Quantum Wronskian approach to six-point gluon scattering amplitudes at strong coupling

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

Abstract

We study the six-point gluon scattering amplitudes in ...... = 4 super Yang-Mills theory at strong coupling based on the twisted ^sub 4^-symmetric integrable model. The lattice regularization allows us to derive the associated thermodynamic Bethe ansatz (TBA) equations as well as the functional relations among the Q-/T-/Y-functions. The quantum Wronskian relation for the Q-/T-functions plays an important role in determining a series of the expansion coefficients of the T-/Y-functions around the UV limit, including the dependence on the twist parameter. Studying the CFT limit of the TBA equations, we derive the leading analytic expansion of the remainder function for the general kinematics around the limit where the dual Wilson loops become regular-polygonal. We also compare the rescaled remainder functions at strong coupling with those at two, three and four loops, and find that they are close to each other along the trajectories parameterized by the scale parameter of the integrable model.

Published for SISSA by Springer

Received: June 30, 2014 Accepted: August 4, 2014 Published: August 28, 2014

Yasuyuki Hatsuda,a Katsushi Ito,b Yuji Satohc and Junji Suzukid

aDESY Theory Group, DESY Hamburg,

D-22603 Hamburg, Germany

bDepartment of Physics, Tokyo Institute of Technology, Tokyo 152-8551, Japan

cInstitute of Physics, University of Tsukuba,

Ibaraki 305-8571, Japan

dDepartment of Physics, Shizuoka University,

Shizuoka 422-8529, Japan

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We study the six-point gluon scattering amplitudes in N = 4 super Yang-

Mills theory at strong coupling based on the twisted Z4-symmetric integrable model. The lattice regularization allows us to derive the associated thermodynamic Bethe ansatz (TBA)

equations as well as the functional relations among the Q-/T-/Y-functions. The quantum Wronskian relation for the Q-/T-functions plays an important role in determining a series of the expansion coe cients of the T-/Y-functions around the UV limit, including the dependence on the twist parameter. Studying the CFT limit of the TBA equations, we derive the leading analytic expansion of the remainder function for the general kinematics around the limit where the dual Wilson loops become regular-polygonal. We also compare the rescaled remainder functions at strong coupling with those at two, three and four loops, and nd that they are close to each other along the trajectories parameterized by the scale parameter of the integrable model.

Keywords: Integrable Equations in Physics, Scattering Amplitudes, AdS-CFT Correspondence, Bethe Ansatz

ArXiv ePrint: 1406.5904

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP08(2014)162

Web End =10.1007/JHEP08(2014)162

Quantum Wronskian approach to six-point gluon scattering amplitudes at strong coupling

JHEP08(2014)162

Contents

1 Introduction 1

2 Y-system and TBA for scattering amplitudes at strong coupling 32.1 Hitchin system and Stokes data 32.2 Y-functions, TBA and evaluation of area 5

3 T-functions from the lattice regularization and their scaling limit 7

4 Expansions of T-/Y-functions around CFT limit 114.1 General argument 114.2 Quantum Wronskian and CFT limit 12

5 Application to six-point amplitudes at strong coupling 16

6 Comparison with perturbative results 18

7 Conclusions 21

A Non-linear integral equation 22

1 Introduction

The gluon scattering amplitudes in N = 4 super Yang-Mills theory are a subject of great

interest in recent years. In the planar limit, they are dