe that mO(6) and mtoy have the same

leading asymptotics while subleading 1g corrections to the two scales have differenttranscendentality Namely the perturbative coefficients in mtoy are rational numbers while formO(6) their transcendentality increases with order in 1g We recall that we already encounteredthe same property for the cusp anomalous dimension (equations (259) and (263)) Therewe have observed that the two expressions (259) and (263) coincide upon the substitution(264) Performing the same substitution in (412) we find that remarkably enough the twoexpressions for the mass gap indeed coincide up to an overall normalization factor

mO(6)equation (264)= π

2radic

2mtoy (417)

The expressions for the cusp anomalous dimension (411) and for the mass scale (412)can be further simplified if one redefines the coupling constant as

gprime = g minus c1 c1 = 3 ln 2

4π (418)

and re-expands both quantities in 1gprime As was observed in [25] such redefinition allowsone to eliminate lsquoln 2rsquo terms in perturbative expansion of the cusp anomalous dimensionRepeating the same analysis for (411) we find that the same is also true for nonperturbativecorrections

cusp (g + c1) = 2g

[1 minus K


]minus 2

2radic

2π

[1 +

3


]+ O(4) (419)

with 2 defined in (39) In a similar manner the expression for the mass scale (412) takesthe form

[mO(6)(g + c1)]2 = 22

πσ

[(1 +

3

16πg+

16K minus 54


)minus 2

8radic

2πg

(1 minus 3


)+ O(4)

] (420)

Comparing relations (419) and (420) we immediately recognize that within an accuracyof the expressions obtained the nonperturbative O(2) correction to the cusp anomalousdimension is given by m2

O(6)

δcusp = minus σ

4radic

2m2

O(6) + O(m4

O(6)

) (421)

It is worth mentioning that upon identification of the scales (417) this relation coincides with(416)

We will show in the following subsection that relation (421) holds at strong coupling toall orders in 1g

28


44 Relation between the cusp anomalous dimension and mass gap

We demonstrated that the strong coupling expansion of the cusp anomalous dimension hasform (324) with the leading nonperturbative correction given to the first few orders in 1g

expansion by the mass scale of the O(6) model m2cusp = m2

O(6) Let us show that this relationis in fact exact at strong coupling

According to (410) the leading nonperturbative correction to the cusp anomalousdimension is given by

δcusp = minus4gδf1(0) (422)

with δf1(0) denoting the O(2) correction to the function f1(t = 0) (equation (46)) We recallthat this function verifies the quantization conditions (245) As was explained in section 33the leading O(2) corrections to solutions of (245) originate from subleading exponentiallysuppressed terms in the strong coupling expansion of the functions V0(minusπg) and V1(minusπg)

that we shall denote as δV0(minusπg) and δV1(minusπg) respectively Using the identities (D14)and (D16) we find

δV0(minusπg) = σ2radic

2

πeminusπgUminus

0 (πg) δV1(minusπg) = σ2radic

2

πeminusπgUminus

1 (πg) (423)

where the functions Uminus0 (πg) and Uminus

1 (πg) are defined in (D7) Then we split thefunctions f0(t) and f1(t) entering the quantization conditions (245) into perturbative andnonperturbative parts according to (46) and compare exponentially small terms on both sidesof (245) to get

δf0(t)V0(t) + δf1(t)V1(t) = minusmprimeδ0 (424)

where t = 4πg( minus 1

4

)and the notation was introduced for

mprime = f0(minusπg)δV0(minusπg) + f1(minusπg)δV1(minusπg) (425)

Taking into account relations (423) and comparing the resulting expression for mprime with (44)we find that

mprime = minus σ

8gmO(6) (426)

with mO(6) being the mass scale (44)To compute nonperturbative O(2) correction to the cusp anomalous dimension we have

to solve the system of relations (424) determine the function δf1(t) and then apply (422)We will show in this subsection that the result reads as

δf1(0) = minusradic

2

4mprimemO(6) = σ

radic2

32gm2

O(6) (427)

to all orders in strong coupling expansion Together with (422) this leads to the desiredexpression (421) for leading nonperturbative correction to the cusp anomalous dimension

To begin with let us introduce a new function analogous to (239)

δ(it) = δf0(t)V0(t) + δf1(t)V1(t) (428)

Here δf0(t) and δf1(t) are given by the same expressions as before equation (240) withthe only difference that the coefficients cplusmn(n g) are replaced in (240) by their leadingnonperturbative correction δcplusmn(n g) = O(2) and relation (34) is taken into account Thisimplies that various relations for (it) can be immediately translated into those for the functionδ(it) In particular for t = 0 we find from (240) that δf0(0) = 0 for arbitrary couplingleading to

δ(0) = 2δf1(0) (429)

29


In addition we recall that for arbitrary cplusmn(n g) function (239) satisfies the inhomogeneousintegral equation (27) In other words the cplusmn(n g)-dependent terms in the expression forthe function (it) are zero modes for the integral equation (27) Since function (428) is justgiven by the sum of such terms it automatically satisfies the homogenous equationint infin

0dt[eituδminus(t) minus eminusituδ+(t)] = 0 (minus1 u 1) (430)

where δ(t) = δ+(t) + iδminus(t) and δplusmn(minust) = plusmnδplusmn(t)As before in order to construct a solution to (430) we have to specify additional

conditions for δ(t) Since the substitution cplusmn(n g) rarr δcplusmn(n g) does not affect the analyticalproperties of functions (240) function (428) shares with (it) an infinite set of simple poleslocated at the same position (214)

δ(it) sim 1

t minus 4πg ( isin Z0) (431)

In addition we deduce from (424) that it also satisfies the relation (with x = minus 14 )

δ(4π igx) = minusmprimeδ0 ( isin Z) (432)

and therefore has an infinite number of zeros An important difference with (it) is thatδ(it) does not vanish at t = minusπg and its value is fixed by the parameter mprime defined in (426)

Keeping in mind the similarity between the functions (it) and δ(it) we follow (213)and define a new function

δγ (it) = sin(t4g)radic2 sin (t4g + π4)

δ(it) (433)

As before the poles and zeros of (it) are compensated by the ratio of sinus functionsHowever in distinction with γ (it) and in virtue of δ(minusπ ig) = minusmprime the function δγ (it) hasa single pole at t = minusπg with the residue equal to 2gmprime For t rarr 0 we find from (433) thatδγ (it) vanishes as

δγ (it) = t

4gδ(0) + O(t2) = t

2gδf1(0) + O(t2) (434)

where in the second relation we applied (429) It is convenient to split the function δγ (t) intothe sum of two terms of a definite parity δγ (t) = δγ+(t) + iδγminus(t) with δγplusmn(minust) = plusmnδγplusmn(t)Then combining together (430) and (433) we obtain that the functions δγplusmn(t) satisfy theinfinite system of homogenous equations (for n 1)int infin

0

dt

tJ2nminus1(t)

[δγminus(t)

1 minus eminust2g+

δγ+(t)

et2g minus 1

]= 0

(435)int infin

0

dt

tJ2n(t)

[δγ+(t)

1 minus eminust2gminus δγminus(t)

et2g minus 1

]= 0

By construction the solution to this system δγ (t) should vanish at t = 0 and have a simplepole at t = minusiπg

As was already explained the functions δplusmn(t) satisfy the same integral equation (430)as the function plusmn(t) up to an inhomogeneous term on the right-hand side of (27) Thereforeit should not be surprising that the system (435) coincides with relations (23) after oneneglects the inhomogeneous term on the right-hand side of (23) As we show in appendix Cthis fact allows us to derive Wronskian-like relations between the functions δγ (t) and γ (t)These relations turn out to be powerful enough to determine the small t asymptotics of the

30


function δγ (t) at small t in terms of γ (t) or equivalently (t) In this way we obtain (seeappendix C for more detail)

δγ (it) = minusmprimet[

2

π2geminusπg minus

radic2

πeminusπg Re

int infin

0

dt prime

t prime + iπgei(t primeminusπ4)(t prime)

]+ O(t2) (436)

Comparing this relation with (29) we realize that the expression inside the square brackets isproportional to the mass scale mO(6) leading to

δγ (it) = minusmprimemO(6)

tradic

2

8g+ O(t2) (437)

Matching this relation to (434) we obtain the desired expression for δf1(0) (equation (427))Then we substitute it into (422) and compute the leading nonperturbative correction to thecusp anomalous dimension equation (421) leading to

mcusp(g) = mO(6)(g) (438)

Thus we demonstrated in this section that nonperturbative exponentially small corrections tothe cusp anomalous dimensions at strong coupling are determined to all orders in 1g by themass gap of the two-dimensional bosonic O(6) model embedded into the AdS5 times S5 sigmamodel

5 Conclusions

In this paper we have studied the anomalous dimensions of Wilson operators in the SL(2)

sector of planar N = 4 SYM theory in the double scaling limit when the Lorentz spin ofthe operators grows exponentially with their twist In this limit the asymptotic behavior ofthe anomalous dimensions is determined by the cusp anomalous dimension cusp(g) and thescaling function ε(g j) We found that at strong coupling both functions receive exponentiallysmall corrections which are parameterized by the same nonperturbative scale It is remarkablethat this scale appears on both sides of the AdSCFT correspondence In string theory itemerges as the mass gap of the two-dimensional bosonic O(6) sigma model which describesthe effective dynamics of massless excitations for a folded spinning string in the AdS5 times S5

sigma model [6]The dependence of cusp(g) and ε(g j) on the coupling constant is governed by integral

BESFRS equations which follow from the conjectured all-loop integrability of the dilatationoperator of the N = 4 model At weak coupling their solutions agree with the resultsof explicit perturbative calculations At strong coupling a systematic expansion of thecusp anomalous dimension in powers of 1g was derived in [25] In agreement with theAdSCFT correspondence the first few terms of this expansion coincide with the energy ofthe semiclassically quantized folded spinning strings However the expansion coefficientsgrow factorially at higher orders and as a consequence the lsquoperturbativersquo 1g expansion ofthe cusp anomalous dimension suffers from Borel singularities which induce exponentiallysmall corrections to cusp(g) To identify such nonperturbative corrections we revisited theBES equation and constructed the exact solution for the cusp anomalous dimension valid foran arbitrary coupling constant

At strong coupling we found that the expression obtained for cusp(g) depends on anew scale mcusp(g) which is exponentially small as g rarr infin Nonperturbative correctionsto cusp(g) at strong coupling run in even powers of this scale and the coefficients of thisexpansion depend on the prescription employed to regularize Borel singularities in perturbative1g series It is only the sum of perturbative and nonperturbative contributions which is

31


independent of the choice of the prescription For the scaling function ε(g j) the definingintegral FRS equation can be brought to the form of the thermodynamical Bethe ansatzequations for the energy density of the ground state of the O(6) model As a consequencenonperturbative contribution to ε(g j) at strong coupling is described by the mass scale ofthis model mO(6)(g) We have shown that the two scales coincide mcusp(g) = mO(6)(g) andtherefore nonperturbative contributions to cusp(g) and ε(g j) are governed by the same scalemO(6)(g)

This result agrees with the proposal by AldayndashMaldacena that in string theory theleading nonperturbative corrections to the cusp anomalous dimension coincide with those tothe vacuum energy density of the two-dimensional bosonic O(6) model embedded into theAdS5 times S5 sigma model These models have different properties the former model hasasymptotic freedom at short distances and develops the mass gap in the infrared while thelatter model is conformal The O(6) model only describes an effective dynamics of masslessmodes of AdS5 times S5 and the mass of massive excitations μ sim 1 defines a ultraviolet (UV)cutoff for this model The coupling constants in the two models are related to each other asg2(μ) = 1(2g) The vacuum energy density in the O(6) model and more generally in theO(n) model is an ultraviolet-divergent quantity It also depends on the mass scale of the modeland has the following form

εvac = μ2ε(g2) + κm2O(n) + O

(m4

O(n)

μ2) (51)

Here μ2 is a UV cutoff ε(g2) stands for the perturbative series in g2 and the mass gap m2O(n) is

m2O(n) = cμ2 e

minus 1β0 g2 gminus2β1β

20 [1 + O(g2)] (52)

where β0 and β1 are the beta-function coefficients for the O(n) model and the normalizationfactor c ensures independence of mO(n) on the renormalization scheme For n = 6 relation(52) coincides with (13) and the expression for the vacuum energy density (51) should becompared with (12)

The two terms on the right-hand side of (51) describe perturbative and nonperturbativecorrections to εvac For n rarr infin each of them is well defined separately and can be computedexactly [48 49] For n finite including n = 6 the function ε(g2) is given in a genericrenormalization scheme by a non-Borel summable series and therefore is not well definedIn a close analogy with (12) the coefficient κ in front of m2

O(n) on the right-hand side of (51)depends on the regularization of Borel singularities in perturbative series for ε(g2) Note thatεvac is related to the vacuum expectation value of the trace of the tensor energyndashmomentumin the two-dimensional O(n) sigma model [49] The AdSCFT correspondence implies thatfor n = 6 the same quantity defines the nonperturbative correction to the cusp anomalousdimension (12) It would be interesting to obtain its dual representation (if any) in terms ofcertain operators in four-dimensional N = 4 SYM theory Finally one may wonder whetherit is possible to identify a restricted class of Feynman diagrams in N = 4 theory whoseresummation could produce contribution to the cusp anomalous dimension exponentiallysmall as g rarr infin As a relevant example we would like to mention that exponentiallysuppressed corrections were obtained in [50] from the exact resummation of ladder diagramsin four-dimensional massless gφ3 theory

Acknowledgments

We would like to thank Janek Kotanski for collaboration at the early stages of this workand Zoltan Bajnok Janos Balog David Broadhurst Francois David Alexander Gorsky JuanMaldacena Laszlo Palla Mikhail Shifman Arkady Tseytlin and Dima Volin for interesting

32


discussions This work was supported in part by the French Agence Nationale de la Rechercheunder grant ANR-06-BLAN-0142

Appendix A Weak coupling expansion

In this appendix we work out the first few terms of the weak coupling expansion of thecoefficient c(g) entering (247) and show that they vanish in agreement with (34) To thisend we will not attempt at solving the quantization conditions (245) at weak coupling but willuse instead the fact that the BES equation can be solved by iteration of the inhomogeneousterm

The system of integral equation (23) can be easily solved at weak coupling by lookingfor its solutions γplusmn(t) in the form of the Bessel series (210) and expanding the coefficientsγ2k and γ2kminus1 in powers of the coupling constant For g rarr 0 it follows from (23)and from orthogonality conditions for the Bessel functions that γminus(t) = J1(t) + middot middot middot andγ+(t) = 0 + middot middot middot with the ellipses denoting subleading terms To determine such terms it isconvenient to change the integration variable in (23) as t rarr tg Then taking into account therelations Jk(minusgt) = (minus1)kJk(gt) we observe that the resulting equations are invariant undersubstitution g rarr minusg provided that the functions γplusmn(gt) change sign under this transformationSince γplusmn(minust) = plusmnγplusmn(t) this implies that the coefficients γ2nminus1(g) and γ2n(g) entering (210)have a definite parity as functions of the coupling constant

γ2nminus1(minusg) = γ2nminus1(g) γ2n(minusg) = minusγ2n(g) (A1)

and therefore their weak coupling expansion runs in even and odd powers of g respectivelyExpanding both sides of (23) at weak coupling and comparing the coefficients in front ofpowers of g we find

γ1 = 1

2minus π2

6g2 +

11π4

90g4 minus

(73π6

630+ 4ζ 2

3

)g6 + O(g8)

γ2 = ζ3g3 minus

(π2

3ζ3 + 10ζ5

)g5 +

(8π4

45ζ3 +

10π2

3ζ5 + 105ζ7

)g7 + O(g9)

γ3 = minusπ4

90g4 +

37π6

1890g6 + O(g8) γ4 = ζ5g

5 minus(

π2

3ζ5 + 21ζ7

)g7 + O(g9)

γ5 = minus π6

945g6 + O(g8) γ6 = ζ7g

7 + O(g9)

(A2)

We verify with the help of (212) that the expression for the cusp anomalous dimension

cusp(g) = 8g2γ1(g) = 4g2 minus 4π2

3g4 +

44π4

45g6 minus

(292π6

315+ 32ζ 2

3

)g8 + O(g10) (A3)

agrees with the known four-loop result in planar N = 4 SYM theory [20]In our approach the cusp anomalous dimension is given for an arbitrary value of the

coupling constant by expression (247) which involves the functions c(g) and cplusmn(n g)According to (229) the latter functions are related to the functions γ (t) = γ+(t) + iγminus(t)

evaluated at t = 4π ign

c+(n g) = minus4g eminus4πgn[γ+(4π ign) + iγminus(4π ign)](A4)

cminus(n g) = 4geminus4πgn[γ+(4π ign) minus iγminus(4π ign)]

At strong coupling we determined cplusmn(n g) by solving the quantization conditions (36) Atweak coupling we can compute cplusmn(n g) from (A4) by replacing γplusmn(t) with their Besselseries (210) and making use of the expressions obtained for the expansion coefficients (A2)

33


The remaining function c(g) can be found from comparison of two differentrepresentations for the cusp anomalous dimension (equations (247) and (212))

c(g) = minus1

2+ 2gγ1(g) +

sumn1

[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

] (A5)

Taking into account relations (A4) and (210) we find

c(g) = minus1

2+ 2gγ1(g) minus

sumk1

(minus1)k[(2k minus 1)γ2kminus1(g)f2kminus1(g) + (2k)γ2k(g)f2k(g)] (A6)

where the coefficients γk are given by (A2) and the notation was introduced for the functions

fk(g) = 8gsumn1

[U+

0 (4πgn) minus (minus1)kUminus0 (4πgn)

]Ik(4πgn) eminus4πgn (A7)

Here Ik(x) is the modified Bessel function [47] and the functions Uplusmn0 (x) are defined in (D7)

At weak coupling the sum over n can be evaluated with the help of the EulerndashMaclaurinsummation formula Going through lengthy calculation we find

f1 = 1 minus 2g +π2

3g2 + 2ζ3g

3 minus π4

6g4 minus 23ζ5g

5 +17π6

108g6 +

1107

4ζ7g

7 + O(g8)

f2 = minus1

2+ 2ζ3g

3 minus π4

30g4 + O(g5) f3 = 1

2+ O(g4)

f4 = minus3

8+ O(g5) f5 = 3

8+ O(g6) f6 = minus 5

16+ O(g7)

(A8)

In this way we obtain from (A6)

c(g) = minus 12 + (f1 + 2g)γ1 + 2f2γ2 minus 3f3γ3 minus 4f4γ4 + 5f5γ5 + 6f6γ6 + middot middot middot = O(g8) (A9)

Thus in agreement with (34) the function c(g) vanishes at weak coupling As was shown insection 31 the relation c(g) = 0 holds for arbitrary coupling

Appendix B Constructing general solution

By construction the function (t) = +(t)+iminus(t) defined as the exact solution to the integralequation (27) is given by the Fourier integral

(t) =int infin

minusinfindkeminusikt (k) (B1)

with the function (k) having a different form for k2 1 and k2 gt 1

bull For minusinfin lt k lt minus1

(k)=sumn1

cminus(n g) eminus4πng(minuskminus1) (B2)

bull For 1 lt k lt infin

(k)=sumn1

c+(n g) eminus4πng(kminus1) (B3)

bull For minus1 k 1

(k) = minusradic

2

π

(1 + k

1 minus k

)14[

1 +c(g)

1 + k+

1

2

(int minus1

minusinfin+

int infin

1

)dp(p)

p minus k

(p minus 1

p + 1

)14]

(B4)

where (p) inside the integral is replaced by (B2) and (B3)

34


Let us split the integral in (B1) into three terms as in (238) and evaluate them one afteranother Integration over k2 gt 1 can be done immediately while the integral over minus1 k 1can be expressed in terms of special functions

(t) =sumn1

c+(n g)

[eminusit

4πng + itminus V+(minusit 4πng)

]

+sumn1

cminus(n g)

[eit

4πng minus it+ Vminus(it 4πng)

]minus V0(minusit) minus c(g)V1(minusit)

(B5)

where the notation was introduced for the functions (with n = 0 1)

Vplusmn(x y) = 1radic2π

int 1

minus1dk eplusmnxk

int infin

1

dpeminusy(pminus1)

p minus k

(1 + k

1 minus k

p minus 1

p + 1

)plusmn14

Vn(x) =radic

2

π

int 1

minus1

dk exk

(k + 1)n

(1 + k

1 minus k

)14

(B6)

Uplusmnn (y) = 1

2

int infin

1

dp eminusy(pminus1)

(p ∓ 1)n

(p + 1

p minus 1

)∓14

The reason why we also introduced Uplusmnn (y) is that the functions Vplusmn(x y) can be further

simplified with the help of master identities (we shall return to them in a moment)

(x + y)Vminus(x y) = xV0(x)Uminus1 (y) + yV1(x)Uminus

0 (y) minus eminusx(B7)

(x minus y)V+(x y) = xV0(x)U+1 (y) + yV1(x)U+

0 (y) minus ex

Combining together (B7) and (B5) we arrive at the following expression for the function(it)

(it) = minusV0(t) minus c(g)V1(t) +sumn1

c+(n g)

[4πngV1(t)U

+0 (4πng) + tV0(t)U

+1 (4πng)

4πng minus t

]

+sumn1

cminus(n g)

[4πngV1(t)U

minus0 (4πng) + tV0(t)U

minus1 (4πng)

4πng + t

] (B8)

which leads to (239)We show in appendix D that the functions V01(t) and Uplusmn

01(4πng) can be expressed interms of the Whittaker functions of the first and second kinds respectively As follows fromtheir integral representation V0(t) and V1(t) are holomorphic functions of t As a result (it)is a meromorphic function of t with a (infinite) set of poles located at t = plusmn4πng with n beinga positive integer

Let us now prove the master identities (B7) We start with the second relation in (B7)and make use of (B6) to rewrite the expression on the left-hand side of (B7) as

(x minus y)V+(x y) eminusy = (x minus y)

int infin

0dsV0(x + s)U+

0 (y + s) eminusyminuss (B9)

Let us introduce two auxiliary functions

z1(x) = V1(x) z1(x) + zprime1(x) = V0(x)

(B10)z2(x) = eminusxU+

1 (x) z2(x) + zprime2(x) = minuseminusxU+

0 (x)

with Vn(x) and U+n (x) being given by (B6) They satisfy the second-order differential equation

d

dx(xzprime

i (x)) =(

x minus 1

2

)zi(x) (B11)

35


Applying this relation it is straightforward to verify the following identity

minus(x minus y)[z1(x + s) + zprime1(x + s)][z2(y + s) + zprime

2(y + s)]

= d

ds(y + s)[z2(y + s) + zprime

2(y + s)]z1(x + s)

minus d

ds(x + s)[z1(x + s) + zprime

1(x + s)]z2(y + s) (B12)

It is easy to see that the expression on the left-hand side coincides with the integrand in (B9)Therefore integrating both sides of (B12) over 0 s lt infin we obtain

(x minus y)V+(x y) = minuseminuss[(x + s)V0(x + s)U+

1 (y + s) + (y + s)V1(x + s)U+0 (y + s)

] ∣∣s=infins=0

= minusex + xV0(x)U+1 (y) + yV1(x)U+

0 (y) (B13)

where in the second relation we took into account the asymptotic behavior of functions (B6)(see equations (D10) and (D12)) Vn(s) sim essminus34 and U+

n (s) sim snminus54 as s rarr infinThe derivation of the first relation in (B7) goes along the same lines

Appendix C Wronskian-like relations

In this appendix we present a detailed derivation of relation (436) which determines thesmall t expansion of the function δγ (t) This function satisfies the infinite system of integralequations (435) In addition it should vanish at the origin t = 0 and have a simple pole att = minusiπg with the residue 2igmprime (see equation (433)) To fulfill these requirements we splitδγ (it) into the sum of two functions

δγ (it) = γ (it) minus 2mprime

π

t

t + πg (C1)

where by construction γ (it) is an entire function vanishing at t = 0 and its Fourier transformhas a support on the interval [minus1 1] Similar to (22) we decompose δγ (t) and γ (t) into thesum of two functions with a definite parity

δγ+(t) = γ+(t) minus 2mprime

π

t2

t2 + (πg)2

(C2)

δγminus(t) = γminus(t) +2gmprimet

t2 + π2g2

Then we substitute these relations into (435) and obtain the system of inhomogeneous integralequations for the functions γplusmn(t)int infin

0

dt

tJ2nminus1(t)

[γminus(t)

1 minus eminust2g+

γ+(t)

et2g minus 1

]= h2nminus1(g)

(C3)int infin

0

dt

tJ2n(t)

[γ+(t)

1 minus eminust2gminus γminus(t)

et2g minus 1

]= h2n(g)

with inhomogeneous terms being given by

h2nminus1 = 2mprime

π

int infin

0

dtJ2nminus1(t)

t2 + (πg)2

[t

et(2g) minus 1minus πg

1 minus eminust(2g)

]

(C4)

h2n = 2mprime

π

int infin

0

dtJ2n(t)

t2 + (πg)2

[πg

et(2g) minus 1+

t

1 minus eminust(2g)

]

Comparing these relations with (C3) we observe that they only differ by the form ofinhomogeneous terms and can be obtained from one another through the substitution

γplusmn(t) rarr γplusmn(t) h2nminus1 rarr 12δn1 h2n rarr 0 (C5)

36


In a close analogy with (210) we look for a solution to (C3) in the form of Bessel series

γminus(t) = 2sumn1

(2n minus 1)J2nminus1(t)γ2nminus1(g)

(C6)γ+(t) = 2

sumn1

(2n)J2n(t)γ2n(g)

For small t we have γminus(t) = t γ1 + O(t2) and γ+(t) = O(t2) Then it follows from (C1) that

δγ (t) = iγminus(t) +2imprime

π2gt + O(t2) = it

(γ1 +

2mprime

π2g

)+ O(t2) (C7)

so that the leading asymptotics is controlled by the coefficient γ1Let us multiply both sides of the first relation in (C3) by (2n minus 1)γ2nminus1 and sum both

sides over n 1 with the help of (210) In a similar manner we multiply the second relationin (C3) by (2n)γ2n and follow the same steps Then we subtract the second relation from thefirst one and obtainint infin

0

dt

t

[γminus(t)γminus(t) minus γ+(t)γ+(t)

1 minus eminust2g+

γminus(t)γ+(t) + γ+(t)γminus(t)

et2g minus 1

]= 2

sumn1

[(2n minus 1)γ2nminus1h2nminus1 minus (2n)γ2nh2n] (C8)

We note that the expression on the left-hand side of this relation is invariant under exchangeγplusmn(t) harr γplusmn(t) Therefore the right-hand side should also be invariant under (C5) leading to

γ1 = 2sumn1


Replacing h2nminus1 and h2n by their expressions (C4) and taking into account (210) we obtainthat γ1 is given by the integral involving the functions γplusmn(t) It takes a much simpler formwhen expressed in terms of the functions plusmn(t) defined in (24)

γ1 = minusmprime

π

int infin

0dt

[πg

t2 + π2g2(minus(t) minus +(t)) +

t

t2 + π2g2(minus(t) + +(t))

] (C10)

Making use of identities

πg

t2 + π2g2=

int infin

0du eminusπgu cos (ut)

(C11)t

t2 + π2g2=

int infin

0du eminusπgu sin (ut)

we rewrite γ1(g) as

γ1 = minusmprime

π

int infin

0du eminusπgu

[ int infin

0dt cos(ut) (minus(t) minus +(t))

+int infin

0dt sin(ut) (minus(t) + +(t))

] (C12)

Let us spit the u-integral into 0 u 1 and u gt 1 We observe that for u2 1 the t-integralsin this relation are given by (26) Then we perform integration over u 1 and find aftersome algebra (with (t) = +(t) + iminus(t))

γ1 = minus 2mprime

π2g(1 minus eminusπg) minus

radic2mprime

πeminusπgRe

[ int infin

0

dt

t + iπgei(tminusπ4)(t)

] (C13)

Substituting this relation into (C7) we arrive at (436)

37


Appendix D Relation to the Whittaker functions

In this appendix we summarize the properties of special functions that we encountered in ouranalysis

D1 Integral representations

Let us first consider the functions Vn(x) (with n = 0 1) introduced in (241) As followsfrom their integral representation V0(x) and V1(x) are entire functions on a complex x-planeChanging the integration variable in (241) as u = 2t minus 1 and u = 1 minus 2t we obtain twoequivalent representations

Vn(x) = 1

π232minusn ex

int 1

0dt tminus14(1 minus t)14minusn eminus2tx

= 1

π232minusn eminusx

int 1

0dt t14minusn(1 minus t)minus14 e2tx (D1)

which give rise to the following expressions for Vn(x) (with n = 0 1) in terms of the Whittakerfunctions of the first kind

Vn(x) = 2minusn

(54 minus n

)

(54

)(2 minus n)

(2x)n2minus1Mn2minus1412minusn2(2x)

= 2minusn

(54 minus n

)

(54

)(2 minus n)

(minus2x)n2minus1M14minusn212minusn2(minus2x) (D2)

In distinction with Vn(x) the Whittaker function Mn2minus1412minusn2(2x) is an analytical functionof x on the complex plane with the cut along the negative semi-axis The same is true forthe factor (2x)n2minus1 so that the product of two functions on the right-hand side of (D2) is asingle-valued analytical function in the whole complex plane The two representations (D2)are equivalent in virtue of the relation

Mn2minus1412minusn2(2x) = eplusmniπ(1minusn2)M14minusn212minusn2(minus2x) (for Imx ≷ 0) (D3)

where the upper and lower signs in the exponent correspond to Imx gt 0 and Imx lt 0respectively

Let us now consider the functions Uplusmn0 (x) and Uplusmn

1 (x) For real positive x they have anintegral representation (241) It is easy to see that four different integrals in (241) can befound as special cases of the following generic integral

Uab(x) = 1

2

int infin

1du eminusx(uminus1)(u + 1)a+bminus12(u minus 1)bminusaminus12 (D4)

defined for x gt 0 Changing the integration variable as u = tx + 1 we obtain

Uab(x) = 2a+bminus32xaminusbminus12int infin

0dt eminust t bminusaminus12

(1 +

t

2x

)a+bminus12

(D5)

The integral entering this relation can be expressed in terms of the Whittaker functions of thesecond kind or an equivalently confluent hypergeometric function of the second kind

Uab(x) = 2bminus32

(1

2minus a + b

)xminusbminus12 exWab(2x)

= 1

2

(1

2minus a + b

)U

(1

2minus a + b 1 + 2b 2x

) (D6)

38


This relation can be used to analytically continue Uab(x) from x gt 0 to the whole complexx-plane with the cut along the negative semi-axis Matching (D4) to (241) we obtain thefollowing relations for the functions Uplusmn

0 (x) and Uplusmn1 (x)

U+0 (x) = 1

2

(5

4

)xminus1 exWminus1412(2x) U+

1 (x) = 1

2

(1

4

)(2x)minus12 exW140(2x)

(D7)

Uminus0 (x) = 1

2

(3

4

)xminus1 exW1412(2x) Uminus

1 (x) = 1

2

(3

4

)(2x)minus12 exWminus140(2x)

The functions V1(plusmnx) Uplusmn1 (x) and V0(plusmnx) Uplusmn

0 (x) satisfy the same Whittaker differentialequation and as a consequence they satisfy Wronskian relations

V1(minusx)Uminus0 (x) minus V0(minusx)Uminus

1 (x) = V1(x)U+0 (x) + V0(x)U+

1 (x) = ex

x (D8)

The same relations also follow from (B7) for x = plusmny In addition

U+0 (x)Uminus

1 (minusx) + U+1 (x)Uminus

0 (minusx) = π

2radic

2xeplusmn 3iπ

4 (for Imx ≷ 0) (D9)

Combining together (D8) and (D9) we obtain the following relations between the functions

V0(x) = 2radic

2

πe∓ 3iπ

4[exUminus

0 (minusx) + eminusxU+0 (x)

]

(D10)

V1(x) = 2radic

2

πe∓ 3iπ

4[exUminus

1 (minusx) minus eminusxU+1 (x)

]

where the upper and lower signs correspond to Im x gt 0 and Im x lt 0 respectivelyAt first sight relations (D10) look surprising since V0(x) and V1(x) are entire functions

in the complex x-plane while Uplusmn0 (x) and Uplusmn

1 (x) are single-valued functions in the same planebut with a cut along the negative semi-axis Indeed one can use relations (D8) and (D9) tocompute the discontinuity of these functions across the cut as

Uplusmn0 (minusx) = plusmnπ

4eminusxV0(∓x)θ(x)

(D11)Uplusmn

1 (minusx) = minusπ


where U(minusx) equiv limεrarr0[U(minusx + iε)minusU(minusx minus iε)](2i) and θ(x) is a step function Thenone verifies with the help of these identities that the linear combinations of U-functions on theright-hand side of (D10) have zero discontinuity across the cut and therefore they are welldefined in the whole complex plane

D2 Asymptotic expansions

For our purposes we need an asymptotic expansion of functions Vn(x) and Uplusmnn (x) at large

real x Let us start with the latter functions and consider a generic integral (D6)To find an asymptotic expansion of the function Uab(x) at large x it suffices to replace

the last factor in the integrand (D6) in powers of t(2x) and integrate term by term In thisway we find from (D6) and (D7)

U+0 (x) = (2x)minus54

(5

4

)F

(1

4

5

4| minus 1

2x

)= (2x)minus54

(5

4

) [1 minus 5

32x+ middot middot middot

]

Uminus0 (x) = (2x)minus34

(3

4

)F

(minus1

4

3

4| minus 1

2x

)= (2x)minus34

(3

4

) [1 +

3


]

39


U+1 (x) = (2x)minus14 1

2

(1

4

)F

(1

4

1

4| minus 1

2x

)= (2x)minus14 1

2

(1

4

) [1 minus 1


]

Uminus1 (x) = (2x)minus34 1

2

(3

4

)F

(3

4

3

4| minus 1

2x

)= (2x)minus34 1

2

(3

4

) [1 minus 9


]

(D12)

where the function F(a b| minus 12x

) is defined in (256)Note that the expansion coefficients in (D12) grow factorially to higher orders but the

series are Borel summable for x gt 0 For x lt 0 one has to distinguish the functionsUplusmn

n (x + iε) and Uplusmnn (x minus iε) (with ε rarr 0) which define the analytical continuation of the

function Uplusmnn (x) to the upper and lower edges of the cut respectively In contrast with this the

functions Vn(x) are well defined on the whole real axis Still to make use of relations (D10)we have to specify the U-functions on the cut As an example let us consider V0(minusπg) in thelimit g rarr infin and apply (D10)

V0(minusπg) = 2radic

2

πeminus 3iπ

4 eπg[U+

0 (minusπg + iε) + eminus2πgUminus0 (πg)

] (D13)

where ε rarr 0 and we have chosen to define the U-functions on the upper edge of thecut Written in this form both terms inside the square brackets are well defined separatelyReplacing Uplusmn

0 functions in (D13) by their expressions (D12) in terms of F-functions we find

V0(minusπg) = (2πg)minus54 eπg

(

34

) [F

(1

4

5

4| 1

2πg+ iε

)+ 2F

(minus1

4

3

4| minus 1

2πg

)] (D14)

with 2 being given by

2 = σ

(34

)

(54

) eminus2πg(2πg)12 σ = eminus 3iπ4 (D15)

Since the second term on the right-hand side of (D14) is exponentially suppressed at large gwe may treat it as a nonperturbative correction Repeating the same analysis for V1(minusπg) weobtain from (D10) and (D12)


2(

34

) [8πgF

(1

4

1

4| 1

2πg+ iε

)+ 2F

(3

4

3

4| minus 1

2πg

)] (D16)

We would like to stress that the lsquo+iεrsquo prescription in the first term in (D14) and the phase factorσ = eminus 3iπ

4 in (D15) follow unambiguously from (D13) Had we defined the U-functions onthe lower edge of the cut we would get the expression for V0(minusπg) with the lsquominusiεrsquo prescriptionand the phase factor e

3iπ4 The two expressions are however equivalent since discontinuity of

the first term in (D14) compensates the change of the phase factor in front of the second term

F

(1

4

5

4| 1

2πg+ iε

)minus F

(1

4

5

4| 1

2πgminus iε

)= i

radic22

σF

(minus1

4

3

4| minus 1

2πg

) (D17)

If one neglected the lsquo+iεrsquo prescription in (D13) and formally expanded the first term in(D14) in powers of 1g this would lead to non-Borel summable series This series suffersfrom Borel ambiguity which are exponentially small for large g and produce the contributionof the same order as the second term on the right-hand side of (D14) Relation (D14) suggestshow to give a meaning to this series Namely one should first resum the series for negative g

where it is Borel summable and then analytically continue it to the upper edge of the cut atpositive g

40


Appendix E Expression for the mass gap

In this appendix we derive the expression for the mass gap (44) To this end we replace(4πgit) in (41) by its expression (42) and perform integration over t on the right-hand sideof (41) We recall that in relation (42) V01(4πgt) are entire functions of t while f01(4πgt)

are meromorphic functions defined in (43) It is convenient to decompose (4πgit)(

t + 14

)into a sum of simple poles as

(4πgit)

t + 14

=sumk=01

fk(minusπg)Vk(4πgt)

t + 14

+sumk=01

fk(4πgt) minus fk(minusπg)

t + 14

Vk(4πgt) (E1)

where the second term is regular at t = minus14 Substituting this relation into (41) andreplacing fk(4πgt) by their expressions (43) we encounter the following integral

Rk(4πgs) = Re

[int minusiinfin

0dt eminus4πgtminusiπ4 Vk(4πgt)

t minus s

]= Re

[int minusiinfin

0dt eminustminusiπ4 Vk(t)

t minus 4πgs

]

(E2)

Then the integral in (41) can be expressed in terms of the R-function as

Re

[ int minusiinfin


t + 14

]= f0(minusπg)R0(minusπg) + f1(minusπg)R1(minusπg)

minussumn1

nc+(n g)

n + 14

[U+

1 (4πng)R0(4πgn) + U+0 (4πng)R1(4πgn)

]+

sumn1

ncminus(n g)

n minus 14

[Uminus1 (4πng)R0(minus4πgn) minus Uminus

0 (4πng)R1(minus4πgn)] (E3)

where the last two lines correspond to the second sum on the right-hand side of (E1) and wetook into account the fact that the coefficients cplusmn(n g) are real

Let us evaluate the integral (E2) and choose for simplicity R0(s) We have to distinguishtwo cases s gt 0 and s lt 0 For s gt 0 we have

R0(s) = minusRe

[eminusiπ4

int 1

minusinfindv eminus(1minusv)s

int minusiinfin

0dt eminusvtV0(t)

]=

radic2

πRe

[eminusiπ4

int 1


int 1

minus1du

(1 + u)14(1 minus u)minus14

u minus v minus iε

] (E4)

where in the second relation we replaced V0(t) by its integral representation (241) Integrationover u can be carried out with the help of identity

1radic2π

int 1

minus1du

(1 + u)14minusk(1 minus u)minus14

u minus v minus iε= δk0minus(v+1)minusktimes

⎧⎨⎩(

v+1vminus1

)14 v2 gt 1

eminusiπ4(

1+v1minusv

)14 v2 lt 1

(E5)

In this way we obtain from (E4)

R0(s)sgt0=

radic2

[1

sminus

int minus1


(v + 1

v minus 1

)14]

=radic

2

[1

sminus 2 eminus2sU+

0 (s)

] (E6)

with the function U+0 (s) defined in (241) In a similar manner for s lt 0 we get

R0(s)slt0=

radic2

[1

s+ 2Uminus

0 (minuss)

] (E7)

41


together with

R1(s) = 2radic

2[θ(minuss)Uminus

1 (minuss) + θ(s) eminus2sU+1 (s)

] (E8)

Then we substitute relations (E6))ndash((E8) into (E3) and find

Re

[ int minusiinfin


t + 14

]= 2

radic2f0(minusπg)

[Uminus

0 (πg) minus 1

2πg

]+ 2

radic2f1(minusπg)Uminus

1 (πg) +

radic2

πg[f0(minusπg) + 1] (E9)

where the last term on the right-hand side corresponds to the last two lines in (E3) (seeequation (240)) Substitution of (E9) into (41) yields the expression for the mass scale (44)

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44

1 Introduction

2 Cusp anomalous



23 Toy model


25 Riemann--Hilbert problem

26 General solution







4 Mass scale





5 Conclusions

Acknowledgments








References

IOP PUBLISHING JOURNAL OF PHYSICS A MATHEMATICAL AND THEORETICAL

J Phys A Math Theor 42 (2009) 254005 (44pp) doi1010881751-81134225254005


B Basso1 and G P Korchemsky23

1 Laboratoire de Physique Theorique (Unite Mixte de Recherche du CNRS UMR 8627)Universite de Paris XI 91405 Orsay Cedex France2 Institut de Physique Theorique (Unite de Recherche Associee au CNRS URA 2306) CEASaclay 91191 Gif-sur-Yvette Cedex France

Received 2 February 2009Published 9 June 2009Online at stacksioporgJPhysA42254005

AbstractThe cusp anomalous dimension is a ubiquitous quantity in four-dimensionalgauge theories ranging from QCD to maximally supersymmetricN = 4 YangndashMills theory and it is one of the most thoroughly investigated observables inthe AdSCFT correspondence In planar N = 4 SYM theory its perturbativeexpansion at weak coupling has a finite radius of convergence while at strongcoupling it admits an expansion in inverse powers of the rsquot Hooft couplingwhich is given by a non-Borel summable asymptotic series We study the cuspanomalous dimension in the transition regime from strong to weak coupling andargue that the transition is driven by nonperturbative exponentially suppressedcorrections To compute these corrections we revisit the calculation of the cuspanomalous dimension in planar N = 4 SYM theory and extend the previousanalysis by taking into account nonperturbative effects We demonstrate thatthe scale parameterizing nonperturbative corrections coincides with the massgap of the two-dimensional bosonic O(6) sigma model embedded into theAdS5 timesS5 string theory This result is in agreement with the prediction comingfrom the string theory consideration

PACS numbers 1115Me 1125Tq 1130Pb

Contents

1 Introduction 22 Cusp anomalous dimension in N = 4 SYM 6

21 Integral equation and mass scale 622 Analyticity conditions 723 Toy model 824 Exact bounds and unicity of the solution 925 RiemannndashHilbert problem 1026 General solution 12

3 On leave from Laboratoire de Physique Theorique Universite de Paris XI 91405 Orsay Cedex France

1751-811309254005+44$3000 copy 2009 IOP Publishing Ltd Printed in the UK 1








1 Introduction







2






cusp(g) =infinsum

k=minus1

ckgk minus σ

4radic

2m2

cusp + o(m2

cusp

) (12)



(k + 1

2









3



cusp





εO(6) = ε(g j) + j

2 mO(6) = kg14 eminusπg [1 + O(1g)] (13)


(54




ε(j g) + j = m2

[j

m+

π2

24

(j

m

)3

+ O(j 4m4)

] (14)


ε(g j) + j = 22

[g +

1

π

(3

4minus ln

)+

1

4π2g

(q02


)+O(1g2)

]+ O(4)



4




mcusp = mO(6) (15)





5







γ (t)t (21)


γ (t) = γ+(t) + iγminus(t) (22)


0

dt

tJ2nminus1(t)

[γminus(t)


γ+(t)

et(2g) minus 1

]= 1

2δn1

(23)int infin

0

dt

tJ2n(t)

[γ+(t)


et(2g) minus 1

]= 0



(t) =(

1 + i cotht

4g






(26)int infin





6



cos(ut) = 2sumn1


cos ϕ

J2nminus1(t)

t

(28)sin(ut) = 2

sumn1

(2n)sin(2nϕ)

cos ϕ

J2n(t)

t





mO(6) = 8radic

2


πeminusπgRe

[int infin

0

dt ei(tminusπ4)


] (29)






γminus(t) = 2sumn1


(2n)J2n(t)γ2n (210)

7




0

dt


int infin

0

dt

tJ2n(t)γ+(t) (211)



cusp(g) = 8g2γ1(g) (212)



(it) = γ (it)sin

(t

4g+ π

4

)sin

(t

4g

)sin

(π4

) = γ (it)radic

2infinprod

k=minusinfin


4

)t minus 4πgk

(213)



4



23 Toy model






t (216)



2t+ 1


2tminus 1


8









γ1 =int infin

0

dt

t

(γ+(t))2 + (γminus(t))2



γ1 int infin

0

dt


1 0 (218)


0 γ1 12 (219)


0 cusp(g) 4g2 (220)



cusp(g) = 4g2

[1 minus 1

3π2g2 +

11

45π4g4 minus 2

(73

630π6 + 4ζ 2

3


] (221)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus1 minus K

16π2gminus2 minus

(3K ln 2

64π3+

27ζ3

2048π3


] (222)




9


0

02

04

06

08

1

05 1 15 2 25 3



0 =int infin

0

dt

t

(γ(0)+ (t))2 + (γ

(0)minus (t))2





(k) =int infin

minusinfin

dt

2πeikt(t) γ (k) =

int infin

minusinfin

dt

2πeiktγ (t) (224)


γ (k) = 0 for k2 gt 1 (225)

10





(k) =int infin

minusinfin

dt

2πeikt

sinh(

t4g

+ iπ4

)sinh

(t

4g

)sin

(π4

)γ (t) (227)




sumn1










c(toy)+ (n g) = c

(toy)minus (n g) = 0 (231)



+(t) =int infin


(232)




11



minusint infin

minusinfin

dk(k)



(u) +1

πminusint 1

minus1

dk(k)



φ(u) = minus 1

π

(2 +

int minus1

minusinfin

dk(k)

k minus u+

int infin

1

dk(k)

k minus u

) (235)



(k) = 1

2φ(k) minus 1

2π

(1 + k

1 minus k

)14

minusint 1

minus1

duφ(u)

u minus k

(1 minus u

1 + u

)14

minusradic

2

π

(1 + k

1 minus k

)14c

1 + k (236)


(k)k21= minus

radic2

π

(1 + k

1 minus k

)14[

1 +c

1 + k+

1

2

int infin

minusinfin

dp(p)

p minus k

(p minus 1

p + 1

)14

θ(p2 minus 1)

] (237)



26 General solution


(t) =int 1


int minus1


int infin



(it) = f0(t)V0(t) + f1(t)V1(t) (239)

12




t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

4πng + t

]

(240)


4πng

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

4πng + t

]


Vn(x) =radic

2

π

int 1


(241)

Uplusmnn (x) = 1

2

int infin




f(toy)


1 (t) = minusc(toy)(g) (242)







(4π ig

( minus 1

4

)) = 0 isin Z (244)



4

) (245)



cusp(g) = 2g[1 minus 2f1(0)] (246)

13



cusp(g) = 2g


[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

]⎫⎬⎭ (247)






V1(minusπg) (248)



[1 minus 2

V0(minusπg)

V1(minusπg)

] (249)


(toy)cusp (g) = 2g


Mminus140(2πg)

] (250)







couplings


(toy)cusp (g) = 3

2πg2 minus 1

2π2g3 minus 1

64π3g4 +

5

64π4g5 minus 11

512π5g6 minus 3

512π6g7 + O(g8) (251)



14






(

34

)[F

(1

4

5

4|α + i0

)+ 2F

(minus1

4

3

4| minus α

)] α = 1(2πg)

(252)



(34

)

(54



F

(1

4

5

4| minus α

)= αminus54U+

0 (1(2α))

(5

4

)

(254)

F

(minus1

4

3

4| minus α

)= αminus34Uminus

0 (1(2α))

(3

4

)



4 34 | minus α



4 and b = 34 )


(minusα)k

k

(a + k)(b + k)

(a)(b)= 1 minus αab + O(α2) (255)



(a)

int infin





(a)

int infin


15







F(

14 5

4 |α+)

+ 2F(minus 1

4 34 | minus α

)F

(14 1

4 |α+)

+ 142αF

(34 3

4 | minus α) (258)


(toy)cusp (g)



4α24C4(α) + O(6) (259)


C0 = 1 minus αF

(14 5

4 |α+)

F(

14 1

4 |α+) C2 = 1[

F(

14 1

4 |α+)]2 C4 = F

(34 3

4 | minus α)[

F(

14 1

4 |α+)]3 (260)



16



4α2 minus 3

8α3 minus 61

64α4 minus 433

128α5 + O(α6)

C2 = 1 minus 1

8α minus 11

128α2 minus 151

1024α3 minus 13085

32768α4 + O(α5) (261)

C4 = 1 minus 3

4α minus 27

32α2 minus 317

128α3 + O(α4)















17




2α minus K

4α2 minus

(3K ln 2

8+

27ζ3

256

)α3 + (263)



3 ln 2

2rarr 1

K

2rarr 1

2

9ζ3

32rarr 1

3 (264)



exp

[3 ln 2

2τ minus K

2τ 2 +

9ζ3

32τ 3 +

]rarr exp

[τ minus τ 2

2+

τ 3

3+

] (265)


(

14

)

(1 + τ

4

)

(34 minus τ

4

)

(34

)

(1 minus τ

4

)

(14 + τ

4

) rarr (1 + τ) (266)



cusp (g)




k (k + 1

2



(toy)cusp (g)

18















plusmn1 (4πng)

n ∓ x

(32)





V0(4πgx)=



(33)


19



g 01 02 04 06 08 10 12 14 16 18

Numer 01976 03616 05843 07096 07825 08276 08576 08787 08944 09065Exact 01939 03584 05821 07080 07813 08267 08568 08781 08938 09059


nmax 10 20 30 40 50 60 70 infinNumer 08305 08286 08279 08276 08274 08273 08272 08267


c(g) = 0 (34)











20








1 =sumn1

c+(n g)

[nU+

0 (4πng)r(x) + U+1 (4πng)x

n minus x

]

+sumn1

cminus(n g)

[nUminus


n + x

] (36)





4 isin Z) (37)




[aplusmn(n) +

bplusmn(n)


]+ 2

[αplusmn(n) +

βplusmn(n)


]+ O(4)

(38)

21




(34

)

(54

) (39)






4 )

2x

(5

4

) sumn1

a+(n)

n minus x

= 1 ( 1)

(310)

minus2x

(3

4

) sumn1

aminus(n)

n + x

= 1 ( 0)


a+(n) = 2(n + 1

4

)(n + 1)2

(14

)

(311)

aminus(n) = (n + 3

4

)2(n + 1)2

(34

)


4 )

2x

(5

4

) sumn1

b+(n)

n minus x

= minus 3

32x

minus 3π

64minus 15

32ln 2 ( 1)

(312)

minus 2x

(3

4

) sumn1

bminus(n)

n + x

= minus 5

32x

minus 5π

64+

9

32ln 2 ( 0)


b+(n) = minusa+(n)

(3 ln 2

4+

3

32n


(3 ln 2

4+

5

32n

) (313)




(2πn)minus1

[

(5

4

) (a+(n) +

b+(n)


)(1 minus 5

128πgn+

)

+

(3

4

) (aminus(n) +

bminus(n)


)(1 +

3


)]+ O(2) (314)

22



cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K

16π2g2+ O(1g3)

]+ O(2) (315)




4 + O(gminus1)]

+ O(4) (316)



2x

(5

4

) sumn1

α+(n)

n minus x

= 0 ( 1)

(317)

minus2x

(3

4

) sumn1

αminus(n)

n + x

= π

2radic

2δ0 ( 0)


α+(n) = 0(318)



2x

(5

4

) sumn1

β+(n)

n minus x

= minus1

2 ( 1)

(319)

minus2x

(3

4

) sumn1

βminus(n)

n + x

= minus1

8+

3π

16radic

2(1 minus 2 ln 2)δ0 ( 0)


β+(n) = minus1

2a+(n)

(320)


(1

4minus 3 ln 2

4+

1

32n

)



23




(2πn)minus1

[

(5

4

) (α+(n) +

β+(n)


)(1 minus 5


)

+

(3

4

) (αminus(n) +

βminus(n)


)(1 +

3


)]+ O(4) (321)


δcusp(g) = minus2

π

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (322)




m2cusp = 4

radic2

πσ2

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (323)


cusp(g) =[

2g minus 3 ln 2

2πminus K

8π2g+ O(1g2)

]minus σ

4radic

2m2

cusp + O(m4

cusp

) (324)


4 Mass scale




mO(6) = 8radic

2


πeminusπgRe

[int minusiinfin


t + 14

] (41)



24



f0(4πgt) =sumn1

t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

n + t

]minus 1

(43)

f1(4πgt) =sumn1

n

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

n + t

]





2




mO(6) = 16radic

2

π2

f1(minusπg)

V0(minusπg) (45)







fn(4πgt) = f (PT)

n (4πgt) + δfn(4πgt) (n = 0 1) (46)

where f (PT)



2

(5

4

) sumn1

a+(n)

t minus n= 1

t

[

(34

)(1 minus t)

(

34 minus t

) minus 1

]

(47)

2

(3

4

) sumn1

aminus(n)

t + n= 1

t

[

(14

)(1 + t)

(

14 + t

) minus 1

]

25



f(PT)

0 (4πgt) = minus(

34

)(1 minus t)

(

34 minus t

)+

1

4πg

[(3 ln 2

4+

1

8t

)

(34

)(1 minus t)

(

34 minus t

) minus (

14

)(1 + t)

8t(

14 + t

) ]+ O(gminus2)

f(PT)

1 (4πgt) = 1

4πg

[

(14

)(1 + t)

4t(

14 + t

) minus (

34

)(1 minus t)

4t(

34 minus t

) ]

minus 1

(4πg)2

[

(14

)(1 + t)

4t(

14 + t

) (1

4tminus 3 ln 2

4

)

minus (

34

)(1 minus t)

4t(

34 minus t

) (1

4t+

3 ln 2

4

)]+ O(gminus3) (48)


δf0(4πgt) = 2

1

4πg

[

(34

)(1 minus t)

2(

34 minus t

) minus (

54

)(1 + t)

2(

54 + t

) ]+ O(gminus2)


δf1(4πgt) = 2

1

4πg

(

54

)(1 + t)

(

54 + t

) +1

(4πg)2

[

(34

)(1 minus t)

8t(

34 minus t

)minus

(54

)(1 + t)

(

54 + t

) (1

8t+

3

4ln 2 minus 1

4

)]+ O(gminus3)





1 (0) minus 4gδf1(0) (410)

Replacing f(PT)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K


]minus 2

π

[1 +

3 minus 6 ln 2


]+ O(4) (411)



mO(6) =radic

2

(

54


[1 +

3 minus 6 ln 2

32πg+



]minus 2

8πg



]+ O(4)

(412)


(PT)


26




fn(4πgt)

(34 minus t

)

(34

)(1 minus t)

rarr f (toy)n (4πgt) (n = 0 1) (413)





(toy)


mtoy = 16radic

2

π2

f(toy)

1 (minusπg)

V0(minusπg)= 16

radic2

π2

1

V1(minusπg) (414)




mtoy = 4

π(

54


[1 minus 1

32πgminus 23


]minus 2

8πg

[1 minus 11


]+ O(4)

(415)


0 and f(toy)


27






32radic

2σm2

toy + O(m4

toy

) (416)




2radic

2mtoy (417)



4π (418)


cusp (g + c1) = 2g

[1 minus K


]minus 2

2radic

2π

[1 +

3


]+ O(4) (419)


[mO(6)(g + c1)]2 = 22

πσ

[(1 +

3

16πg+

16K minus 54


)minus 2

8radic

2πg

(1 minus 3


)+ O(4)

] (420)


O(6)

δcusp = minus σ

4radic

2m2

O(6) + O(m4

O(6)

) (421)



28











2

πeminusπgUminus


2

πeminusπgUminus

1 (πg) (423)





4




mprime = minus σ

8gmO(6) (426)




2

4mprimemO(6) = σ

radic2

32gm2

O(6) (427)



δ(it) = δf0(t)V0(t) + δf1(t)V1(t) (428)


δ(0) = 2δf1(0) (429)

29






δ(it) sim 1







δ(it) (433)


δγ (it) = t

4gδ(0) + O(t2) = t

2gδf1(0) + O(t2) (434)


0

dt

tJ2nminus1(t)

[δγminus(t)

1 minus eminust2g+

δγ+(t)

et2g minus 1

]= 0

(435)int infin

0

dt

tJ2n(t)

[δγ+(t)


et2g minus 1

]= 0



30




2

π2geminusπg minus

radic2

πeminusπg Re

int infin

0

dt prime


]+ O(t2) (436)



tradic

2

8g+ O(t2) (437)


mcusp(g) = mO(6)(g) (438)


5 Conclusions






31





(m4

O(n)

μ2) (51)


m2O(n) = cμ2 e


20 [1 + O(g2)] (52)




Acknowledgments


32








γ1 = 1

2minus π2

6g2 +

11π4

90g4 minus

(73π6

630+ 4ζ 2

3

)g6 + O(g8)

γ2 = ζ3g3 minus

(π2

3ζ3 + 10ζ5

)g5 +

(8π4

45ζ3 +

10π2

3ζ5 + 105ζ7

)g7 + O(g9)

γ3 = minusπ4

90g4 +

37π6

1890g6 + O(g8) γ4 = ζ5g

5 minus(

π2

3ζ5 + 21ζ7

)g7 + O(g9)

γ5 = minus π6

945g6 + O(g8) γ6 = ζ7g

7 + O(g9)

(A2)



3g4 +

44π4

45g6 minus

(292π6

315+ 32ζ 2

3

)g8 + O(g10) (A3)







33



c(g) = minus1

2+ 2gγ1(g) +

sumn1

[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

] (A5)


c(g) = minus1

2+ 2gγ1(g) minus

sumk1



fk(g) = 8gsumn1

[U+






3g2 + 2ζ3g

3 minus π4

6g4 minus 23ζ5g

5 +17π6

108g6 +

1107

4ζ7g

7 + O(g8)

f2 = minus1

2+ 2ζ3g

3 minus π4

30g4 + O(g5) f3 = 1

2+ O(g4)

f4 = minus3

8+ O(g5) f5 = 3

8+ O(g6) f6 = minus 5

16+ O(g7)

(A8)






(t) =int infin




(k)=sumn1



(k)=sumn1


bull For minus1 k 1

(k) = minusradic

2

π

(1 + k

1 minus k

)14[

1 +c(g)

1 + k+

1

2

(int minus1

minusinfin+

int infin

1

)dp(p)

p minus k

(p minus 1

p + 1

)14]

(B4)


34



(t) =sumn1

c+(n g)

[eminusit


]

+sumn1

cminus(n g)

[eit



(B5)



int 1

minus1dk eplusmnxk

int infin

1

dpeminusy(pminus1)

p minus k

(1 + k

1 minus k

p minus 1

p + 1

)plusmn14

Vn(x) =radic

2

π

int 1

minus1

dk exk

(k + 1)n

(1 + k

1 minus k

)14

(B6)

Uplusmnn (y) = 1

2

int infin

1

dp eminusy(pminus1)

(p ∓ 1)n

(p + 1

p minus 1

)∓14






0 (y) minus ex



c+(n g)

[4πngV1(t)U

+0 (4πng) + tV0(t)U

+1 (4πng)

4πng minus t

]

+sumn1

cminus(n g)

[4πngV1(t)U


minus1 (4πng)

4πng + t

] (B8)





int infin

0dsV0(x + s)U+






0 (x)


d

dx(xzprime

i (x)) =(

x minus 1

2

)zi(x) (B11)

35




2(y + s)]

= d


2(y + s)]z1(x + s)

minus d


1(x + s)]z2(y + s) (B12)



1 (y + s) + (y + s)V1(x + s)U+0 (y + s)

] ∣∣s=infins=0


0 (y) (B13)






π

t

t + πg (C1)



π

t2

t2 + (πg)2

(C2)


t2 + π2g2


0

dt

tJ2nminus1(t)

[γminus(t)

1 minus eminust2g+

γ+(t)

et2g minus 1

]= h2nminus1(g)

(C3)int infin

0

dt

tJ2n(t)

[γ+(t)


et2g minus 1

]= h2n(g)


h2nminus1 = 2mprime

π

int infin

0

dtJ2nminus1(t)

t2 + (πg)2

[t


1 minus eminust(2g)

]

(C4)

h2n = 2mprime

π

int infin

0

dtJ2n(t)

t2 + (πg)2

[πg

et(2g) minus 1+

t

1 minus eminust(2g)

]



36



γminus(t) = 2sumn1


(C6)γ+(t) = 2

sumn1

(2n)J2n(t)γ2n(g)



π2gt + O(t2) = it

(γ1 +

2mprime

π2g

)+ O(t2) (C7)



0

dt

t


1 minus eminust2g+


et2g minus 1

]= 2

sumn1



γ1 = 2sumn1



γ1 = minusmprime

π

int infin

0dt

[πg


t

t2 + π2g2(minus(t) + +(t))

] (C10)


πg

t2 + π2g2=

int infin


(C11)t

t2 + π2g2=

int infin



γ1 = minusmprime

π

int infin

0du eminusπgu

[ int infin


+int infin


] (C12)


γ1 = minus 2mprime


radic2mprime

πeminusπgRe

[ int infin

0

dt


] (C13)


37






Vn(x) = 1

π232minusn ex

int 1


= 1

π232minusn eminusx

int 1



Vn(x) = 2minusn

(54 minus n

)

(54

)(2 minus n)


= 2minusn

(54 minus n

)

(54

)(2 minus n)







Uab(x) = 1

2

int infin





(1 +

t

2x

)a+bminus12

(D5)


Uab(x) = 2bminus32

(1

2minus a + b


= 1

2

(1

2minus a + b

)U

(1


) (D6)

38




U+0 (x) = 1

2

(5

4


1 (x) = 1

2

(1

4


(D7)

Uminus0 (x) = 1

2

(3

4


1 (x) = 1

2

(3

4





1 (x) = V1(x)U+0 (x) + V0(x)U+

1 (x) = ex

x (D8)


U+0 (x)Uminus


0 (minusx) = π

2radic

2xeplusmn 3iπ

4 (for Imx ≷ 0) (D9)


V0(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]

(D10)

V1(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]






(D11)Uplusmn








U+0 (x) = (2x)minus54

(5

4

)F

(1

4

5

4| minus 1

2x

)= (2x)minus54

(5

4

) [1 minus 5


]


(3

4

)F

(minus1

4

3

4| minus 1

2x

)= (2x)minus34

(3

4

) [1 +

3


]

39


U+1 (x) = (2x)minus14 1

2

(1

4

)F

(1

4

1

4| minus 1

2x

)= (2x)minus14 1

2

(1

4

) [1 minus 1


]


2

(3

4

)F

(3

4

3

4| minus 1

2x

)= (2x)minus34 1

2

(3

4

) [1 minus 9


]

(D12)








2

πeminus 3iπ

4 eπg[U+


] (D13)




(

34

) [F

(1

4

5

4| 1

2πg+ iε

)+ 2F

(minus1

4

3

4| minus 1

2πg

)] (D14)


2 = σ

(34

)

(54




2(

34

) [8πgF

(1

4

1

4| 1

2πg+ iε

)+ 2F

(3

4

3

4| minus 1

2πg

)] (D16)





F

(1

4

5

4| 1

2πg+ iε

)minus F

(1

4

5

4| 1

2πgminus iε

)= i

radic22

σF

(minus1

4

3

4| minus 1

2πg

) (D17)



40





t + 14


(4πgit)

t + 14

=sumk=01


t + 14

+sumk=01


t + 14

Vk(4πgt) (E1)


Rk(4πgs) = Re

[int minusiinfin


t minus s

]= Re

[int minusiinfin


t minus 4πgs

]

(E2)


Re

[ int minusiinfin


t + 14


minussumn1

nc+(n g)

n + 14

[U+


]+

sumn1

ncminus(n g)

n minus 14





R0(s) = minusRe

[eminusiπ4

int 1


int minusiinfin

0dt eminusvtV0(t)

]=

radic2

πRe

[eminusiπ4

int 1


int 1

minus1du


u minus v minus iε

] (E4)


1radic2π

int 1

minus1du



⎧⎨⎩(

v+1vminus1

)14 v2 gt 1

eminusiπ4(

1+v1minusv

)14 v2 lt 1

(E5)


R0(s)sgt0=

radic2

[1

sminus

int minus1


(v + 1

v minus 1

)14]

=radic

2

[1

sminus 2 eminus2sU+

0 (s)

] (E6)


R0(s)slt0=

radic2

[1

s+ 2Uminus

0 (minuss)

] (E7)

41


together with

R1(s) = 2radic

2[θ(minuss)Uminus


] (E8)


Re

[ int minusiinfin


t + 14

]= 2

radic2f0(minusπg)

[Uminus

0 (πg) minus 1

2πg

]+ 2


1 (πg) +

radic2



References



















42





























Ann Phys 167 227

43

























44

1 Introduction

2 Cusp anomalous



23 Toy model



26 General solution







4 Mass scale





5 Conclusions

Acknowledgments








References








1 Introduction







2






cusp(g) =infinsum

k=minus1

ckgk minus σ

4radic

2m2

cusp + o(m2

cusp

) (12)



(k + 1

2









3



cusp





εO(6) = ε(g j) + j

2 mO(6) = kg14 eminusπg [1 + O(1g)] (13)


(54




ε(j g) + j = m2

[j

m+

π2

24

(j

m

)3

+ O(j 4m4)

] (14)


ε(g j) + j = 22

[g +

1

π

(3

4minus ln

)+

1

4π2g

(q02


)+O(1g2)

]+ O(4)



4




mcusp = mO(6) (15)





5







γ (t)t (21)


γ (t) = γ+(t) + iγminus(t) (22)


0

dt

tJ2nminus1(t)

[γminus(t)


γ+(t)

et(2g) minus 1

]= 1

2δn1

(23)int infin

0

dt

tJ2n(t)

[γ+(t)


et(2g) minus 1

]= 0



(t) =(

1 + i cotht

4g






(26)int infin





6



cos(ut) = 2sumn1


cos ϕ

J2nminus1(t)

t

(28)sin(ut) = 2

sumn1

(2n)sin(2nϕ)

cos ϕ

J2n(t)

t





mO(6) = 8radic

2


πeminusπgRe

[int infin

0

dt ei(tminusπ4)


] (29)






γminus(t) = 2sumn1


(2n)J2n(t)γ2n (210)

7




0

dt


int infin

0

dt

tJ2n(t)γ+(t) (211)



cusp(g) = 8g2γ1(g) (212)



(it) = γ (it)sin

(t

4g+ π

4

)sin

(t

4g

)sin

(π4

) = γ (it)radic

2infinprod

k=minusinfin


4

)t minus 4πgk

(213)



4



23 Toy model






t (216)



2t+ 1


2tminus 1


8









γ1 =int infin

0

dt

t

(γ+(t))2 + (γminus(t))2



γ1 int infin

0

dt


1 0 (218)


0 γ1 12 (219)


0 cusp(g) 4g2 (220)



cusp(g) = 4g2

[1 minus 1

3π2g2 +

11

45π4g4 minus 2

(73

630π6 + 4ζ 2

3


] (221)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus1 minus K

16π2gminus2 minus

(3K ln 2

64π3+

27ζ3

2048π3


] (222)




9


0

02

04

06

08

1

05 1 15 2 25 3



0 =int infin

0

dt

t

(γ(0)+ (t))2 + (γ

(0)minus (t))2





(k) =int infin

minusinfin

dt

2πeikt(t) γ (k) =

int infin

minusinfin

dt

2πeiktγ (t) (224)


γ (k) = 0 for k2 gt 1 (225)

10





(k) =int infin

minusinfin

dt

2πeikt

sinh(

t4g

+ iπ4

)sinh

(t

4g

)sin

(π4

)γ (t) (227)




sumn1










c(toy)+ (n g) = c

(toy)minus (n g) = 0 (231)



+(t) =int infin


(232)




11



minusint infin

minusinfin

dk(k)



(u) +1

πminusint 1

minus1

dk(k)



φ(u) = minus 1

π

(2 +

int minus1

minusinfin

dk(k)

k minus u+

int infin

1

dk(k)

k minus u

) (235)



(k) = 1

2φ(k) minus 1

2π

(1 + k

1 minus k

)14

minusint 1

minus1

duφ(u)

u minus k

(1 minus u

1 + u

)14

minusradic

2

π

(1 + k

1 minus k

)14c

1 + k (236)


(k)k21= minus

radic2

π

(1 + k

1 minus k

)14[

1 +c

1 + k+

1

2

int infin

minusinfin

dp(p)

p minus k

(p minus 1

p + 1

)14

θ(p2 minus 1)

] (237)



26 General solution


(t) =int 1


int minus1


int infin



(it) = f0(t)V0(t) + f1(t)V1(t) (239)

12




t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

4πng + t

]

(240)


4πng

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

4πng + t

]


Vn(x) =radic

2

π

int 1


(241)

Uplusmnn (x) = 1

2

int infin




f(toy)


1 (t) = minusc(toy)(g) (242)







(4π ig

( minus 1

4

)) = 0 isin Z (244)



4

) (245)



cusp(g) = 2g[1 minus 2f1(0)] (246)

13



cusp(g) = 2g


[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

]⎫⎬⎭ (247)






V1(minusπg) (248)



[1 minus 2

V0(minusπg)

V1(minusπg)

] (249)


(toy)cusp (g) = 2g


Mminus140(2πg)

] (250)







couplings


(toy)cusp (g) = 3

2πg2 minus 1

2π2g3 minus 1

64π3g4 +

5

64π4g5 minus 11

512π5g6 minus 3

512π6g7 + O(g8) (251)



14






(

34

)[F

(1

4

5

4|α + i0

)+ 2F

(minus1

4

3

4| minus α

)] α = 1(2πg)

(252)



(34

)

(54



F

(1

4

5

4| minus α

)= αminus54U+

0 (1(2α))

(5

4

)

(254)

F

(minus1

4

3

4| minus α

)= αminus34Uminus

0 (1(2α))

(3

4

)



4 34 | minus α



4 and b = 34 )


(minusα)k

k

(a + k)(b + k)

(a)(b)= 1 minus αab + O(α2) (255)



(a)

int infin





(a)

int infin


15







F(

14 5

4 |α+)

+ 2F(minus 1

4 34 | minus α

)F

(14 1

4 |α+)

+ 142αF

(34 3

4 | minus α) (258)


(toy)cusp (g)



4α24C4(α) + O(6) (259)


C0 = 1 minus αF

(14 5

4 |α+)

F(

14 1

4 |α+) C2 = 1[

F(

14 1

4 |α+)]2 C4 = F

(34 3

4 | minus α)[

F(

14 1

4 |α+)]3 (260)



16



4α2 minus 3

8α3 minus 61

64α4 minus 433

128α5 + O(α6)

C2 = 1 minus 1

8α minus 11

128α2 minus 151

1024α3 minus 13085

32768α4 + O(α5) (261)

C4 = 1 minus 3

4α minus 27

32α2 minus 317

128α3 + O(α4)















17




2α minus K

4α2 minus

(3K ln 2

8+

27ζ3

256

)α3 + (263)



3 ln 2

2rarr 1

K

2rarr 1

2

9ζ3

32rarr 1

3 (264)



exp

[3 ln 2

2τ minus K

2τ 2 +

9ζ3

32τ 3 +

]rarr exp

[τ minus τ 2

2+

τ 3

3+

] (265)


(

14

)

(1 + τ

4

)

(34 minus τ

4

)

(34

)

(1 minus τ

4

)

(14 + τ

4

) rarr (1 + τ) (266)



cusp (g)




k (k + 1

2



(toy)cusp (g)

18















plusmn1 (4πng)

n ∓ x

(32)





V0(4πgx)=



(33)


19



g 01 02 04 06 08 10 12 14 16 18

Numer 01976 03616 05843 07096 07825 08276 08576 08787 08944 09065Exact 01939 03584 05821 07080 07813 08267 08568 08781 08938 09059


nmax 10 20 30 40 50 60 70 infinNumer 08305 08286 08279 08276 08274 08273 08272 08267


c(g) = 0 (34)











20








1 =sumn1

c+(n g)

[nU+

0 (4πng)r(x) + U+1 (4πng)x

n minus x

]

+sumn1

cminus(n g)

[nUminus


n + x

] (36)





4 isin Z) (37)




[aplusmn(n) +

bplusmn(n)


]+ 2

[αplusmn(n) +

βplusmn(n)


]+ O(4)

(38)

21




(34

)

(54

) (39)






4 )

2x

(5

4

) sumn1

a+(n)

n minus x

= 1 ( 1)

(310)

minus2x

(3

4

) sumn1

aminus(n)

n + x

= 1 ( 0)


a+(n) = 2(n + 1

4

)(n + 1)2

(14

)

(311)

aminus(n) = (n + 3

4

)2(n + 1)2

(34

)


4 )

2x

(5

4

) sumn1

b+(n)

n minus x

= minus 3

32x

minus 3π

64minus 15

32ln 2 ( 1)

(312)

minus 2x

(3

4

) sumn1

bminus(n)

n + x

= minus 5

32x

minus 5π

64+

9

32ln 2 ( 0)


b+(n) = minusa+(n)

(3 ln 2

4+

3

32n


(3 ln 2

4+

5

32n

) (313)




(2πn)minus1

[

(5

4

) (a+(n) +

b+(n)


)(1 minus 5

128πgn+

)

+

(3

4

) (aminus(n) +

bminus(n)


)(1 +

3


)]+ O(2) (314)

22



cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K

16π2g2+ O(1g3)

]+ O(2) (315)




4 + O(gminus1)]

+ O(4) (316)



2x

(5

4

) sumn1

α+(n)

n minus x

= 0 ( 1)

(317)

minus2x

(3

4

) sumn1

αminus(n)

n + x

= π

2radic

2δ0 ( 0)


α+(n) = 0(318)



2x

(5

4

) sumn1

β+(n)

n minus x

= minus1

2 ( 1)

(319)

minus2x

(3

4

) sumn1

βminus(n)

n + x

= minus1

8+

3π

16radic

2(1 minus 2 ln 2)δ0 ( 0)


β+(n) = minus1

2a+(n)

(320)


(1

4minus 3 ln 2

4+

1

32n

)



23




(2πn)minus1

[

(5

4

) (α+(n) +

β+(n)


)(1 minus 5


)

+

(3

4

) (αminus(n) +

βminus(n)


)(1 +

3


)]+ O(4) (321)


δcusp(g) = minus2

π

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (322)




m2cusp = 4

radic2

πσ2

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (323)


cusp(g) =[

2g minus 3 ln 2

2πminus K

8π2g+ O(1g2)

]minus σ

4radic

2m2

cusp + O(m4

cusp

) (324)


4 Mass scale




mO(6) = 8radic

2


πeminusπgRe

[int minusiinfin


t + 14

] (41)



24



f0(4πgt) =sumn1

t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

n + t

]minus 1

(43)

f1(4πgt) =sumn1

n

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

n + t

]





2




mO(6) = 16radic

2

π2

f1(minusπg)

V0(minusπg) (45)







fn(4πgt) = f (PT)

n (4πgt) + δfn(4πgt) (n = 0 1) (46)

where f (PT)



2

(5

4

) sumn1

a+(n)

t minus n= 1

t

[

(34

)(1 minus t)

(

34 minus t

) minus 1

]

(47)

2

(3

4

) sumn1

aminus(n)

t + n= 1

t

[

(14

)(1 + t)

(

14 + t

) minus 1

]

25



f(PT)

0 (4πgt) = minus(

34

)(1 minus t)

(

34 minus t

)+

1

4πg

[(3 ln 2

4+

1

8t

)

(34

)(1 minus t)

(

34 minus t

) minus (

14

)(1 + t)

8t(

14 + t

) ]+ O(gminus2)

f(PT)

1 (4πgt) = 1

4πg

[

(14

)(1 + t)

4t(

14 + t

) minus (

34

)(1 minus t)

4t(

34 minus t

) ]

minus 1

(4πg)2

[

(14

)(1 + t)

4t(

14 + t

) (1

4tminus 3 ln 2

4

)

minus (

34

)(1 minus t)

4t(

34 minus t

) (1

4t+

3 ln 2

4

)]+ O(gminus3) (48)


δf0(4πgt) = 2

1

4πg

[

(34

)(1 minus t)

2(

34 minus t

) minus (

54

)(1 + t)

2(

54 + t

) ]+ O(gminus2)


δf1(4πgt) = 2

1

4πg

(

54

)(1 + t)

(

54 + t

) +1

(4πg)2

[

(34

)(1 minus t)

8t(

34 minus t

)minus

(54

)(1 + t)

(

54 + t

) (1

8t+

3

4ln 2 minus 1

4

)]+ O(gminus3)





1 (0) minus 4gδf1(0) (410)

Replacing f(PT)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K


]minus 2

π

[1 +

3 minus 6 ln 2


]+ O(4) (411)



mO(6) =radic

2

(

54


[1 +

3 minus 6 ln 2

32πg+



]minus 2

8πg



]+ O(4)

(412)


(PT)


26




fn(4πgt)

(34 minus t

)

(34

)(1 minus t)

rarr f (toy)n (4πgt) (n = 0 1) (413)





(toy)


mtoy = 16radic

2

π2

f(toy)

1 (minusπg)

V0(minusπg)= 16

radic2

π2

1

V1(minusπg) (414)




mtoy = 4

π(

54


[1 minus 1

32πgminus 23


]minus 2

8πg

[1 minus 11


]+ O(4)

(415)


0 and f(toy)


27






32radic

2σm2

toy + O(m4

toy

) (416)




2radic

2mtoy (417)



4π (418)


cusp (g + c1) = 2g

[1 minus K


]minus 2

2radic

2π

[1 +

3


]+ O(4) (419)


[mO(6)(g + c1)]2 = 22

πσ

[(1 +

3

16πg+

16K minus 54


)minus 2

8radic

2πg

(1 minus 3


)+ O(4)

] (420)


O(6)

δcusp = minus σ

4radic

2m2

O(6) + O(m4

O(6)

) (421)



28











2

πeminusπgUminus


2

πeminusπgUminus

1 (πg) (423)





4




mprime = minus σ

8gmO(6) (426)




2

4mprimemO(6) = σ

radic2

32gm2

O(6) (427)



δ(it) = δf0(t)V0(t) + δf1(t)V1(t) (428)


δ(0) = 2δf1(0) (429)

29






δ(it) sim 1







δ(it) (433)


δγ (it) = t

4gδ(0) + O(t2) = t

2gδf1(0) + O(t2) (434)


0

dt

tJ2nminus1(t)

[δγminus(t)

1 minus eminust2g+

δγ+(t)

et2g minus 1

]= 0

(435)int infin

0

dt

tJ2n(t)

[δγ+(t)


et2g minus 1

]= 0



30




2

π2geminusπg minus

radic2

πeminusπg Re

int infin

0

dt prime


]+ O(t2) (436)



tradic

2

8g+ O(t2) (437)


mcusp(g) = mO(6)(g) (438)


5 Conclusions






31





(m4

O(n)

μ2) (51)


m2O(n) = cμ2 e


20 [1 + O(g2)] (52)




Acknowledgments


32








γ1 = 1

2minus π2

6g2 +

11π4

90g4 minus

(73π6

630+ 4ζ 2

3

)g6 + O(g8)

γ2 = ζ3g3 minus

(π2

3ζ3 + 10ζ5

)g5 +

(8π4

45ζ3 +

10π2

3ζ5 + 105ζ7

)g7 + O(g9)

γ3 = minusπ4

90g4 +

37π6

1890g6 + O(g8) γ4 = ζ5g

5 minus(

π2

3ζ5 + 21ζ7

)g7 + O(g9)

γ5 = minus π6

945g6 + O(g8) γ6 = ζ7g

7 + O(g9)

(A2)



3g4 +

44π4

45g6 minus

(292π6

315+ 32ζ 2

3

)g8 + O(g10) (A3)







33



c(g) = minus1

2+ 2gγ1(g) +

sumn1

[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

] (A5)


c(g) = minus1

2+ 2gγ1(g) minus

sumk1



fk(g) = 8gsumn1

[U+






3g2 + 2ζ3g

3 minus π4

6g4 minus 23ζ5g

5 +17π6

108g6 +

1107

4ζ7g

7 + O(g8)

f2 = minus1

2+ 2ζ3g

3 minus π4

30g4 + O(g5) f3 = 1

2+ O(g4)

f4 = minus3

8+ O(g5) f5 = 3

8+ O(g6) f6 = minus 5

16+ O(g7)

(A8)






(t) =int infin




(k)=sumn1



(k)=sumn1


bull For minus1 k 1

(k) = minusradic

2

π

(1 + k

1 minus k

)14[

1 +c(g)

1 + k+

1

2

(int minus1

minusinfin+

int infin

1

)dp(p)

p minus k

(p minus 1

p + 1

)14]

(B4)


34



(t) =sumn1

c+(n g)

[eminusit


]

+sumn1

cminus(n g)

[eit



(B5)



int 1

minus1dk eplusmnxk

int infin

1

dpeminusy(pminus1)

p minus k

(1 + k

1 minus k

p minus 1

p + 1

)plusmn14

Vn(x) =radic

2

π

int 1

minus1

dk exk

(k + 1)n

(1 + k

1 minus k

)14

(B6)

Uplusmnn (y) = 1

2

int infin

1

dp eminusy(pminus1)

(p ∓ 1)n

(p + 1

p minus 1

)∓14






0 (y) minus ex



c+(n g)

[4πngV1(t)U

+0 (4πng) + tV0(t)U

+1 (4πng)

4πng minus t

]

+sumn1

cminus(n g)

[4πngV1(t)U


minus1 (4πng)

4πng + t

] (B8)





int infin

0dsV0(x + s)U+






0 (x)


d

dx(xzprime

i (x)) =(

x minus 1

2

)zi(x) (B11)

35




2(y + s)]

= d


2(y + s)]z1(x + s)

minus d


1(x + s)]z2(y + s) (B12)



1 (y + s) + (y + s)V1(x + s)U+0 (y + s)

] ∣∣s=infins=0


0 (y) (B13)






π

t

t + πg (C1)



π

t2

t2 + (πg)2

(C2)


t2 + π2g2


0

dt

tJ2nminus1(t)

[γminus(t)

1 minus eminust2g+

γ+(t)

et2g minus 1

]= h2nminus1(g)

(C3)int infin

0

dt

tJ2n(t)

[γ+(t)


et2g minus 1

]= h2n(g)


h2nminus1 = 2mprime

π

int infin

0

dtJ2nminus1(t)

t2 + (πg)2

[t


1 minus eminust(2g)

]

(C4)

h2n = 2mprime

π

int infin

0

dtJ2n(t)

t2 + (πg)2

[πg

et(2g) minus 1+

t

1 minus eminust(2g)

]



36



γminus(t) = 2sumn1


(C6)γ+(t) = 2

sumn1

(2n)J2n(t)γ2n(g)



π2gt + O(t2) = it

(γ1 +

2mprime

π2g

)+ O(t2) (C7)



0

dt

t


1 minus eminust2g+


et2g minus 1

]= 2

sumn1



γ1 = 2sumn1



γ1 = minusmprime

π

int infin

0dt

[πg


t

t2 + π2g2(minus(t) + +(t))

] (C10)


πg

t2 + π2g2=

int infin


(C11)t

t2 + π2g2=

int infin



γ1 = minusmprime

π

int infin

0du eminusπgu

[ int infin


+int infin


] (C12)


γ1 = minus 2mprime


radic2mprime

πeminusπgRe

[ int infin

0

dt


] (C13)


37






Vn(x) = 1

π232minusn ex

int 1


= 1

π232minusn eminusx

int 1



Vn(x) = 2minusn

(54 minus n

)

(54

)(2 minus n)


= 2minusn

(54 minus n

)

(54

)(2 minus n)







Uab(x) = 1

2

int infin





(1 +

t

2x

)a+bminus12

(D5)


Uab(x) = 2bminus32

(1

2minus a + b


= 1

2

(1

2minus a + b

)U

(1


) (D6)

38




U+0 (x) = 1

2

(5

4


1 (x) = 1

2

(1

4


(D7)

Uminus0 (x) = 1

2

(3

4


1 (x) = 1

2

(3

4





1 (x) = V1(x)U+0 (x) + V0(x)U+

1 (x) = ex

x (D8)


U+0 (x)Uminus


0 (minusx) = π

2radic

2xeplusmn 3iπ

4 (for Imx ≷ 0) (D9)


V0(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]

(D10)

V1(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]






(D11)Uplusmn








U+0 (x) = (2x)minus54

(5

4

)F

(1

4

5

4| minus 1

2x

)= (2x)minus54

(5

4

) [1 minus 5


]


(3

4

)F

(minus1

4

3

4| minus 1

2x

)= (2x)minus34

(3

4

) [1 +

3


]

39


U+1 (x) = (2x)minus14 1

2

(1

4

)F

(1

4

1

4| minus 1

2x

)= (2x)minus14 1

2

(1

4

) [1 minus 1


]


2

(3

4

)F

(3

4

3

4| minus 1

2x

)= (2x)minus34 1

2

(3

4

) [1 minus 9


]

(D12)








2

πeminus 3iπ

4 eπg[U+


] (D13)




(

34

) [F

(1

4

5

4| 1

2πg+ iε

)+ 2F

(minus1

4

3

4| minus 1

2πg

)] (D14)


2 = σ

(34

)

(54




2(

34

) [8πgF

(1

4

1

4| 1

2πg+ iε

)+ 2F

(3

4

3

4| minus 1

2πg

)] (D16)





F

(1

4

5

4| 1

2πg+ iε

)minus F

(1

4

5

4| 1

2πgminus iε

)= i

radic22

σF

(minus1

4

3

4| minus 1

2πg

) (D17)



40





t + 14


(4πgit)

t + 14

=sumk=01


t + 14

+sumk=01


t + 14

Vk(4πgt) (E1)


Rk(4πgs) = Re

[int minusiinfin


t minus s

]= Re

[int minusiinfin


t minus 4πgs

]

(E2)


Re

[ int minusiinfin


t + 14


minussumn1

nc+(n g)

n + 14

[U+


]+

sumn1

ncminus(n g)

n minus 14





R0(s) = minusRe

[eminusiπ4

int 1


int minusiinfin

0dt eminusvtV0(t)

]=

radic2

πRe

[eminusiπ4

int 1


int 1

minus1du


u minus v minus iε

] (E4)


1radic2π

int 1

minus1du



⎧⎨⎩(

v+1vminus1

)14 v2 gt 1

eminusiπ4(

1+v1minusv

)14 v2 lt 1

(E5)


R0(s)sgt0=

radic2

[1

sminus

int minus1


(v + 1

v minus 1

)14]

=radic

2

[1

sminus 2 eminus2sU+

0 (s)

] (E6)


R0(s)slt0=

radic2

[1

s+ 2Uminus

0 (minuss)

] (E7)

41


together with

R1(s) = 2radic

2[θ(minuss)Uminus


] (E8)


Re

[ int minusiinfin


t + 14

]= 2

radic2f0(minusπg)

[Uminus

0 (πg) minus 1

2πg

]+ 2


1 (πg) +

radic2



References



















42





























Ann Phys 167 227

43

























44

1 Introduction

2 Cusp anomalous



23 Toy model



26 General solution







4 Mass scale





5 Conclusions

Acknowledgments








References






cusp(g) =infinsum

k=minus1

ckgk minus σ

4radic

2m2

cusp + o(m2

cusp

) (12)



(k + 1

2









3



cusp





εO(6) = ε(g j) + j

2 mO(6) = kg14 eminusπg [1 + O(1g)] (13)


(54




ε(j g) + j = m2

[j

m+

π2

24

(j

m

)3

+ O(j 4m4)

] (14)


ε(g j) + j = 22

[g +

1

π

(3

4minus ln

)+

1

4π2g

(q02


)+O(1g2)

]+ O(4)



4




mcusp = mO(6) (15)





5







γ (t)t (21)


γ (t) = γ+(t) + iγminus(t) (22)


0

dt

tJ2nminus1(t)

[γminus(t)


γ+(t)

et(2g) minus 1

]= 1

2δn1

(23)int infin

0

dt

tJ2n(t)

[γ+(t)


et(2g) minus 1

]= 0



(t) =(

1 + i cotht

4g






(26)int infin





6



cos(ut) = 2sumn1


cos ϕ

J2nminus1(t)

t

(28)sin(ut) = 2

sumn1

(2n)sin(2nϕ)

cos ϕ

J2n(t)

t





mO(6) = 8radic

2


πeminusπgRe

[int infin

0

dt ei(tminusπ4)


] (29)






γminus(t) = 2sumn1


(2n)J2n(t)γ2n (210)

7




0

dt


int infin

0

dt

tJ2n(t)γ+(t) (211)



cusp(g) = 8g2γ1(g) (212)



(it) = γ (it)sin

(t

4g+ π

4

)sin

(t

4g

)sin

(π4

) = γ (it)radic

2infinprod

k=minusinfin


4

)t minus 4πgk

(213)



4



23 Toy model






t (216)



2t+ 1


2tminus 1


8









γ1 =int infin

0

dt

t

(γ+(t))2 + (γminus(t))2



γ1 int infin

0

dt


1 0 (218)


0 γ1 12 (219)


0 cusp(g) 4g2 (220)



cusp(g) = 4g2

[1 minus 1

3π2g2 +

11

45π4g4 minus 2

(73

630π6 + 4ζ 2

3


] (221)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus1 minus K

16π2gminus2 minus

(3K ln 2

64π3+

27ζ3

2048π3


] (222)




9


0

02

04

06

08

1

05 1 15 2 25 3



0 =int infin

0

dt

t

(γ(0)+ (t))2 + (γ

(0)minus (t))2





(k) =int infin

minusinfin

dt

2πeikt(t) γ (k) =

int infin

minusinfin

dt

2πeiktγ (t) (224)


γ (k) = 0 for k2 gt 1 (225)

10





(k) =int infin

minusinfin

dt

2πeikt

sinh(

t4g

+ iπ4

)sinh

(t

4g

)sin

(π4

)γ (t) (227)




sumn1










c(toy)+ (n g) = c

(toy)minus (n g) = 0 (231)



+(t) =int infin


(232)




11



minusint infin

minusinfin

dk(k)



(u) +1

πminusint 1

minus1

dk(k)



φ(u) = minus 1

π

(2 +

int minus1

minusinfin

dk(k)

k minus u+

int infin

1

dk(k)

k minus u

) (235)



(k) = 1

2φ(k) minus 1

2π

(1 + k

1 minus k

)14

minusint 1

minus1

duφ(u)

u minus k

(1 minus u

1 + u

)14

minusradic

2

π

(1 + k

1 minus k

)14c

1 + k (236)


(k)k21= minus

radic2

π

(1 + k

1 minus k

)14[

1 +c

1 + k+

1

2

int infin

minusinfin

dp(p)

p minus k

(p minus 1

p + 1

)14

θ(p2 minus 1)

] (237)



26 General solution


(t) =int 1


int minus1


int infin



(it) = f0(t)V0(t) + f1(t)V1(t) (239)

12




t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

4πng + t

]

(240)


4πng

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

4πng + t

]


Vn(x) =radic

2

π

int 1


(241)

Uplusmnn (x) = 1

2

int infin




f(toy)


1 (t) = minusc(toy)(g) (242)







(4π ig

( minus 1

4

)) = 0 isin Z (244)



4

) (245)



cusp(g) = 2g[1 minus 2f1(0)] (246)

13



cusp(g) = 2g


[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

]⎫⎬⎭ (247)






V1(minusπg) (248)



[1 minus 2

V0(minusπg)

V1(minusπg)

] (249)


(toy)cusp (g) = 2g


Mminus140(2πg)

] (250)







couplings


(toy)cusp (g) = 3

2πg2 minus 1

2π2g3 minus 1

64π3g4 +

5

64π4g5 minus 11

512π5g6 minus 3

512π6g7 + O(g8) (251)



14






(

34

)[F

(1

4

5

4|α + i0

)+ 2F

(minus1

4

3

4| minus α

)] α = 1(2πg)

(252)



(34

)

(54



F

(1

4

5

4| minus α

)= αminus54U+

0 (1(2α))

(5

4

)

(254)

F

(minus1

4

3

4| minus α

)= αminus34Uminus

0 (1(2α))

(3

4

)



4 34 | minus α



4 and b = 34 )


(minusα)k

k

(a + k)(b + k)

(a)(b)= 1 minus αab + O(α2) (255)



(a)

int infin





(a)

int infin


15







F(

14 5

4 |α+)

+ 2F(minus 1

4 34 | minus α

)F

(14 1

4 |α+)

+ 142αF

(34 3

4 | minus α) (258)


(toy)cusp (g)



4α24C4(α) + O(6) (259)


C0 = 1 minus αF

(14 5

4 |α+)

F(

14 1

4 |α+) C2 = 1[

F(

14 1

4 |α+)]2 C4 = F

(34 3

4 | minus α)[

F(

14 1

4 |α+)]3 (260)



16



4α2 minus 3

8α3 minus 61

64α4 minus 433

128α5 + O(α6)

C2 = 1 minus 1

8α minus 11

128α2 minus 151

1024α3 minus 13085

32768α4 + O(α5) (261)

C4 = 1 minus 3

4α minus 27

32α2 minus 317

128α3 + O(α4)















17




2α minus K

4α2 minus

(3K ln 2

8+

27ζ3

256

)α3 + (263)



3 ln 2

2rarr 1

K

2rarr 1

2

9ζ3

32rarr 1

3 (264)



exp

[3 ln 2

2τ minus K

2τ 2 +

9ζ3

32τ 3 +

]rarr exp

[τ minus τ 2

2+

τ 3

3+

] (265)


(

14

)

(1 + τ

4

)

(34 minus τ

4

)

(34

)

(1 minus τ

4

)

(14 + τ

4

) rarr (1 + τ) (266)



cusp (g)




k (k + 1

2



(toy)cusp (g)

18















plusmn1 (4πng)

n ∓ x

(32)





V0(4πgx)=



(33)


19



g 01 02 04 06 08 10 12 14 16 18

Numer 01976 03616 05843 07096 07825 08276 08576 08787 08944 09065Exact 01939 03584 05821 07080 07813 08267 08568 08781 08938 09059


nmax 10 20 30 40 50 60 70 infinNumer 08305 08286 08279 08276 08274 08273 08272 08267


c(g) = 0 (34)











20








1 =sumn1

c+(n g)

[nU+

0 (4πng)r(x) + U+1 (4πng)x

n minus x

]

+sumn1

cminus(n g)

[nUminus


n + x

] (36)





4 isin Z) (37)




[aplusmn(n) +

bplusmn(n)


]+ 2

[αplusmn(n) +

βplusmn(n)


]+ O(4)

(38)

21




(34

)

(54

) (39)






4 )

2x

(5

4

) sumn1

a+(n)

n minus x

= 1 ( 1)

(310)

minus2x

(3

4

) sumn1

aminus(n)

n + x

= 1 ( 0)


a+(n) = 2(n + 1

4

)(n + 1)2

(14

)

(311)

aminus(n) = (n + 3

4

)2(n + 1)2

(34

)


4 )

2x

(5

4

) sumn1

b+(n)

n minus x

= minus 3

32x

minus 3π

64minus 15

32ln 2 ( 1)

(312)

minus 2x

(3

4

) sumn1

bminus(n)

n + x

= minus 5

32x

minus 5π

64+

9

32ln 2 ( 0)


b+(n) = minusa+(n)

(3 ln 2

4+

3

32n


(3 ln 2

4+

5

32n

) (313)




(2πn)minus1

[

(5

4

) (a+(n) +

b+(n)


)(1 minus 5

128πgn+

)

+

(3

4

) (aminus(n) +

bminus(n)


)(1 +

3


)]+ O(2) (314)

22



cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K

16π2g2+ O(1g3)

]+ O(2) (315)




4 + O(gminus1)]

+ O(4) (316)



2x

(5

4

) sumn1

α+(n)

n minus x

= 0 ( 1)

(317)

minus2x

(3

4

) sumn1

αminus(n)

n + x

= π

2radic

2δ0 ( 0)


α+(n) = 0(318)



2x

(5

4

) sumn1

β+(n)

n minus x

= minus1

2 ( 1)

(319)

minus2x

(3

4

) sumn1

βminus(n)

n + x

= minus1

8+

3π

16radic

2(1 minus 2 ln 2)δ0 ( 0)


β+(n) = minus1

2a+(n)

(320)


(1

4minus 3 ln 2

4+

1

32n

)



23




(2πn)minus1

[

(5

4

) (α+(n) +

β+(n)


)(1 minus 5


)

+

(3

4

) (αminus(n) +

βminus(n)


)(1 +

3


)]+ O(4) (321)


δcusp(g) = minus2

π

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (322)




m2cusp = 4

radic2

πσ2

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (323)


cusp(g) =[

2g minus 3 ln 2

2πminus K

8π2g+ O(1g2)

]minus σ

4radic

2m2

cusp + O(m4

cusp

) (324)


4 Mass scale




mO(6) = 8radic

2


πeminusπgRe

[int minusiinfin


t + 14

] (41)



24



f0(4πgt) =sumn1

t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

n + t

]minus 1

(43)

f1(4πgt) =sumn1

n

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

n + t

]





2




mO(6) = 16radic

2

π2

f1(minusπg)

V0(minusπg) (45)







fn(4πgt) = f (PT)

n (4πgt) + δfn(4πgt) (n = 0 1) (46)

where f (PT)



2

(5

4

) sumn1

a+(n)

t minus n= 1

t

[

(34

)(1 minus t)

(

34 minus t

) minus 1

]

(47)

2

(3

4

) sumn1

aminus(n)

t + n= 1

t

[

(14

)(1 + t)

(

14 + t

) minus 1

]

25



f(PT)

0 (4πgt) = minus(

34

)(1 minus t)

(

34 minus t

)+

1

4πg

[(3 ln 2

4+

1

8t

)

(34

)(1 minus t)

(

34 minus t

) minus (

14

)(1 + t)

8t(

14 + t

) ]+ O(gminus2)

f(PT)

1 (4πgt) = 1

4πg

[

(14

)(1 + t)

4t(

14 + t

) minus (

34

)(1 minus t)

4t(

34 minus t

) ]

minus 1

(4πg)2

[

(14

)(1 + t)

4t(

14 + t

) (1

4tminus 3 ln 2

4

)

minus (

34

)(1 minus t)

4t(

34 minus t

) (1

4t+

3 ln 2

4

)]+ O(gminus3) (48)


δf0(4πgt) = 2

1

4πg

[

(34

)(1 minus t)

2(

34 minus t

) minus (

54

)(1 + t)

2(

54 + t

) ]+ O(gminus2)


δf1(4πgt) = 2

1

4πg

(

54

)(1 + t)

(

54 + t

) +1

(4πg)2

[

(34

)(1 minus t)

8t(

34 minus t

)minus

(54

)(1 + t)

(

54 + t

) (1

8t+

3

4ln 2 minus 1

4

)]+ O(gminus3)





1 (0) minus 4gδf1(0) (410)

Replacing f(PT)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K


]minus 2

π

[1 +

3 minus 6 ln 2


]+ O(4) (411)



mO(6) =radic

2

(

54


[1 +

3 minus 6 ln 2

32πg+



]minus 2

8πg



]+ O(4)

(412)


(PT)


26




fn(4πgt)

(34 minus t

)

(34

)(1 minus t)

rarr f (toy)n (4πgt) (n = 0 1) (413)





(toy)


mtoy = 16radic

2

π2

f(toy)

1 (minusπg)

V0(minusπg)= 16

radic2

π2

1

V1(minusπg) (414)




mtoy = 4

π(

54


[1 minus 1

32πgminus 23


]minus 2

8πg

[1 minus 11


]+ O(4)

(415)


0 and f(toy)


27






32radic

2σm2

toy + O(m4

toy

) (416)




2radic

2mtoy (417)



4π (418)


cusp (g + c1) = 2g

[1 minus K


]minus 2

2radic

2π

[1 +

3


]+ O(4) (419)


[mO(6)(g + c1)]2 = 22

πσ

[(1 +

3

16πg+

16K minus 54


)minus 2

8radic

2πg

(1 minus 3


)+ O(4)

] (420)


O(6)

δcusp = minus σ

4radic

2m2

O(6) + O(m4

O(6)

) (421)



28











2

πeminusπgUminus


2

πeminusπgUminus

1 (πg) (423)





4




mprime = minus σ

8gmO(6) (426)




2

4mprimemO(6) = σ

radic2

32gm2

O(6) (427)



δ(it) = δf0(t)V0(t) + δf1(t)V1(t) (428)


δ(0) = 2δf1(0) (429)

29






δ(it) sim 1







δ(it) (433)


δγ (it) = t

4gδ(0) + O(t2) = t

2gδf1(0) + O(t2) (434)


0

dt

tJ2nminus1(t)

[δγminus(t)

1 minus eminust2g+

δγ+(t)

et2g minus 1

]= 0

(435)int infin

0

dt

tJ2n(t)

[δγ+(t)


et2g minus 1

]= 0



30




2

π2geminusπg minus

radic2

πeminusπg Re

int infin

0

dt prime


]+ O(t2) (436)



tradic

2

8g+ O(t2) (437)


mcusp(g) = mO(6)(g) (438)


5 Conclusions






31





(m4

O(n)

μ2) (51)


m2O(n) = cμ2 e


20 [1 + O(g2)] (52)




Acknowledgments


32








γ1 = 1

2minus π2

6g2 +

11π4

90g4 minus

(73π6

630+ 4ζ 2

3

)g6 + O(g8)

γ2 = ζ3g3 minus

(π2

3ζ3 + 10ζ5

)g5 +

(8π4

45ζ3 +

10π2

3ζ5 + 105ζ7

)g7 + O(g9)

γ3 = minusπ4

90g4 +

37π6

1890g6 + O(g8) γ4 = ζ5g

5 minus(

π2

3ζ5 + 21ζ7

)g7 + O(g9)

γ5 = minus π6

945g6 + O(g8) γ6 = ζ7g

7 + O(g9)

(A2)



3g4 +

44π4

45g6 minus

(292π6

315+ 32ζ 2

3

)g8 + O(g10) (A3)







33



c(g) = minus1

2+ 2gγ1(g) +

sumn1

[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

] (A5)


c(g) = minus1

2+ 2gγ1(g) minus

sumk1



fk(g) = 8gsumn1

[U+






3g2 + 2ζ3g

3 minus π4

6g4 minus 23ζ5g

5 +17π6

108g6 +

1107

4ζ7g

7 + O(g8)

f2 = minus1

2+ 2ζ3g

3 minus π4

30g4 + O(g5) f3 = 1

2+ O(g4)

f4 = minus3

8+ O(g5) f5 = 3

8+ O(g6) f6 = minus 5

16+ O(g7)

(A8)






(t) =int infin




(k)=sumn1



(k)=sumn1


bull For minus1 k 1

(k) = minusradic

2

π

(1 + k

1 minus k

)14[

1 +c(g)

1 + k+

1

2

(int minus1

minusinfin+

int infin

1

)dp(p)

p minus k

(p minus 1

p + 1

)14]

(B4)


34



(t) =sumn1

c+(n g)

[eminusit


]

+sumn1

cminus(n g)

[eit



(B5)



int 1

minus1dk eplusmnxk

int infin

1

dpeminusy(pminus1)

p minus k

(1 + k

1 minus k

p minus 1

p + 1

)plusmn14

Vn(x) =radic

2

π

int 1

minus1

dk exk

(k + 1)n

(1 + k

1 minus k

)14

(B6)

Uplusmnn (y) = 1

2

int infin

1

dp eminusy(pminus1)

(p ∓ 1)n

(p + 1

p minus 1

)∓14






0 (y) minus ex



c+(n g)

[4πngV1(t)U

+0 (4πng) + tV0(t)U

+1 (4πng)

4πng minus t

]

+sumn1

cminus(n g)

[4πngV1(t)U


minus1 (4πng)

4πng + t

] (B8)





int infin

0dsV0(x + s)U+






0 (x)


d

dx(xzprime

i (x)) =(

x minus 1

2

)zi(x) (B11)

35




2(y + s)]

= d


2(y + s)]z1(x + s)

minus d


1(x + s)]z2(y + s) (B12)



1 (y + s) + (y + s)V1(x + s)U+0 (y + s)

] ∣∣s=infins=0


0 (y) (B13)






π

t

t + πg (C1)



π

t2

t2 + (πg)2

(C2)


t2 + π2g2


0

dt

tJ2nminus1(t)

[γminus(t)

1 minus eminust2g+

γ+(t)

et2g minus 1

]= h2nminus1(g)

(C3)int infin

0

dt

tJ2n(t)

[γ+(t)


et2g minus 1

]= h2n(g)


h2nminus1 = 2mprime

π

int infin

0

dtJ2nminus1(t)

t2 + (πg)2

[t


1 minus eminust(2g)

]

(C4)

h2n = 2mprime

π

int infin

0

dtJ2n(t)

t2 + (πg)2

[πg

et(2g) minus 1+

t

1 minus eminust(2g)

]



36



γminus(t) = 2sumn1


(C6)γ+(t) = 2

sumn1

(2n)J2n(t)γ2n(g)



π2gt + O(t2) = it

(γ1 +

2mprime

π2g

)+ O(t2) (C7)



0

dt

t


1 minus eminust2g+


et2g minus 1

]= 2

sumn1



γ1 = 2sumn1



γ1 = minusmprime

π

int infin

0dt

[πg


t

t2 + π2g2(minus(t) + +(t))

] (C10)


πg

t2 + π2g2=

int infin


(C11)t

t2 + π2g2=

int infin



γ1 = minusmprime

π

int infin

0du eminusπgu

[ int infin


+int infin


] (C12)


γ1 = minus 2mprime


radic2mprime

πeminusπgRe

[ int infin

0

dt


] (C13)


37






Vn(x) = 1

π232minusn ex

int 1


= 1

π232minusn eminusx

int 1



Vn(x) = 2minusn

(54 minus n

)

(54

)(2 minus n)


= 2minusn

(54 minus n

)

(54

)(2 minus n)







Uab(x) = 1

2

int infin





(1 +

t

2x

)a+bminus12

(D5)


Uab(x) = 2bminus32

(1

2minus a + b


= 1

2

(1

2minus a + b

)U

(1


) (D6)

38




U+0 (x) = 1

2

(5

4


1 (x) = 1

2

(1

4


(D7)

Uminus0 (x) = 1

2

(3

4


1 (x) = 1

2

(3

4





1 (x) = V1(x)U+0 (x) + V0(x)U+

1 (x) = ex

x (D8)


U+0 (x)Uminus


0 (minusx) = π

2radic

2xeplusmn 3iπ

4 (for Imx ≷ 0) (D9)


V0(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]

(D10)

V1(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]






(D11)Uplusmn








U+0 (x) = (2x)minus54

(5

4

)F

(1

4

5

4| minus 1

2x

)= (2x)minus54

(5

4

) [1 minus 5


]


(3

4

)F

(minus1

4

3

4| minus 1

2x

)= (2x)minus34

(3

4

) [1 +

3


]

39


U+1 (x) = (2x)minus14 1

2

(1

4

)F

(1

4

1

4| minus 1

2x

)= (2x)minus14 1

2

(1

4

) [1 minus 1


]


2

(3

4

)F

(3

4

3

4| minus 1

2x

)= (2x)minus34 1

2

(3

4

) [1 minus 9


]

(D12)








2

πeminus 3iπ

4 eπg[U+


] (D13)




(

34

) [F

(1

4

5

4| 1

2πg+ iε

)+ 2F

(minus1

4

3

4| minus 1

2πg

)] (D14)


2 = σ

(34

)

(54




2(

34

) [8πgF

(1

4

1

4| 1

2πg+ iε

)+ 2F

(3

4

3

4| minus 1

2πg

)] (D16)





F

(1

4

5

4| 1

2πg+ iε

)minus F

(1

4

5

4| 1

2πgminus iε

)= i

radic22

σF

(minus1

4

3

4| minus 1

2πg

) (D17)



40





t + 14


(4πgit)

t + 14

=sumk=01


t + 14

+sumk=01


t + 14

Vk(4πgt) (E1)


Rk(4πgs) = Re

[int minusiinfin


t minus s

]= Re

[int minusiinfin


t minus 4πgs

]

(E2)


Re

[ int minusiinfin


t + 14


minussumn1

nc+(n g)

n + 14

[U+


]+

sumn1

ncminus(n g)

n minus 14





R0(s) = minusRe

[eminusiπ4

int 1


int minusiinfin

0dt eminusvtV0(t)

]=

radic2

πRe

[eminusiπ4

int 1


int 1

minus1du


u minus v minus iε

] (E4)


1radic2π

int 1

minus1du



⎧⎨⎩(

v+1vminus1

)14 v2 gt 1

eminusiπ4(

1+v1minusv

)14 v2 lt 1

(E5)


R0(s)sgt0=

radic2

[1

sminus

int minus1


(v + 1

v minus 1

)14]

=radic

2

[1

sminus 2 eminus2sU+

0 (s)

] (E6)


R0(s)slt0=

radic2

[1

s+ 2Uminus

0 (minuss)

] (E7)

41


together with

R1(s) = 2radic

2[θ(minuss)Uminus


] (E8)


Re

[ int minusiinfin


t + 14

]= 2

radic2f0(minusπg)

[Uminus

0 (πg) minus 1

2πg

]+ 2


1 (πg) +

radic2



References



















42





























Ann Phys 167 227

43

























44

1 Introduction

2 Cusp anomalous



23 Toy model



26 General solution







4 Mass scale





5 Conclusions

Acknowledgments








References



cusp





εO(6) = ε(g j) + j

2 mO(6) = kg14 eminusπg [1 + O(1g)] (13)


(54




ε(j g) + j = m2

[j

m+

π2

24

(j

m

)3

+ O(j 4m4)

] (14)


ε(g j) + j = 22

[g +

1

π

(3

4minus ln

)+

1

4π2g

(q02


)+O(1g2)

]+ O(4)



4




mcusp = mO(6) (15)





5







γ (t)t (21)


γ (t) = γ+(t) + iγminus(t) (22)


0

dt

tJ2nminus1(t)

[γminus(t)


γ+(t)

et(2g) minus 1

]= 1

2δn1

(23)int infin

0

dt

tJ2n(t)

[γ+(t)


et(2g) minus 1

]= 0



(t) =(

1 + i cotht

4g






(26)int infin





6



cos(ut) = 2sumn1


cos ϕ

J2nminus1(t)

t

(28)sin(ut) = 2

sumn1

(2n)sin(2nϕ)

cos ϕ

J2n(t)

t





mO(6) = 8radic

2


πeminusπgRe

[int infin

0

dt ei(tminusπ4)


] (29)






γminus(t) = 2sumn1


(2n)J2n(t)γ2n (210)

7




0

dt


int infin

0

dt

tJ2n(t)γ+(t) (211)



cusp(g) = 8g2γ1(g) (212)



(it) = γ (it)sin

(t

4g+ π

4

)sin

(t

4g

)sin

(π4

) = γ (it)radic

2infinprod

k=minusinfin


4

)t minus 4πgk

(213)



4



23 Toy model






t (216)



2t+ 1


2tminus 1


8









γ1 =int infin

0

dt

t

(γ+(t))2 + (γminus(t))2



γ1 int infin

0

dt


1 0 (218)


0 γ1 12 (219)


0 cusp(g) 4g2 (220)



cusp(g) = 4g2

[1 minus 1

3π2g2 +

11

45π4g4 minus 2

(73

630π6 + 4ζ 2

3


] (221)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus1 minus K

16π2gminus2 minus

(3K ln 2

64π3+

27ζ3

2048π3


] (222)




9


0

02

04

06

08

1

05 1 15 2 25 3



0 =int infin

0

dt

t

(γ(0)+ (t))2 + (γ

(0)minus (t))2





(k) =int infin

minusinfin

dt

2πeikt(t) γ (k) =

int infin

minusinfin

dt

2πeiktγ (t) (224)


γ (k) = 0 for k2 gt 1 (225)

10





(k) =int infin

minusinfin

dt

2πeikt

sinh(

t4g

+ iπ4

)sinh

(t

4g

)sin

(π4

)γ (t) (227)




sumn1










c(toy)+ (n g) = c

(toy)minus (n g) = 0 (231)



+(t) =int infin


(232)




11



minusint infin

minusinfin

dk(k)



(u) +1

πminusint 1

minus1

dk(k)



φ(u) = minus 1

π

(2 +

int minus1

minusinfin

dk(k)

k minus u+

int infin

1

dk(k)

k minus u

) (235)



(k) = 1

2φ(k) minus 1

2π

(1 + k

1 minus k

)14

minusint 1

minus1

duφ(u)

u minus k

(1 minus u

1 + u

)14

minusradic

2

π

(1 + k

1 minus k

)14c

1 + k (236)


(k)k21= minus

radic2

π

(1 + k

1 minus k

)14[

1 +c

1 + k+

1

2

int infin

minusinfin

dp(p)

p minus k

(p minus 1

p + 1

)14

θ(p2 minus 1)

] (237)



26 General solution


(t) =int 1


int minus1


int infin



(it) = f0(t)V0(t) + f1(t)V1(t) (239)

12




t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

4πng + t

]

(240)


4πng

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

4πng + t

]


Vn(x) =radic

2

π

int 1


(241)

Uplusmnn (x) = 1

2

int infin




f(toy)


1 (t) = minusc(toy)(g) (242)







(4π ig

( minus 1

4

)) = 0 isin Z (244)



4

) (245)



cusp(g) = 2g[1 minus 2f1(0)] (246)

13



cusp(g) = 2g


[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

]⎫⎬⎭ (247)






V1(minusπg) (248)



[1 minus 2

V0(minusπg)

V1(minusπg)

] (249)


(toy)cusp (g) = 2g


Mminus140(2πg)

] (250)







couplings


(toy)cusp (g) = 3

2πg2 minus 1

2π2g3 minus 1

64π3g4 +

5

64π4g5 minus 11

512π5g6 minus 3

512π6g7 + O(g8) (251)



14






(

34

)[F

(1

4

5

4|α + i0

)+ 2F

(minus1

4

3

4| minus α

)] α = 1(2πg)

(252)



(34

)

(54



F

(1

4

5

4| minus α

)= αminus54U+

0 (1(2α))

(5

4

)

(254)

F

(minus1

4

3

4| minus α

)= αminus34Uminus

0 (1(2α))

(3

4

)



4 34 | minus α



4 and b = 34 )


(minusα)k

k

(a + k)(b + k)

(a)(b)= 1 minus αab + O(α2) (255)



(a)

int infin





(a)

int infin


15







F(

14 5

4 |α+)

+ 2F(minus 1

4 34 | minus α

)F

(14 1

4 |α+)

+ 142αF

(34 3

4 | minus α) (258)


(toy)cusp (g)



4α24C4(α) + O(6) (259)


C0 = 1 minus αF

(14 5

4 |α+)

F(

14 1

4 |α+) C2 = 1[

F(

14 1

4 |α+)]2 C4 = F

(34 3

4 | minus α)[

F(

14 1

4 |α+)]3 (260)



16



4α2 minus 3

8α3 minus 61

64α4 minus 433

128α5 + O(α6)

C2 = 1 minus 1

8α minus 11

128α2 minus 151

1024α3 minus 13085

32768α4 + O(α5) (261)

C4 = 1 minus 3

4α minus 27

32α2 minus 317

128α3 + O(α4)















17




2α minus K

4α2 minus

(3K ln 2

8+

27ζ3

256

)α3 + (263)



3 ln 2

2rarr 1

K

2rarr 1

2

9ζ3

32rarr 1

3 (264)



exp

[3 ln 2

2τ minus K

2τ 2 +

9ζ3

32τ 3 +

]rarr exp

[τ minus τ 2

2+

τ 3

3+

] (265)


(

14

)

(1 + τ

4

)

(34 minus τ

4

)

(34

)

(1 minus τ

4

)

(14 + τ

4

) rarr (1 + τ) (266)



cusp (g)




k (k + 1

2



(toy)cusp (g)

18















plusmn1 (4πng)

n ∓ x

(32)





V0(4πgx)=



(33)


19



g 01 02 04 06 08 10 12 14 16 18

Numer 01976 03616 05843 07096 07825 08276 08576 08787 08944 09065Exact 01939 03584 05821 07080 07813 08267 08568 08781 08938 09059


nmax 10 20 30 40 50 60 70 infinNumer 08305 08286 08279 08276 08274 08273 08272 08267


c(g) = 0 (34)











20








1 =sumn1

c+(n g)

[nU+

0 (4πng)r(x) + U+1 (4πng)x

n minus x

]

+sumn1

cminus(n g)

[nUminus


n + x

] (36)





4 isin Z) (37)




[aplusmn(n) +

bplusmn(n)


]+ 2

[αplusmn(n) +

βplusmn(n)


]+ O(4)

(38)

21




(34

)

(54

) (39)






4 )

2x

(5

4

) sumn1

a+(n)

n minus x

= 1 ( 1)

(310)

minus2x

(3

4

) sumn1

aminus(n)

n + x

= 1 ( 0)


a+(n) = 2(n + 1

4

)(n + 1)2

(14

)

(311)

aminus(n) = (n + 3

4

)2(n + 1)2

(34

)


4 )

2x

(5

4

) sumn1

b+(n)

n minus x

= minus 3

32x

minus 3π

64minus 15

32ln 2 ( 1)

(312)

minus 2x

(3

4

) sumn1

bminus(n)

n + x

= minus 5

32x

minus 5π

64+

9

32ln 2 ( 0)


b+(n) = minusa+(n)

(3 ln 2

4+

3

32n


(3 ln 2

4+

5

32n

) (313)




(2πn)minus1

[

(5

4

) (a+(n) +

b+(n)


)(1 minus 5

128πgn+

)

+

(3

4

) (aminus(n) +

bminus(n)


)(1 +

3


)]+ O(2) (314)

22



cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K

16π2g2+ O(1g3)

]+ O(2) (315)




4 + O(gminus1)]

+ O(4) (316)



2x

(5

4

) sumn1

α+(n)

n minus x

= 0 ( 1)

(317)

minus2x

(3

4

) sumn1

αminus(n)

n + x

= π

2radic

2δ0 ( 0)


α+(n) = 0(318)



2x

(5

4

) sumn1

β+(n)

n minus x

= minus1

2 ( 1)

(319)

minus2x

(3

4

) sumn1

βminus(n)

n + x

= minus1

8+

3π

16radic

2(1 minus 2 ln 2)δ0 ( 0)


β+(n) = minus1

2a+(n)

(320)


(1

4minus 3 ln 2

4+

1

32n

)



23




(2πn)minus1

[

(5

4

) (α+(n) +

β+(n)


)(1 minus 5


)

+

(3

4

) (αminus(n) +

βminus(n)


)(1 +

3


)]+ O(4) (321)


δcusp(g) = minus2

π

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (322)




m2cusp = 4

radic2

πσ2

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (323)


cusp(g) =[

2g minus 3 ln 2

2πminus K

8π2g+ O(1g2)

]minus σ

4radic

2m2

cusp + O(m4

cusp

) (324)


4 Mass scale




mO(6) = 8radic

2


πeminusπgRe

[int minusiinfin


t + 14

] (41)



24



f0(4πgt) =sumn1

t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

n + t

]minus 1

(43)

f1(4πgt) =sumn1

n

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

n + t

]





2




mO(6) = 16radic

2

π2

f1(minusπg)

V0(minusπg) (45)







fn(4πgt) = f (PT)

n (4πgt) + δfn(4πgt) (n = 0 1) (46)

where f (PT)



2

(5

4

) sumn1

a+(n)

t minus n= 1

t

[

(34

)(1 minus t)

(

34 minus t

) minus 1

]

(47)

2

(3

4

) sumn1

aminus(n)

t + n= 1

t

[

(14

)(1 + t)

(

14 + t

) minus 1

]

25



f(PT)

0 (4πgt) = minus(

34

)(1 minus t)

(

34 minus t

)+

1

4πg

[(3 ln 2

4+

1

8t

)

(34

)(1 minus t)

(

34 minus t

) minus (

14

)(1 + t)

8t(

14 + t

) ]+ O(gminus2)

f(PT)

1 (4πgt) = 1

4πg

[

(14

)(1 + t)

4t(

14 + t

) minus (

34

)(1 minus t)

4t(

34 minus t

) ]

minus 1

(4πg)2

[

(14

)(1 + t)

4t(

14 + t

) (1

4tminus 3 ln 2

4

)

minus (

34

)(1 minus t)

4t(

34 minus t

) (1

4t+

3 ln 2

4

)]+ O(gminus3) (48)


δf0(4πgt) = 2

1

4πg

[

(34

)(1 minus t)

2(

34 minus t

) minus (

54

)(1 + t)

2(

54 + t

) ]+ O(gminus2)


δf1(4πgt) = 2

1

4πg

(

54

)(1 + t)

(

54 + t

) +1

(4πg)2

[

(34

)(1 minus t)

8t(

34 minus t

)minus

(54

)(1 + t)

(

54 + t

) (1

8t+

3

4ln 2 minus 1

4

)]+ O(gminus3)





1 (0) minus 4gδf1(0) (410)

Replacing f(PT)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K


]minus 2

π

[1 +

3 minus 6 ln 2


]+ O(4) (411)



mO(6) =radic

2

(

54


[1 +

3 minus 6 ln 2

32πg+



]minus 2

8πg



]+ O(4)

(412)


(PT)


26




fn(4πgt)

(34 minus t

)

(34

)(1 minus t)

rarr f (toy)n (4πgt) (n = 0 1) (413)





(toy)


mtoy = 16radic

2

π2

f(toy)

1 (minusπg)

V0(minusπg)= 16

radic2

π2

1

V1(minusπg) (414)




mtoy = 4

π(

54


[1 minus 1

32πgminus 23


]minus 2

8πg

[1 minus 11


]+ O(4)

(415)


0 and f(toy)


27






32radic

2σm2

toy + O(m4

toy

) (416)




2radic

2mtoy (417)



4π (418)


cusp (g + c1) = 2g

[1 minus K


]minus 2

2radic

2π

[1 +

3


]+ O(4) (419)


[mO(6)(g + c1)]2 = 22

πσ

[(1 +

3

16πg+

16K minus 54


)minus 2

8radic

2πg

(1 minus 3


)+ O(4)

] (420)


O(6)

δcusp = minus σ

4radic

2m2

O(6) + O(m4

O(6)

) (421)



28











2

πeminusπgUminus


2

πeminusπgUminus

1 (πg) (423)





4




mprime = minus σ

8gmO(6) (426)




2

4mprimemO(6) = σ

radic2

32gm2

O(6) (427)



δ(it) = δf0(t)V0(t) + δf1(t)V1(t) (428)


δ(0) = 2δf1(0) (429)

29






δ(it) sim 1







δ(it) (433)


δγ (it) = t

4gδ(0) + O(t2) = t

2gδf1(0) + O(t2) (434)


0

dt

tJ2nminus1(t)

[δγminus(t)

1 minus eminust2g+

δγ+(t)

et2g minus 1

]= 0

(435)int infin

0

dt

tJ2n(t)

[δγ+(t)


et2g minus 1

]= 0



30




2

π2geminusπg minus

radic2

πeminusπg Re

int infin

0

dt prime


]+ O(t2) (436)



tradic

2

8g+ O(t2) (437)


mcusp(g) = mO(6)(g) (438)


5 Conclusions






31





(m4

O(n)

μ2) (51)


m2O(n) = cμ2 e


20 [1 + O(g2)] (52)




Acknowledgments


32








γ1 = 1

2minus π2

6g2 +

11π4

90g4 minus

(73π6

630+ 4ζ 2

3

)g6 + O(g8)

γ2 = ζ3g3 minus

(π2

3ζ3 + 10ζ5

)g5 +

(8π4

45ζ3 +

10π2

3ζ5 + 105ζ7

)g7 + O(g9)

γ3 = minusπ4

90g4 +

37π6

1890g6 + O(g8) γ4 = ζ5g

5 minus(

π2

3ζ5 + 21ζ7

)g7 + O(g9)

γ5 = minus π6

945g6 + O(g8) γ6 = ζ7g

7 + O(g9)

(A2)



3g4 +

44π4

45g6 minus

(292π6

315+ 32ζ 2

3

)g8 + O(g10) (A3)







33



c(g) = minus1

2+ 2gγ1(g) +

sumn1

[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

] (A5)


c(g) = minus1

2+ 2gγ1(g) minus

sumk1



fk(g) = 8gsumn1

[U+






3g2 + 2ζ3g

3 minus π4

6g4 minus 23ζ5g

5 +17π6

108g6 +

1107

4ζ7g

7 + O(g8)

f2 = minus1

2+ 2ζ3g

3 minus π4

30g4 + O(g5) f3 = 1

2+ O(g4)

f4 = minus3

8+ O(g5) f5 = 3

8+ O(g6) f6 = minus 5

16+ O(g7)

(A8)






(t) =int infin




(k)=sumn1



(k)=sumn1


bull For minus1 k 1

(k) = minusradic

2

π

(1 + k

1 minus k

)14[

1 +c(g)

1 + k+

1

2

(int minus1

minusinfin+

int infin

1

)dp(p)

p minus k

(p minus 1

p + 1

)14]

(B4)


34



(t) =sumn1

c+(n g)

[eminusit


]

+sumn1

cminus(n g)

[eit



(B5)



int 1

minus1dk eplusmnxk

int infin

1

dpeminusy(pminus1)

p minus k

(1 + k

1 minus k

p minus 1

p + 1

)plusmn14

Vn(x) =radic

2

π

int 1

minus1

dk exk

(k + 1)n

(1 + k

1 minus k

)14

(B6)

Uplusmnn (y) = 1

2

int infin

1

dp eminusy(pminus1)

(p ∓ 1)n

(p + 1

p minus 1

)∓14






0 (y) minus ex



c+(n g)

[4πngV1(t)U

+0 (4πng) + tV0(t)U

+1 (4πng)

4πng minus t

]

+sumn1

cminus(n g)

[4πngV1(t)U


minus1 (4πng)

4πng + t

] (B8)





int infin

0dsV0(x + s)U+






0 (x)


d

dx(xzprime

i (x)) =(

x minus 1

2

)zi(x) (B11)

35




2(y + s)]

= d


2(y + s)]z1(x + s)

minus d


1(x + s)]z2(y + s) (B12)



1 (y + s) + (y + s)V1(x + s)U+0 (y + s)

] ∣∣s=infins=0


0 (y) (B13)






π

t

t + πg (C1)



π

t2

t2 + (πg)2

(C2)


t2 + π2g2


0

dt

tJ2nminus1(t)

[γminus(t)

1 minus eminust2g+

γ+(t)

et2g minus 1

]= h2nminus1(g)

(C3)int infin

0

dt

tJ2n(t)

[γ+(t)


et2g minus 1

]= h2n(g)


h2nminus1 = 2mprime

π

int infin

0

dtJ2nminus1(t)

t2 + (πg)2

[t


1 minus eminust(2g)

]

(C4)

h2n = 2mprime

π

int infin

0

dtJ2n(t)

t2 + (πg)2

[πg

et(2g) minus 1+

t

1 minus eminust(2g)

]



36



γminus(t) = 2sumn1


(C6)γ+(t) = 2

sumn1

(2n)J2n(t)γ2n(g)



π2gt + O(t2) = it

(γ1 +

2mprime

π2g

)+ O(t2) (C7)



0

dt

t


1 minus eminust2g+


et2g minus 1

]= 2

sumn1



γ1 = 2sumn1



γ1 = minusmprime

π

int infin

0dt

[πg


t

t2 + π2g2(minus(t) + +(t))

] (C10)


πg

t2 + π2g2=

int infin


(C11)t

t2 + π2g2=

int infin



γ1 = minusmprime

π

int infin

0du eminusπgu

[ int infin


+int infin


] (C12)


γ1 = minus 2mprime


radic2mprime

πeminusπgRe

[ int infin

0

dt


] (C13)


37






Vn(x) = 1

π232minusn ex

int 1


= 1

π232minusn eminusx

int 1



Vn(x) = 2minusn

(54 minus n

)

(54

)(2 minus n)


= 2minusn

(54 minus n

)

(54

)(2 minus n)







Uab(x) = 1

2

int infin





(1 +

t

2x

)a+bminus12

(D5)


Uab(x) = 2bminus32

(1

2minus a + b


= 1

2

(1

2minus a + b

)U

(1


) (D6)

38




U+0 (x) = 1

2

(5

4


1 (x) = 1

2

(1

4


(D7)

Uminus0 (x) = 1

2

(3

4


1 (x) = 1

2

(3

4





1 (x) = V1(x)U+0 (x) + V0(x)U+

1 (x) = ex

x (D8)


U+0 (x)Uminus


0 (minusx) = π

2radic

2xeplusmn 3iπ

4 (for Imx ≷ 0) (D9)


V0(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]

(D10)

V1(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]






(D11)Uplusmn








U+0 (x) = (2x)minus54

(5

4

)F

(1

4

5

4| minus 1

2x

)= (2x)minus54

(5

4

) [1 minus 5


]


(3

4

)F

(minus1

4

3

4| minus 1

2x

)= (2x)minus34

(3

4

) [1 +

3


]

39


U+1 (x) = (2x)minus14 1

2

(1

4

)F

(1

4

1

4| minus 1

2x

)= (2x)minus14 1

2

(1

4

) [1 minus 1


]


2

(3

4

)F

(3

4

3

4| minus 1

2x

)= (2x)minus34 1

2

(3

4

) [1 minus 9


]

(D12)








2

πeminus 3iπ

4 eπg[U+


] (D13)




(

34

) [F

(1

4

5

4| 1

2πg+ iε

)+ 2F

(minus1

4

3

4| minus 1

2πg

)] (D14)


2 = σ

(34

)

(54




2(

34

) [8πgF

(1

4

1

4| 1

2πg+ iε

)+ 2F

(3

4

3

4| minus 1

2πg

)] (D16)





F

(1

4

5

4| 1

2πg+ iε

)minus F

(1

4

5

4| 1

2πgminus iε

)= i

radic22

σF

(minus1

4

3

4| minus 1

2πg

) (D17)



40





t + 14


(4πgit)

t + 14

=sumk=01


t + 14

+sumk=01


t + 14

Vk(4πgt) (E1)


Rk(4πgs) = Re

[int minusiinfin


t minus s

]= Re

[int minusiinfin


t minus 4πgs

]

(E2)


Re

[ int minusiinfin


t + 14


minussumn1

nc+(n g)

n + 14

[U+


]+

sumn1

ncminus(n g)

n minus 14





R0(s) = minusRe

[eminusiπ4

int 1


int minusiinfin

0dt eminusvtV0(t)

]=

radic2

πRe

[eminusiπ4

int 1


int 1

minus1du


u minus v minus iε

] (E4)


1radic2π

int 1

minus1du



⎧⎨⎩(

v+1vminus1

)14 v2 gt 1

eminusiπ4(

1+v1minusv

)14 v2 lt 1

(E5)


R0(s)sgt0=

radic2

[1

sminus

int minus1


(v + 1

v minus 1

)14]

=radic

2

[1

sminus 2 eminus2sU+

0 (s)

] (E6)


R0(s)slt0=

radic2

[1

s+ 2Uminus

0 (minuss)

] (E7)

41


together with

R1(s) = 2radic

2[θ(minuss)Uminus


] (E8)


Re

[ int minusiinfin


t + 14

]= 2

radic2f0(minusπg)

[Uminus

0 (πg) minus 1

2πg

]+ 2


1 (πg) +

radic2



References



















42





























Ann Phys 167 227

43

























44

1 Introduction

2 Cusp anomalous



23 Toy model



26 General solution







4 Mass scale





5 Conclusions

Acknowledgments








References




mcusp = mO(6) (15)





5







γ (t)t (21)


γ (t) = γ+(t) + iγminus(t) (22)


0

dt

tJ2nminus1(t)

[γminus(t)


γ+(t)

et(2g) minus 1

]= 1

2δn1

(23)int infin

0

dt

tJ2n(t)

[γ+(t)


et(2g) minus 1

]= 0



(t) =(

1 + i cotht

4g






(26)int infin





6



cos(ut) = 2sumn1


cos ϕ

J2nminus1(t)

t

(28)sin(ut) = 2

sumn1

(2n)sin(2nϕ)

cos ϕ

J2n(t)

t





mO(6) = 8radic

2


πeminusπgRe

[int infin

0

dt ei(tminusπ4)


] (29)






γminus(t) = 2sumn1


(2n)J2n(t)γ2n (210)

7




0

dt


int infin

0

dt

tJ2n(t)γ+(t) (211)



cusp(g) = 8g2γ1(g) (212)



(it) = γ (it)sin

(t

4g+ π

4

)sin

(t

4g

)sin

(π4

) = γ (it)radic

2infinprod

k=minusinfin


4

)t minus 4πgk

(213)



4



23 Toy model






t (216)



2t+ 1


2tminus 1


8









γ1 =int infin

0

dt

t

(γ+(t))2 + (γminus(t))2



γ1 int infin

0

dt


1 0 (218)


0 γ1 12 (219)


0 cusp(g) 4g2 (220)



cusp(g) = 4g2

[1 minus 1

3π2g2 +

11

45π4g4 minus 2

(73

630π6 + 4ζ 2

3


] (221)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus1 minus K

16π2gminus2 minus

(3K ln 2

64π3+

27ζ3

2048π3


] (222)




9


0

02

04

06

08

1

05 1 15 2 25 3



0 =int infin

0

dt

t

(γ(0)+ (t))2 + (γ

(0)minus (t))2





(k) =int infin

minusinfin

dt

2πeikt(t) γ (k) =

int infin

minusinfin

dt

2πeiktγ (t) (224)


γ (k) = 0 for k2 gt 1 (225)

10





(k) =int infin

minusinfin

dt

2πeikt

sinh(

t4g

+ iπ4

)sinh

(t

4g

)sin

(π4

)γ (t) (227)




sumn1










c(toy)+ (n g) = c

(toy)minus (n g) = 0 (231)



+(t) =int infin


(232)




11



minusint infin

minusinfin

dk(k)



(u) +1

πminusint 1

minus1

dk(k)



φ(u) = minus 1

π

(2 +

int minus1

minusinfin

dk(k)

k minus u+

int infin

1

dk(k)

k minus u

) (235)



(k) = 1

2φ(k) minus 1

2π

(1 + k

1 minus k

)14

minusint 1

minus1

duφ(u)

u minus k

(1 minus u

1 + u

)14

minusradic

2

π

(1 + k

1 minus k

)14c

1 + k (236)


(k)k21= minus

radic2

π

(1 + k

1 minus k

)14[

1 +c

1 + k+

1

2

int infin

minusinfin

dp(p)

p minus k

(p minus 1

p + 1

)14

θ(p2 minus 1)

] (237)



26 General solution


(t) =int 1


int minus1


int infin



(it) = f0(t)V0(t) + f1(t)V1(t) (239)

12




t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

4πng + t

]

(240)


4πng

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

4πng + t

]


Vn(x) =radic

2

π

int 1


(241)

Uplusmnn (x) = 1

2

int infin




f(toy)


1 (t) = minusc(toy)(g) (242)







(4π ig

( minus 1

4

)) = 0 isin Z (244)



4

) (245)



cusp(g) = 2g[1 minus 2f1(0)] (246)

13



cusp(g) = 2g


[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

]⎫⎬⎭ (247)






V1(minusπg) (248)



[1 minus 2

V0(minusπg)

V1(minusπg)

] (249)


(toy)cusp (g) = 2g


Mminus140(2πg)

] (250)







couplings


(toy)cusp (g) = 3

2πg2 minus 1

2π2g3 minus 1

64π3g4 +

5

64π4g5 minus 11

512π5g6 minus 3

512π6g7 + O(g8) (251)



14






(

34

)[F

(1

4

5

4|α + i0

)+ 2F

(minus1

4

3

4| minus α

)] α = 1(2πg)

(252)



(34

)

(54



F

(1

4

5

4| minus α

)= αminus54U+

0 (1(2α))

(5

4

)

(254)

F

(minus1

4

3

4| minus α

)= αminus34Uminus

0 (1(2α))

(3

4

)



4 34 | minus α



4 and b = 34 )


(minusα)k

k

(a + k)(b + k)

(a)(b)= 1 minus αab + O(α2) (255)



(a)

int infin





(a)

int infin


15







F(

14 5

4 |α+)

+ 2F(minus 1

4 34 | minus α

)F

(14 1

4 |α+)

+ 142αF

(34 3

4 | minus α) (258)


(toy)cusp (g)



4α24C4(α) + O(6) (259)


C0 = 1 minus αF

(14 5

4 |α+)

F(

14 1

4 |α+) C2 = 1[

F(

14 1

4 |α+)]2 C4 = F

(34 3

4 | minus α)[

F(

14 1

4 |α+)]3 (260)



16



4α2 minus 3

8α3 minus 61

64α4 minus 433

128α5 + O(α6)

C2 = 1 minus 1

8α minus 11

128α2 minus 151

1024α3 minus 13085

32768α4 + O(α5) (261)

C4 = 1 minus 3

4α minus 27

32α2 minus 317

128α3 + O(α4)















17




2α minus K

4α2 minus

(3K ln 2

8+

27ζ3

256

)α3 + (263)



3 ln 2

2rarr 1

K

2rarr 1

2

9ζ3

32rarr 1

3 (264)



exp

[3 ln 2

2τ minus K

2τ 2 +

9ζ3

32τ 3 +

]rarr exp

[τ minus τ 2

2+

τ 3

3+

] (265)


(

14

)

(1 + τ

4

)

(34 minus τ

4

)

(34

)

(1 minus τ

4

)

(14 + τ

4

) rarr (1 + τ) (266)



cusp (g)




k (k + 1

2



(toy)cusp (g)

18















plusmn1 (4πng)

n ∓ x

(32)





V0(4πgx)=



(33)


19



g 01 02 04 06 08 10 12 14 16 18

Numer 01976 03616 05843 07096 07825 08276 08576 08787 08944 09065Exact 01939 03584 05821 07080 07813 08267 08568 08781 08938 09059


nmax 10 20 30 40 50 60 70 infinNumer 08305 08286 08279 08276 08274 08273 08272 08267


c(g) = 0 (34)











20








1 =sumn1

c+(n g)

[nU+

0 (4πng)r(x) + U+1 (4πng)x

n minus x

]

+sumn1

cminus(n g)

[nUminus


n + x

] (36)





4 isin Z) (37)




[aplusmn(n) +

bplusmn(n)


]+ 2

[αplusmn(n) +

βplusmn(n)


]+ O(4)

(38)

21




(34

)

(54

) (39)






4 )

2x

(5

4

) sumn1

a+(n)

n minus x

= 1 ( 1)

(310)

minus2x

(3

4

) sumn1

aminus(n)

n + x

= 1 ( 0)


a+(n) = 2(n + 1

4

)(n + 1)2

(14

)

(311)

aminus(n) = (n + 3

4

)2(n + 1)2

(34

)


4 )

2x

(5

4

) sumn1

b+(n)

n minus x

= minus 3

32x

minus 3π

64minus 15

32ln 2 ( 1)

(312)

minus 2x

(3

4

) sumn1

bminus(n)

n + x

= minus 5

32x

minus 5π

64+

9

32ln 2 ( 0)


b+(n) = minusa+(n)

(3 ln 2

4+

3

32n


(3 ln 2

4+

5

32n

) (313)




(2πn)minus1

[

(5

4

) (a+(n) +

b+(n)


)(1 minus 5

128πgn+

)

+

(3

4

) (aminus(n) +

bminus(n)


)(1 +

3


)]+ O(2) (314)

22



cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K

16π2g2+ O(1g3)

]+ O(2) (315)




4 + O(gminus1)]

+ O(4) (316)



2x

(5

4

) sumn1

α+(n)

n minus x

= 0 ( 1)

(317)

minus2x

(3

4

) sumn1

αminus(n)

n + x

= π

2radic

2δ0 ( 0)


α+(n) = 0(318)



2x

(5

4

) sumn1

β+(n)

n minus x

= minus1

2 ( 1)

(319)

minus2x

(3

4

) sumn1

βminus(n)

n + x

= minus1

8+

3π

16radic

2(1 minus 2 ln 2)δ0 ( 0)


β+(n) = minus1

2a+(n)

(320)


(1

4minus 3 ln 2

4+

1

32n

)



23




(2πn)minus1

[

(5

4

) (α+(n) +

β+(n)


)(1 minus 5


)

+

(3

4

) (αminus(n) +

βminus(n)


)(1 +

3


)]+ O(4) (321)


δcusp(g) = minus2

π

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (322)




m2cusp = 4

radic2

πσ2

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (323)


cusp(g) =[

2g minus 3 ln 2

2πminus K

8π2g+ O(1g2)

]minus σ

4radic

2m2

cusp + O(m4

cusp

) (324)


4 Mass scale




mO(6) = 8radic

2


πeminusπgRe

[int minusiinfin


t + 14

] (41)



24



f0(4πgt) =sumn1

t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

n + t

]minus 1

(43)

f1(4πgt) =sumn1

n

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

n + t

]





2




mO(6) = 16radic

2

π2

f1(minusπg)

V0(minusπg) (45)







fn(4πgt) = f (PT)

n (4πgt) + δfn(4πgt) (n = 0 1) (46)

where f (PT)



2

(5

4

) sumn1

a+(n)

t minus n= 1

t

[

(34

)(1 minus t)

(

34 minus t

) minus 1

]

(47)

2

(3

4

) sumn1

aminus(n)

t + n= 1

t

[

(14

)(1 + t)

(

14 + t

) minus 1

]

25



f(PT)

0 (4πgt) = minus(

34

)(1 minus t)

(

34 minus t

)+

1

4πg

[(3 ln 2

4+

1

8t

)

(34

)(1 minus t)

(

34 minus t

) minus (

14

)(1 + t)

8t(

14 + t

) ]+ O(gminus2)

f(PT)

1 (4πgt) = 1

4πg

[

(14

)(1 + t)

4t(

14 + t

) minus (

34

)(1 minus t)

4t(

34 minus t

) ]

minus 1

(4πg)2

[

(14

)(1 + t)

4t(

14 + t

) (1

4tminus 3 ln 2

4

)

minus (

34

)(1 minus t)

4t(

34 minus t

) (1

4t+

3 ln 2

4

)]+ O(gminus3) (48)


δf0(4πgt) = 2

1

4πg

[

(34

)(1 minus t)

2(

34 minus t

) minus (

54

)(1 + t)

2(

54 + t

) ]+ O(gminus2)


δf1(4πgt) = 2

1

4πg

(

54

)(1 + t)

(

54 + t

) +1

(4πg)2

[

(34

)(1 minus t)

8t(

34 minus t

)minus

(54

)(1 + t)

(

54 + t

) (1

8t+

3

4ln 2 minus 1

4

)]+ O(gminus3)





1 (0) minus 4gδf1(0) (410)

Replacing f(PT)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K


]minus 2

π

[1 +

3 minus 6 ln 2


]+ O(4) (411)



mO(6) =radic

2

(

54


[1 +

3 minus 6 ln 2

32πg+



]minus 2

8πg



]+ O(4)

(412)


(PT)


26




fn(4πgt)

(34 minus t

)

(34

)(1 minus t)

rarr f (toy)n (4πgt) (n = 0 1) (413)





(toy)


mtoy = 16radic

2

π2

f(toy)

1 (minusπg)

V0(minusπg)= 16

radic2

π2

1

V1(minusπg) (414)




mtoy = 4

π(

54


[1 minus 1

32πgminus 23


]minus 2

8πg

[1 minus 11


]+ O(4)

(415)


0 and f(toy)


27






32radic

2σm2

toy + O(m4

toy

) (416)




2radic

2mtoy (417)



4π (418)


cusp (g + c1) = 2g

[1 minus K


]minus 2

2radic

2π

[1 +

3


]+ O(4) (419)


[mO(6)(g + c1)]2 = 22

πσ

[(1 +

3

16πg+

16K minus 54


)minus 2

8radic

2πg

(1 minus 3


)+ O(4)

] (420)


O(6)

δcusp = minus σ

4radic

2m2

O(6) + O(m4

O(6)

) (421)



28











2

πeminusπgUminus


2

πeminusπgUminus

1 (πg) (423)





4




mprime = minus σ

8gmO(6) (426)




2

4mprimemO(6) = σ

radic2

32gm2

O(6) (427)



δ(it) = δf0(t)V0(t) + δf1(t)V1(t) (428)


δ(0) = 2δf1(0) (429)

29






δ(it) sim 1







δ(it) (433)


δγ (it) = t

4gδ(0) + O(t2) = t

2gδf1(0) + O(t2) (434)


0

dt

tJ2nminus1(t)

[δγminus(t)

1 minus eminust2g+

δγ+(t)

et2g minus 1

]= 0

(435)int infin

0

dt

tJ2n(t)

[δγ+(t)


et2g minus 1

]= 0



30




2

π2geminusπg minus

radic2

πeminusπg Re

int infin

0

dt prime


]+ O(t2) (436)



tradic

2

8g+ O(t2) (437)


mcusp(g) = mO(6)(g) (438)


5 Conclusions






31





(m4

O(n)

μ2) (51)


m2O(n) = cμ2 e


20 [1 + O(g2)] (52)




Acknowledgments


32








γ1 = 1

2minus π2

6g2 +

11π4

90g4 minus

(73π6

630+ 4ζ 2

3

)g6 + O(g8)

γ2 = ζ3g3 minus

(π2

3ζ3 + 10ζ5

)g5 +

(8π4

45ζ3 +

10π2

3ζ5 + 105ζ7

)g7 + O(g9)

γ3 = minusπ4

90g4 +

37π6

1890g6 + O(g8) γ4 = ζ5g

5 minus(

π2

3ζ5 + 21ζ7

)g7 + O(g9)

γ5 = minus π6

945g6 + O(g8) γ6 = ζ7g

7 + O(g9)

(A2)



3g4 +

44π4

45g6 minus

(292π6

315+ 32ζ 2

3

)g8 + O(g10) (A3)







33



c(g) = minus1

2+ 2gγ1(g) +

sumn1

[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

] (A5)


c(g) = minus1

2+ 2gγ1(g) minus

sumk1



fk(g) = 8gsumn1

[U+






3g2 + 2ζ3g

3 minus π4

6g4 minus 23ζ5g

5 +17π6

108g6 +

1107

4ζ7g

7 + O(g8)

f2 = minus1

2+ 2ζ3g

3 minus π4

30g4 + O(g5) f3 = 1

2+ O(g4)

f4 = minus3

8+ O(g5) f5 = 3

8+ O(g6) f6 = minus 5

16+ O(g7)

(A8)






(t) =int infin




(k)=sumn1



(k)=sumn1


bull For minus1 k 1

(k) = minusradic

2

π

(1 + k

1 minus k

)14[

1 +c(g)

1 + k+

1

2

(int minus1

minusinfin+

int infin

1

)dp(p)

p minus k

(p minus 1

p + 1

)14]

(B4)


34



(t) =sumn1

c+(n g)

[eminusit


]

+sumn1

cminus(n g)

[eit



(B5)



int 1

minus1dk eplusmnxk

int infin

1

dpeminusy(pminus1)

p minus k

(1 + k

1 minus k

p minus 1

p + 1

)plusmn14

Vn(x) =radic

2

π

int 1

minus1

dk exk

(k + 1)n

(1 + k

1 minus k

)14

(B6)

Uplusmnn (y) = 1

2

int infin

1

dp eminusy(pminus1)

(p ∓ 1)n

(p + 1

p minus 1

)∓14






0 (y) minus ex



c+(n g)

[4πngV1(t)U

+0 (4πng) + tV0(t)U

+1 (4πng)

4πng minus t

]

+sumn1

cminus(n g)

[4πngV1(t)U


minus1 (4πng)

4πng + t

] (B8)





int infin

0dsV0(x + s)U+






0 (x)


d

dx(xzprime

i (x)) =(

x minus 1

2

)zi(x) (B11)

35




2(y + s)]

= d


2(y + s)]z1(x + s)

minus d


1(x + s)]z2(y + s) (B12)



1 (y + s) + (y + s)V1(x + s)U+0 (y + s)

] ∣∣s=infins=0


0 (y) (B13)






π

t

t + πg (C1)



π

t2

t2 + (πg)2

(C2)


t2 + π2g2


0

dt

tJ2nminus1(t)

[γminus(t)

1 minus eminust2g+

γ+(t)

et2g minus 1

]= h2nminus1(g)

(C3)int infin

0

dt

tJ2n(t)

[γ+(t)


et2g minus 1

]= h2n(g)


h2nminus1 = 2mprime

π

int infin

0

dtJ2nminus1(t)

t2 + (πg)2

[t


1 minus eminust(2g)

]

(C4)

h2n = 2mprime

π

int infin

0

dtJ2n(t)

t2 + (πg)2

[πg

et(2g) minus 1+

t

1 minus eminust(2g)

]



36



γminus(t) = 2sumn1


(C6)γ+(t) = 2

sumn1

(2n)J2n(t)γ2n(g)



π2gt + O(t2) = it

(γ1 +

2mprime

π2g

)+ O(t2) (C7)



0

dt

t


1 minus eminust2g+


et2g minus 1

]= 2

sumn1



γ1 = 2sumn1



γ1 = minusmprime

π

int infin

0dt

[πg


t

t2 + π2g2(minus(t) + +(t))

] (C10)


πg

t2 + π2g2=

int infin


(C11)t

t2 + π2g2=

int infin



γ1 = minusmprime

π

int infin

0du eminusπgu

[ int infin


+int infin


] (C12)


γ1 = minus 2mprime


radic2mprime

πeminusπgRe

[ int infin

0

dt


] (C13)


37






Vn(x) = 1

π232minusn ex

int 1


= 1

π232minusn eminusx

int 1



Vn(x) = 2minusn

(54 minus n

)

(54

)(2 minus n)


= 2minusn

(54 minus n

)

(54

)(2 minus n)







Uab(x) = 1

2

int infin





(1 +

t

2x

)a+bminus12

(D5)


Uab(x) = 2bminus32

(1

2minus a + b


= 1

2

(1

2minus a + b

)U

(1


) (D6)

38




U+0 (x) = 1

2

(5

4


1 (x) = 1

2

(1

4


(D7)

Uminus0 (x) = 1

2

(3

4


1 (x) = 1

2

(3

4





1 (x) = V1(x)U+0 (x) + V0(x)U+

1 (x) = ex

x (D8)


U+0 (x)Uminus


0 (minusx) = π

2radic

2xeplusmn 3iπ

4 (for Imx ≷ 0) (D9)


V0(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]

(D10)

V1(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]






(D11)Uplusmn








U+0 (x) = (2x)minus54

(5

4

)F

(1

4

5

4| minus 1

2x

)= (2x)minus54

(5

4

) [1 minus 5


]


(3

4

)F

(minus1

4

3

4| minus 1

2x

)= (2x)minus34

(3

4

) [1 +

3


]

39


U+1 (x) = (2x)minus14 1

2

(1

4

)F

(1

4

1

4| minus 1

2x

)= (2x)minus14 1

2

(1

4

) [1 minus 1


]


2

(3

4

)F

(3

4

3

4| minus 1

2x

)= (2x)minus34 1

2

(3

4

) [1 minus 9


]

(D12)








2

πeminus 3iπ

4 eπg[U+


] (D13)




(

34

) [F

(1

4

5

4| 1

2πg+ iε

)+ 2F

(minus1

4

3

4| minus 1

2πg

)] (D14)


2 = σ

(34

)

(54




2(

34

) [8πgF

(1

4

1

4| 1

2πg+ iε

)+ 2F

(3

4

3

4| minus 1

2πg

)] (D16)





F

(1

4

5

4| 1

2πg+ iε

)minus F

(1

4

5

4| 1

2πgminus iε

)= i

radic22

σF

(minus1

4

3

4| minus 1

2πg

) (D17)



40





t + 14


(4πgit)

t + 14

=sumk=01


t + 14

+sumk=01


t + 14

Vk(4πgt) (E1)


Rk(4πgs) = Re

[int minusiinfin


t minus s

]= Re

[int minusiinfin


t minus 4πgs

]

(E2)


Re

[ int minusiinfin


t + 14


minussumn1

nc+(n g)

n + 14

[U+


]+

sumn1

ncminus(n g)

n minus 14





R0(s) = minusRe

[eminusiπ4

int 1


int minusiinfin

0dt eminusvtV0(t)

]=

radic2

πRe

[eminusiπ4

int 1


int 1

minus1du


u minus v minus iε

] (E4)


1radic2π

int 1

minus1du



⎧⎨⎩(

v+1vminus1

)14 v2 gt 1

eminusiπ4(

1+v1minusv

)14 v2 lt 1

(E5)


R0(s)sgt0=

radic2

[1

sminus

int minus1


(v + 1

v minus 1

)14]

=radic

2

[1

sminus 2 eminus2sU+

0 (s)

] (E6)


R0(s)slt0=

radic2

[1

s+ 2Uminus

0 (minuss)

] (E7)

41


together with

R1(s) = 2radic

2[θ(minuss)Uminus


] (E8)


Re

[ int minusiinfin


t + 14

]= 2

radic2f0(minusπg)

[Uminus

0 (πg) minus 1

2πg

]+ 2


1 (πg) +

radic2



References



















42





























Ann Phys 167 227

43

























44

1 Introduction

2 Cusp anomalous



23 Toy model



26 General solution







4 Mass scale





5 Conclusions

Acknowledgments








References







γ (t)t (21)


γ (t) = γ+(t) + iγminus(t) (22)


0

dt

tJ2nminus1(t)

[γminus(t)


γ+(t)

et(2g) minus 1

]= 1

2δn1

(23)int infin

0

dt

tJ2n(t)

[γ+(t)


et(2g) minus 1

]= 0



(t) =(

1 + i cotht

4g






(26)int infin





6



cos(ut) = 2sumn1


cos ϕ

J2nminus1(t)

t

(28)sin(ut) = 2

sumn1

(2n)sin(2nϕ)

cos ϕ

J2n(t)

t





mO(6) = 8radic

2


πeminusπgRe

[int infin

0

dt ei(tminusπ4)


] (29)






γminus(t) = 2sumn1


(2n)J2n(t)γ2n (210)

7




0

dt


int infin

0

dt

tJ2n(t)γ+(t) (211)



cusp(g) = 8g2γ1(g) (212)



(it) = γ (it)sin

(t

4g+ π

4

)sin

(t

4g

)sin

(π4

) = γ (it)radic

2infinprod

k=minusinfin


4

)t minus 4πgk

(213)



4



23 Toy model






t (216)



2t+ 1


2tminus 1


8









γ1 =int infin

0

dt

t

(γ+(t))2 + (γminus(t))2



γ1 int infin

0

dt


1 0 (218)


0 γ1 12 (219)


0 cusp(g) 4g2 (220)



cusp(g) = 4g2

[1 minus 1

3π2g2 +

11

45π4g4 minus 2

(73

630π6 + 4ζ 2

3


] (221)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus1 minus K

16π2gminus2 minus

(3K ln 2

64π3+

27ζ3

2048π3


] (222)




9


0

02

04

06

08

1

05 1 15 2 25 3



0 =int infin

0

dt

t

(γ(0)+ (t))2 + (γ

(0)minus (t))2





(k) =int infin

minusinfin

dt

2πeikt(t) γ (k) =

int infin

minusinfin

dt

2πeiktγ (t) (224)


γ (k) = 0 for k2 gt 1 (225)

10





(k) =int infin

minusinfin

dt

2πeikt

sinh(

t4g

+ iπ4

)sinh

(t

4g

)sin

(π4

)γ (t) (227)




sumn1










c(toy)+ (n g) = c

(toy)minus (n g) = 0 (231)



+(t) =int infin


(232)




11



minusint infin

minusinfin

dk(k)



(u) +1

πminusint 1

minus1

dk(k)



φ(u) = minus 1

π

(2 +

int minus1

minusinfin

dk(k)

k minus u+

int infin

1

dk(k)

k minus u

) (235)



(k) = 1

2φ(k) minus 1

2π

(1 + k

1 minus k

)14

minusint 1

minus1

duφ(u)

u minus k

(1 minus u

1 + u

)14

minusradic

2

π

(1 + k

1 minus k

)14c

1 + k (236)


(k)k21= minus

radic2

π

(1 + k

1 minus k

)14[

1 +c

1 + k+

1

2

int infin

minusinfin

dp(p)

p minus k

(p minus 1

p + 1

)14

θ(p2 minus 1)

] (237)



26 General solution


(t) =int 1


int minus1


int infin



(it) = f0(t)V0(t) + f1(t)V1(t) (239)

12




t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

4πng + t

]

(240)


4πng

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

4πng + t

]


Vn(x) =radic

2

π

int 1


(241)

Uplusmnn (x) = 1

2

int infin




f(toy)


1 (t) = minusc(toy)(g) (242)







(4π ig

( minus 1

4

)) = 0 isin Z (244)



4

) (245)



cusp(g) = 2g[1 minus 2f1(0)] (246)

13



cusp(g) = 2g


[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

]⎫⎬⎭ (247)






V1(minusπg) (248)



[1 minus 2

V0(minusπg)

V1(minusπg)

] (249)


(toy)cusp (g) = 2g


Mminus140(2πg)

] (250)







couplings


(toy)cusp (g) = 3

2πg2 minus 1

2π2g3 minus 1

64π3g4 +

5

64π4g5 minus 11

512π5g6 minus 3

512π6g7 + O(g8) (251)



14






(

34

)[F

(1

4

5

4|α + i0

)+ 2F

(minus1

4

3

4| minus α

)] α = 1(2πg)

(252)



(34

)

(54



F

(1

4

5

4| minus α

)= αminus54U+

0 (1(2α))

(5

4

)

(254)

F

(minus1

4

3

4| minus α

)= αminus34Uminus

0 (1(2α))

(3

4

)



4 34 | minus α



4 and b = 34 )


(minusα)k

k

(a + k)(b + k)

(a)(b)= 1 minus αab + O(α2) (255)



(a)

int infin





(a)

int infin


15







F(

14 5

4 |α+)

+ 2F(minus 1

4 34 | minus α

)F

(14 1

4 |α+)

+ 142αF

(34 3

4 | minus α) (258)


(toy)cusp (g)



4α24C4(α) + O(6) (259)


C0 = 1 minus αF

(14 5

4 |α+)

F(

14 1

4 |α+) C2 = 1[

F(

14 1

4 |α+)]2 C4 = F

(34 3

4 | minus α)[

F(

14 1

4 |α+)]3 (260)



16



4α2 minus 3

8α3 minus 61

64α4 minus 433

128α5 + O(α6)

C2 = 1 minus 1

8α minus 11

128α2 minus 151

1024α3 minus 13085

32768α4 + O(α5) (261)

C4 = 1 minus 3

4α minus 27

32α2 minus 317

128α3 + O(α4)















17




2α minus K

4α2 minus

(3K ln 2

8+

27ζ3

256

)α3 + (263)



3 ln 2

2rarr 1

K

2rarr 1

2

9ζ3

32rarr 1

3 (264)



exp

[3 ln 2

2τ minus K

2τ 2 +

9ζ3

32τ 3 +

]rarr exp

[τ minus τ 2

2+

τ 3

3+

] (265)


(

14

)

(1 + τ

4

)

(34 minus τ

4

)

(34

)

(1 minus τ

4

)

(14 + τ

4

) rarr (1 + τ) (266)



cusp (g)




k (k + 1

2



(toy)cusp (g)

18















plusmn1 (4πng)

n ∓ x

(32)





V0(4πgx)=



(33)


19



g 01 02 04 06 08 10 12 14 16 18

Numer 01976 03616 05843 07096 07825 08276 08576 08787 08944 09065Exact 01939 03584 05821 07080 07813 08267 08568 08781 08938 09059


nmax 10 20 30 40 50 60 70 infinNumer 08305 08286 08279 08276 08274 08273 08272 08267


c(g) = 0 (34)











20








1 =sumn1

c+(n g)

[nU+

0 (4πng)r(x) + U+1 (4πng)x

n minus x

]

+sumn1

cminus(n g)

[nUminus


n + x

] (36)





4 isin Z) (37)




[aplusmn(n) +

bplusmn(n)


]+ 2

[αplusmn(n) +

βplusmn(n)


]+ O(4)

(38)

21




(34

)

(54

) (39)






4 )

2x

(5

4

) sumn1

a+(n)

n minus x

= 1 ( 1)

(310)

minus2x

(3

4

) sumn1

aminus(n)

n + x

= 1 ( 0)


a+(n) = 2(n + 1

4

)(n + 1)2

(14

)

(311)

aminus(n) = (n + 3

4

)2(n + 1)2

(34

)


4 )

2x

(5

4

) sumn1

b+(n)

n minus x

= minus 3

32x

minus 3π

64minus 15

32ln 2 ( 1)

(312)

minus 2x

(3

4

) sumn1

bminus(n)

n + x

= minus 5

32x

minus 5π

64+

9

32ln 2 ( 0)


b+(n) = minusa+(n)

(3 ln 2

4+

3

32n


(3 ln 2

4+

5

32n

) (313)




(2πn)minus1

[

(5

4

) (a+(n) +

b+(n)


)(1 minus 5

128πgn+

)

+

(3

4

) (aminus(n) +

bminus(n)


)(1 +

3


)]+ O(2) (314)

22



cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K

16π2g2+ O(1g3)

]+ O(2) (315)




4 + O(gminus1)]

+ O(4) (316)



2x

(5

4

) sumn1

α+(n)

n minus x

= 0 ( 1)

(317)

minus2x

(3

4

) sumn1

αminus(n)

n + x

= π

2radic

2δ0 ( 0)


α+(n) = 0(318)



2x

(5

4

) sumn1

β+(n)

n minus x

= minus1

2 ( 1)

(319)

minus2x

(3

4

) sumn1

βminus(n)

n + x

= minus1

8+

3π

16radic

2(1 minus 2 ln 2)δ0 ( 0)


β+(n) = minus1

2a+(n)

(320)


(1

4minus 3 ln 2

4+

1

32n

)



23




(2πn)minus1

[

(5

4

) (α+(n) +

β+(n)


)(1 minus 5


)

+

(3

4

) (αminus(n) +

βminus(n)


)(1 +

3


)]+ O(4) (321)


δcusp(g) = minus2

π

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (322)




m2cusp = 4

radic2

πσ2

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (323)


cusp(g) =[

2g minus 3 ln 2

2πminus K

8π2g+ O(1g2)

]minus σ

4radic

2m2

cusp + O(m4

cusp

) (324)


4 Mass scale




mO(6) = 8radic

2


πeminusπgRe

[int minusiinfin


t + 14

] (41)



24



f0(4πgt) =sumn1

t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

n + t

]minus 1

(43)

f1(4πgt) =sumn1

n

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

n + t

]





2




mO(6) = 16radic

2

π2

f1(minusπg)

V0(minusπg) (45)







fn(4πgt) = f (PT)

n (4πgt) + δfn(4πgt) (n = 0 1) (46)

where f (PT)



2

(5

4

) sumn1

a+(n)

t minus n= 1

t

[

(34

)(1 minus t)

(

34 minus t

) minus 1

]

(47)

2

(3

4

) sumn1

aminus(n)

t + n= 1

t

[

(14

)(1 + t)

(

14 + t

) minus 1

]

25



f(PT)

0 (4πgt) = minus(

34

)(1 minus t)

(

34 minus t

)+

1

4πg

[(3 ln 2

4+

1

8t

)

(34

)(1 minus t)

(

34 minus t

) minus (

14

)(1 + t)

8t(

14 + t

) ]+ O(gminus2)

f(PT)

1 (4πgt) = 1

4πg

[

(14

)(1 + t)

4t(

14 + t

) minus (

34

)(1 minus t)

4t(

34 minus t

) ]

minus 1

(4πg)2

[

(14

)(1 + t)

4t(

14 + t

) (1

4tminus 3 ln 2

4

)

minus (

34

)(1 minus t)

4t(

34 minus t

) (1

4t+

3 ln 2

4

)]+ O(gminus3) (48)


δf0(4πgt) = 2

1

4πg

[

(34

)(1 minus t)

2(

34 minus t

) minus (

54

)(1 + t)

2(

54 + t

) ]+ O(gminus2)


δf1(4πgt) = 2

1

4πg

(

54

)(1 + t)

(

54 + t

) +1

(4πg)2

[

(34

)(1 minus t)

8t(

34 minus t

)minus

(54

)(1 + t)

(

54 + t

) (1

8t+

3

4ln 2 minus 1

4

)]+ O(gminus3)





1 (0) minus 4gδf1(0) (410)

Replacing f(PT)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K


]minus 2

π

[1 +

3 minus 6 ln 2


]+ O(4) (411)



mO(6) =radic

2

(

54


[1 +

3 minus 6 ln 2

32πg+



]minus 2

8πg



]+ O(4)

(412)


(PT)


26




fn(4πgt)

(34 minus t

)

(34

)(1 minus t)

rarr f (toy)n (4πgt) (n = 0 1) (413)





(toy)


mtoy = 16radic

2

π2

f(toy)

1 (minusπg)

V0(minusπg)= 16

radic2

π2

1

V1(minusπg) (414)




mtoy = 4

π(

54


[1 minus 1

32πgminus 23


]minus 2

8πg

[1 minus 11


]+ O(4)

(415)


0 and f(toy)


27






32radic

2σm2

toy + O(m4

toy

) (416)




2radic

2mtoy (417)



4π (418)


cusp (g + c1) = 2g

[1 minus K


]minus 2

2radic

2π

[1 +

3


]+ O(4) (419)


[mO(6)(g + c1)]2 = 22

πσ

[(1 +

3

16πg+

16K minus 54


)minus 2

8radic

2πg

(1 minus 3


)+ O(4)

] (420)


O(6)

δcusp = minus σ

4radic

2m2

O(6) + O(m4

O(6)

) (421)



28











2

πeminusπgUminus


2

πeminusπgUminus

1 (πg) (423)





4




mprime = minus σ

8gmO(6) (426)




2

4mprimemO(6) = σ

radic2

32gm2

O(6) (427)



δ(it) = δf0(t)V0(t) + δf1(t)V1(t) (428)


δ(0) = 2δf1(0) (429)

29






δ(it) sim 1







δ(it) (433)


δγ (it) = t

4gδ(0) + O(t2) = t

2gδf1(0) + O(t2) (434)


0

dt

tJ2nminus1(t)

[δγminus(t)

1 minus eminust2g+

δγ+(t)

et2g minus 1

]= 0

(435)int infin

0

dt

tJ2n(t)

[δγ+(t)


et2g minus 1

]= 0



30




2

π2geminusπg minus

radic2

πeminusπg Re

int infin

0

dt prime


]+ O(t2) (436)



tradic

2

8g+ O(t2) (437)


mcusp(g) = mO(6)(g) (438)


5 Conclusions






31





(m4

O(n)

μ2) (51)


m2O(n) = cμ2 e


20 [1 + O(g2)] (52)




Acknowledgments


32








γ1 = 1

2minus π2

6g2 +

11π4

90g4 minus

(73π6

630+ 4ζ 2

3

)g6 + O(g8)

γ2 = ζ3g3 minus

(π2

3ζ3 + 10ζ5

)g5 +

(8π4

45ζ3 +

10π2

3ζ5 + 105ζ7

)g7 + O(g9)

γ3 = minusπ4

90g4 +

37π6

1890g6 + O(g8) γ4 = ζ5g

5 minus(

π2

3ζ5 + 21ζ7

)g7 + O(g9)

γ5 = minus π6

945g6 + O(g8) γ6 = ζ7g

7 + O(g9)

(A2)



3g4 +

44π4

45g6 minus

(292π6

315+ 32ζ 2

3

)g8 + O(g10) (A3)







33



c(g) = minus1

2+ 2gγ1(g) +

sumn1

[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

] (A5)


c(g) = minus1

2+ 2gγ1(g) minus

sumk1



fk(g) = 8gsumn1

[U+






3g2 + 2ζ3g

3 minus π4

6g4 minus 23ζ5g

5 +17π6

108g6 +

1107

4ζ7g

7 + O(g8)

f2 = minus1

2+ 2ζ3g

3 minus π4

30g4 + O(g5) f3 = 1

2+ O(g4)

f4 = minus3

8+ O(g5) f5 = 3

8+ O(g6) f6 = minus 5

16+ O(g7)

(A8)






(t) =int infin




(k)=sumn1



(k)=sumn1


bull For minus1 k 1

(k) = minusradic

2

π

(1 + k

1 minus k

)14[

1 +c(g)

1 + k+

1

2

(int minus1

minusinfin+

int infin

1

)dp(p)

p minus k

(p minus 1

p + 1

)14]

(B4)


34



(t) =sumn1

c+(n g)

[eminusit


]

+sumn1

cminus(n g)

[eit



(B5)



int 1

minus1dk eplusmnxk

int infin

1

dpeminusy(pminus1)

p minus k

(1 + k

1 minus k

p minus 1

p + 1

)plusmn14

Vn(x) =radic

2

π

int 1

minus1

dk exk

(k + 1)n

(1 + k

1 minus k

)14

(B6)

Uplusmnn (y) = 1

2

int infin

1

dp eminusy(pminus1)

(p ∓ 1)n

(p + 1

p minus 1

)∓14






0 (y) minus ex



c+(n g)

[4πngV1(t)U

+0 (4πng) + tV0(t)U

+1 (4πng)

4πng minus t

]

+sumn1

cminus(n g)

[4πngV1(t)U


minus1 (4πng)

4πng + t

] (B8)





int infin

0dsV0(x + s)U+






0 (x)


d

dx(xzprime

i (x)) =(

x minus 1

2

)zi(x) (B11)

35




2(y + s)]

= d


2(y + s)]z1(x + s)

minus d


1(x + s)]z2(y + s) (B12)



1 (y + s) + (y + s)V1(x + s)U+0 (y + s)

] ∣∣s=infins=0


0 (y) (B13)






π

t

t + πg (C1)



π

t2

t2 + (πg)2

(C2)


t2 + π2g2


0

dt

tJ2nminus1(t)

[γminus(t)

1 minus eminust2g+

γ+(t)

et2g minus 1

]= h2nminus1(g)

(C3)int infin

0

dt

tJ2n(t)

[γ+(t)


et2g minus 1

]= h2n(g)


h2nminus1 = 2mprime

π

int infin

0

dtJ2nminus1(t)

t2 + (πg)2

[t


1 minus eminust(2g)

]

(C4)

h2n = 2mprime

π

int infin

0

dtJ2n(t)

t2 + (πg)2

[πg

et(2g) minus 1+

t

1 minus eminust(2g)

]



36



γminus(t) = 2sumn1


(C6)γ+(t) = 2

sumn1

(2n)J2n(t)γ2n(g)



π2gt + O(t2) = it

(γ1 +

2mprime

π2g

)+ O(t2) (C7)



0

dt

t


1 minus eminust2g+


et2g minus 1

]= 2

sumn1



γ1 = 2sumn1



γ1 = minusmprime

π

int infin

0dt

[πg


t

t2 + π2g2(minus(t) + +(t))

] (C10)


πg

t2 + π2g2=

int infin


(C11)t

t2 + π2g2=

int infin



γ1 = minusmprime

π

int infin

0du eminusπgu

[ int infin


+int infin


] (C12)


γ1 = minus 2mprime


radic2mprime

πeminusπgRe

[ int infin

0

dt


] (C13)


37






Vn(x) = 1

π232minusn ex

int 1


= 1

π232minusn eminusx

int 1



Vn(x) = 2minusn

(54 minus n

)

(54

)(2 minus n)


= 2minusn

(54 minus n

)

(54

)(2 minus n)







Uab(x) = 1

2

int infin





(1 +

t

2x

)a+bminus12

(D5)


Uab(x) = 2bminus32

(1

2minus a + b


= 1

2

(1

2minus a + b

)U

(1


) (D6)

38




U+0 (x) = 1

2

(5

4


1 (x) = 1

2

(1

4


(D7)

Uminus0 (x) = 1

2

(3

4


1 (x) = 1

2

(3

4





1 (x) = V1(x)U+0 (x) + V0(x)U+

1 (x) = ex

x (D8)


U+0 (x)Uminus


0 (minusx) = π

2radic

2xeplusmn 3iπ

4 (for Imx ≷ 0) (D9)


V0(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]

(D10)

V1(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]






(D11)Uplusmn








U+0 (x) = (2x)minus54

(5

4

)F

(1

4

5

4| minus 1

2x

)= (2x)minus54

(5

4

) [1 minus 5


]


(3

4

)F

(minus1

4

3

4| minus 1

2x

)= (2x)minus34

(3

4

) [1 +

3


]

39


U+1 (x) = (2x)minus14 1

2

(1

4

)F

(1

4

1

4| minus 1

2x

)= (2x)minus14 1

2

(1

4

) [1 minus 1


]


2

(3

4

)F

(3

4

3

4| minus 1

2x

)= (2x)minus34 1

2

(3

4

) [1 minus 9


]

(D12)








2

πeminus 3iπ

4 eπg[U+


] (D13)




(

34

) [F

(1

4

5

4| 1

2πg+ iε

)+ 2F

(minus1

4

3

4| minus 1

2πg

)] (D14)


2 = σ

(34

)

(54




2(

34

) [8πgF

(1

4

1

4| 1

2πg+ iε

)+ 2F

(3

4

3

4| minus 1

2πg

)] (D16)





F

(1

4

5

4| 1

2πg+ iε

)minus F

(1

4

5

4| 1

2πgminus iε

)= i

radic22

σF

(minus1

4

3

4| minus 1

2πg

) (D17)



40





t + 14


(4πgit)

t + 14

=sumk=01


t + 14

+sumk=01


t + 14

Vk(4πgt) (E1)


Rk(4πgs) = Re

[int minusiinfin


t minus s

]= Re

[int minusiinfin


t minus 4πgs

]

(E2)


Re

[ int minusiinfin


t + 14


minussumn1

nc+(n g)

n + 14

[U+


]+

sumn1

ncminus(n g)

n minus 14





R0(s) = minusRe

[eminusiπ4

int 1


int minusiinfin

0dt eminusvtV0(t)

]=

radic2

πRe

[eminusiπ4

int 1


int 1

minus1du


u minus v minus iε

] (E4)


1radic2π

int 1

minus1du



⎧⎨⎩(

v+1vminus1

)14 v2 gt 1

eminusiπ4(

1+v1minusv

)14 v2 lt 1

(E5)


R0(s)sgt0=

radic2

[1

sminus

int minus1


(v + 1

v minus 1

)14]

=radic

2

[1

sminus 2 eminus2sU+

0 (s)

] (E6)


R0(s)slt0=

radic2

[1

s+ 2Uminus

0 (minuss)

] (E7)

41


together with

R1(s) = 2radic

2[θ(minuss)Uminus


] (E8)


Re

[ int minusiinfin


t + 14

]= 2

radic2f0(minusπg)

[Uminus

0 (πg) minus 1

2πg

]+ 2


1 (πg) +

radic2



References



















42





























Ann Phys 167 227

43

























44

1 Introduction

2 Cusp anomalous



23 Toy model



26 General solution







4 Mass scale





5 Conclusions

Acknowledgments








References



cos(ut) = 2sumn1


cos ϕ

J2nminus1(t)

t

(28)sin(ut) = 2

sumn1

(2n)sin(2nϕ)

cos ϕ

J2n(t)

t





mO(6) = 8radic

2


πeminusπgRe

[int infin

0

dt ei(tminusπ4)


] (29)






γminus(t) = 2sumn1


(2n)J2n(t)γ2n (210)

7




0

dt


int infin

0

dt

tJ2n(t)γ+(t) (211)



cusp(g) = 8g2γ1(g) (212)



(it) = γ (it)sin

(t

4g+ π

4

)sin

(t

4g

)sin

(π4

) = γ (it)radic

2infinprod

k=minusinfin


4

)t minus 4πgk

(213)



4



23 Toy model






t (216)



2t+ 1


2tminus 1


8









γ1 =int infin

0

dt

t

(γ+(t))2 + (γminus(t))2



γ1 int infin

0

dt


1 0 (218)


0 γ1 12 (219)


0 cusp(g) 4g2 (220)



cusp(g) = 4g2

[1 minus 1

3π2g2 +

11

45π4g4 minus 2

(73

630π6 + 4ζ 2

3


] (221)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus1 minus K

16π2gminus2 minus

(3K ln 2

64π3+

27ζ3

2048π3


] (222)




9


0

02

04

06

08

1

05 1 15 2 25 3



0 =int infin

0

dt

t

(γ(0)+ (t))2 + (γ

(0)minus (t))2





(k) =int infin

minusinfin

dt

2πeikt(t) γ (k) =

int infin

minusinfin

dt

2πeiktγ (t) (224)


γ (k) = 0 for k2 gt 1 (225)

10





(k) =int infin

minusinfin

dt

2πeikt

sinh(

t4g

+ iπ4

)sinh

(t

4g

)sin

(π4

)γ (t) (227)




sumn1










c(toy)+ (n g) = c

(toy)minus (n g) = 0 (231)



+(t) =int infin


(232)




11



minusint infin

minusinfin

dk(k)



(u) +1

πminusint 1

minus1

dk(k)



φ(u) = minus 1

π

(2 +

int minus1

minusinfin

dk(k)

k minus u+

int infin

1

dk(k)

k minus u

) (235)



(k) = 1

2φ(k) minus 1

2π

(1 + k

1 minus k

)14

minusint 1

minus1

duφ(u)

u minus k

(1 minus u

1 + u

)14

minusradic

2

π

(1 + k

1 minus k

)14c

1 + k (236)


(k)k21= minus

radic2

π

(1 + k

1 minus k

)14[

1 +c

1 + k+

1

2

int infin

minusinfin

dp(p)

p minus k

(p minus 1

p + 1

)14

θ(p2 minus 1)

] (237)



26 General solution


(t) =int 1


int minus1


int infin



(it) = f0(t)V0(t) + f1(t)V1(t) (239)

12




t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

4πng + t

]

(240)


4πng

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

4πng + t

]


Vn(x) =radic

2

π

int 1


(241)

Uplusmnn (x) = 1

2

int infin




f(toy)


1 (t) = minusc(toy)(g) (242)







(4π ig

( minus 1

4

)) = 0 isin Z (244)



4

) (245)



cusp(g) = 2g[1 minus 2f1(0)] (246)

13



cusp(g) = 2g


[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

]⎫⎬⎭ (247)






V1(minusπg) (248)



[1 minus 2

V0(minusπg)

V1(minusπg)

] (249)


(toy)cusp (g) = 2g


Mminus140(2πg)

] (250)







couplings


(toy)cusp (g) = 3

2πg2 minus 1

2π2g3 minus 1

64π3g4 +

5

64π4g5 minus 11

512π5g6 minus 3

512π6g7 + O(g8) (251)



14






(

34

)[F

(1

4

5

4|α + i0

)+ 2F

(minus1

4

3

4| minus α

)] α = 1(2πg)

(252)



(34

)

(54



F

(1

4

5

4| minus α

)= αminus54U+

0 (1(2α))

(5

4

)

(254)

F

(minus1

4

3

4| minus α

)= αminus34Uminus

0 (1(2α))

(3

4

)



4 34 | minus α



4 and b = 34 )


(minusα)k

k

(a + k)(b + k)

(a)(b)= 1 minus αab + O(α2) (255)



(a)

int infin





(a)

int infin


15







F(

14 5

4 |α+)

+ 2F(minus 1

4 34 | minus α

)F

(14 1

4 |α+)

+ 142αF

(34 3

4 | minus α) (258)


(toy)cusp (g)



4α24C4(α) + O(6) (259)


C0 = 1 minus αF

(14 5

4 |α+)

F(

14 1

4 |α+) C2 = 1[

F(

14 1

4 |α+)]2 C4 = F

(34 3

4 | minus α)[

F(

14 1

4 |α+)]3 (260)



16



4α2 minus 3

8α3 minus 61

64α4 minus 433

128α5 + O(α6)

C2 = 1 minus 1

8α minus 11

128α2 minus 151

1024α3 minus 13085

32768α4 + O(α5) (261)

C4 = 1 minus 3

4α minus 27

32α2 minus 317

128α3 + O(α4)















17




2α minus K

4α2 minus

(3K ln 2

8+

27ζ3

256

)α3 + (263)



3 ln 2

2rarr 1

K

2rarr 1

2

9ζ3

32rarr 1

3 (264)



exp

[3 ln 2

2τ minus K

2τ 2 +

9ζ3

32τ 3 +

]rarr exp

[τ minus τ 2

2+

τ 3

3+

] (265)


(

14

)

(1 + τ

4

)

(34 minus τ

4

)

(34

)

(1 minus τ

4

)

(14 + τ

4

) rarr (1 + τ) (266)



cusp (g)




k (k + 1

2



(toy)cusp (g)

18















plusmn1 (4πng)

n ∓ x

(32)





V0(4πgx)=



(33)


19



g 01 02 04 06 08 10 12 14 16 18

Numer 01976 03616 05843 07096 07825 08276 08576 08787 08944 09065Exact 01939 03584 05821 07080 07813 08267 08568 08781 08938 09059


nmax 10 20 30 40 50 60 70 infinNumer 08305 08286 08279 08276 08274 08273 08272 08267


c(g) = 0 (34)











20








1 =sumn1

c+(n g)

[nU+

0 (4πng)r(x) + U+1 (4πng)x

n minus x

]

+sumn1

cminus(n g)

[nUminus


n + x

] (36)





4 isin Z) (37)




[aplusmn(n) +

bplusmn(n)


]+ 2

[αplusmn(n) +

βplusmn(n)


]+ O(4)

(38)

21




(34

)

(54

) (39)






4 )

2x

(5

4

) sumn1

a+(n)

n minus x

= 1 ( 1)

(310)

minus2x

(3

4

) sumn1

aminus(n)

n + x

= 1 ( 0)


a+(n) = 2(n + 1

4

)(n + 1)2

(14

)

(311)

aminus(n) = (n + 3

4

)2(n + 1)2

(34

)


4 )

2x

(5

4

) sumn1

b+(n)

n minus x

= minus 3

32x

minus 3π

64minus 15

32ln 2 ( 1)

(312)

minus 2x

(3

4

) sumn1

bminus(n)

n + x

= minus 5

32x

minus 5π

64+

9

32ln 2 ( 0)


b+(n) = minusa+(n)

(3 ln 2

4+

3

32n


(3 ln 2

4+

5

32n

) (313)




(2πn)minus1

[

(5

4

) (a+(n) +

b+(n)


)(1 minus 5

128πgn+

)

+

(3

4

) (aminus(n) +

bminus(n)


)(1 +

3


)]+ O(2) (314)

22



cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K

16π2g2+ O(1g3)

]+ O(2) (315)




4 + O(gminus1)]

+ O(4) (316)



2x

(5

4

) sumn1

α+(n)

n minus x

= 0 ( 1)

(317)

minus2x

(3

4

) sumn1

αminus(n)

n + x

= π

2radic

2δ0 ( 0)


α+(n) = 0(318)



2x

(5

4

) sumn1

β+(n)

n minus x

= minus1

2 ( 1)

(319)

minus2x

(3

4

) sumn1

βminus(n)

n + x

= minus1

8+

3π

16radic

2(1 minus 2 ln 2)δ0 ( 0)


β+(n) = minus1

2a+(n)

(320)


(1

4minus 3 ln 2

4+

1

32n

)



23




(2πn)minus1

[

(5

4

) (α+(n) +

β+(n)


)(1 minus 5


)

+

(3

4

) (αminus(n) +

βminus(n)


)(1 +

3


)]+ O(4) (321)


δcusp(g) = minus2

π

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (322)




m2cusp = 4

radic2

πσ2

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (323)


cusp(g) =[

2g minus 3 ln 2

2πminus K

8π2g+ O(1g2)

]minus σ

4radic

2m2

cusp + O(m4

cusp

) (324)


4 Mass scale




mO(6) = 8radic

2


πeminusπgRe

[int minusiinfin


t + 14

] (41)



24



f0(4πgt) =sumn1

t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

n + t

]minus 1

(43)

f1(4πgt) =sumn1

n

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

n + t

]





2




mO(6) = 16radic

2

π2

f1(minusπg)

V0(minusπg) (45)







fn(4πgt) = f (PT)

n (4πgt) + δfn(4πgt) (n = 0 1) (46)

where f (PT)



2

(5

4

) sumn1

a+(n)

t minus n= 1

t

[

(34

)(1 minus t)

(

34 minus t

) minus 1

]

(47)

2

(3

4

) sumn1

aminus(n)

t + n= 1

t

[

(14

)(1 + t)

(

14 + t

) minus 1

]

25



f(PT)

0 (4πgt) = minus(

34

)(1 minus t)

(

34 minus t

)+

1

4πg

[(3 ln 2

4+

1

8t

)

(34

)(1 minus t)

(

34 minus t

) minus (

14

)(1 + t)

8t(

14 + t

) ]+ O(gminus2)

f(PT)

1 (4πgt) = 1

4πg

[

(14

)(1 + t)

4t(

14 + t

) minus (

34

)(1 minus t)

4t(

34 minus t

) ]

minus 1

(4πg)2

[

(14

)(1 + t)

4t(

14 + t

) (1

4tminus 3 ln 2

4

)

minus (

34

)(1 minus t)

4t(

34 minus t

) (1

4t+

3 ln 2

4

)]+ O(gminus3) (48)


δf0(4πgt) = 2

1

4πg

[

(34

)(1 minus t)

2(

34 minus t

) minus (

54

)(1 + t)

2(

54 + t

) ]+ O(gminus2)


δf1(4πgt) = 2

1

4πg

(

54

)(1 + t)

(

54 + t

) +1

(4πg)2

[

(34

)(1 minus t)

8t(

34 minus t

)minus

(54

)(1 + t)

(

54 + t

) (1

8t+

3

4ln 2 minus 1

4

)]+ O(gminus3)





1 (0) minus 4gδf1(0) (410)

Replacing f(PT)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K


]minus 2

π

[1 +

3 minus 6 ln 2


]+ O(4) (411)



mO(6) =radic

2

(

54


[1 +

3 minus 6 ln 2

32πg+



]minus 2

8πg



]+ O(4)

(412)


(PT)


26




fn(4πgt)

(34 minus t

)

(34

)(1 minus t)

rarr f (toy)n (4πgt) (n = 0 1) (413)





(toy)


mtoy = 16radic

2

π2

f(toy)

1 (minusπg)

V0(minusπg)= 16

radic2

π2

1

V1(minusπg) (414)




mtoy = 4

π(

54


[1 minus 1

32πgminus 23


]minus 2

8πg

[1 minus 11


]+ O(4)

(415)


0 and f(toy)


27






32radic

2σm2

toy + O(m4

toy

) (416)




2radic

2mtoy (417)



4π (418)


cusp (g + c1) = 2g

[1 minus K


]minus 2

2radic

2π

[1 +

3


]+ O(4) (419)


[mO(6)(g + c1)]2 = 22

πσ

[(1 +

3

16πg+

16K minus 54


)minus 2

8radic

2πg

(1 minus 3


)+ O(4)

] (420)


O(6)

δcusp = minus σ

4radic

2m2

O(6) + O(m4

O(6)

) (421)



28











2

πeminusπgUminus


2

πeminusπgUminus

1 (πg) (423)





4




mprime = minus σ

8gmO(6) (426)




2

4mprimemO(6) = σ

radic2

32gm2

O(6) (427)



δ(it) = δf0(t)V0(t) + δf1(t)V1(t) (428)


δ(0) = 2δf1(0) (429)

29






δ(it) sim 1







δ(it) (433)


δγ (it) = t

4gδ(0) + O(t2) = t

2gδf1(0) + O(t2) (434)


0

dt

tJ2nminus1(t)

[δγminus(t)

1 minus eminust2g+

δγ+(t)

et2g minus 1

]= 0

(435)int infin

0

dt

tJ2n(t)

[δγ+(t)


et2g minus 1

]= 0



30




2

π2geminusπg minus

radic2

πeminusπg Re

int infin

0

dt prime


]+ O(t2) (436)



tradic

2

8g+ O(t2) (437)


mcusp(g) = mO(6)(g) (438)


5 Conclusions






31





(m4

O(n)

μ2) (51)


m2O(n) = cμ2 e


20 [1 + O(g2)] (52)




Acknowledgments


32








γ1 = 1

2minus π2

6g2 +

11π4

90g4 minus

(73π6

630+ 4ζ 2

3

)g6 + O(g8)

γ2 = ζ3g3 minus

(π2

3ζ3 + 10ζ5

)g5 +

(8π4

45ζ3 +

10π2

3ζ5 + 105ζ7

)g7 + O(g9)

γ3 = minusπ4

90g4 +

37π6

1890g6 + O(g8) γ4 = ζ5g

5 minus(

π2

3ζ5 + 21ζ7

)g7 + O(g9)

γ5 = minus π6

945g6 + O(g8) γ6 = ζ7g

7 + O(g9)

(A2)



3g4 +

44π4

45g6 minus

(292π6

315+ 32ζ 2

3

)g8 + O(g10) (A3)







33



c(g) = minus1

2+ 2gγ1(g) +

sumn1

[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

] (A5)


c(g) = minus1

2+ 2gγ1(g) minus

sumk1



fk(g) = 8gsumn1

[U+






3g2 + 2ζ3g

3 minus π4

6g4 minus 23ζ5g

5 +17π6

108g6 +

1107

4ζ7g

7 + O(g8)

f2 = minus1

2+ 2ζ3g

3 minus π4

30g4 + O(g5) f3 = 1

2+ O(g4)

f4 = minus3

8+ O(g5) f5 = 3

8+ O(g6) f6 = minus 5

16+ O(g7)

(A8)






(t) =int infin




(k)=sumn1



(k)=sumn1


bull For minus1 k 1

(k) = minusradic

2

π

(1 + k

1 minus k

)14[

1 +c(g)

1 + k+

1

2

(int minus1

minusinfin+

int infin

1

)dp(p)

p minus k

(p minus 1

p + 1

)14]

(B4)


34



(t) =sumn1

c+(n g)

[eminusit


]

+sumn1

cminus(n g)

[eit



(B5)



int 1

minus1dk eplusmnxk

int infin

1

dpeminusy(pminus1)

p minus k

(1 + k

1 minus k

p minus 1

p + 1

)plusmn14

Vn(x) =radic

2

π

int 1

minus1

dk exk

(k + 1)n

(1 + k

1 minus k

)14

(B6)

Uplusmnn (y) = 1

2

int infin

1

dp eminusy(pminus1)

(p ∓ 1)n

(p + 1

p minus 1

)∓14






0 (y) minus ex



c+(n g)

[4πngV1(t)U

+0 (4πng) + tV0(t)U

+1 (4πng)

4πng minus t

]

+sumn1

cminus(n g)

[4πngV1(t)U


minus1 (4πng)

4πng + t

] (B8)





int infin

0dsV0(x + s)U+






0 (x)


d

dx(xzprime

i (x)) =(

x minus 1

2

)zi(x) (B11)

35




2(y + s)]

= d


2(y + s)]z1(x + s)

minus d


1(x + s)]z2(y + s) (B12)



1 (y + s) + (y + s)V1(x + s)U+0 (y + s)

] ∣∣s=infins=0


0 (y) (B13)






π

t

t + πg (C1)



π

t2

t2 + (πg)2

(C2)


t2 + π2g2


0

dt

tJ2nminus1(t)

[γminus(t)

1 minus eminust2g+

γ+(t)

et2g minus 1

]= h2nminus1(g)

(C3)int infin

0

dt

tJ2n(t)

[γ+(t)


et2g minus 1

]= h2n(g)


h2nminus1 = 2mprime

π

int infin

0

dtJ2nminus1(t)

t2 + (πg)2

[t


1 minus eminust(2g)

]

(C4)

h2n = 2mprime

π

int infin

0

dtJ2n(t)

t2 + (πg)2

[πg

et(2g) minus 1+

t

1 minus eminust(2g)

]



36



γminus(t) = 2sumn1


(C6)γ+(t) = 2

sumn1

(2n)J2n(t)γ2n(g)



π2gt + O(t2) = it

(γ1 +

2mprime

π2g

)+ O(t2) (C7)



0

dt

t


1 minus eminust2g+


et2g minus 1

]= 2

sumn1



γ1 = 2sumn1



γ1 = minusmprime

π

int infin

0dt

[πg


t

t2 + π2g2(minus(t) + +(t))

] (C10)


πg

t2 + π2g2=

int infin


(C11)t

t2 + π2g2=

int infin



γ1 = minusmprime

π

int infin

0du eminusπgu

[ int infin


+int infin


] (C12)


γ1 = minus 2mprime


radic2mprime

πeminusπgRe

[ int infin

0

dt


] (C13)


37






Vn(x) = 1

π232minusn ex

int 1


= 1

π232minusn eminusx

int 1



Vn(x) = 2minusn

(54 minus n

)

(54

)(2 minus n)


= 2minusn

(54 minus n

)

(54

)(2 minus n)







Uab(x) = 1

2

int infin





(1 +

t

2x

)a+bminus12

(D5)


Uab(x) = 2bminus32

(1

2minus a + b


= 1

2

(1

2minus a + b

)U

(1


) (D6)

38




U+0 (x) = 1

2

(5

4


1 (x) = 1

2

(1

4


(D7)

Uminus0 (x) = 1

2

(3

4


1 (x) = 1

2

(3

4





1 (x) = V1(x)U+0 (x) + V0(x)U+

1 (x) = ex

x (D8)


U+0 (x)Uminus


0 (minusx) = π

2radic

2xeplusmn 3iπ

4 (for Imx ≷ 0) (D9)


V0(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]

(D10)

V1(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]






(D11)Uplusmn








U+0 (x) = (2x)minus54

(5

4

)F

(1

4

5

4| minus 1

2x

)= (2x)minus54

(5

4

) [1 minus 5


]


(3

4

)F

(minus1

4

3

4| minus 1

2x

)= (2x)minus34

(3

4

) [1 +

3


]

39


U+1 (x) = (2x)minus14 1

2

(1

4

)F

(1

4

1

4| minus 1

2x

)= (2x)minus14 1

2

(1

4

) [1 minus 1


]


2

(3

4

)F

(3

4

3

4| minus 1

2x

)= (2x)minus34 1

2

(3

4

) [1 minus 9


]

(D12)








2

πeminus 3iπ

4 eπg[U+


] (D13)




(

34

) [F

(1

4

5

4| 1

2πg+ iε

)+ 2F

(minus1

4

3

4| minus 1

2πg

)] (D14)


2 = σ

(34

)

(54




2(

34

) [8πgF

(1

4

1

4| 1

2πg+ iε

)+ 2F

(3

4

3

4| minus 1

2πg

)] (D16)





F

(1

4

5

4| 1

2πg+ iε

)minus F

(1

4

5

4| 1

2πgminus iε

)= i

radic22

σF

(minus1

4

3

4| minus 1

2πg

) (D17)



40





t + 14


(4πgit)

t + 14

=sumk=01


t + 14

+sumk=01


t + 14

Vk(4πgt) (E1)


Rk(4πgs) = Re

[int minusiinfin


t minus s

]= Re

[int minusiinfin


t minus 4πgs

]

(E2)


Re

[ int minusiinfin


t + 14


minussumn1

nc+(n g)

n + 14

[U+


]+

sumn1

ncminus(n g)

n minus 14





R0(s) = minusRe

[eminusiπ4

int 1


int minusiinfin

0dt eminusvtV0(t)

]=

radic2

πRe

[eminusiπ4

int 1


int 1

minus1du


u minus v minus iε

] (E4)


1radic2π

int 1

minus1du



⎧⎨⎩(

v+1vminus1

)14 v2 gt 1

eminusiπ4(

1+v1minusv

)14 v2 lt 1

(E5)


R0(s)sgt0=

radic2

[1

sminus

int minus1


(v + 1

v minus 1

)14]

=radic

2

[1

sminus 2 eminus2sU+

0 (s)

] (E6)


R0(s)slt0=

radic2

[1

s+ 2Uminus

0 (minuss)

] (E7)

41


together with

R1(s) = 2radic

2[θ(minuss)Uminus


] (E8)


Re

[ int minusiinfin


t + 14

]= 2

radic2f0(minusπg)

[Uminus

0 (πg) minus 1

2πg

]+ 2


1 (πg) +

radic2



References



















42





























Ann Phys 167 227

43

























44

1 Introduction

2 Cusp anomalous



23 Toy model



26 General solution







4 Mass scale





5 Conclusions

Acknowledgments








References




0

dt


int infin

0

dt

tJ2n(t)γ+(t) (211)



cusp(g) = 8g2γ1(g) (212)



(it) = γ (it)sin

(t

4g+ π

4

)sin

(t

4g

)sin

(π4

) = γ (it)radic

2infinprod

k=minusinfin


4

)t minus 4πgk

(213)



4



23 Toy model






t (216)



2t+ 1


2tminus 1


8









γ1 =int infin

0

dt

t

(γ+(t))2 + (γminus(t))2



γ1 int infin

0

dt


1 0 (218)


0 γ1 12 (219)


0 cusp(g) 4g2 (220)



cusp(g) = 4g2

[1 minus 1

3π2g2 +

11

45π4g4 minus 2

(73

630π6 + 4ζ 2

3


] (221)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus1 minus K

16π2gminus2 minus

(3K ln 2

64π3+

27ζ3

2048π3


] (222)




9


0

02

04

06

08

1

05 1 15 2 25 3



0 =int infin

0

dt

t

(γ(0)+ (t))2 + (γ

(0)minus (t))2





(k) =int infin

minusinfin

dt

2πeikt(t) γ (k) =

int infin

minusinfin

dt

2πeiktγ (t) (224)


γ (k) = 0 for k2 gt 1 (225)

10





(k) =int infin

minusinfin

dt

2πeikt

sinh(

t4g

+ iπ4

)sinh

(t

4g

)sin

(π4

)γ (t) (227)




sumn1










c(toy)+ (n g) = c

(toy)minus (n g) = 0 (231)



+(t) =int infin


(232)




11



minusint infin

minusinfin

dk(k)



(u) +1

πminusint 1

minus1

dk(k)



φ(u) = minus 1

π

(2 +

int minus1

minusinfin

dk(k)

k minus u+

int infin

1

dk(k)

k minus u

) (235)



(k) = 1

2φ(k) minus 1

2π

(1 + k

1 minus k

)14

minusint 1

minus1

duφ(u)

u minus k

(1 minus u

1 + u

)14

minusradic

2

π

(1 + k

1 minus k

)14c

1 + k (236)


(k)k21= minus

radic2

π

(1 + k

1 minus k

)14[

1 +c

1 + k+

1

2

int infin

minusinfin

dp(p)

p minus k

(p minus 1

p + 1

)14

θ(p2 minus 1)

] (237)



26 General solution


(t) =int 1


int minus1


int infin



(it) = f0(t)V0(t) + f1(t)V1(t) (239)

12




t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

4πng + t

]

(240)


4πng

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

4πng + t

]


Vn(x) =radic

2

π

int 1


(241)

Uplusmnn (x) = 1

2

int infin




f(toy)


1 (t) = minusc(toy)(g) (242)







(4π ig

( minus 1

4

)) = 0 isin Z (244)



4

) (245)



cusp(g) = 2g[1 minus 2f1(0)] (246)

13



cusp(g) = 2g


[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

]⎫⎬⎭ (247)






V1(minusπg) (248)



[1 minus 2

V0(minusπg)

V1(minusπg)

] (249)


(toy)cusp (g) = 2g


Mminus140(2πg)

] (250)







couplings


(toy)cusp (g) = 3

2πg2 minus 1

2π2g3 minus 1

64π3g4 +

5

64π4g5 minus 11

512π5g6 minus 3

512π6g7 + O(g8) (251)



14






(

34

)[F

(1

4

5

4|α + i0

)+ 2F

(minus1

4

3

4| minus α

)] α = 1(2πg)

(252)



(34

)

(54



F

(1

4

5

4| minus α

)= αminus54U+

0 (1(2α))

(5

4

)

(254)

F

(minus1

4

3

4| minus α

)= αminus34Uminus

0 (1(2α))

(3

4

)



4 34 | minus α



4 and b = 34 )


(minusα)k

k

(a + k)(b + k)

(a)(b)= 1 minus αab + O(α2) (255)



(a)

int infin





(a)

int infin


15







F(

14 5

4 |α+)

+ 2F(minus 1

4 34 | minus α

)F

(14 1

4 |α+)

+ 142αF

(34 3

4 | minus α) (258)


(toy)cusp (g)



4α24C4(α) + O(6) (259)


C0 = 1 minus αF

(14 5

4 |α+)

F(

14 1

4 |α+) C2 = 1[

F(

14 1

4 |α+)]2 C4 = F

(34 3

4 | minus α)[

F(

14 1

4 |α+)]3 (260)



16



4α2 minus 3

8α3 minus 61

64α4 minus 433

128α5 + O(α6)

C2 = 1 minus 1

8α minus 11

128α2 minus 151

1024α3 minus 13085

32768α4 + O(α5) (261)

C4 = 1 minus 3

4α minus 27

32α2 minus 317

128α3 + O(α4)















17




2α minus K

4α2 minus

(3K ln 2

8+

27ζ3

256

)α3 + (263)



3 ln 2

2rarr 1

K

2rarr 1

2

9ζ3

32rarr 1

3 (264)



exp

[3 ln 2

2τ minus K

2τ 2 +

9ζ3

32τ 3 +

]rarr exp

[τ minus τ 2

2+

τ 3

3+

] (265)


(

14

)

(1 + τ

4

)

(34 minus τ

4

)

(34

)

(1 minus τ

4

)

(14 + τ

4

) rarr (1 + τ) (266)



cusp (g)




k (k + 1

2



(toy)cusp (g)

18















plusmn1 (4πng)

n ∓ x

(32)





V0(4πgx)=



(33)


19



g 01 02 04 06 08 10 12 14 16 18

Numer 01976 03616 05843 07096 07825 08276 08576 08787 08944 09065Exact 01939 03584 05821 07080 07813 08267 08568 08781 08938 09059


nmax 10 20 30 40 50 60 70 infinNumer 08305 08286 08279 08276 08274 08273 08272 08267


c(g) = 0 (34)











20








1 =sumn1

c+(n g)

[nU+

0 (4πng)r(x) + U+1 (4πng)x

n minus x

]

+sumn1

cminus(n g)

[nUminus


n + x

] (36)





4 isin Z) (37)




[aplusmn(n) +

bplusmn(n)


]+ 2

[αplusmn(n) +

βplusmn(n)


]+ O(4)

(38)

21




(34

)

(54

) (39)






4 )

2x

(5

4

) sumn1

a+(n)

n minus x

= 1 ( 1)

(310)

minus2x

(3

4

) sumn1

aminus(n)

n + x

= 1 ( 0)


a+(n) = 2(n + 1

4

)(n + 1)2

(14

)

(311)

aminus(n) = (n + 3

4

)2(n + 1)2

(34

)


4 )

2x

(5

4

) sumn1

b+(n)

n minus x

= minus 3

32x

minus 3π

64minus 15

32ln 2 ( 1)

(312)

minus 2x

(3

4

) sumn1

bminus(n)

n + x

= minus 5

32x

minus 5π

64+

9

32ln 2 ( 0)


b+(n) = minusa+(n)

(3 ln 2

4+

3

32n


(3 ln 2

4+

5

32n

) (313)




(2πn)minus1

[

(5

4

) (a+(n) +

b+(n)


)(1 minus 5

128πgn+

)

+

(3

4

) (aminus(n) +

bminus(n)


)(1 +

3


)]+ O(2) (314)

22



cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K

16π2g2+ O(1g3)

]+ O(2) (315)




4 + O(gminus1)]

+ O(4) (316)



2x

(5

4

) sumn1

α+(n)

n minus x

= 0 ( 1)

(317)

minus2x

(3

4

) sumn1

αminus(n)

n + x

= π

2radic

2δ0 ( 0)


α+(n) = 0(318)



2x

(5

4

) sumn1

β+(n)

n minus x

= minus1

2 ( 1)

(319)

minus2x

(3

4

) sumn1

βminus(n)

n + x

= minus1

8+

3π

16radic

2(1 minus 2 ln 2)δ0 ( 0)


β+(n) = minus1

2a+(n)

(320)


(1

4minus 3 ln 2

4+

1

32n

)



23




(2πn)minus1

[

(5

4

) (α+(n) +

β+(n)


)(1 minus 5


)

+

(3

4

) (αminus(n) +

βminus(n)


)(1 +

3


)]+ O(4) (321)


δcusp(g) = minus2

π

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (322)




m2cusp = 4

radic2

πσ2

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (323)


cusp(g) =[

2g minus 3 ln 2

2πminus K

8π2g+ O(1g2)

]minus σ

4radic

2m2

cusp + O(m4

cusp

) (324)


4 Mass scale




mO(6) = 8radic

2


πeminusπgRe

[int minusiinfin


t + 14

] (41)



24



f0(4πgt) =sumn1

t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

n + t

]minus 1

(43)

f1(4πgt) =sumn1

n

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

n + t

]





2




mO(6) = 16radic

2

π2

f1(minusπg)

V0(minusπg) (45)







fn(4πgt) = f (PT)

n (4πgt) + δfn(4πgt) (n = 0 1) (46)

where f (PT)



2

(5

4

) sumn1

a+(n)

t minus n= 1

t

[

(34

)(1 minus t)

(

34 minus t

) minus 1

]

(47)

2

(3

4

) sumn1

aminus(n)

t + n= 1

t

[

(14

)(1 + t)

(

14 + t

) minus 1

]

25



f(PT)

0 (4πgt) = minus(

34

)(1 minus t)

(

34 minus t

)+

1

4πg

[(3 ln 2

4+

1

8t

)

(34

)(1 minus t)

(

34 minus t

) minus (

14

)(1 + t)

8t(

14 + t

) ]+ O(gminus2)

f(PT)

1 (4πgt) = 1

4πg

[

(14

)(1 + t)

4t(

14 + t

) minus (

34

)(1 minus t)

4t(

34 minus t

) ]

minus 1

(4πg)2

[

(14

)(1 + t)

4t(

14 + t

) (1

4tminus 3 ln 2

4

)

minus (

34

)(1 minus t)

4t(

34 minus t

) (1

4t+

3 ln 2

4

)]+ O(gminus3) (48)


δf0(4πgt) = 2

1

4πg

[

(34

)(1 minus t)

2(

34 minus t

) minus (

54

)(1 + t)

2(

54 + t

) ]+ O(gminus2)


δf1(4πgt) = 2

1

4πg

(

54

)(1 + t)

(

54 + t

) +1

(4πg)2

[

(34

)(1 minus t)

8t(

34 minus t

)minus

(54

)(1 + t)

(

54 + t

) (1

8t+

3

4ln 2 minus 1

4

)]+ O(gminus3)





1 (0) minus 4gδf1(0) (410)

Replacing f(PT)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K


]minus 2

π

[1 +

3 minus 6 ln 2


]+ O(4) (411)



mO(6) =radic

2

(

54


[1 +

3 minus 6 ln 2

32πg+



]minus 2

8πg



]+ O(4)

(412)


(PT)


26




fn(4πgt)

(34 minus t

)

(34

)(1 minus t)

rarr f (toy)n (4πgt) (n = 0 1) (413)





(toy)


mtoy = 16radic

2

π2

f(toy)

1 (minusπg)

V0(minusπg)= 16

radic2

π2

1

V1(minusπg) (414)




mtoy = 4

π(

54


[1 minus 1

32πgminus 23


]minus 2

8πg

[1 minus 11


]+ O(4)

(415)


0 and f(toy)


27






32radic

2σm2

toy + O(m4

toy

) (416)




2radic

2mtoy (417)



4π (418)


cusp (g + c1) = 2g

[1 minus K


]minus 2

2radic

2π

[1 +

3


]+ O(4) (419)


[mO(6)(g + c1)]2 = 22

πσ

[(1 +

3

16πg+

16K minus 54


)minus 2

8radic

2πg

(1 minus 3


)+ O(4)

] (420)


O(6)

δcusp = minus σ

4radic

2m2

O(6) + O(m4

O(6)

) (421)



28











2

πeminusπgUminus


2

πeminusπgUminus

1 (πg) (423)





4




mprime = minus σ

8gmO(6) (426)




2

4mprimemO(6) = σ

radic2

32gm2

O(6) (427)



δ(it) = δf0(t)V0(t) + δf1(t)V1(t) (428)


δ(0) = 2δf1(0) (429)

29






δ(it) sim 1







δ(it) (433)


δγ (it) = t

4gδ(0) + O(t2) = t

2gδf1(0) + O(t2) (434)


0

dt

tJ2nminus1(t)

[δγminus(t)

1 minus eminust2g+

δγ+(t)

et2g minus 1

]= 0

(435)int infin

0

dt

tJ2n(t)

[δγ+(t)


et2g minus 1

]= 0



30




2

π2geminusπg minus

radic2

πeminusπg Re

int infin

0

dt prime


]+ O(t2) (436)



tradic

2

8g+ O(t2) (437)


mcusp(g) = mO(6)(g) (438)


5 Conclusions






31





(m4

O(n)

μ2) (51)


m2O(n) = cμ2 e


20 [1 + O(g2)] (52)




Acknowledgments


32








γ1 = 1

2minus π2

6g2 +

11π4

90g4 minus

(73π6

630+ 4ζ 2

3

)g6 + O(g8)

γ2 = ζ3g3 minus

(π2

3ζ3 + 10ζ5

)g5 +

(8π4

45ζ3 +

10π2

3ζ5 + 105ζ7

)g7 + O(g9)

γ3 = minusπ4

90g4 +

37π6

1890g6 + O(g8) γ4 = ζ5g

5 minus(

π2

3ζ5 + 21ζ7

)g7 + O(g9)

γ5 = minus π6

945g6 + O(g8) γ6 = ζ7g

7 + O(g9)

(A2)



3g4 +

44π4

45g6 minus

(292π6

315+ 32ζ 2

3

)g8 + O(g10) (A3)







33



c(g) = minus1

2+ 2gγ1(g) +

sumn1

[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

] (A5)


c(g) = minus1

2+ 2gγ1(g) minus

sumk1



fk(g) = 8gsumn1

[U+






3g2 + 2ζ3g

3 minus π4

6g4 minus 23ζ5g

5 +17π6

108g6 +

1107

4ζ7g

7 + O(g8)

f2 = minus1

2+ 2ζ3g

3 minus π4

30g4 + O(g5) f3 = 1

2+ O(g4)

f4 = minus3

8+ O(g5) f5 = 3

8+ O(g6) f6 = minus 5

16+ O(g7)

(A8)






(t) =int infin




(k)=sumn1



(k)=sumn1


bull For minus1 k 1

(k) = minusradic

2

π

(1 + k

1 minus k

)14[

1 +c(g)

1 + k+

1

2

(int minus1

minusinfin+

int infin

1

)dp(p)

p minus k

(p minus 1

p + 1

)14]

(B4)


34



(t) =sumn1

c+(n g)

[eminusit


]

+sumn1

cminus(n g)

[eit



(B5)



int 1

minus1dk eplusmnxk

int infin

1

dpeminusy(pminus1)

p minus k

(1 + k

1 minus k

p minus 1

p + 1

)plusmn14

Vn(x) =radic

2

π

int 1

minus1

dk exk

(k + 1)n

(1 + k

1 minus k

)14

(B6)

Uplusmnn (y) = 1

2

int infin

1

dp eminusy(pminus1)

(p ∓ 1)n

(p + 1

p minus 1

)∓14






0 (y) minus ex



c+(n g)

[4πngV1(t)U

+0 (4πng) + tV0(t)U

+1 (4πng)

4πng minus t

]

+sumn1

cminus(n g)

[4πngV1(t)U


minus1 (4πng)

4πng + t

] (B8)





int infin

0dsV0(x + s)U+






0 (x)


d

dx(xzprime

i (x)) =(

x minus 1

2

)zi(x) (B11)

35




2(y + s)]

= d


2(y + s)]z1(x + s)

minus d


1(x + s)]z2(y + s) (B12)



1 (y + s) + (y + s)V1(x + s)U+0 (y + s)

] ∣∣s=infins=0


0 (y) (B13)






π

t

t + πg (C1)



π

t2

t2 + (πg)2

(C2)


t2 + π2g2


0

dt

tJ2nminus1(t)

[γminus(t)

1 minus eminust2g+

γ+(t)

et2g minus 1

]= h2nminus1(g)

(C3)int infin

0

dt

tJ2n(t)

[γ+(t)


et2g minus 1

]= h2n(g)


h2nminus1 = 2mprime

π

int infin

0

dtJ2nminus1(t)

t2 + (πg)2

[t


1 minus eminust(2g)

]

(C4)

h2n = 2mprime

π

int infin

0

dtJ2n(t)

t2 + (πg)2

[πg

et(2g) minus 1+

t

1 minus eminust(2g)

]



36



γminus(t) = 2sumn1


(C6)γ+(t) = 2

sumn1

(2n)J2n(t)γ2n(g)



π2gt + O(t2) = it

(γ1 +

2mprime

π2g

)+ O(t2) (C7)



0

dt

t


1 minus eminust2g+


et2g minus 1

]= 2

sumn1



γ1 = 2sumn1



γ1 = minusmprime

π

int infin

0dt

[πg


t

t2 + π2g2(minus(t) + +(t))

] (C10)


πg

t2 + π2g2=

int infin


(C11)t

t2 + π2g2=

int infin



γ1 = minusmprime

π

int infin

0du eminusπgu

[ int infin


+int infin


] (C12)


γ1 = minus 2mprime


radic2mprime

πeminusπgRe

[ int infin

0

dt


] (C13)


37






Vn(x) = 1

π232minusn ex

int 1


= 1

π232minusn eminusx

int 1



Vn(x) = 2minusn

(54 minus n

)

(54

)(2 minus n)


= 2minusn

(54 minus n

)

(54

)(2 minus n)







Uab(x) = 1

2

int infin





(1 +

t

2x

)a+bminus12

(D5)


Uab(x) = 2bminus32

(1

2minus a + b


= 1

2

(1

2minus a + b

)U

(1


) (D6)

38




U+0 (x) = 1

2

(5

4


1 (x) = 1

2

(1

4


(D7)

Uminus0 (x) = 1

2

(3

4


1 (x) = 1

2

(3

4





1 (x) = V1(x)U+0 (x) + V0(x)U+

1 (x) = ex

x (D8)


U+0 (x)Uminus


0 (minusx) = π

2radic

2xeplusmn 3iπ

4 (for Imx ≷ 0) (D9)


V0(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]

(D10)

V1(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]






(D11)Uplusmn








U+0 (x) = (2x)minus54

(5

4

)F

(1

4

5

4| minus 1

2x

)= (2x)minus54

(5

4

) [1 minus 5


]


(3

4

)F

(minus1

4

3

4| minus 1

2x

)= (2x)minus34

(3

4

) [1 +

3


]

39


U+1 (x) = (2x)minus14 1

2

(1

4

)F

(1

4

1

4| minus 1

2x

)= (2x)minus14 1

2

(1

4

) [1 minus 1


]


2

(3

4

)F

(3

4

3

4| minus 1

2x

)= (2x)minus34 1

2

(3

4

) [1 minus 9


]

(D12)








2

πeminus 3iπ

4 eπg[U+


] (D13)




(

34

) [F

(1

4

5

4| 1

2πg+ iε

)+ 2F

(minus1

4

3

4| minus 1

2πg

)] (D14)


2 = σ

(34

)

(54




2(

34

) [8πgF

(1

4

1

4| 1

2πg+ iε

)+ 2F

(3

4

3

4| minus 1

2πg

)] (D16)





F

(1

4

5

4| 1

2πg+ iε

)minus F

(1

4

5

4| 1

2πgminus iε

)= i

radic22

σF

(minus1

4

3

4| minus 1

2πg

) (D17)



40





t + 14


(4πgit)

t + 14

=sumk=01


t + 14

+sumk=01


t + 14

Vk(4πgt) (E1)


Rk(4πgs) = Re

[int minusiinfin


t minus s

]= Re

[int minusiinfin


t minus 4πgs

]

(E2)


Re

[ int minusiinfin


t + 14


minussumn1

nc+(n g)

n + 14

[U+


]+

sumn1

ncminus(n g)

n minus 14





R0(s) = minusRe

[eminusiπ4

int 1


int minusiinfin

0dt eminusvtV0(t)

]=

radic2

πRe

[eminusiπ4

int 1


int 1

minus1du


u minus v minus iε

] (E4)


1radic2π

int 1

minus1du



⎧⎨⎩(

v+1vminus1

)14 v2 gt 1

eminusiπ4(

1+v1minusv

)14 v2 lt 1

(E5)


R0(s)sgt0=

radic2

[1

sminus

int minus1


(v + 1

v minus 1

)14]

=radic

2

[1

sminus 2 eminus2sU+

0 (s)

] (E6)


R0(s)slt0=

radic2

[1

s+ 2Uminus

0 (minuss)

] (E7)

41


together with

R1(s) = 2radic

2[θ(minuss)Uminus


] (E8)


Re

[ int minusiinfin


t + 14

]= 2

radic2f0(minusπg)

[Uminus

0 (πg) minus 1

2πg

]+ 2


1 (πg) +

radic2



References



















42





























Ann Phys 167 227

43

























44

1 Introduction

2 Cusp anomalous



23 Toy model



26 General solution







4 Mass scale





5 Conclusions

Acknowledgments








References









γ1 =int infin

0

dt

t

(γ+(t))2 + (γminus(t))2



γ1 int infin

0

dt


1 0 (218)


0 γ1 12 (219)


0 cusp(g) 4g2 (220)



cusp(g) = 4g2

[1 minus 1

3π2g2 +

11

45π4g4 minus 2

(73

630π6 + 4ζ 2

3


] (221)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus1 minus K

16π2gminus2 minus

(3K ln 2

64π3+

27ζ3

2048π3


] (222)




9


0

02

04

06

08

1

05 1 15 2 25 3



0 =int infin

0

dt

t

(γ(0)+ (t))2 + (γ

(0)minus (t))2





(k) =int infin

minusinfin

dt

2πeikt(t) γ (k) =

int infin

minusinfin

dt

2πeiktγ (t) (224)


γ (k) = 0 for k2 gt 1 (225)

10





(k) =int infin

minusinfin

dt

2πeikt

sinh(

t4g

+ iπ4

)sinh

(t

4g

)sin

(π4

)γ (t) (227)




sumn1










c(toy)+ (n g) = c

(toy)minus (n g) = 0 (231)



+(t) =int infin


(232)




11



minusint infin

minusinfin

dk(k)



(u) +1

πminusint 1

minus1

dk(k)



φ(u) = minus 1

π

(2 +

int minus1

minusinfin

dk(k)

k minus u+

int infin

1

dk(k)

k minus u

) (235)



(k) = 1

2φ(k) minus 1

2π

(1 + k

1 minus k

)14

minusint 1

minus1

duφ(u)

u minus k

(1 minus u

1 + u

)14

minusradic

2

π

(1 + k

1 minus k

)14c

1 + k (236)


(k)k21= minus

radic2

π

(1 + k

1 minus k

)14[

1 +c

1 + k+

1

2

int infin

minusinfin

dp(p)

p minus k

(p minus 1

p + 1

)14

θ(p2 minus 1)

] (237)



26 General solution


(t) =int 1


int minus1


int infin



(it) = f0(t)V0(t) + f1(t)V1(t) (239)

12




t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

4πng + t

]

(240)


4πng

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

4πng + t

]


Vn(x) =radic

2

π

int 1


(241)

Uplusmnn (x) = 1

2

int infin




f(toy)


1 (t) = minusc(toy)(g) (242)







(4π ig

( minus 1

4

)) = 0 isin Z (244)



4

) (245)



cusp(g) = 2g[1 minus 2f1(0)] (246)

13



cusp(g) = 2g


[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

]⎫⎬⎭ (247)






V1(minusπg) (248)



[1 minus 2

V0(minusπg)

V1(minusπg)

] (249)


(toy)cusp (g) = 2g


Mminus140(2πg)

] (250)







couplings


(toy)cusp (g) = 3

2πg2 minus 1

2π2g3 minus 1

64π3g4 +

5

64π4g5 minus 11

512π5g6 minus 3

512π6g7 + O(g8) (251)



14






(

34

)[F

(1

4

5

4|α + i0

)+ 2F

(minus1

4

3

4| minus α

)] α = 1(2πg)

(252)



(34

)

(54



F

(1

4

5

4| minus α

)= αminus54U+

0 (1(2α))

(5

4

)

(254)

F

(minus1

4

3

4| minus α

)= αminus34Uminus

0 (1(2α))

(3

4

)



4 34 | minus α



4 and b = 34 )


(minusα)k

k

(a + k)(b + k)

(a)(b)= 1 minus αab + O(α2) (255)



(a)

int infin





(a)

int infin


15







F(

14 5

4 |α+)

+ 2F(minus 1

4 34 | minus α

)F

(14 1

4 |α+)

+ 142αF

(34 3

4 | minus α) (258)


(toy)cusp (g)



4α24C4(α) + O(6) (259)


C0 = 1 minus αF

(14 5

4 |α+)

F(

14 1

4 |α+) C2 = 1[

F(

14 1

4 |α+)]2 C4 = F

(34 3

4 | minus α)[

F(

14 1

4 |α+)]3 (260)



16



4α2 minus 3

8α3 minus 61

64α4 minus 433

128α5 + O(α6)

C2 = 1 minus 1

8α minus 11

128α2 minus 151

1024α3 minus 13085

32768α4 + O(α5) (261)

C4 = 1 minus 3

4α minus 27

32α2 minus 317

128α3 + O(α4)















17




2α minus K

4α2 minus

(3K ln 2

8+

27ζ3

256

)α3 + (263)



3 ln 2

2rarr 1

K

2rarr 1

2

9ζ3

32rarr 1

3 (264)



exp

[3 ln 2

2τ minus K

2τ 2 +

9ζ3

32τ 3 +

]rarr exp

[τ minus τ 2

2+

τ 3

3+

] (265)


(

14

)

(1 + τ

4

)

(34 minus τ

4

)

(34

)

(1 minus τ

4

)

(14 + τ

4

) rarr (1 + τ) (266)



cusp (g)




k (k + 1

2



(toy)cusp (g)

18















plusmn1 (4πng)

n ∓ x

(32)





V0(4πgx)=



(33)


19



g 01 02 04 06 08 10 12 14 16 18

Numer 01976 03616 05843 07096 07825 08276 08576 08787 08944 09065Exact 01939 03584 05821 07080 07813 08267 08568 08781 08938 09059


nmax 10 20 30 40 50 60 70 infinNumer 08305 08286 08279 08276 08274 08273 08272 08267


c(g) = 0 (34)











20








1 =sumn1

c+(n g)

[nU+

0 (4πng)r(x) + U+1 (4πng)x

n minus x

]

+sumn1

cminus(n g)

[nUminus


n + x

] (36)





4 isin Z) (37)




[aplusmn(n) +

bplusmn(n)


]+ 2

[αplusmn(n) +

βplusmn(n)


]+ O(4)

(38)

21




(34

)

(54

) (39)






4 )

2x

(5

4

) sumn1

a+(n)

n minus x

= 1 ( 1)

(310)

minus2x

(3

4

) sumn1

aminus(n)

n + x

= 1 ( 0)


a+(n) = 2(n + 1

4

)(n + 1)2

(14

)

(311)

aminus(n) = (n + 3

4

)2(n + 1)2

(34

)


4 )

2x

(5

4

) sumn1

b+(n)

n minus x

= minus 3

32x

minus 3π

64minus 15

32ln 2 ( 1)

(312)

minus 2x

(3

4

) sumn1

bminus(n)

n + x

= minus 5

32x

minus 5π

64+

9

32ln 2 ( 0)


b+(n) = minusa+(n)

(3 ln 2

4+

3

32n


(3 ln 2

4+

5

32n

) (313)




(2πn)minus1

[

(5

4

) (a+(n) +

b+(n)


)(1 minus 5

128πgn+

)

+

(3

4

) (aminus(n) +

bminus(n)


)(1 +

3


)]+ O(2) (314)

22



cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K

16π2g2+ O(1g3)

]+ O(2) (315)




4 + O(gminus1)]

+ O(4) (316)



2x

(5

4

) sumn1

α+(n)

n minus x

= 0 ( 1)

(317)

minus2x

(3

4

) sumn1

αminus(n)

n + x

= π

2radic

2δ0 ( 0)


α+(n) = 0(318)



2x

(5

4

) sumn1

β+(n)

n minus x

= minus1

2 ( 1)

(319)

minus2x

(3

4

) sumn1

βminus(n)

n + x

= minus1

8+

3π

16radic

2(1 minus 2 ln 2)δ0 ( 0)


β+(n) = minus1

2a+(n)

(320)


(1

4minus 3 ln 2

4+

1

32n

)



23




(2πn)minus1

[

(5

4

) (α+(n) +

β+(n)


)(1 minus 5


)

+

(3

4

) (αminus(n) +

βminus(n)


)(1 +

3


)]+ O(4) (321)


δcusp(g) = minus2

π

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (322)




m2cusp = 4

radic2

πσ2

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (323)


cusp(g) =[

2g minus 3 ln 2

2πminus K

8π2g+ O(1g2)

]minus σ

4radic

2m2

cusp + O(m4

cusp

) (324)


4 Mass scale




mO(6) = 8radic

2


πeminusπgRe

[int minusiinfin


t + 14

] (41)



24



f0(4πgt) =sumn1

t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

n + t

]minus 1

(43)

f1(4πgt) =sumn1

n

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

n + t

]





2




mO(6) = 16radic

2

π2

f1(minusπg)

V0(minusπg) (45)







fn(4πgt) = f (PT)

n (4πgt) + δfn(4πgt) (n = 0 1) (46)

where f (PT)



2

(5

4

) sumn1

a+(n)

t minus n= 1

t

[

(34

)(1 minus t)

(

34 minus t

) minus 1

]

(47)

2

(3

4

) sumn1

aminus(n)

t + n= 1

t

[

(14

)(1 + t)

(

14 + t

) minus 1

]

25



f(PT)

0 (4πgt) = minus(

34

)(1 minus t)

(

34 minus t

)+

1

4πg

[(3 ln 2

4+

1

8t

)

(34

)(1 minus t)

(

34 minus t

) minus (

14

)(1 + t)

8t(

14 + t

) ]+ O(gminus2)

f(PT)

1 (4πgt) = 1

4πg

[

(14

)(1 + t)

4t(

14 + t

) minus (

34

)(1 minus t)

4t(

34 minus t

) ]

minus 1

(4πg)2

[

(14

)(1 + t)

4t(

14 + t

) (1

4tminus 3 ln 2

4

)

minus (

34

)(1 minus t)

4t(

34 minus t

) (1

4t+

3 ln 2

4

)]+ O(gminus3) (48)


δf0(4πgt) = 2

1

4πg

[

(34

)(1 minus t)

2(

34 minus t

) minus (

54

)(1 + t)

2(

54 + t

) ]+ O(gminus2)


δf1(4πgt) = 2

1

4πg

(

54

)(1 + t)

(

54 + t

) +1

(4πg)2

[

(34

)(1 minus t)

8t(

34 minus t

)minus

(54

)(1 + t)

(

54 + t

) (1

8t+

3

4ln 2 minus 1

4

)]+ O(gminus3)





1 (0) minus 4gδf1(0) (410)

Replacing f(PT)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K


]minus 2

π

[1 +

3 minus 6 ln 2


]+ O(4) (411)



mO(6) =radic

2

(

54


[1 +

3 minus 6 ln 2

32πg+



]minus 2

8πg



]+ O(4)

(412)


(PT)


26




fn(4πgt)

(34 minus t

)

(34

)(1 minus t)

rarr f (toy)n (4πgt) (n = 0 1) (413)





(toy)


mtoy = 16radic

2

π2

f(toy)

1 (minusπg)

V0(minusπg)= 16

radic2

π2

1

V1(minusπg) (414)




mtoy = 4

π(

54


[1 minus 1

32πgminus 23


]minus 2

8πg

[1 minus 11


]+ O(4)

(415)


0 and f(toy)


27






32radic

2σm2

toy + O(m4

toy

) (416)




2radic

2mtoy (417)



4π (418)


cusp (g + c1) = 2g

[1 minus K


]minus 2

2radic

2π

[1 +

3


]+ O(4) (419)


[mO(6)(g + c1)]2 = 22

πσ

[(1 +

3

16πg+

16K minus 54


)minus 2

8radic

2πg

(1 minus 3


)+ O(4)

] (420)


O(6)

δcusp = minus σ

4radic

2m2

O(6) + O(m4

O(6)

) (421)



28











2

πeminusπgUminus


2

πeminusπgUminus

1 (πg) (423)





4




mprime = minus σ

8gmO(6) (426)




2

4mprimemO(6) = σ

radic2

32gm2

O(6) (427)



δ(it) = δf0(t)V0(t) + δf1(t)V1(t) (428)


δ(0) = 2δf1(0) (429)

29






δ(it) sim 1







δ(it) (433)


δγ (it) = t

4gδ(0) + O(t2) = t

2gδf1(0) + O(t2) (434)


0

dt

tJ2nminus1(t)

[δγminus(t)

1 minus eminust2g+

δγ+(t)

et2g minus 1

]= 0

(435)int infin

0

dt

tJ2n(t)

[δγ+(t)


et2g minus 1

]= 0



30




2

π2geminusπg minus

radic2

πeminusπg Re

int infin

0

dt prime


]+ O(t2) (436)



tradic

2

8g+ O(t2) (437)


mcusp(g) = mO(6)(g) (438)


5 Conclusions






31





(m4

O(n)

μ2) (51)


m2O(n) = cμ2 e


20 [1 + O(g2)] (52)




Acknowledgments


32








γ1 = 1

2minus π2

6g2 +

11π4

90g4 minus

(73π6

630+ 4ζ 2

3

)g6 + O(g8)

γ2 = ζ3g3 minus

(π2

3ζ3 + 10ζ5

)g5 +

(8π4

45ζ3 +

10π2

3ζ5 + 105ζ7

)g7 + O(g9)

γ3 = minusπ4

90g4 +

37π6

1890g6 + O(g8) γ4 = ζ5g

5 minus(

π2

3ζ5 + 21ζ7

)g7 + O(g9)

γ5 = minus π6

945g6 + O(g8) γ6 = ζ7g

7 + O(g9)

(A2)



3g4 +

44π4

45g6 minus

(292π6

315+ 32ζ 2

3

)g8 + O(g10) (A3)







33



c(g) = minus1

2+ 2gγ1(g) +

sumn1

[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

] (A5)


c(g) = minus1

2+ 2gγ1(g) minus

sumk1



fk(g) = 8gsumn1

[U+






3g2 + 2ζ3g

3 minus π4

6g4 minus 23ζ5g

5 +17π6

108g6 +

1107

4ζ7g

7 + O(g8)

f2 = minus1

2+ 2ζ3g

3 minus π4

30g4 + O(g5) f3 = 1

2+ O(g4)

f4 = minus3

8+ O(g5) f5 = 3

8+ O(g6) f6 = minus 5

16+ O(g7)

(A8)






(t) =int infin




(k)=sumn1



(k)=sumn1


bull For minus1 k 1

(k) = minusradic

2

π

(1 + k

1 minus k

)14[

1 +c(g)

1 + k+

1

2

(int minus1

minusinfin+

int infin

1

)dp(p)

p minus k

(p minus 1

p + 1

)14]

(B4)


34



(t) =sumn1

c+(n g)

[eminusit


]

+sumn1

cminus(n g)

[eit



(B5)



int 1

minus1dk eplusmnxk

int infin

1

dpeminusy(pminus1)

p minus k

(1 + k

1 minus k

p minus 1

p + 1

)plusmn14

Vn(x) =radic

2

π

int 1

minus1

dk exk

(k + 1)n

(1 + k

1 minus k

)14

(B6)

Uplusmnn (y) = 1

2

int infin

1

dp eminusy(pminus1)

(p ∓ 1)n

(p + 1

p minus 1

)∓14






0 (y) minus ex



c+(n g)

[4πngV1(t)U

+0 (4πng) + tV0(t)U

+1 (4πng)

4πng minus t

]

+sumn1

cminus(n g)

[4πngV1(t)U


minus1 (4πng)

4πng + t

] (B8)





int infin

0dsV0(x + s)U+






0 (x)


d

dx(xzprime

i (x)) =(

x minus 1

2

)zi(x) (B11)

35




2(y + s)]

= d


2(y + s)]z1(x + s)

minus d


1(x + s)]z2(y + s) (B12)



1 (y + s) + (y + s)V1(x + s)U+0 (y + s)

] ∣∣s=infins=0


0 (y) (B13)






π

t

t + πg (C1)



π

t2

t2 + (πg)2

(C2)


t2 + π2g2


0

dt

tJ2nminus1(t)

[γminus(t)

1 minus eminust2g+

γ+(t)

et2g minus 1

]= h2nminus1(g)

(C3)int infin

0

dt

tJ2n(t)

[γ+(t)


et2g minus 1

]= h2n(g)


h2nminus1 = 2mprime

π

int infin

0

dtJ2nminus1(t)

t2 + (πg)2

[t


1 minus eminust(2g)

]

(C4)

h2n = 2mprime

π

int infin

0

dtJ2n(t)

t2 + (πg)2

[πg

et(2g) minus 1+

t

1 minus eminust(2g)

]



36



γminus(t) = 2sumn1


(C6)γ+(t) = 2

sumn1

(2n)J2n(t)γ2n(g)



π2gt + O(t2) = it

(γ1 +

2mprime

π2g

)+ O(t2) (C7)



0

dt

t


1 minus eminust2g+


et2g minus 1

]= 2

sumn1



γ1 = 2sumn1



γ1 = minusmprime

π

int infin

0dt

[πg


t

t2 + π2g2(minus(t) + +(t))

] (C10)


πg

t2 + π2g2=

int infin


(C11)t

t2 + π2g2=

int infin



γ1 = minusmprime

π

int infin

0du eminusπgu

[ int infin


+int infin


] (C12)


γ1 = minus 2mprime


radic2mprime

πeminusπgRe

[ int infin

0

dt


] (C13)


37






Vn(x) = 1

π232minusn ex

int 1


= 1

π232minusn eminusx

int 1



Vn(x) = 2minusn

(54 minus n

)

(54

)(2 minus n)


= 2minusn

(54 minus n

)

(54

)(2 minus n)







Uab(x) = 1

2

int infin





(1 +

t

2x

)a+bminus12

(D5)


Uab(x) = 2bminus32

(1

2minus a + b


= 1

2

(1

2minus a + b

)U

(1


) (D6)

38




U+0 (x) = 1

2

(5

4


1 (x) = 1

2

(1

4


(D7)

Uminus0 (x) = 1

2

(3

4


1 (x) = 1

2

(3

4





1 (x) = V1(x)U+0 (x) + V0(x)U+

1 (x) = ex

x (D8)


U+0 (x)Uminus


0 (minusx) = π

2radic

2xeplusmn 3iπ

4 (for Imx ≷ 0) (D9)


V0(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]

(D10)

V1(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]






(D11)Uplusmn








U+0 (x) = (2x)minus54

(5

4

)F

(1

4

5

4| minus 1

2x

)= (2x)minus54

(5

4

) [1 minus 5


]


(3

4

)F

(minus1

4

3

4| minus 1

2x

)= (2x)minus34

(3

4

) [1 +

3


]

39


U+1 (x) = (2x)minus14 1

2

(1

4

)F

(1

4

1

4| minus 1

2x

)= (2x)minus14 1

2

(1

4

) [1 minus 1


]


2

(3

4

)F

(3

4

3

4| minus 1

2x

)= (2x)minus34 1

2

(3

4

) [1 minus 9


]

(D12)








2

πeminus 3iπ

4 eπg[U+


] (D13)




(

34

) [F

(1

4

5

4| 1

2πg+ iε

)+ 2F

(minus1

4

3

4| minus 1

2πg

)] (D14)


2 = σ

(34

)

(54




2(

34

) [8πgF

(1

4

1

4| 1

2πg+ iε

)+ 2F

(3

4

3

4| minus 1

2πg

)] (D16)





F

(1

4

5

4| 1

2πg+ iε

)minus F

(1

4

5

4| 1

2πgminus iε

)= i

radic22

σF

(minus1

4

3

4| minus 1

2πg

) (D17)



40





t + 14


(4πgit)

t + 14

=sumk=01


t + 14

+sumk=01


t + 14

Vk(4πgt) (E1)


Rk(4πgs) = Re

[int minusiinfin


t minus s

]= Re

[int minusiinfin


t minus 4πgs

]

(E2)


Re

[ int minusiinfin


t + 14


minussumn1

nc+(n g)

n + 14

[U+


]+

sumn1

ncminus(n g)

n minus 14





R0(s) = minusRe

[eminusiπ4

int 1


int minusiinfin

0dt eminusvtV0(t)

]=

radic2

πRe

[eminusiπ4

int 1


int 1

minus1du


u minus v minus iε

] (E4)


1radic2π

int 1

minus1du



⎧⎨⎩(

v+1vminus1

)14 v2 gt 1

eminusiπ4(

1+v1minusv

)14 v2 lt 1

(E5)


R0(s)sgt0=

radic2

[1

sminus

int minus1


(v + 1

v minus 1

)14]

=radic

2

[1

sminus 2 eminus2sU+

0 (s)

] (E6)


R0(s)slt0=

radic2

[1

s+ 2Uminus

0 (minuss)

] (E7)

41


together with

R1(s) = 2radic

2[θ(minuss)Uminus


] (E8)


Re

[ int minusiinfin


t + 14

]= 2

radic2f0(minusπg)

[Uminus

0 (πg) minus 1

2πg

]+ 2


1 (πg) +

radic2



References



















42





























Ann Phys 167 227

43

























44

1 Introduction

2 Cusp anomalous



23 Toy model



26 General solution







4 Mass scale





5 Conclusions

Acknowledgments








References


0

02

04

06

08

1

05 1 15 2 25 3



0 =int infin

0

dt

t

(γ(0)+ (t))2 + (γ

(0)minus (t))2





(k) =int infin

minusinfin

dt

2πeikt(t) γ (k) =

int infin

minusinfin

dt

2πeiktγ (t) (224)


γ (k) = 0 for k2 gt 1 (225)

10





(k) =int infin

minusinfin

dt

2πeikt

sinh(

t4g

+ iπ4

)sinh

(t

4g

)sin

(π4

)γ (t) (227)




sumn1










c(toy)+ (n g) = c

(toy)minus (n g) = 0 (231)



+(t) =int infin


(232)




11



minusint infin

minusinfin

dk(k)



(u) +1

πminusint 1

minus1

dk(k)



φ(u) = minus 1

π

(2 +

int minus1

minusinfin

dk(k)

k minus u+

int infin

1

dk(k)

k minus u

) (235)



(k) = 1

2φ(k) minus 1

2π

(1 + k

1 minus k

)14

minusint 1

minus1

duφ(u)

u minus k

(1 minus u

1 + u

)14

minusradic

2

π

(1 + k

1 minus k

)14c

1 + k (236)


(k)k21= minus

radic2

π

(1 + k

1 minus k

)14[

1 +c

1 + k+

1

2

int infin

minusinfin

dp(p)

p minus k

(p minus 1

p + 1

)14

θ(p2 minus 1)

] (237)



26 General solution


(t) =int 1


int minus1


int infin



(it) = f0(t)V0(t) + f1(t)V1(t) (239)

12




t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

4πng + t

]

(240)


4πng

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

4πng + t

]


Vn(x) =radic

2

π

int 1


(241)

Uplusmnn (x) = 1

2

int infin




f(toy)


1 (t) = minusc(toy)(g) (242)







(4π ig

( minus 1

4

)) = 0 isin Z (244)



4

) (245)



cusp(g) = 2g[1 minus 2f1(0)] (246)

13



cusp(g) = 2g


[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

]⎫⎬⎭ (247)






V1(minusπg) (248)



[1 minus 2

V0(minusπg)

V1(minusπg)

] (249)


(toy)cusp (g) = 2g


Mminus140(2πg)

] (250)







couplings


(toy)cusp (g) = 3

2πg2 minus 1

2π2g3 minus 1

64π3g4 +

5

64π4g5 minus 11

512π5g6 minus 3

512π6g7 + O(g8) (251)



14






(

34

)[F

(1

4

5

4|α + i0

)+ 2F

(minus1

4

3

4| minus α

)] α = 1(2πg)

(252)



(34

)

(54



F

(1

4

5

4| minus α

)= αminus54U+

0 (1(2α))

(5

4

)

(254)

F

(minus1

4

3

4| minus α

)= αminus34Uminus

0 (1(2α))

(3

4

)



4 34 | minus α



4 and b = 34 )


(minusα)k

k

(a + k)(b + k)

(a)(b)= 1 minus αab + O(α2) (255)



(a)

int infin





(a)

int infin


15







F(

14 5

4 |α+)

+ 2F(minus 1

4 34 | minus α

)F

(14 1

4 |α+)

+ 142αF

(34 3

4 | minus α) (258)


(toy)cusp (g)



4α24C4(α) + O(6) (259)


C0 = 1 minus αF

(14 5

4 |α+)

F(

14 1

4 |α+) C2 = 1[

F(

14 1

4 |α+)]2 C4 = F

(34 3

4 | minus α)[

F(

14 1

4 |α+)]3 (260)



16



4α2 minus 3

8α3 minus 61

64α4 minus 433

128α5 + O(α6)

C2 = 1 minus 1

8α minus 11

128α2 minus 151

1024α3 minus 13085

32768α4 + O(α5) (261)

C4 = 1 minus 3

4α minus 27

32α2 minus 317

128α3 + O(α4)















17




2α minus K

4α2 minus

(3K ln 2

8+

27ζ3

256

)α3 + (263)



3 ln 2

2rarr 1

K

2rarr 1

2

9ζ3

32rarr 1

3 (264)



exp

[3 ln 2

2τ minus K

2τ 2 +

9ζ3

32τ 3 +

]rarr exp

[τ minus τ 2

2+

τ 3

3+

] (265)


(

14

)

(1 + τ

4

)

(34 minus τ

4

)

(34

)

(1 minus τ

4

)

(14 + τ

4

) rarr (1 + τ) (266)



cusp (g)




k (k + 1

2



(toy)cusp (g)

18















plusmn1 (4πng)

n ∓ x

(32)





V0(4πgx)=



(33)


19



g 01 02 04 06 08 10 12 14 16 18

Numer 01976 03616 05843 07096 07825 08276 08576 08787 08944 09065Exact 01939 03584 05821 07080 07813 08267 08568 08781 08938 09059


nmax 10 20 30 40 50 60 70 infinNumer 08305 08286 08279 08276 08274 08273 08272 08267


c(g) = 0 (34)











20








1 =sumn1

c+(n g)

[nU+

0 (4πng)r(x) + U+1 (4πng)x

n minus x

]

+sumn1

cminus(n g)

[nUminus


n + x

] (36)





4 isin Z) (37)




[aplusmn(n) +

bplusmn(n)


]+ 2

[αplusmn(n) +

βplusmn(n)


]+ O(4)

(38)

21




(34

)

(54

) (39)






4 )

2x

(5

4

) sumn1

a+(n)

n minus x

= 1 ( 1)

(310)

minus2x

(3

4

) sumn1

aminus(n)

n + x

= 1 ( 0)


a+(n) = 2(n + 1

4

)(n + 1)2

(14

)

(311)

aminus(n) = (n + 3

4

)2(n + 1)2

(34

)


4 )

2x

(5

4

) sumn1

b+(n)

n minus x

= minus 3

32x

minus 3π

64minus 15

32ln 2 ( 1)

(312)

minus 2x

(3

4

) sumn1

bminus(n)

n + x

= minus 5

32x

minus 5π

64+

9

32ln 2 ( 0)


b+(n) = minusa+(n)

(3 ln 2

4+

3

32n


(3 ln 2

4+

5

32n

) (313)




(2πn)minus1

[

(5

4

) (a+(n) +

b+(n)


)(1 minus 5

128πgn+

)

+

(3

4

) (aminus(n) +

bminus(n)


)(1 +

3


)]+ O(2) (314)

22



cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K

16π2g2+ O(1g3)

]+ O(2) (315)




4 + O(gminus1)]

+ O(4) (316)



2x

(5

4

) sumn1

α+(n)

n minus x

= 0 ( 1)

(317)

minus2x

(3

4

) sumn1

αminus(n)

n + x

= π

2radic

2δ0 ( 0)


α+(n) = 0(318)



2x

(5

4

) sumn1

β+(n)

n minus x

= minus1

2 ( 1)

(319)

minus2x

(3

4

) sumn1

βminus(n)

n + x

= minus1

8+

3π

16radic

2(1 minus 2 ln 2)δ0 ( 0)


β+(n) = minus1

2a+(n)

(320)


(1

4minus 3 ln 2

4+

1

32n

)



23




(2πn)minus1

[

(5

4

) (α+(n) +

β+(n)


)(1 minus 5


)

+

(3

4

) (αminus(n) +

βminus(n)


)(1 +

3


)]+ O(4) (321)


δcusp(g) = minus2

π

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (322)




m2cusp = 4

radic2

πσ2

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (323)


cusp(g) =[

2g minus 3 ln 2

2πminus K

8π2g+ O(1g2)

]minus σ

4radic

2m2

cusp + O(m4

cusp

) (324)


4 Mass scale




mO(6) = 8radic

2


πeminusπgRe

[int minusiinfin


t + 14

] (41)



24



f0(4πgt) =sumn1

t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

n + t

]minus 1

(43)

f1(4πgt) =sumn1

n

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

n + t

]





2




mO(6) = 16radic

2

π2

f1(minusπg)

V0(minusπg) (45)







fn(4πgt) = f (PT)

n (4πgt) + δfn(4πgt) (n = 0 1) (46)

where f (PT)



2

(5

4

) sumn1

a+(n)

t minus n= 1

t

[

(34

)(1 minus t)

(

34 minus t

) minus 1

]

(47)

2

(3

4

) sumn1

aminus(n)

t + n= 1

t

[

(14

)(1 + t)

(

14 + t

) minus 1

]

25



f(PT)

0 (4πgt) = minus(

34

)(1 minus t)

(

34 minus t

)+

1

4πg

[(3 ln 2

4+

1

8t

)

(34

)(1 minus t)

(

34 minus t

) minus (

14

)(1 + t)

8t(

14 + t

) ]+ O(gminus2)

f(PT)

1 (4πgt) = 1

4πg

[

(14

)(1 + t)

4t(

14 + t

) minus (

34

)(1 minus t)

4t(

34 minus t

) ]

minus 1

(4πg)2

[

(14

)(1 + t)

4t(

14 + t

) (1

4tminus 3 ln 2

4

)

minus (

34

)(1 minus t)

4t(

34 minus t

) (1

4t+

3 ln 2

4

)]+ O(gminus3) (48)


δf0(4πgt) = 2

1

4πg

[

(34

)(1 minus t)

2(

34 minus t

) minus (

54

)(1 + t)

2(

54 + t

) ]+ O(gminus2)


δf1(4πgt) = 2

1

4πg

(

54

)(1 + t)

(

54 + t

) +1

(4πg)2

[

(34

)(1 minus t)

8t(

34 minus t

)minus

(54

)(1 + t)

(

54 + t

) (1

8t+

3

4ln 2 minus 1

4

)]+ O(gminus3)





1 (0) minus 4gδf1(0) (410)

Replacing f(PT)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K


]minus 2

π

[1 +

3 minus 6 ln 2


]+ O(4) (411)



mO(6) =radic

2

(

54


[1 +

3 minus 6 ln 2

32πg+



]minus 2

8πg



]+ O(4)

(412)


(PT)


26




fn(4πgt)

(34 minus t

)

(34

)(1 minus t)

rarr f (toy)n (4πgt) (n = 0 1) (413)





(toy)


mtoy = 16radic

2

π2

f(toy)

1 (minusπg)

V0(minusπg)= 16

radic2

π2

1

V1(minusπg) (414)




mtoy = 4

π(

54


[1 minus 1

32πgminus 23


]minus 2

8πg

[1 minus 11


]+ O(4)

(415)


0 and f(toy)


27






32radic

2σm2

toy + O(m4

toy

) (416)




2radic

2mtoy (417)



4π (418)


cusp (g + c1) = 2g

[1 minus K


]minus 2

2radic

2π

[1 +

3


]+ O(4) (419)


[mO(6)(g + c1)]2 = 22

πσ

[(1 +

3

16πg+

16K minus 54


)minus 2

8radic

2πg

(1 minus 3


)+ O(4)

] (420)


O(6)

δcusp = minus σ

4radic

2m2

O(6) + O(m4

O(6)

) (421)



28











2

πeminusπgUminus


2

πeminusπgUminus

1 (πg) (423)





4




mprime = minus σ

8gmO(6) (426)




2

4mprimemO(6) = σ

radic2

32gm2

O(6) (427)



δ(it) = δf0(t)V0(t) + δf1(t)V1(t) (428)


δ(0) = 2δf1(0) (429)

29






δ(it) sim 1







δ(it) (433)


δγ (it) = t

4gδ(0) + O(t2) = t

2gδf1(0) + O(t2) (434)


0

dt

tJ2nminus1(t)

[δγminus(t)

1 minus eminust2g+

δγ+(t)

et2g minus 1

]= 0

(435)int infin

0

dt

tJ2n(t)

[δγ+(t)


et2g minus 1

]= 0



30




2

π2geminusπg minus

radic2

πeminusπg Re

int infin

0

dt prime


]+ O(t2) (436)



tradic

2

8g+ O(t2) (437)


mcusp(g) = mO(6)(g) (438)


5 Conclusions






31





(m4

O(n)

μ2) (51)


m2O(n) = cμ2 e


20 [1 + O(g2)] (52)




Acknowledgments


32








γ1 = 1

2minus π2

6g2 +

11π4

90g4 minus

(73π6

630+ 4ζ 2

3

)g6 + O(g8)

γ2 = ζ3g3 minus

(π2

3ζ3 + 10ζ5

)g5 +

(8π4

45ζ3 +

10π2

3ζ5 + 105ζ7

)g7 + O(g9)

γ3 = minusπ4

90g4 +

37π6

1890g6 + O(g8) γ4 = ζ5g

5 minus(

π2

3ζ5 + 21ζ7

)g7 + O(g9)

γ5 = minus π6

945g6 + O(g8) γ6 = ζ7g

7 + O(g9)

(A2)



3g4 +

44π4

45g6 minus

(292π6

315+ 32ζ 2

3

)g8 + O(g10) (A3)







33



c(g) = minus1

2+ 2gγ1(g) +

sumn1

[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

] (A5)


c(g) = minus1

2+ 2gγ1(g) minus

sumk1



fk(g) = 8gsumn1

[U+






3g2 + 2ζ3g

3 minus π4

6g4 minus 23ζ5g

5 +17π6

108g6 +

1107

4ζ7g

7 + O(g8)

f2 = minus1

2+ 2ζ3g

3 minus π4

30g4 + O(g5) f3 = 1

2+ O(g4)

f4 = minus3

8+ O(g5) f5 = 3

8+ O(g6) f6 = minus 5

16+ O(g7)

(A8)






(t) =int infin




(k)=sumn1



(k)=sumn1


bull For minus1 k 1

(k) = minusradic

2

π

(1 + k

1 minus k

)14[

1 +c(g)

1 + k+

1

2

(int minus1

minusinfin+

int infin

1

)dp(p)

p minus k

(p minus 1

p + 1

)14]

(B4)


34



(t) =sumn1

c+(n g)

[eminusit


]

+sumn1

cminus(n g)

[eit



(B5)



int 1

minus1dk eplusmnxk

int infin

1

dpeminusy(pminus1)

p minus k

(1 + k

1 minus k

p minus 1

p + 1

)plusmn14

Vn(x) =radic

2

π

int 1

minus1

dk exk

(k + 1)n

(1 + k

1 minus k

)14

(B6)

Uplusmnn (y) = 1

2

int infin

1

dp eminusy(pminus1)

(p ∓ 1)n

(p + 1

p minus 1

)∓14






0 (y) minus ex



c+(n g)

[4πngV1(t)U

+0 (4πng) + tV0(t)U

+1 (4πng)

4πng minus t

]

+sumn1

cminus(n g)

[4πngV1(t)U


minus1 (4πng)

4πng + t

] (B8)





int infin

0dsV0(x + s)U+






0 (x)


d

dx(xzprime

i (x)) =(

x minus 1

2

)zi(x) (B11)

35




2(y + s)]

= d


2(y + s)]z1(x + s)

minus d


1(x + s)]z2(y + s) (B12)



1 (y + s) + (y + s)V1(x + s)U+0 (y + s)

] ∣∣s=infins=0


0 (y) (B13)






π

t

t + πg (C1)



π

t2

t2 + (πg)2

(C2)


t2 + π2g2


0

dt

tJ2nminus1(t)

[γminus(t)

1 minus eminust2g+

γ+(t)

et2g minus 1

]= h2nminus1(g)

(C3)int infin

0

dt

tJ2n(t)

[γ+(t)


et2g minus 1

]= h2n(g)


h2nminus1 = 2mprime

π

int infin

0

dtJ2nminus1(t)

t2 + (πg)2

[t


1 minus eminust(2g)

]

(C4)

h2n = 2mprime

π

int infin

0

dtJ2n(t)

t2 + (πg)2

[πg

et(2g) minus 1+

t

1 minus eminust(2g)

]



36



γminus(t) = 2sumn1


(C6)γ+(t) = 2

sumn1

(2n)J2n(t)γ2n(g)



π2gt + O(t2) = it

(γ1 +

2mprime

π2g

)+ O(t2) (C7)



0

dt

t


1 minus eminust2g+


et2g minus 1

]= 2

sumn1



γ1 = 2sumn1



γ1 = minusmprime

π

int infin

0dt

[πg


t

t2 + π2g2(minus(t) + +(t))

] (C10)


πg

t2 + π2g2=

int infin


(C11)t

t2 + π2g2=

int infin



γ1 = minusmprime

π

int infin

0du eminusπgu

[ int infin


+int infin


] (C12)


γ1 = minus 2mprime


radic2mprime

πeminusπgRe

[ int infin

0

dt


] (C13)


37






Vn(x) = 1

π232minusn ex

int 1


= 1

π232minusn eminusx

int 1



Vn(x) = 2minusn

(54 minus n

)

(54

)(2 minus n)


= 2minusn

(54 minus n

)

(54

)(2 minus n)







Uab(x) = 1

2

int infin





(1 +

t

2x

)a+bminus12

(D5)


Uab(x) = 2bminus32

(1

2minus a + b


= 1

2

(1

2minus a + b

)U

(1


) (D6)

38




U+0 (x) = 1

2

(5

4


1 (x) = 1

2

(1

4


(D7)

Uminus0 (x) = 1

2

(3

4


1 (x) = 1

2

(3

4





1 (x) = V1(x)U+0 (x) + V0(x)U+

1 (x) = ex

x (D8)


U+0 (x)Uminus


0 (minusx) = π

2radic

2xeplusmn 3iπ

4 (for Imx ≷ 0) (D9)


V0(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]

(D10)

V1(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]






(D11)Uplusmn








U+0 (x) = (2x)minus54

(5

4

)F

(1

4

5

4| minus 1

2x

)= (2x)minus54

(5

4

) [1 minus 5


]


(3

4

)F

(minus1

4

3

4| minus 1

2x

)= (2x)minus34

(3

4

) [1 +

3


]

39


U+1 (x) = (2x)minus14 1

2

(1

4

)F

(1

4

1

4| minus 1

2x

)= (2x)minus14 1

2

(1

4

) [1 minus 1


]


2

(3

4

)F

(3

4

3

4| minus 1

2x

)= (2x)minus34 1

2

(3

4

) [1 minus 9


]

(D12)








2

πeminus 3iπ

4 eπg[U+


] (D13)




(

34

) [F

(1

4

5

4| 1

2πg+ iε

)+ 2F

(minus1

4

3

4| minus 1

2πg

)] (D14)


2 = σ

(34

)

(54




2(

34

) [8πgF

(1

4

1

4| 1

2πg+ iε

)+ 2F

(3

4

3

4| minus 1

2πg

)] (D16)





F

(1

4

5

4| 1

2πg+ iε

)minus F

(1

4

5

4| 1

2πgminus iε

)= i

radic22

σF

(minus1

4

3

4| minus 1

2πg

) (D17)



40





t + 14


(4πgit)

t + 14

=sumk=01


t + 14

+sumk=01


t + 14

Vk(4πgt) (E1)


Rk(4πgs) = Re

[int minusiinfin


t minus s

]= Re

[int minusiinfin


t minus 4πgs

]

(E2)


Re

[ int minusiinfin


t + 14


minussumn1

nc+(n g)

n + 14

[U+


]+

sumn1

ncminus(n g)

n minus 14





R0(s) = minusRe

[eminusiπ4

int 1


int minusiinfin

0dt eminusvtV0(t)

]=

radic2

πRe

[eminusiπ4

int 1


int 1

minus1du


u minus v minus iε

] (E4)


1radic2π

int 1

minus1du



⎧⎨⎩(

v+1vminus1

)14 v2 gt 1

eminusiπ4(

1+v1minusv

)14 v2 lt 1

(E5)


R0(s)sgt0=

radic2

[1

sminus

int minus1


(v + 1

v minus 1

)14]

=radic

2

[1

sminus 2 eminus2sU+

0 (s)

] (E6)


R0(s)slt0=

radic2

[1

s+ 2Uminus

0 (minuss)

] (E7)

41


together with

R1(s) = 2radic

2[θ(minuss)Uminus


] (E8)


Re

[ int minusiinfin


t + 14

]= 2

radic2f0(minusπg)

[Uminus

0 (πg) minus 1

2πg

]+ 2


1 (πg) +

radic2



References



















42





























Ann Phys 167 227

43

























44

1 Introduction

2 Cusp anomalous



23 Toy model



26 General solution







4 Mass scale





5 Conclusions

Acknowledgments








References





(k) =int infin

minusinfin

dt

2πeikt

sinh(

t4g

+ iπ4

)sinh

(t

4g

)sin

(π4

)γ (t) (227)




sumn1










c(toy)+ (n g) = c

(toy)minus (n g) = 0 (231)



+(t) =int infin


(232)




11



minusint infin

minusinfin

dk(k)



(u) +1

πminusint 1

minus1

dk(k)



φ(u) = minus 1

π

(2 +

int minus1

minusinfin

dk(k)

k minus u+

int infin

1

dk(k)

k minus u

) (235)



(k) = 1

2φ(k) minus 1

2π

(1 + k

1 minus k

)14

minusint 1

minus1

duφ(u)

u minus k

(1 minus u

1 + u

)14

minusradic

2

π

(1 + k

1 minus k

)14c

1 + k (236)


(k)k21= minus

radic2

π

(1 + k

1 minus k

)14[

1 +c

1 + k+

1

2

int infin

minusinfin

dp(p)

p minus k

(p minus 1

p + 1

)14

θ(p2 minus 1)

] (237)



26 General solution


(t) =int 1


int minus1


int infin



(it) = f0(t)V0(t) + f1(t)V1(t) (239)

12




t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

4πng + t

]

(240)


4πng

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

4πng + t

]


Vn(x) =radic

2

π

int 1


(241)

Uplusmnn (x) = 1

2

int infin




f(toy)


1 (t) = minusc(toy)(g) (242)







(4π ig

( minus 1

4

)) = 0 isin Z (244)



4

) (245)



cusp(g) = 2g[1 minus 2f1(0)] (246)

13



cusp(g) = 2g


[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

]⎫⎬⎭ (247)






V1(minusπg) (248)



[1 minus 2

V0(minusπg)

V1(minusπg)

] (249)


(toy)cusp (g) = 2g


Mminus140(2πg)

] (250)







couplings


(toy)cusp (g) = 3

2πg2 minus 1

2π2g3 minus 1

64π3g4 +

5

64π4g5 minus 11

512π5g6 minus 3

512π6g7 + O(g8) (251)



14






(

34

)[F

(1

4

5

4|α + i0

)+ 2F

(minus1

4

3

4| minus α

)] α = 1(2πg)

(252)



(34

)

(54



F

(1

4

5

4| minus α

)= αminus54U+

0 (1(2α))

(5

4

)

(254)

F

(minus1

4

3

4| minus α

)= αminus34Uminus

0 (1(2α))

(3

4

)



4 34 | minus α



4 and b = 34 )


(minusα)k

k

(a + k)(b + k)

(a)(b)= 1 minus αab + O(α2) (255)



(a)

int infin





(a)

int infin


15







F(

14 5

4 |α+)

+ 2F(minus 1

4 34 | minus α

)F

(14 1

4 |α+)

+ 142αF

(34 3

4 | minus α) (258)


(toy)cusp (g)



4α24C4(α) + O(6) (259)


C0 = 1 minus αF

(14 5

4 |α+)

F(

14 1

4 |α+) C2 = 1[

F(

14 1

4 |α+)]2 C4 = F

(34 3

4 | minus α)[

F(

14 1

4 |α+)]3 (260)



16



4α2 minus 3

8α3 minus 61

64α4 minus 433

128α5 + O(α6)

C2 = 1 minus 1

8α minus 11

128α2 minus 151

1024α3 minus 13085

32768α4 + O(α5) (261)

C4 = 1 minus 3

4α minus 27

32α2 minus 317

128α3 + O(α4)















17




2α minus K

4α2 minus

(3K ln 2

8+

27ζ3

256

)α3 + (263)



3 ln 2

2rarr 1

K

2rarr 1

2

9ζ3

32rarr 1

3 (264)



exp

[3 ln 2

2τ minus K

2τ 2 +

9ζ3

32τ 3 +

]rarr exp

[τ minus τ 2

2+

τ 3

3+

] (265)


(

14

)

(1 + τ

4

)

(34 minus τ

4

)

(34

)

(1 minus τ

4

)

(14 + τ

4

) rarr (1 + τ) (266)



cusp (g)




k (k + 1

2



(toy)cusp (g)

18















plusmn1 (4πng)

n ∓ x

(32)





V0(4πgx)=



(33)


19



g 01 02 04 06 08 10 12 14 16 18

Numer 01976 03616 05843 07096 07825 08276 08576 08787 08944 09065Exact 01939 03584 05821 07080 07813 08267 08568 08781 08938 09059


nmax 10 20 30 40 50 60 70 infinNumer 08305 08286 08279 08276 08274 08273 08272 08267


c(g) = 0 (34)











20








1 =sumn1

c+(n g)

[nU+

0 (4πng)r(x) + U+1 (4πng)x

n minus x

]

+sumn1

cminus(n g)

[nUminus


n + x

] (36)





4 isin Z) (37)




[aplusmn(n) +

bplusmn(n)


]+ 2

[αplusmn(n) +

βplusmn(n)


]+ O(4)

(38)

21




(34

)

(54

) (39)






4 )

2x

(5

4

) sumn1

a+(n)

n minus x

= 1 ( 1)

(310)

minus2x

(3

4

) sumn1

aminus(n)

n + x

= 1 ( 0)


a+(n) = 2(n + 1

4

)(n + 1)2

(14

)

(311)

aminus(n) = (n + 3

4

)2(n + 1)2

(34

)


4 )

2x

(5

4

) sumn1

b+(n)

n minus x

= minus 3

32x

minus 3π

64minus 15

32ln 2 ( 1)

(312)

minus 2x

(3

4

) sumn1

bminus(n)

n + x

= minus 5

32x

minus 5π

64+

9

32ln 2 ( 0)


b+(n) = minusa+(n)

(3 ln 2

4+

3

32n


(3 ln 2

4+

5

32n

) (313)




(2πn)minus1

[

(5

4

) (a+(n) +

b+(n)


)(1 minus 5

128πgn+

)

+

(3

4

) (aminus(n) +

bminus(n)


)(1 +

3


)]+ O(2) (314)

22



cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K

16π2g2+ O(1g3)

]+ O(2) (315)




4 + O(gminus1)]

+ O(4) (316)



2x

(5

4

) sumn1

α+(n)

n minus x

= 0 ( 1)

(317)

minus2x

(3

4

) sumn1

αminus(n)

n + x

= π

2radic

2δ0 ( 0)


α+(n) = 0(318)



2x

(5

4

) sumn1

β+(n)

n minus x

= minus1

2 ( 1)

(319)

minus2x

(3

4

) sumn1

βminus(n)

n + x

= minus1

8+

3π

16radic

2(1 minus 2 ln 2)δ0 ( 0)


β+(n) = minus1

2a+(n)

(320)


(1

4minus 3 ln 2

4+

1

32n

)



23




(2πn)minus1

[

(5

4

) (α+(n) +

β+(n)


)(1 minus 5


)

+

(3

4

) (αminus(n) +

βminus(n)


)(1 +

3


)]+ O(4) (321)


δcusp(g) = minus2

π

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (322)




m2cusp = 4

radic2

πσ2

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (323)


cusp(g) =[

2g minus 3 ln 2

2πminus K

8π2g+ O(1g2)

]minus σ

4radic

2m2

cusp + O(m4

cusp

) (324)


4 Mass scale




mO(6) = 8radic

2


πeminusπgRe

[int minusiinfin


t + 14

] (41)



24



f0(4πgt) =sumn1

t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

n + t

]minus 1

(43)

f1(4πgt) =sumn1

n

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

n + t

]





2




mO(6) = 16radic

2

π2

f1(minusπg)

V0(minusπg) (45)







fn(4πgt) = f (PT)

n (4πgt) + δfn(4πgt) (n = 0 1) (46)

where f (PT)



2

(5

4

) sumn1

a+(n)

t minus n= 1

t

[

(34

)(1 minus t)

(

34 minus t

) minus 1

]

(47)

2

(3

4

) sumn1

aminus(n)

t + n= 1

t

[

(14

)(1 + t)

(

14 + t

) minus 1

]

25



f(PT)

0 (4πgt) = minus(

34

)(1 minus t)

(

34 minus t

)+

1

4πg

[(3 ln 2

4+

1

8t

)

(34

)(1 minus t)

(

34 minus t

) minus (

14

)(1 + t)

8t(

14 + t

) ]+ O(gminus2)

f(PT)

1 (4πgt) = 1

4πg

[

(14

)(1 + t)

4t(

14 + t

) minus (

34

)(1 minus t)

4t(

34 minus t

) ]

minus 1

(4πg)2

[

(14

)(1 + t)

4t(

14 + t

) (1

4tminus 3 ln 2

4

)

minus (

34

)(1 minus t)

4t(

34 minus t

) (1

4t+

3 ln 2

4

)]+ O(gminus3) (48)


δf0(4πgt) = 2

1

4πg

[

(34

)(1 minus t)

2(

34 minus t

) minus (

54

)(1 + t)

2(

54 + t

) ]+ O(gminus2)


δf1(4πgt) = 2

1

4πg

(

54

)(1 + t)

(

54 + t

) +1

(4πg)2

[

(34

)(1 minus t)

8t(

34 minus t

)minus

(54

)(1 + t)

(

54 + t

) (1

8t+

3

4ln 2 minus 1

4

)]+ O(gminus3)





1 (0) minus 4gδf1(0) (410)

Replacing f(PT)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K


]minus 2

π

[1 +

3 minus 6 ln 2


]+ O(4) (411)



mO(6) =radic

2

(

54


[1 +

3 minus 6 ln 2

32πg+



]minus 2

8πg



]+ O(4)

(412)


(PT)


26




fn(4πgt)

(34 minus t

)

(34

)(1 minus t)

rarr f (toy)n (4πgt) (n = 0 1) (413)





(toy)


mtoy = 16radic

2

π2

f(toy)

1 (minusπg)

V0(minusπg)= 16

radic2

π2

1

V1(minusπg) (414)




mtoy = 4

π(

54


[1 minus 1

32πgminus 23


]minus 2

8πg

[1 minus 11


]+ O(4)

(415)


0 and f(toy)


27






32radic

2σm2

toy + O(m4

toy

) (416)




2radic

2mtoy (417)



4π (418)


cusp (g + c1) = 2g

[1 minus K


]minus 2

2radic

2π

[1 +

3


]+ O(4) (419)


[mO(6)(g + c1)]2 = 22

πσ

[(1 +

3

16πg+

16K minus 54


)minus 2

8radic

2πg

(1 minus 3


)+ O(4)

] (420)


O(6)

δcusp = minus σ

4radic

2m2

O(6) + O(m4

O(6)

) (421)



28











2

πeminusπgUminus


2

πeminusπgUminus

1 (πg) (423)





4




mprime = minus σ

8gmO(6) (426)




2

4mprimemO(6) = σ

radic2

32gm2

O(6) (427)



δ(it) = δf0(t)V0(t) + δf1(t)V1(t) (428)


δ(0) = 2δf1(0) (429)

29






δ(it) sim 1







δ(it) (433)


δγ (it) = t

4gδ(0) + O(t2) = t

2gδf1(0) + O(t2) (434)


0

dt

tJ2nminus1(t)

[δγminus(t)

1 minus eminust2g+

δγ+(t)

et2g minus 1

]= 0

(435)int infin

0

dt

tJ2n(t)

[δγ+(t)


et2g minus 1

]= 0



30




2

π2geminusπg minus

radic2

πeminusπg Re

int infin

0

dt prime


]+ O(t2) (436)



tradic

2

8g+ O(t2) (437)


mcusp(g) = mO(6)(g) (438)


5 Conclusions






31





(m4

O(n)

μ2) (51)


m2O(n) = cμ2 e


20 [1 + O(g2)] (52)




Acknowledgments


32








γ1 = 1

2minus π2

6g2 +

11π4

90g4 minus

(73π6

630+ 4ζ 2

3

)g6 + O(g8)

γ2 = ζ3g3 minus

(π2

3ζ3 + 10ζ5

)g5 +

(8π4

45ζ3 +

10π2

3ζ5 + 105ζ7

)g7 + O(g9)

γ3 = minusπ4

90g4 +

37π6

1890g6 + O(g8) γ4 = ζ5g

5 minus(

π2

3ζ5 + 21ζ7

)g7 + O(g9)

γ5 = minus π6

945g6 + O(g8) γ6 = ζ7g

7 + O(g9)

(A2)



3g4 +

44π4

45g6 minus

(292π6

315+ 32ζ 2

3

)g8 + O(g10) (A3)







33



c(g) = minus1

2+ 2gγ1(g) +

sumn1

[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

] (A5)


c(g) = minus1

2+ 2gγ1(g) minus

sumk1



fk(g) = 8gsumn1

[U+






3g2 + 2ζ3g

3 minus π4

6g4 minus 23ζ5g

5 +17π6

108g6 +

1107

4ζ7g

7 + O(g8)

f2 = minus1

2+ 2ζ3g

3 minus π4

30g4 + O(g5) f3 = 1

2+ O(g4)

f4 = minus3

8+ O(g5) f5 = 3

8+ O(g6) f6 = minus 5

16+ O(g7)

(A8)






(t) =int infin




(k)=sumn1



(k)=sumn1


bull For minus1 k 1

(k) = minusradic

2

π

(1 + k

1 minus k

)14[

1 +c(g)

1 + k+

1

2

(int minus1

minusinfin+

int infin

1

)dp(p)

p minus k

(p minus 1

p + 1

)14]

(B4)


34



(t) =sumn1

c+(n g)

[eminusit


]

+sumn1

cminus(n g)

[eit



(B5)



int 1

minus1dk eplusmnxk

int infin

1

dpeminusy(pminus1)

p minus k

(1 + k

1 minus k

p minus 1

p + 1

)plusmn14

Vn(x) =radic

2

π

int 1

minus1

dk exk

(k + 1)n

(1 + k

1 minus k

)14

(B6)

Uplusmnn (y) = 1

2

int infin

1

dp eminusy(pminus1)

(p ∓ 1)n

(p + 1

p minus 1

)∓14






0 (y) minus ex



c+(n g)

[4πngV1(t)U

+0 (4πng) + tV0(t)U

+1 (4πng)

4πng minus t

]

+sumn1

cminus(n g)

[4πngV1(t)U


minus1 (4πng)

4πng + t

] (B8)





int infin

0dsV0(x + s)U+






0 (x)


d

dx(xzprime

i (x)) =(

x minus 1

2

)zi(x) (B11)

35




2(y + s)]

= d


2(y + s)]z1(x + s)

minus d


1(x + s)]z2(y + s) (B12)



1 (y + s) + (y + s)V1(x + s)U+0 (y + s)

] ∣∣s=infins=0


0 (y) (B13)






π

t

t + πg (C1)



π

t2

t2 + (πg)2

(C2)


t2 + π2g2


0

dt

tJ2nminus1(t)

[γminus(t)

1 minus eminust2g+

γ+(t)

et2g minus 1

]= h2nminus1(g)

(C3)int infin

0

dt

tJ2n(t)

[γ+(t)


et2g minus 1

]= h2n(g)


h2nminus1 = 2mprime

π

int infin

0

dtJ2nminus1(t)

t2 + (πg)2

[t


1 minus eminust(2g)

]

(C4)

h2n = 2mprime

π

int infin

0

dtJ2n(t)

t2 + (πg)2

[πg

et(2g) minus 1+

t

1 minus eminust(2g)

]



36



γminus(t) = 2sumn1


(C6)γ+(t) = 2

sumn1

(2n)J2n(t)γ2n(g)



π2gt + O(t2) = it

(γ1 +

2mprime

π2g

)+ O(t2) (C7)



0

dt

t


1 minus eminust2g+


et2g minus 1

]= 2

sumn1



γ1 = 2sumn1



γ1 = minusmprime

π

int infin

0dt

[πg


t

t2 + π2g2(minus(t) + +(t))

] (C10)


πg

t2 + π2g2=

int infin


(C11)t

t2 + π2g2=

int infin



γ1 = minusmprime

π

int infin

0du eminusπgu

[ int infin


+int infin


] (C12)


γ1 = minus 2mprime


radic2mprime

πeminusπgRe

[ int infin

0

dt


] (C13)


37






Vn(x) = 1

π232minusn ex

int 1


= 1

π232minusn eminusx

int 1



Vn(x) = 2minusn

(54 minus n

)

(54

)(2 minus n)


= 2minusn

(54 minus n

)

(54

)(2 minus n)







Uab(x) = 1

2

int infin





(1 +

t

2x

)a+bminus12

(D5)


Uab(x) = 2bminus32

(1

2minus a + b


= 1

2

(1

2minus a + b

)U

(1


) (D6)

38




U+0 (x) = 1

2

(5

4


1 (x) = 1

2

(1

4


(D7)

Uminus0 (x) = 1

2

(3

4


1 (x) = 1

2

(3

4





1 (x) = V1(x)U+0 (x) + V0(x)U+

1 (x) = ex

x (D8)


U+0 (x)Uminus


0 (minusx) = π

2radic

2xeplusmn 3iπ

4 (for Imx ≷ 0) (D9)


V0(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]

(D10)

V1(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]






(D11)Uplusmn








U+0 (x) = (2x)minus54

(5

4

)F

(1

4

5

4| minus 1

2x

)= (2x)minus54

(5

4

) [1 minus 5


]


(3

4

)F

(minus1

4

3

4| minus 1

2x

)= (2x)minus34

(3

4

) [1 +

3


]

39


U+1 (x) = (2x)minus14 1

2

(1

4

)F

(1

4

1

4| minus 1

2x

)= (2x)minus14 1

2

(1

4

) [1 minus 1


]


2

(3

4

)F

(3

4

3

4| minus 1

2x

)= (2x)minus34 1

2

(3

4

) [1 minus 9


]

(D12)








2

πeminus 3iπ

4 eπg[U+


] (D13)




(

34

) [F

(1

4

5

4| 1

2πg+ iε

)+ 2F

(minus1

4

3

4| minus 1

2πg

)] (D14)


2 = σ

(34

)

(54




2(

34

) [8πgF

(1

4

1

4| 1

2πg+ iε

)+ 2F

(3

4

3

4| minus 1

2πg

)] (D16)





F

(1

4

5

4| 1

2πg+ iε

)minus F

(1

4

5

4| 1

2πgminus iε

)= i

radic22

σF

(minus1

4

3

4| minus 1

2πg

) (D17)



40





t + 14


(4πgit)

t + 14

=sumk=01


t + 14

+sumk=01


t + 14

Vk(4πgt) (E1)


Rk(4πgs) = Re

[int minusiinfin


t minus s

]= Re

[int minusiinfin


t minus 4πgs

]

(E2)


Re

[ int minusiinfin


t + 14


minussumn1

nc+(n g)

n + 14

[U+


]+

sumn1

ncminus(n g)

n minus 14





R0(s) = minusRe

[eminusiπ4

int 1


int minusiinfin

0dt eminusvtV0(t)

]=

radic2

πRe

[eminusiπ4

int 1


int 1

minus1du


u minus v minus iε

] (E4)


1radic2π

int 1

minus1du



⎧⎨⎩(

v+1vminus1

)14 v2 gt 1

eminusiπ4(

1+v1minusv

)14 v2 lt 1

(E5)


R0(s)sgt0=

radic2

[1

sminus

int minus1


(v + 1

v minus 1

)14]

=radic

2

[1

sminus 2 eminus2sU+

0 (s)

] (E6)


R0(s)slt0=

radic2

[1

s+ 2Uminus

0 (minuss)

] (E7)

41


together with

R1(s) = 2radic

2[θ(minuss)Uminus


] (E8)


Re

[ int minusiinfin


t + 14

]= 2

radic2f0(minusπg)

[Uminus

0 (πg) minus 1

2πg

]+ 2


1 (πg) +

radic2



References



















42





























Ann Phys 167 227

43

























44

1 Introduction

2 Cusp anomalous



23 Toy model



26 General solution







4 Mass scale





5 Conclusions

Acknowledgments








References



minusint infin

minusinfin

dk(k)



(u) +1

πminusint 1

minus1

dk(k)



φ(u) = minus 1

π

(2 +

int minus1

minusinfin

dk(k)

k minus u+

int infin

1

dk(k)

k minus u

) (235)



(k) = 1

2φ(k) minus 1

2π

(1 + k

1 minus k

)14

minusint 1

minus1

duφ(u)

u minus k

(1 minus u

1 + u

)14

minusradic

2

π

(1 + k

1 minus k

)14c

1 + k (236)


(k)k21= minus

radic2

π

(1 + k

1 minus k

)14[

1 +c

1 + k+

1

2

int infin

minusinfin

dp(p)

p minus k

(p minus 1

p + 1

)14

θ(p2 minus 1)

] (237)



26 General solution


(t) =int 1


int minus1


int infin



(it) = f0(t)V0(t) + f1(t)V1(t) (239)

12




t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

4πng + t

]

(240)


4πng

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

4πng + t

]


Vn(x) =radic

2

π

int 1


(241)

Uplusmnn (x) = 1

2

int infin




f(toy)


1 (t) = minusc(toy)(g) (242)







(4π ig

( minus 1

4

)) = 0 isin Z (244)



4

) (245)



cusp(g) = 2g[1 minus 2f1(0)] (246)

13



cusp(g) = 2g


[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

]⎫⎬⎭ (247)






V1(minusπg) (248)



[1 minus 2

V0(minusπg)

V1(minusπg)

] (249)


(toy)cusp (g) = 2g


Mminus140(2πg)

] (250)







couplings


(toy)cusp (g) = 3

2πg2 minus 1

2π2g3 minus 1

64π3g4 +

5

64π4g5 minus 11

512π5g6 minus 3

512π6g7 + O(g8) (251)



14






(

34

)[F

(1

4

5

4|α + i0

)+ 2F

(minus1

4

3

4| minus α

)] α = 1(2πg)

(252)



(34

)

(54



F

(1

4

5

4| minus α

)= αminus54U+

0 (1(2α))

(5

4

)

(254)

F

(minus1

4

3

4| minus α

)= αminus34Uminus

0 (1(2α))

(3

4

)



4 34 | minus α



4 and b = 34 )


(minusα)k

k

(a + k)(b + k)

(a)(b)= 1 minus αab + O(α2) (255)



(a)

int infin





(a)

int infin


15







F(

14 5

4 |α+)

+ 2F(minus 1

4 34 | minus α

)F

(14 1

4 |α+)

+ 142αF

(34 3

4 | minus α) (258)


(toy)cusp (g)



4α24C4(α) + O(6) (259)


C0 = 1 minus αF

(14 5

4 |α+)

F(

14 1

4 |α+) C2 = 1[

F(

14 1

4 |α+)]2 C4 = F

(34 3

4 | minus α)[

F(

14 1

4 |α+)]3 (260)



16



4α2 minus 3

8α3 minus 61

64α4 minus 433

128α5 + O(α6)

C2 = 1 minus 1

8α minus 11

128α2 minus 151

1024α3 minus 13085

32768α4 + O(α5) (261)

C4 = 1 minus 3

4α minus 27

32α2 minus 317

128α3 + O(α4)















17




2α minus K

4α2 minus

(3K ln 2

8+

27ζ3

256

)α3 + (263)



3 ln 2

2rarr 1

K

2rarr 1

2

9ζ3

32rarr 1

3 (264)



exp

[3 ln 2

2τ minus K

2τ 2 +

9ζ3

32τ 3 +

]rarr exp

[τ minus τ 2

2+

τ 3

3+

] (265)


(

14

)

(1 + τ

4

)

(34 minus τ

4

)

(34

)

(1 minus τ

4

)

(14 + τ

4

) rarr (1 + τ) (266)



cusp (g)




k (k + 1

2



(toy)cusp (g)

18















plusmn1 (4πng)

n ∓ x

(32)





V0(4πgx)=



(33)


19



g 01 02 04 06 08 10 12 14 16 18

Numer 01976 03616 05843 07096 07825 08276 08576 08787 08944 09065Exact 01939 03584 05821 07080 07813 08267 08568 08781 08938 09059


nmax 10 20 30 40 50 60 70 infinNumer 08305 08286 08279 08276 08274 08273 08272 08267


c(g) = 0 (34)











20








1 =sumn1

c+(n g)

[nU+

0 (4πng)r(x) + U+1 (4πng)x

n minus x

]

+sumn1

cminus(n g)

[nUminus


n + x

] (36)





4 isin Z) (37)




[aplusmn(n) +

bplusmn(n)


]+ 2

[αplusmn(n) +

βplusmn(n)


]+ O(4)

(38)

21




(34

)

(54

) (39)






4 )

2x

(5

4

) sumn1

a+(n)

n minus x

= 1 ( 1)

(310)

minus2x

(3

4

) sumn1

aminus(n)

n + x

= 1 ( 0)


a+(n) = 2(n + 1

4

)(n + 1)2

(14

)

(311)

aminus(n) = (n + 3

4

)2(n + 1)2

(34

)


4 )

2x

(5

4

) sumn1

b+(n)

n minus x

= minus 3

32x

minus 3π

64minus 15

32ln 2 ( 1)

(312)

minus 2x

(3

4

) sumn1

bminus(n)

n + x

= minus 5

32x

minus 5π

64+

9

32ln 2 ( 0)


b+(n) = minusa+(n)

(3 ln 2

4+

3

32n


(3 ln 2

4+

5

32n

) (313)




(2πn)minus1

[

(5

4

) (a+(n) +

b+(n)


)(1 minus 5

128πgn+

)

+

(3

4

) (aminus(n) +

bminus(n)


)(1 +

3


)]+ O(2) (314)

22



cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K

16π2g2+ O(1g3)

]+ O(2) (315)




4 + O(gminus1)]

+ O(4) (316)



2x

(5

4

) sumn1

α+(n)

n minus x

= 0 ( 1)

(317)

minus2x

(3

4

) sumn1

αminus(n)

n + x

= π

2radic

2δ0 ( 0)


α+(n) = 0(318)



2x

(5

4

) sumn1

β+(n)

n minus x

= minus1

2 ( 1)

(319)

minus2x

(3

4

) sumn1

βminus(n)

n + x

= minus1

8+

3π

16radic

2(1 minus 2 ln 2)δ0 ( 0)


β+(n) = minus1

2a+(n)

(320)


(1

4minus 3 ln 2

4+

1

32n

)



23




(2πn)minus1

[

(5

4

) (α+(n) +

β+(n)


)(1 minus 5


)

+

(3

4

) (αminus(n) +

βminus(n)


)(1 +

3


)]+ O(4) (321)


δcusp(g) = minus2

π

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (322)




m2cusp = 4

radic2

πσ2

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (323)


cusp(g) =[

2g minus 3 ln 2

2πminus K

8π2g+ O(1g2)

]minus σ

4radic

2m2

cusp + O(m4

cusp

) (324)


4 Mass scale




mO(6) = 8radic

2


πeminusπgRe

[int minusiinfin


t + 14

] (41)



24



f0(4πgt) =sumn1

t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

n + t

]minus 1

(43)

f1(4πgt) =sumn1

n

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

n + t

]





2




mO(6) = 16radic

2

π2

f1(minusπg)

V0(minusπg) (45)







fn(4πgt) = f (PT)

n (4πgt) + δfn(4πgt) (n = 0 1) (46)

where f (PT)



2

(5

4

) sumn1

a+(n)

t minus n= 1

t

[

(34

)(1 minus t)

(

34 minus t

) minus 1

]

(47)

2

(3

4

) sumn1

aminus(n)

t + n= 1

t

[

(14

)(1 + t)

(

14 + t

) minus 1

]

25



f(PT)

0 (4πgt) = minus(

34

)(1 minus t)

(

34 minus t

)+

1

4πg

[(3 ln 2

4+

1

8t

)

(34

)(1 minus t)

(

34 minus t

) minus (

14

)(1 + t)

8t(

14 + t

) ]+ O(gminus2)

f(PT)

1 (4πgt) = 1

4πg

[

(14

)(1 + t)

4t(

14 + t

) minus (

34

)(1 minus t)

4t(

34 minus t

) ]

minus 1

(4πg)2

[

(14

)(1 + t)

4t(

14 + t

) (1

4tminus 3 ln 2

4

)

minus (

34

)(1 minus t)

4t(

34 minus t

) (1

4t+

3 ln 2

4

)]+ O(gminus3) (48)


δf0(4πgt) = 2

1

4πg

[

(34

)(1 minus t)

2(

34 minus t

) minus (

54

)(1 + t)

2(

54 + t

) ]+ O(gminus2)


δf1(4πgt) = 2

1

4πg

(

54

)(1 + t)

(

54 + t

) +1

(4πg)2

[

(34

)(1 minus t)

8t(

34 minus t

)minus

(54

)(1 + t)

(

54 + t

) (1

8t+

3

4ln 2 minus 1

4

)]+ O(gminus3)





1 (0) minus 4gδf1(0) (410)

Replacing f(PT)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K


]minus 2

π

[1 +

3 minus 6 ln 2


]+ O(4) (411)



mO(6) =radic

2

(

54


[1 +

3 minus 6 ln 2

32πg+



]minus 2

8πg



]+ O(4)

(412)


(PT)


26




fn(4πgt)

(34 minus t

)

(34

)(1 minus t)

rarr f (toy)n (4πgt) (n = 0 1) (413)





(toy)


mtoy = 16radic

2

π2

f(toy)

1 (minusπg)

V0(minusπg)= 16

radic2

π2

1

V1(minusπg) (414)




mtoy = 4

π(

54


[1 minus 1

32πgminus 23


]minus 2

8πg

[1 minus 11


]+ O(4)

(415)


0 and f(toy)


27






32radic

2σm2

toy + O(m4

toy

) (416)




2radic

2mtoy (417)



4π (418)


cusp (g + c1) = 2g

[1 minus K


]minus 2

2radic

2π

[1 +

3


]+ O(4) (419)


[mO(6)(g + c1)]2 = 22

πσ

[(1 +

3

16πg+

16K minus 54


)minus 2

8radic

2πg

(1 minus 3


)+ O(4)

] (420)


O(6)

δcusp = minus σ

4radic

2m2

O(6) + O(m4

O(6)

) (421)



28











2

πeminusπgUminus


2

πeminusπgUminus

1 (πg) (423)





4




mprime = minus σ

8gmO(6) (426)




2

4mprimemO(6) = σ

radic2

32gm2

O(6) (427)



δ(it) = δf0(t)V0(t) + δf1(t)V1(t) (428)


δ(0) = 2δf1(0) (429)

29






δ(it) sim 1







δ(it) (433)


δγ (it) = t

4gδ(0) + O(t2) = t

2gδf1(0) + O(t2) (434)


0

dt

tJ2nminus1(t)

[δγminus(t)

1 minus eminust2g+

δγ+(t)

et2g minus 1

]= 0

(435)int infin

0

dt

tJ2n(t)

[δγ+(t)


et2g minus 1

]= 0



30




2

π2geminusπg minus

radic2

πeminusπg Re

int infin

0

dt prime


]+ O(t2) (436)



tradic

2

8g+ O(t2) (437)


mcusp(g) = mO(6)(g) (438)


5 Conclusions






31





(m4

O(n)

μ2) (51)


m2O(n) = cμ2 e


20 [1 + O(g2)] (52)




Acknowledgments


32








γ1 = 1

2minus π2

6g2 +

11π4

90g4 minus

(73π6

630+ 4ζ 2

3

)g6 + O(g8)

γ2 = ζ3g3 minus

(π2

3ζ3 + 10ζ5

)g5 +

(8π4

45ζ3 +

10π2

3ζ5 + 105ζ7

)g7 + O(g9)

γ3 = minusπ4

90g4 +

37π6

1890g6 + O(g8) γ4 = ζ5g

5 minus(

π2

3ζ5 + 21ζ7

)g7 + O(g9)

γ5 = minus π6

945g6 + O(g8) γ6 = ζ7g

7 + O(g9)

(A2)



3g4 +

44π4

45g6 minus

(292π6

315+ 32ζ 2

3

)g8 + O(g10) (A3)







33



c(g) = minus1

2+ 2gγ1(g) +

sumn1

[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

] (A5)


c(g) = minus1

2+ 2gγ1(g) minus

sumk1



fk(g) = 8gsumn1

[U+






3g2 + 2ζ3g

3 minus π4

6g4 minus 23ζ5g

5 +17π6

108g6 +

1107

4ζ7g

7 + O(g8)

f2 = minus1

2+ 2ζ3g

3 minus π4

30g4 + O(g5) f3 = 1

2+ O(g4)

f4 = minus3

8+ O(g5) f5 = 3

8+ O(g6) f6 = minus 5

16+ O(g7)

(A8)






(t) =int infin




(k)=sumn1



(k)=sumn1


bull For minus1 k 1

(k) = minusradic

2

π

(1 + k

1 minus k

)14[

1 +c(g)

1 + k+

1

2

(int minus1

minusinfin+

int infin

1

)dp(p)

p minus k

(p minus 1

p + 1

)14]

(B4)


34



(t) =sumn1

c+(n g)

[eminusit


]

+sumn1

cminus(n g)

[eit



(B5)



int 1

minus1dk eplusmnxk

int infin

1

dpeminusy(pminus1)

p minus k

(1 + k

1 minus k

p minus 1

p + 1

)plusmn14

Vn(x) =radic

2

π

int 1

minus1

dk exk

(k + 1)n

(1 + k

1 minus k

)14

(B6)

Uplusmnn (y) = 1

2

int infin

1

dp eminusy(pminus1)

(p ∓ 1)n

(p + 1

p minus 1

)∓14






0 (y) minus ex



c+(n g)

[4πngV1(t)U

+0 (4πng) + tV0(t)U

+1 (4πng)

4πng minus t

]

+sumn1

cminus(n g)

[4πngV1(t)U


minus1 (4πng)

4πng + t

] (B8)





int infin

0dsV0(x + s)U+






0 (x)


d

dx(xzprime

i (x)) =(

x minus 1

2

)zi(x) (B11)

35




2(y + s)]

= d


2(y + s)]z1(x + s)

minus d


1(x + s)]z2(y + s) (B12)



1 (y + s) + (y + s)V1(x + s)U+0 (y + s)

] ∣∣s=infins=0


0 (y) (B13)






π

t

t + πg (C1)



π

t2

t2 + (πg)2

(C2)


t2 + π2g2


0

dt

tJ2nminus1(t)

[γminus(t)

1 minus eminust2g+

γ+(t)

et2g minus 1

]= h2nminus1(g)

(C3)int infin

0

dt

tJ2n(t)

[γ+(t)


et2g minus 1

]= h2n(g)


h2nminus1 = 2mprime

π

int infin

0

dtJ2nminus1(t)

t2 + (πg)2

[t


1 minus eminust(2g)

]

(C4)

h2n = 2mprime

π

int infin

0

dtJ2n(t)

t2 + (πg)2

[πg

et(2g) minus 1+

t

1 minus eminust(2g)

]



36



γminus(t) = 2sumn1


(C6)γ+(t) = 2

sumn1

(2n)J2n(t)γ2n(g)



π2gt + O(t2) = it

(γ1 +

2mprime

π2g

)+ O(t2) (C7)



0

dt

t


1 minus eminust2g+


et2g minus 1

]= 2

sumn1



γ1 = 2sumn1



γ1 = minusmprime

π

int infin

0dt

[πg


t

t2 + π2g2(minus(t) + +(t))

] (C10)


πg

t2 + π2g2=

int infin


(C11)t

t2 + π2g2=

int infin



γ1 = minusmprime

π

int infin

0du eminusπgu

[ int infin


+int infin


] (C12)


γ1 = minus 2mprime


radic2mprime

πeminusπgRe

[ int infin

0

dt


] (C13)


37






Vn(x) = 1

π232minusn ex

int 1


= 1

π232minusn eminusx

int 1



Vn(x) = 2minusn

(54 minus n

)

(54

)(2 minus n)


= 2minusn

(54 minus n

)

(54

)(2 minus n)







Uab(x) = 1

2

int infin





(1 +

t

2x

)a+bminus12

(D5)


Uab(x) = 2bminus32

(1

2minus a + b


= 1

2

(1

2minus a + b

)U

(1


) (D6)

38




U+0 (x) = 1

2

(5

4


1 (x) = 1

2

(1

4


(D7)

Uminus0 (x) = 1

2

(3

4


1 (x) = 1

2

(3

4





1 (x) = V1(x)U+0 (x) + V0(x)U+

1 (x) = ex

x (D8)


U+0 (x)Uminus


0 (minusx) = π

2radic

2xeplusmn 3iπ

4 (for Imx ≷ 0) (D9)


V0(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]

(D10)

V1(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]






(D11)Uplusmn








U+0 (x) = (2x)minus54

(5

4

)F

(1

4

5

4| minus 1

2x

)= (2x)minus54

(5

4

) [1 minus 5


]


(3

4

)F

(minus1

4

3

4| minus 1

2x

)= (2x)minus34

(3

4

) [1 +

3


]

39


U+1 (x) = (2x)minus14 1

2

(1

4

)F

(1

4

1

4| minus 1

2x

)= (2x)minus14 1

2

(1

4

) [1 minus 1


]


2

(3

4

)F

(3

4

3

4| minus 1

2x

)= (2x)minus34 1

2

(3

4

) [1 minus 9


]

(D12)








2

πeminus 3iπ

4 eπg[U+


] (D13)




(

34

) [F

(1

4

5

4| 1

2πg+ iε

)+ 2F

(minus1

4

3

4| minus 1

2πg

)] (D14)


2 = σ

(34

)

(54




2(

34

) [8πgF

(1

4

1

4| 1

2πg+ iε

)+ 2F

(3

4

3

4| minus 1

2πg

)] (D16)





F

(1

4

5

4| 1

2πg+ iε

)minus F

(1

4

5

4| 1

2πgminus iε

)= i

radic22

σF

(minus1

4

3

4| minus 1

2πg

) (D17)



40





t + 14


(4πgit)

t + 14

=sumk=01


t + 14

+sumk=01


t + 14

Vk(4πgt) (E1)


Rk(4πgs) = Re

[int minusiinfin


t minus s

]= Re

[int minusiinfin


t minus 4πgs

]

(E2)


Re

[ int minusiinfin


t + 14


minussumn1

nc+(n g)

n + 14

[U+


]+

sumn1

ncminus(n g)

n minus 14





R0(s) = minusRe

[eminusiπ4

int 1


int minusiinfin

0dt eminusvtV0(t)

]=

radic2

πRe

[eminusiπ4

int 1


int 1

minus1du


u minus v minus iε

] (E4)


1radic2π

int 1

minus1du



⎧⎨⎩(

v+1vminus1

)14 v2 gt 1

eminusiπ4(

1+v1minusv

)14 v2 lt 1

(E5)


R0(s)sgt0=

radic2

[1

sminus

int minus1


(v + 1

v minus 1

)14]

=radic

2

[1

sminus 2 eminus2sU+

0 (s)

] (E6)


R0(s)slt0=

radic2

[1

s+ 2Uminus

0 (minuss)

] (E7)

41


together with

R1(s) = 2radic

2[θ(minuss)Uminus


] (E8)


Re

[ int minusiinfin


t + 14

]= 2

radic2f0(minusπg)

[Uminus

0 (πg) minus 1

2πg

]+ 2


1 (πg) +

radic2



References



















42





























Ann Phys 167 227

43

























44

1 Introduction

2 Cusp anomalous



23 Toy model



26 General solution







4 Mass scale





5 Conclusions

Acknowledgments








References




t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

4πng + t

]

(240)


4πng

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

4πng + t

]


Vn(x) =radic

2

π

int 1


(241)

Uplusmnn (x) = 1

2

int infin




f(toy)


1 (t) = minusc(toy)(g) (242)







(4π ig

( minus 1

4

)) = 0 isin Z (244)



4

) (245)



cusp(g) = 2g[1 minus 2f1(0)] (246)

13



cusp(g) = 2g


[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

]⎫⎬⎭ (247)






V1(minusπg) (248)



[1 minus 2

V0(minusπg)

V1(minusπg)

] (249)


(toy)cusp (g) = 2g


Mminus140(2πg)

] (250)







couplings


(toy)cusp (g) = 3

2πg2 minus 1

2π2g3 minus 1

64π3g4 +

5

64π4g5 minus 11

512π5g6 minus 3

512π6g7 + O(g8) (251)



14






(

34

)[F

(1

4

5

4|α + i0

)+ 2F

(minus1

4

3

4| minus α

)] α = 1(2πg)

(252)



(34

)

(54



F

(1

4

5

4| minus α

)= αminus54U+

0 (1(2α))

(5

4

)

(254)

F

(minus1

4

3

4| minus α

)= αminus34Uminus

0 (1(2α))

(3

4

)



4 34 | minus α



4 and b = 34 )


(minusα)k

k

(a + k)(b + k)

(a)(b)= 1 minus αab + O(α2) (255)



(a)

int infin





(a)

int infin


15







F(

14 5

4 |α+)

+ 2F(minus 1

4 34 | minus α

)F

(14 1

4 |α+)

+ 142αF

(34 3

4 | minus α) (258)


(toy)cusp (g)



4α24C4(α) + O(6) (259)


C0 = 1 minus αF

(14 5

4 |α+)

F(

14 1

4 |α+) C2 = 1[

F(

14 1

4 |α+)]2 C4 = F

(34 3

4 | minus α)[

F(

14 1

4 |α+)]3 (260)



16



4α2 minus 3

8α3 minus 61

64α4 minus 433

128α5 + O(α6)

C2 = 1 minus 1

8α minus 11

128α2 minus 151

1024α3 minus 13085

32768α4 + O(α5) (261)

C4 = 1 minus 3

4α minus 27

32α2 minus 317

128α3 + O(α4)















17




2α minus K

4α2 minus

(3K ln 2

8+

27ζ3

256

)α3 + (263)



3 ln 2

2rarr 1

K

2rarr 1

2

9ζ3

32rarr 1

3 (264)



exp

[3 ln 2

2τ minus K

2τ 2 +

9ζ3

32τ 3 +

]rarr exp

[τ minus τ 2

2+

τ 3

3+

] (265)


(

14

)

(1 + τ

4

)

(34 minus τ

4

)

(34

)

(1 minus τ

4

)

(14 + τ

4

) rarr (1 + τ) (266)



cusp (g)




k (k + 1

2



(toy)cusp (g)

18















plusmn1 (4πng)

n ∓ x

(32)





V0(4πgx)=



(33)


19



g 01 02 04 06 08 10 12 14 16 18

Numer 01976 03616 05843 07096 07825 08276 08576 08787 08944 09065Exact 01939 03584 05821 07080 07813 08267 08568 08781 08938 09059


nmax 10 20 30 40 50 60 70 infinNumer 08305 08286 08279 08276 08274 08273 08272 08267


c(g) = 0 (34)











20








1 =sumn1

c+(n g)

[nU+

0 (4πng)r(x) + U+1 (4πng)x

n minus x

]

+sumn1

cminus(n g)

[nUminus


n + x

] (36)





4 isin Z) (37)




[aplusmn(n) +

bplusmn(n)


]+ 2

[αplusmn(n) +

βplusmn(n)


]+ O(4)

(38)

21




(34

)

(54

) (39)






4 )

2x

(5

4

) sumn1

a+(n)

n minus x

= 1 ( 1)

(310)

minus2x

(3

4

) sumn1

aminus(n)

n + x

= 1 ( 0)


a+(n) = 2(n + 1

4

)(n + 1)2

(14

)

(311)

aminus(n) = (n + 3

4

)2(n + 1)2

(34

)


4 )

2x

(5

4

) sumn1

b+(n)

n minus x

= minus 3

32x

minus 3π

64minus 15

32ln 2 ( 1)

(312)

minus 2x

(3

4

) sumn1

bminus(n)

n + x

= minus 5

32x

minus 5π

64+

9

32ln 2 ( 0)


b+(n) = minusa+(n)

(3 ln 2

4+

3

32n


(3 ln 2

4+

5

32n

) (313)




(2πn)minus1

[

(5

4

) (a+(n) +

b+(n)


)(1 minus 5

128πgn+

)

+

(3

4

) (aminus(n) +

bminus(n)


)(1 +

3


)]+ O(2) (314)

22



cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K

16π2g2+ O(1g3)

]+ O(2) (315)




4 + O(gminus1)]

+ O(4) (316)



2x

(5

4

) sumn1

α+(n)

n minus x

= 0 ( 1)

(317)

minus2x

(3

4

) sumn1

αminus(n)

n + x

= π

2radic

2δ0 ( 0)


α+(n) = 0(318)



2x

(5

4

) sumn1

β+(n)

n minus x

= minus1

2 ( 1)

(319)

minus2x

(3

4

) sumn1

βminus(n)

n + x

= minus1

8+

3π

16radic

2(1 minus 2 ln 2)δ0 ( 0)


β+(n) = minus1

2a+(n)

(320)


(1

4minus 3 ln 2

4+

1

32n

)



23




(2πn)minus1

[

(5

4

) (α+(n) +

β+(n)


)(1 minus 5


)

+

(3

4

) (αminus(n) +

βminus(n)


)(1 +

3


)]+ O(4) (321)


δcusp(g) = minus2

π

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (322)




m2cusp = 4

radic2

πσ2

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (323)


cusp(g) =[

2g minus 3 ln 2

2πminus K

8π2g+ O(1g2)

]minus σ

4radic

2m2

cusp + O(m4

cusp

) (324)


4 Mass scale




mO(6) = 8radic

2


πeminusπgRe

[int minusiinfin


t + 14

] (41)



24



f0(4πgt) =sumn1

t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

n + t

]minus 1

(43)

f1(4πgt) =sumn1

n

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

n + t

]





2




mO(6) = 16radic

2

π2

f1(minusπg)

V0(minusπg) (45)







fn(4πgt) = f (PT)

n (4πgt) + δfn(4πgt) (n = 0 1) (46)

where f (PT)



2

(5

4

) sumn1

a+(n)

t minus n= 1

t

[

(34

)(1 minus t)

(

34 minus t

) minus 1

]

(47)

2

(3

4

) sumn1

aminus(n)

t + n= 1

t

[

(14

)(1 + t)

(

14 + t

) minus 1

]

25



f(PT)

0 (4πgt) = minus(

34

)(1 minus t)

(

34 minus t

)+

1

4πg

[(3 ln 2

4+

1

8t

)

(34

)(1 minus t)

(

34 minus t

) minus (

14

)(1 + t)

8t(

14 + t

) ]+ O(gminus2)

f(PT)

1 (4πgt) = 1

4πg

[

(14

)(1 + t)

4t(

14 + t

) minus (

34

)(1 minus t)

4t(

34 minus t

) ]

minus 1

(4πg)2

[

(14

)(1 + t)

4t(

14 + t

) (1

4tminus 3 ln 2

4

)

minus (

34

)(1 minus t)

4t(

34 minus t

) (1

4t+

3 ln 2

4

)]+ O(gminus3) (48)


δf0(4πgt) = 2

1

4πg

[

(34

)(1 minus t)

2(

34 minus t

) minus (

54

)(1 + t)

2(

54 + t

) ]+ O(gminus2)


δf1(4πgt) = 2

1

4πg

(

54

)(1 + t)

(

54 + t

) +1

(4πg)2

[

(34

)(1 minus t)

8t(

34 minus t

)minus

(54

)(1 + t)

(

54 + t

) (1

8t+

3

4ln 2 minus 1

4

)]+ O(gminus3)





1 (0) minus 4gδf1(0) (410)

Replacing f(PT)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K


]minus 2

π

[1 +

3 minus 6 ln 2


]+ O(4) (411)



mO(6) =radic

2

(

54


[1 +

3 minus 6 ln 2

32πg+



]minus 2

8πg



]+ O(4)

(412)


(PT)


26




fn(4πgt)

(34 minus t

)

(34

)(1 minus t)

rarr f (toy)n (4πgt) (n = 0 1) (413)





(toy)


mtoy = 16radic

2

π2

f(toy)

1 (minusπg)

V0(minusπg)= 16

radic2

π2

1

V1(minusπg) (414)




mtoy = 4

π(

54


[1 minus 1

32πgminus 23


]minus 2

8πg

[1 minus 11


]+ O(4)

(415)


0 and f(toy)


27






32radic

2σm2

toy + O(m4

toy

) (416)




2radic

2mtoy (417)



4π (418)


cusp (g + c1) = 2g

[1 minus K


]minus 2

2radic

2π

[1 +

3


]+ O(4) (419)


[mO(6)(g + c1)]2 = 22

πσ

[(1 +

3

16πg+

16K minus 54


)minus 2

8radic

2πg

(1 minus 3


)+ O(4)

] (420)


O(6)

δcusp = minus σ

4radic

2m2

O(6) + O(m4

O(6)

) (421)



28











2

πeminusπgUminus


2

πeminusπgUminus

1 (πg) (423)





4




mprime = minus σ

8gmO(6) (426)




2

4mprimemO(6) = σ

radic2

32gm2

O(6) (427)



δ(it) = δf0(t)V0(t) + δf1(t)V1(t) (428)


δ(0) = 2δf1(0) (429)

29






δ(it) sim 1







δ(it) (433)


δγ (it) = t

4gδ(0) + O(t2) = t

2gδf1(0) + O(t2) (434)


0

dt

tJ2nminus1(t)

[δγminus(t)

1 minus eminust2g+

δγ+(t)

et2g minus 1

]= 0

(435)int infin

0

dt

tJ2n(t)

[δγ+(t)


et2g minus 1

]= 0



30




2

π2geminusπg minus

radic2

πeminusπg Re

int infin

0

dt prime


]+ O(t2) (436)



tradic

2

8g+ O(t2) (437)


mcusp(g) = mO(6)(g) (438)


5 Conclusions






31





(m4

O(n)

μ2) (51)


m2O(n) = cμ2 e


20 [1 + O(g2)] (52)




Acknowledgments


32








γ1 = 1

2minus π2

6g2 +

11π4

90g4 minus

(73π6

630+ 4ζ 2

3

)g6 + O(g8)

γ2 = ζ3g3 minus

(π2

3ζ3 + 10ζ5

)g5 +

(8π4

45ζ3 +

10π2

3ζ5 + 105ζ7

)g7 + O(g9)

γ3 = minusπ4

90g4 +

37π6

1890g6 + O(g8) γ4 = ζ5g

5 minus(

π2

3ζ5 + 21ζ7

)g7 + O(g9)

γ5 = minus π6

945g6 + O(g8) γ6 = ζ7g

7 + O(g9)

(A2)



3g4 +

44π4

45g6 minus

(292π6

315+ 32ζ 2

3

)g8 + O(g10) (A3)







33



c(g) = minus1

2+ 2gγ1(g) +

sumn1

[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

] (A5)


c(g) = minus1

2+ 2gγ1(g) minus

sumk1



fk(g) = 8gsumn1

[U+






3g2 + 2ζ3g

3 minus π4

6g4 minus 23ζ5g

5 +17π6

108g6 +

1107

4ζ7g

7 + O(g8)

f2 = minus1

2+ 2ζ3g

3 minus π4

30g4 + O(g5) f3 = 1

2+ O(g4)

f4 = minus3

8+ O(g5) f5 = 3

8+ O(g6) f6 = minus 5

16+ O(g7)

(A8)






(t) =int infin




(k)=sumn1



(k)=sumn1


bull For minus1 k 1

(k) = minusradic

2

π

(1 + k

1 minus k

)14[

1 +c(g)

1 + k+

1

2

(int minus1

minusinfin+

int infin

1

)dp(p)

p minus k

(p minus 1

p + 1

)14]

(B4)


34



(t) =sumn1

c+(n g)

[eminusit


]

+sumn1

cminus(n g)

[eit



(B5)



int 1

minus1dk eplusmnxk

int infin

1

dpeminusy(pminus1)

p minus k

(1 + k

1 minus k

p minus 1

p + 1

)plusmn14

Vn(x) =radic

2

π

int 1

minus1

dk exk

(k + 1)n

(1 + k

1 minus k

)14

(B6)

Uplusmnn (y) = 1

2

int infin

1

dp eminusy(pminus1)

(p ∓ 1)n

(p + 1

p minus 1

)∓14






0 (y) minus ex



c+(n g)

[4πngV1(t)U

+0 (4πng) + tV0(t)U

+1 (4πng)

4πng minus t

]

+sumn1

cminus(n g)

[4πngV1(t)U


minus1 (4πng)

4πng + t

] (B8)





int infin

0dsV0(x + s)U+






0 (x)


d

dx(xzprime

i (x)) =(

x minus 1

2

)zi(x) (B11)

35




2(y + s)]

= d


2(y + s)]z1(x + s)

minus d


1(x + s)]z2(y + s) (B12)



1 (y + s) + (y + s)V1(x + s)U+0 (y + s)

] ∣∣s=infins=0


0 (y) (B13)






π

t

t + πg (C1)



π

t2

t2 + (πg)2

(C2)


t2 + π2g2


0

dt

tJ2nminus1(t)

[γminus(t)

1 minus eminust2g+

γ+(t)

et2g minus 1

]= h2nminus1(g)

(C3)int infin

0

dt

tJ2n(t)

[γ+(t)


et2g minus 1

]= h2n(g)


h2nminus1 = 2mprime

π

int infin

0

dtJ2nminus1(t)

t2 + (πg)2

[t


1 minus eminust(2g)

]

(C4)

h2n = 2mprime

π

int infin

0

dtJ2n(t)

t2 + (πg)2

[πg

et(2g) minus 1+

t

1 minus eminust(2g)

]



36



γminus(t) = 2sumn1


(C6)γ+(t) = 2

sumn1

(2n)J2n(t)γ2n(g)



π2gt + O(t2) = it

(γ1 +

2mprime

π2g

)+ O(t2) (C7)



0

dt

t


1 minus eminust2g+


et2g minus 1

]= 2

sumn1



γ1 = 2sumn1



γ1 = minusmprime

π

int infin

0dt

[πg


t

t2 + π2g2(minus(t) + +(t))

] (C10)


πg

t2 + π2g2=

int infin


(C11)t

t2 + π2g2=

int infin



γ1 = minusmprime

π

int infin

0du eminusπgu

[ int infin


+int infin


] (C12)


γ1 = minus 2mprime


radic2mprime

πeminusπgRe

[ int infin

0

dt


] (C13)


37






Vn(x) = 1

π232minusn ex

int 1


= 1

π232minusn eminusx

int 1



Vn(x) = 2minusn

(54 minus n

)

(54

)(2 minus n)


= 2minusn

(54 minus n

)

(54

)(2 minus n)







Uab(x) = 1

2

int infin





(1 +

t

2x

)a+bminus12

(D5)


Uab(x) = 2bminus32

(1

2minus a + b


= 1

2

(1

2minus a + b

)U

(1


) (D6)

38




U+0 (x) = 1

2

(5

4


1 (x) = 1

2

(1

4


(D7)

Uminus0 (x) = 1

2

(3

4


1 (x) = 1

2

(3

4





1 (x) = V1(x)U+0 (x) + V0(x)U+

1 (x) = ex

x (D8)


U+0 (x)Uminus


0 (minusx) = π

2radic

2xeplusmn 3iπ

4 (for Imx ≷ 0) (D9)


V0(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]

(D10)

V1(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]






(D11)Uplusmn








U+0 (x) = (2x)minus54

(5

4

)F

(1

4

5

4| minus 1

2x

)= (2x)minus54

(5

4

) [1 minus 5


]


(3

4

)F

(minus1

4

3

4| minus 1

2x

)= (2x)minus34

(3

4

) [1 +

3


]

39


U+1 (x) = (2x)minus14 1

2

(1

4

)F

(1

4

1

4| minus 1

2x

)= (2x)minus14 1

2

(1

4

) [1 minus 1


]


2

(3

4

)F

(3

4

3

4| minus 1

2x

)= (2x)minus34 1

2

(3

4

) [1 minus 9


]

(D12)








2

πeminus 3iπ

4 eπg[U+


] (D13)




(

34

) [F

(1

4

5

4| 1

2πg+ iε

)+ 2F

(minus1

4

3

4| minus 1

2πg

)] (D14)


2 = σ

(34

)

(54




2(

34

) [8πgF

(1

4

1

4| 1

2πg+ iε

)+ 2F

(3

4

3

4| minus 1

2πg

)] (D16)





F

(1

4

5

4| 1

2πg+ iε

)minus F

(1

4

5

4| 1

2πgminus iε

)= i

radic22

σF

(minus1

4

3

4| minus 1

2πg

) (D17)



40





t + 14


(4πgit)

t + 14

=sumk=01


t + 14

+sumk=01


t + 14

Vk(4πgt) (E1)


Rk(4πgs) = Re

[int minusiinfin


t minus s

]= Re

[int minusiinfin


t minus 4πgs

]

(E2)


Re

[ int minusiinfin


t + 14


minussumn1

nc+(n g)

n + 14

[U+


]+

sumn1

ncminus(n g)

n minus 14





R0(s) = minusRe

[eminusiπ4

int 1


int minusiinfin

0dt eminusvtV0(t)

]=

radic2

πRe

[eminusiπ4

int 1


int 1

minus1du


u minus v minus iε

] (E4)


1radic2π

int 1

minus1du



⎧⎨⎩(

v+1vminus1

)14 v2 gt 1

eminusiπ4(

1+v1minusv

)14 v2 lt 1

(E5)


R0(s)sgt0=

radic2

[1

sminus

int minus1


(v + 1

v minus 1

)14]

=radic

2

[1

sminus 2 eminus2sU+

0 (s)

] (E6)


R0(s)slt0=

radic2

[1

s+ 2Uminus

0 (minuss)

] (E7)

41


together with

R1(s) = 2radic

2[θ(minuss)Uminus


] (E8)


Re

[ int minusiinfin


t + 14

]= 2

radic2f0(minusπg)

[Uminus

0 (πg) minus 1

2πg

]+ 2


1 (πg) +

radic2



References



















42





























Ann Phys 167 227

43

























44

1 Introduction

2 Cusp anomalous



23 Toy model



26 General solution







4 Mass scale





5 Conclusions

Acknowledgments








References



cusp(g) = 2g


[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

]⎫⎬⎭ (247)






V1(minusπg) (248)



[1 minus 2

V0(minusπg)

V1(minusπg)

] (249)


(toy)cusp (g) = 2g


Mminus140(2πg)

] (250)







couplings


(toy)cusp (g) = 3

2πg2 minus 1

2π2g3 minus 1

64π3g4 +

5

64π4g5 minus 11

512π5g6 minus 3

512π6g7 + O(g8) (251)



14






(

34

)[F

(1

4

5

4|α + i0

)+ 2F

(minus1

4

3

4| minus α

)] α = 1(2πg)

(252)



(34

)

(54



F

(1

4

5

4| minus α

)= αminus54U+

0 (1(2α))

(5

4

)

(254)

F

(minus1

4

3

4| minus α

)= αminus34Uminus

0 (1(2α))

(3

4

)



4 34 | minus α



4 and b = 34 )


(minusα)k

k

(a + k)(b + k)

(a)(b)= 1 minus αab + O(α2) (255)



(a)

int infin





(a)

int infin


15







F(

14 5

4 |α+)

+ 2F(minus 1

4 34 | minus α

)F

(14 1

4 |α+)

+ 142αF

(34 3

4 | minus α) (258)


(toy)cusp (g)



4α24C4(α) + O(6) (259)


C0 = 1 minus αF

(14 5

4 |α+)

F(

14 1

4 |α+) C2 = 1[

F(

14 1

4 |α+)]2 C4 = F

(34 3

4 | minus α)[

F(

14 1

4 |α+)]3 (260)



16



4α2 minus 3

8α3 minus 61

64α4 minus 433

128α5 + O(α6)

C2 = 1 minus 1

8α minus 11

128α2 minus 151

1024α3 minus 13085

32768α4 + O(α5) (261)

C4 = 1 minus 3

4α minus 27

32α2 minus 317

128α3 + O(α4)















17




2α minus K

4α2 minus

(3K ln 2

8+

27ζ3

256

)α3 + (263)



3 ln 2

2rarr 1

K

2rarr 1

2

9ζ3

32rarr 1

3 (264)



exp

[3 ln 2

2τ minus K

2τ 2 +

9ζ3

32τ 3 +

]rarr exp

[τ minus τ 2

2+

τ 3

3+

] (265)


(

14

)

(1 + τ

4

)

(34 minus τ

4

)

(34

)

(1 minus τ

4

)

(14 + τ

4

) rarr (1 + τ) (266)



cusp (g)




k (k + 1

2



(toy)cusp (g)

18















plusmn1 (4πng)

n ∓ x

(32)





V0(4πgx)=



(33)


19



g 01 02 04 06 08 10 12 14 16 18

Numer 01976 03616 05843 07096 07825 08276 08576 08787 08944 09065Exact 01939 03584 05821 07080 07813 08267 08568 08781 08938 09059


nmax 10 20 30 40 50 60 70 infinNumer 08305 08286 08279 08276 08274 08273 08272 08267


c(g) = 0 (34)











20








1 =sumn1

c+(n g)

[nU+

0 (4πng)r(x) + U+1 (4πng)x

n minus x

]

+sumn1

cminus(n g)

[nUminus


n + x

] (36)





4 isin Z) (37)




[aplusmn(n) +

bplusmn(n)


]+ 2

[αplusmn(n) +

βplusmn(n)


]+ O(4)

(38)

21




(34

)

(54

) (39)






4 )

2x

(5

4

) sumn1

a+(n)

n minus x

= 1 ( 1)

(310)

minus2x

(3

4

) sumn1

aminus(n)

n + x

= 1 ( 0)


a+(n) = 2(n + 1

4

)(n + 1)2

(14

)

(311)

aminus(n) = (n + 3

4

)2(n + 1)2

(34

)


4 )

2x

(5

4

) sumn1

b+(n)

n minus x

= minus 3

32x

minus 3π

64minus 15

32ln 2 ( 1)

(312)

minus 2x

(3

4

) sumn1

bminus(n)

n + x

= minus 5

32x

minus 5π

64+

9

32ln 2 ( 0)


b+(n) = minusa+(n)

(3 ln 2

4+

3

32n


(3 ln 2

4+

5

32n

) (313)




(2πn)minus1

[

(5

4

) (a+(n) +

b+(n)


)(1 minus 5

128πgn+

)

+

(3

4

) (aminus(n) +

bminus(n)


)(1 +

3


)]+ O(2) (314)

22



cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K

16π2g2+ O(1g3)

]+ O(2) (315)




4 + O(gminus1)]

+ O(4) (316)



2x

(5

4

) sumn1

α+(n)

n minus x

= 0 ( 1)

(317)

minus2x

(3

4

) sumn1

αminus(n)

n + x

= π

2radic

2δ0 ( 0)


α+(n) = 0(318)



2x

(5

4

) sumn1

β+(n)

n minus x

= minus1

2 ( 1)

(319)

minus2x

(3

4

) sumn1

βminus(n)

n + x

= minus1

8+

3π

16radic

2(1 minus 2 ln 2)δ0 ( 0)


β+(n) = minus1

2a+(n)

(320)


(1

4minus 3 ln 2

4+

1

32n

)



23




(2πn)minus1

[

(5

4

) (α+(n) +

β+(n)


)(1 minus 5


)

+

(3

4

) (αminus(n) +

βminus(n)


)(1 +

3


)]+ O(4) (321)


δcusp(g) = minus2

π

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (322)




m2cusp = 4

radic2

πσ2

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (323)


cusp(g) =[

2g minus 3 ln 2

2πminus K

8π2g+ O(1g2)

]minus σ

4radic

2m2

cusp + O(m4

cusp

) (324)


4 Mass scale




mO(6) = 8radic

2


πeminusπgRe

[int minusiinfin


t + 14

] (41)



24



f0(4πgt) =sumn1

t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

n + t

]minus 1

(43)

f1(4πgt) =sumn1

n

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

n + t

]





2




mO(6) = 16radic

2

π2

f1(minusπg)

V0(minusπg) (45)







fn(4πgt) = f (PT)

n (4πgt) + δfn(4πgt) (n = 0 1) (46)

where f (PT)



2

(5

4

) sumn1

a+(n)

t minus n= 1

t

[

(34

)(1 minus t)

(

34 minus t

) minus 1

]

(47)

2

(3

4

) sumn1

aminus(n)

t + n= 1

t

[

(14

)(1 + t)

(

14 + t

) minus 1

]

25



f(PT)

0 (4πgt) = minus(

34

)(1 minus t)

(

34 minus t

)+

1

4πg

[(3 ln 2

4+

1

8t

)

(34

)(1 minus t)

(

34 minus t

) minus (

14

)(1 + t)

8t(

14 + t

) ]+ O(gminus2)

f(PT)

1 (4πgt) = 1

4πg

[

(14

)(1 + t)

4t(

14 + t

) minus (

34

)(1 minus t)

4t(

34 minus t

) ]

minus 1

(4πg)2

[

(14

)(1 + t)

4t(

14 + t

) (1

4tminus 3 ln 2

4

)

minus (

34

)(1 minus t)

4t(

34 minus t

) (1

4t+

3 ln 2

4

)]+ O(gminus3) (48)


δf0(4πgt) = 2

1

4πg

[

(34

)(1 minus t)

2(

34 minus t

) minus (

54

)(1 + t)

2(

54 + t

) ]+ O(gminus2)


δf1(4πgt) = 2

1

4πg

(

54

)(1 + t)

(

54 + t

) +1

(4πg)2

[

(34

)(1 minus t)

8t(

34 minus t

)minus

(54

)(1 + t)

(

54 + t

) (1

8t+

3

4ln 2 minus 1

4

)]+ O(gminus3)





1 (0) minus 4gδf1(0) (410)

Replacing f(PT)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K


]minus 2

π

[1 +

3 minus 6 ln 2


]+ O(4) (411)



mO(6) =radic

2

(

54


[1 +

3 minus 6 ln 2

32πg+



]minus 2

8πg



]+ O(4)

(412)


(PT)


26




fn(4πgt)

(34 minus t

)

(34

)(1 minus t)

rarr f (toy)n (4πgt) (n = 0 1) (413)





(toy)


mtoy = 16radic

2

π2

f(toy)

1 (minusπg)

V0(minusπg)= 16

radic2

π2

1

V1(minusπg) (414)




mtoy = 4

π(

54


[1 minus 1

32πgminus 23


]minus 2

8πg

[1 minus 11


]+ O(4)

(415)


0 and f(toy)


27






32radic

2σm2

toy + O(m4

toy

) (416)




2radic

2mtoy (417)



4π (418)


cusp (g + c1) = 2g

[1 minus K


]minus 2

2radic

2π

[1 +

3


]+ O(4) (419)


[mO(6)(g + c1)]2 = 22

πσ

[(1 +

3

16πg+

16K minus 54


)minus 2

8radic

2πg

(1 minus 3


)+ O(4)

] (420)


O(6)

δcusp = minus σ

4radic

2m2

O(6) + O(m4

O(6)

) (421)



28











2

πeminusπgUminus


2

πeminusπgUminus

1 (πg) (423)





4




mprime = minus σ

8gmO(6) (426)




2

4mprimemO(6) = σ

radic2

32gm2

O(6) (427)



δ(it) = δf0(t)V0(t) + δf1(t)V1(t) (428)


δ(0) = 2δf1(0) (429)

29






δ(it) sim 1







δ(it) (433)


δγ (it) = t

4gδ(0) + O(t2) = t

2gδf1(0) + O(t2) (434)


0

dt

tJ2nminus1(t)

[δγminus(t)

1 minus eminust2g+

δγ+(t)

et2g minus 1

]= 0

(435)int infin

0

dt

tJ2n(t)

[δγ+(t)


et2g minus 1

]= 0



30




2

π2geminusπg minus

radic2

πeminusπg Re

int infin

0

dt prime


]+ O(t2) (436)



tradic

2

8g+ O(t2) (437)


mcusp(g) = mO(6)(g) (438)


5 Conclusions






31





(m4

O(n)

μ2) (51)


m2O(n) = cμ2 e


20 [1 + O(g2)] (52)




Acknowledgments


32








γ1 = 1

2minus π2

6g2 +

11π4

90g4 minus

(73π6

630+ 4ζ 2

3

)g6 + O(g8)

γ2 = ζ3g3 minus

(π2

3ζ3 + 10ζ5

)g5 +

(8π4

45ζ3 +

10π2

3ζ5 + 105ζ7

)g7 + O(g9)

γ3 = minusπ4

90g4 +

37π6

1890g6 + O(g8) γ4 = ζ5g

5 minus(

π2

3ζ5 + 21ζ7

)g7 + O(g9)

γ5 = minus π6

945g6 + O(g8) γ6 = ζ7g

7 + O(g9)

(A2)



3g4 +

44π4

45g6 minus

(292π6

315+ 32ζ 2

3

)g8 + O(g10) (A3)







33



c(g) = minus1

2+ 2gγ1(g) +

sumn1

[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

] (A5)


c(g) = minus1

2+ 2gγ1(g) minus

sumk1



fk(g) = 8gsumn1

[U+






3g2 + 2ζ3g

3 minus π4

6g4 minus 23ζ5g

5 +17π6

108g6 +

1107

4ζ7g

7 + O(g8)

f2 = minus1

2+ 2ζ3g

3 minus π4

30g4 + O(g5) f3 = 1

2+ O(g4)

f4 = minus3

8+ O(g5) f5 = 3

8+ O(g6) f6 = minus 5

16+ O(g7)

(A8)






(t) =int infin




(k)=sumn1



(k)=sumn1


bull For minus1 k 1

(k) = minusradic

2

π

(1 + k

1 minus k

)14[

1 +c(g)

1 + k+

1

2

(int minus1

minusinfin+

int infin

1

)dp(p)

p minus k

(p minus 1

p + 1

)14]

(B4)


34



(t) =sumn1

c+(n g)

[eminusit


]

+sumn1

cminus(n g)

[eit



(B5)



int 1

minus1dk eplusmnxk

int infin

1

dpeminusy(pminus1)

p minus k

(1 + k

1 minus k

p minus 1

p + 1

)plusmn14

Vn(x) =radic

2

π

int 1

minus1

dk exk

(k + 1)n

(1 + k

1 minus k

)14

(B6)

Uplusmnn (y) = 1

2

int infin

1

dp eminusy(pminus1)

(p ∓ 1)n

(p + 1

p minus 1

)∓14






0 (y) minus ex



c+(n g)

[4πngV1(t)U

+0 (4πng) + tV0(t)U

+1 (4πng)

4πng minus t

]

+sumn1

cminus(n g)

[4πngV1(t)U


minus1 (4πng)

4πng + t

] (B8)





int infin

0dsV0(x + s)U+






0 (x)


d

dx(xzprime

i (x)) =(

x minus 1

2

)zi(x) (B11)

35




2(y + s)]

= d


2(y + s)]z1(x + s)

minus d


1(x + s)]z2(y + s) (B12)



1 (y + s) + (y + s)V1(x + s)U+0 (y + s)

] ∣∣s=infins=0


0 (y) (B13)






π

t

t + πg (C1)



π

t2

t2 + (πg)2

(C2)


t2 + π2g2


0

dt

tJ2nminus1(t)

[γminus(t)

1 minus eminust2g+

γ+(t)

et2g minus 1

]= h2nminus1(g)

(C3)int infin

0

dt

tJ2n(t)

[γ+(t)


et2g minus 1

]= h2n(g)


h2nminus1 = 2mprime

π

int infin

0

dtJ2nminus1(t)

t2 + (πg)2

[t


1 minus eminust(2g)

]

(C4)

h2n = 2mprime

π

int infin

0

dtJ2n(t)

t2 + (πg)2

[πg

et(2g) minus 1+

t

1 minus eminust(2g)

]



36



γminus(t) = 2sumn1


(C6)γ+(t) = 2

sumn1

(2n)J2n(t)γ2n(g)



π2gt + O(t2) = it

(γ1 +

2mprime

π2g

)+ O(t2) (C7)



0

dt

t


1 minus eminust2g+


et2g minus 1

]= 2

sumn1



γ1 = 2sumn1



γ1 = minusmprime

π

int infin

0dt

[πg


t

t2 + π2g2(minus(t) + +(t))

] (C10)


πg

t2 + π2g2=

int infin


(C11)t

t2 + π2g2=

int infin



γ1 = minusmprime

π

int infin

0du eminusπgu

[ int infin


+int infin


] (C12)


γ1 = minus 2mprime


radic2mprime

πeminusπgRe

[ int infin

0

dt


] (C13)


37






Vn(x) = 1

π232minusn ex

int 1


= 1

π232minusn eminusx

int 1



Vn(x) = 2minusn

(54 minus n

)

(54

)(2 minus n)


= 2minusn

(54 minus n

)

(54

)(2 minus n)







Uab(x) = 1

2

int infin





(1 +

t

2x

)a+bminus12

(D5)


Uab(x) = 2bminus32

(1

2minus a + b


= 1

2

(1

2minus a + b

)U

(1


) (D6)

38




U+0 (x) = 1

2

(5

4


1 (x) = 1

2

(1

4


(D7)

Uminus0 (x) = 1

2

(3

4


1 (x) = 1

2

(3

4





1 (x) = V1(x)U+0 (x) + V0(x)U+

1 (x) = ex

x (D8)


U+0 (x)Uminus


0 (minusx) = π

2radic

2xeplusmn 3iπ

4 (for Imx ≷ 0) (D9)


V0(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]

(D10)

V1(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]






(D11)Uplusmn








U+0 (x) = (2x)minus54

(5

4

)F

(1

4

5

4| minus 1

2x

)= (2x)minus54

(5

4

) [1 minus 5


]


(3

4

)F

(minus1

4

3

4| minus 1

2x

)= (2x)minus34

(3

4

) [1 +

3


]

39


U+1 (x) = (2x)minus14 1

2

(1

4

)F

(1

4

1

4| minus 1

2x

)= (2x)minus14 1

2

(1

4

) [1 minus 1


]


2

(3

4

)F

(3

4

3

4| minus 1

2x

)= (2x)minus34 1

2

(3

4

) [1 minus 9


]

(D12)








2

πeminus 3iπ

4 eπg[U+


] (D13)




(

34

) [F

(1

4

5

4| 1

2πg+ iε

)+ 2F

(minus1

4

3

4| minus 1

2πg

)] (D14)


2 = σ

(34

)

(54




2(

34

) [8πgF

(1

4

1

4| 1

2πg+ iε

)+ 2F

(3

4

3

4| minus 1

2πg

)] (D16)





F

(1

4

5

4| 1

2πg+ iε

)minus F

(1

4

5

4| 1

2πgminus iε

)= i

radic22

σF

(minus1

4

3

4| minus 1

2πg

) (D17)



40





t + 14


(4πgit)

t + 14

=sumk=01


t + 14

+sumk=01


t + 14

Vk(4πgt) (E1)


Rk(4πgs) = Re

[int minusiinfin


t minus s

]= Re

[int minusiinfin


t minus 4πgs

]

(E2)


Re

[ int minusiinfin


t + 14


minussumn1

nc+(n g)

n + 14

[U+


]+

sumn1

ncminus(n g)

n minus 14





R0(s) = minusRe

[eminusiπ4

int 1


int minusiinfin

0dt eminusvtV0(t)

]=

radic2

πRe

[eminusiπ4

int 1


int 1

minus1du


u minus v minus iε

] (E4)


1radic2π

int 1

minus1du



⎧⎨⎩(

v+1vminus1

)14 v2 gt 1

eminusiπ4(

1+v1minusv

)14 v2 lt 1

(E5)


R0(s)sgt0=

radic2

[1

sminus

int minus1


(v + 1

v minus 1

)14]

=radic

2

[1

sminus 2 eminus2sU+

0 (s)

] (E6)


R0(s)slt0=

radic2

[1

s+ 2Uminus

0 (minuss)

] (E7)

41


together with

R1(s) = 2radic

2[θ(minuss)Uminus


] (E8)


Re

[ int minusiinfin


t + 14

]= 2

radic2f0(minusπg)

[Uminus

0 (πg) minus 1

2πg

]+ 2


1 (πg) +

radic2



References



















42





























Ann Phys 167 227

43

























44

1 Introduction

2 Cusp anomalous



23 Toy model



26 General solution







4 Mass scale





5 Conclusions

Acknowledgments








References






(

34

)[F

(1

4

5

4|α + i0

)+ 2F

(minus1

4

3

4| minus α

)] α = 1(2πg)

(252)



(34

)

(54



F

(1

4

5

4| minus α

)= αminus54U+

0 (1(2α))

(5

4

)

(254)

F

(minus1

4

3

4| minus α

)= αminus34Uminus

0 (1(2α))

(3

4

)



4 34 | minus α



4 and b = 34 )


(minusα)k

k

(a + k)(b + k)

(a)(b)= 1 minus αab + O(α2) (255)



(a)

int infin





(a)

int infin


15







F(

14 5

4 |α+)

+ 2F(minus 1

4 34 | minus α

)F

(14 1

4 |α+)

+ 142αF

(34 3

4 | minus α) (258)


(toy)cusp (g)



4α24C4(α) + O(6) (259)


C0 = 1 minus αF

(14 5

4 |α+)

F(

14 1

4 |α+) C2 = 1[

F(

14 1

4 |α+)]2 C4 = F

(34 3

4 | minus α)[

F(

14 1

4 |α+)]3 (260)



16



4α2 minus 3

8α3 minus 61

64α4 minus 433

128α5 + O(α6)

C2 = 1 minus 1

8α minus 11

128α2 minus 151

1024α3 minus 13085

32768α4 + O(α5) (261)

C4 = 1 minus 3

4α minus 27

32α2 minus 317

128α3 + O(α4)















17




2α minus K

4α2 minus

(3K ln 2

8+

27ζ3

256

)α3 + (263)



3 ln 2

2rarr 1

K

2rarr 1

2

9ζ3

32rarr 1

3 (264)



exp

[3 ln 2

2τ minus K

2τ 2 +

9ζ3

32τ 3 +

]rarr exp

[τ minus τ 2

2+

τ 3

3+

] (265)


(

14

)

(1 + τ

4

)

(34 minus τ

4

)

(34

)

(1 minus τ

4

)

(14 + τ

4

) rarr (1 + τ) (266)



cusp (g)




k (k + 1

2



(toy)cusp (g)

18















plusmn1 (4πng)

n ∓ x

(32)





V0(4πgx)=



(33)


19



g 01 02 04 06 08 10 12 14 16 18

Numer 01976 03616 05843 07096 07825 08276 08576 08787 08944 09065Exact 01939 03584 05821 07080 07813 08267 08568 08781 08938 09059


nmax 10 20 30 40 50 60 70 infinNumer 08305 08286 08279 08276 08274 08273 08272 08267


c(g) = 0 (34)











20








1 =sumn1

c+(n g)

[nU+

0 (4πng)r(x) + U+1 (4πng)x

n minus x

]

+sumn1

cminus(n g)

[nUminus


n + x

] (36)





4 isin Z) (37)




[aplusmn(n) +

bplusmn(n)


]+ 2

[αplusmn(n) +

βplusmn(n)


]+ O(4)

(38)

21




(34

)

(54

) (39)






4 )

2x

(5

4

) sumn1

a+(n)

n minus x

= 1 ( 1)

(310)

minus2x

(3

4

) sumn1

aminus(n)

n + x

= 1 ( 0)


a+(n) = 2(n + 1

4

)(n + 1)2

(14

)

(311)

aminus(n) = (n + 3

4

)2(n + 1)2

(34

)


4 )

2x

(5

4

) sumn1

b+(n)

n minus x

= minus 3

32x

minus 3π

64minus 15

32ln 2 ( 1)

(312)

minus 2x

(3

4

) sumn1

bminus(n)

n + x

= minus 5

32x

minus 5π

64+

9

32ln 2 ( 0)


b+(n) = minusa+(n)

(3 ln 2

4+

3

32n


(3 ln 2

4+

5

32n

) (313)




(2πn)minus1

[

(5

4

) (a+(n) +

b+(n)


)(1 minus 5

128πgn+

)

+

(3

4

) (aminus(n) +

bminus(n)


)(1 +

3


)]+ O(2) (314)

22



cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K

16π2g2+ O(1g3)

]+ O(2) (315)




4 + O(gminus1)]

+ O(4) (316)



2x

(5

4

) sumn1

α+(n)

n minus x

= 0 ( 1)

(317)

minus2x

(3

4

) sumn1

αminus(n)

n + x

= π

2radic

2δ0 ( 0)


α+(n) = 0(318)



2x

(5

4

) sumn1

β+(n)

n minus x

= minus1

2 ( 1)

(319)

minus2x

(3

4

) sumn1

βminus(n)

n + x

= minus1

8+

3π

16radic

2(1 minus 2 ln 2)δ0 ( 0)


β+(n) = minus1

2a+(n)

(320)


(1

4minus 3 ln 2

4+

1

32n

)



23




(2πn)minus1

[

(5

4

) (α+(n) +

β+(n)


)(1 minus 5


)

+

(3

4

) (αminus(n) +

βminus(n)


)(1 +

3


)]+ O(4) (321)


δcusp(g) = minus2

π

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (322)




m2cusp = 4

radic2

πσ2

[1 +

3 minus 6 ln 2

16πg+ O(1g2)

]+ O(4) (323)


cusp(g) =[

2g minus 3 ln 2

2πminus K

8π2g+ O(1g2)

]minus σ

4radic

2m2

cusp + O(m4

cusp

) (324)


4 Mass scale




mO(6) = 8radic

2


πeminusπgRe

[int minusiinfin


t + 14

] (41)



24



f0(4πgt) =sumn1

t

[c+(n g)

U+1 (4πng)


Uminus1 (4πng)

n + t

]minus 1

(43)

f1(4πgt) =sumn1

n

[c+(n g)

U+0 (4πng)


Uminus0 (4πng)

n + t

]





2




mO(6) = 16radic

2

π2

f1(minusπg)

V0(minusπg) (45)







fn(4πgt) = f (PT)

n (4πgt) + δfn(4πgt) (n = 0 1) (46)

where f (PT)



2

(5

4

) sumn1

a+(n)

t minus n= 1

t

[

(34

)(1 minus t)

(

34 minus t

) minus 1

]

(47)

2

(3

4

) sumn1

aminus(n)

t + n= 1

t

[

(14

)(1 + t)

(

14 + t

) minus 1

]

25



f(PT)

0 (4πgt) = minus(

34

)(1 minus t)

(

34 minus t

)+

1

4πg

[(3 ln 2

4+

1

8t

)

(34

)(1 minus t)

(

34 minus t

) minus (

14

)(1 + t)

8t(

14 + t

) ]+ O(gminus2)

f(PT)

1 (4πgt) = 1

4πg

[

(14

)(1 + t)

4t(

14 + t

) minus (

34

)(1 minus t)

4t(

34 minus t

) ]

minus 1

(4πg)2

[

(14

)(1 + t)

4t(

14 + t

) (1

4tminus 3 ln 2

4

)

minus (

34

)(1 minus t)

4t(

34 minus t

) (1

4t+

3 ln 2

4

)]+ O(gminus3) (48)


δf0(4πgt) = 2

1

4πg

[

(34

)(1 minus t)

2(

34 minus t

) minus (

54

)(1 + t)

2(

54 + t

) ]+ O(gminus2)


δf1(4πgt) = 2

1

4πg

(

54

)(1 + t)

(

54 + t

) +1

(4πg)2

[

(34

)(1 minus t)

8t(

34 minus t

)minus

(54

)(1 + t)

(

54 + t

) (1

8t+

3

4ln 2 minus 1

4

)]+ O(gminus3)





1 (0) minus 4gδf1(0) (410)

Replacing f(PT)


cusp(g) = 2g

[1 minus 3 ln 2

4πgminus K


]minus 2

π

[1 +

3 minus 6 ln 2


]+ O(4) (411)



mO(6) =radic

2

(

54


[1 +

3 minus 6 ln 2

32πg+



]minus 2

8πg



]+ O(4)

(412)


(PT)


26




fn(4πgt)

(34 minus t

)

(34

)(1 minus t)

rarr f (toy)n (4πgt) (n = 0 1) (413)





(toy)


mtoy = 16radic

2

π2

f(toy)

1 (minusπg)

V0(minusπg)= 16

radic2

π2

1

V1(minusπg) (414)




mtoy = 4

π(

54


[1 minus 1

32πgminus 23


]minus 2

8πg

[1 minus 11


]+ O(4)

(415)


0 and f(toy)


27






32radic

2σm2

toy + O(m4

toy

) (416)




2radic

2mtoy (417)



4π (418)


cusp (g + c1) = 2g

[1 minus K


]minus 2

2radic

2π

[1 +

3


]+ O(4) (419)


[mO(6)(g + c1)]2 = 22

πσ

[(1 +

3

16πg+

16K minus 54


)minus 2

8radic

2πg

(1 minus 3


)+ O(4)

] (420)


O(6)

δcusp = minus σ

4radic

2m2

O(6) + O(m4

O(6)

) (421)



28











2

πeminusπgUminus


2

πeminusπgUminus

1 (πg) (423)





4




mprime = minus σ

8gmO(6) (426)




2

4mprimemO(6) = σ

radic2

32gm2

O(6) (427)



δ(it) = δf0(t)V0(t) + δf1(t)V1(t) (428)


δ(0) = 2δf1(0) (429)

29






δ(it) sim 1







δ(it) (433)


δγ (it) = t

4gδ(0) + O(t2) = t

2gδf1(0) + O(t2) (434)


0

dt

tJ2nminus1(t)

[δγminus(t)

1 minus eminust2g+

δγ+(t)

et2g minus 1

]= 0

(435)int infin

0

dt

tJ2n(t)

[δγ+(t)


et2g minus 1

]= 0



30




2

π2geminusπg minus

radic2

πeminusπg Re

int infin

0

dt prime


]+ O(t2) (436)



tradic

2

8g+ O(t2) (437)


mcusp(g) = mO(6)(g) (438)


5 Conclusions






31





(m4

O(n)

μ2) (51)


m2O(n) = cμ2 e


20 [1 + O(g2)] (52)




Acknowledgments


32








γ1 = 1

2minus π2

6g2 +

11π4

90g4 minus

(73π6

630+ 4ζ 2

3

)g6 + O(g8)

γ2 = ζ3g3 minus

(π2

3ζ3 + 10ζ5

)g5 +

(8π4

45ζ3 +

10π2

3ζ5 + 105ζ7

)g7 + O(g9)

γ3 = minusπ4

90g4 +

37π6

1890g6 + O(g8) γ4 = ζ5g

5 minus(

π2

3ζ5 + 21ζ7

)g7 + O(g9)

γ5 = minus π6

945g6 + O(g8) γ6 = ζ7g

7 + O(g9)

(A2)



3g4 +

44π4

45g6 minus

(292π6

315+ 32ζ 2

3

)g8 + O(g10) (A3)







33



c(g) = minus1

2+ 2gγ1(g) +

sumn1

[cminus(n g)Uminus

0 (4πng) + c+(n g)U+0 (4πng)

] (A5)


c(g) = minus1

2+ 2gγ1(g) minus

sumk1



fk(g) = 8gsumn1

[U+






3g2 + 2ζ3g

3 minus π4

6g4 minus 23ζ5g

5 +17π6

108g6 +

1107

4ζ7g

7 + O(g8)

f2 = minus1

2+ 2ζ3g

3 minus π4

30g4 + O(g5) f3 = 1

2+ O(g4)

f4 = minus3

8+ O(g5) f5 = 3

8+ O(g6) f6 = minus 5

16+ O(g7)

(A8)






(t) =int infin




(k)=sumn1



(k)=sumn1


bull For minus1 k 1

(k) = minusradic

2

π

(1 + k

1 minus k

)14[

1 +c(g)

1 + k+

1

2

(int minus1

minusinfin+

int infin

1

)dp(p)

p minus k

(p minus 1

p + 1

)14]

(B4)


34



(t) =sumn1

c+(n g)

[eminusit


]

+sumn1

cminus(n g)

[eit



(B5)



int 1

minus1dk eplusmnxk

int infin

1

dpeminusy(pminus1)

p minus k

(1 + k

1 minus k

p minus 1

p + 1

)plusmn14

Vn(x) =radic

2

π

int 1

minus1

dk exk

(k + 1)n

(1 + k

1 minus k

)14

(B6)

Uplusmnn (y) = 1

2

int infin

1

dp eminusy(pminus1)

(p ∓ 1)n

(p + 1

p minus 1

)∓14






0 (y) minus ex



c+(n g)

[4πngV1(t)U

+0 (4πng) + tV0(t)U

+1 (4πng)

4πng minus t

]

+sumn1

cminus(n g)

[4πngV1(t)U


minus1 (4πng)

4πng + t

] (B8)





int infin

0dsV0(x + s)U+






0 (x)


d

dx(xzprime

i (x)) =(

x minus 1

2

)zi(x) (B11)

35




2(y + s)]

= d


2(y + s)]z1(x + s)

minus d


1(x + s)]z2(y + s) (B12)



1 (y + s) + (y + s)V1(x + s)U+0 (y + s)

] ∣∣s=infins=0


0 (y) (B13)






π

t

t + πg (C1)



π

t2

t2 + (πg)2

(C2)


t2 + π2g2


0

dt

tJ2nminus1(t)

[γminus(t)

1 minus eminust2g+

γ+(t)

et2g minus 1

]= h2nminus1(g)

(C3)int infin

0

dt

tJ2n(t)

[γ+(t)


et2g minus 1

]= h2n(g)


h2nminus1 = 2mprime

π

int infin

0

dtJ2nminus1(t)

t2 + (πg)2

[t


1 minus eminust(2g)

]

(C4)

h2n = 2mprime

π

int infin

0

dtJ2n(t)

t2 + (πg)2

[πg

et(2g) minus 1+

t

1 minus eminust(2g)

]



36



γminus(t) = 2sumn1


(C6)γ+(t) = 2

sumn1

(2n)J2n(t)γ2n(g)



π2gt + O(t2) = it

(γ1 +

2mprime

π2g

)+ O(t2) (C7)



0

dt

t


1 minus eminust2g+


et2g minus 1

]= 2

sumn1



γ1 = 2sumn1



γ1 = minusmprime

π

int infin

0dt

[πg


t

t2 + π2g2(minus(t) + +(t))

] (C10)


πg

t2 + π2g2=

int infin


(C11)t

t2 + π2g2=

int infin



γ1 = minusmprime

π

int infin

0du eminusπgu

[ int infin


+int infin


] (C12)


γ1 = minus 2mprime


radic2mprime

πeminusπgRe

[ int infin

0

dt


] (C13)


37






Vn(x) = 1

π232minusn ex

int 1


= 1

π232minusn eminusx

int 1



Vn(x) = 2minusn

(54 minus n

)

(54

)(2 minus n)


= 2minusn

(54 minus n

)

(54

)(2 minus n)







Uab(x) = 1

2

int infin





(1 +

t

2x

)a+bminus12

(D5)


Uab(x) = 2bminus32

(1

2minus a + b


= 1

2

(1

2minus a + b

)U

(1


) (D6)

38




U+0 (x) = 1

2

(5

4


1 (x) = 1

2

(1

4


(D7)

Uminus0 (x) = 1

2

(3

4


1 (x) = 1

2

(3

4





1 (x) = V1(x)U+0 (x) + V0(x)U+

1 (x) = ex

x (D8)


U+0 (x)Uminus


0 (minusx) = π

2radic

2xeplusmn 3iπ

4 (for Imx ≷ 0) (D9)


V0(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]

(D10)

V1(x) = 2radic

2

πe∓ 3iπ

4[exUminus


]






(D11)Uplusmn








U+0 (x) = (2x)minus54

(5

4

)F

(1

4

5

4| minus 1

2x

)= (2x)minus54

(5

4

) [1 minus 5


]


(3

4

)F

(minus1

4

3

4| minus 1

2x

)= (2x)minus34

(3

4

) [1 +

3


]

39


U+1 (x) = (2x)minus14 1

2

(1

4

)F

(1

4

1

4| minus 1

2x

)= (2x)minus14 1

2

(1

4

) [1 minus 1


]


2

(3

4

)F

(3

4

3

4| minus 1

2x

)= (2x)minus34 1

2

(3

4

) [1 minus 9


]

(D12)








2

πeminus 3iπ

4 eπg[U+


] (D13)




(

34

) [F

(1

4

5

4| 1

2πg+ iε

)+ 2F

(minus1

4

3

4| minus 1

2πg

)] (D14)


2 = σ

(34

)

(54




2(

34

) [8πgF

(1

4

1

4| 1

2πg+ iε

)+ 2F

(3

4

3

4| minus 1

2πg

)] (D16)





F

(1

4

5

4| 1

2πg+ iε

)minus F

(1

4

5

4| 1

2πgminus iε

)= i

radic22

σF

(minus1

4

3

4| minus 1

2πg

) (D17)



40





t + 14


(4πgit)

t + 14

=sumk=01


t + 14

+sumk=01


t + 14

Vk(4πgt) (E1)


Rk(4πgs) = Re

[int minusiinfin


t minus s

]= Re

[int minusiinfin


t minus 4πgs

]

(E2)


Re

[ int minusiinfin


t + 14


minussumn1

nc+(n g)

n + 14

[U+


]+

sumn1

ncminus(n g)

n minus 14





R0(s) = minusRe

[eminusiπ4

int 1


int minusiinfin

0dt eminusvtV0(t)

]=

radic2

πRe

[eminusiπ4

int 1


int 1

minus1du


u minus v minus iε

] (E4)


1radic2π

int 1

minus1du



⎧⎨⎩(

v+1vminus1

)14 v2 gt 1

eminusiπ4(

1+v1minusv

)14 v2 lt 1

(E5)


R0(s)sgt0=

radic2

[1

sminus

int minus1


(v + 1

v minus 1

)14]

=radic

2

[1

sminus 2 eminus2sU+

0 (s)

] (E6)


R0(s)slt0=

radic2

[1

s+ 2Uminus

0 (minuss)

] (E7)

41


together with

R1(s) = 2radic

2[θ(minuss)Uminus


] (E8)


Re

[ int minusiinfin


t + 14

]= 2

radic2f0(minusπg)

[Uminus

0 (πg) minus 1

2πg

]+ 2


1 (πg) +

radic2



References



















42





























Ann Phys 167 227

43

























44

1 Introduction

2 Cusp anomalous



23 Toy model



26 General solution







4 Mass scale





5 Conclusions

Acknowledgments








References

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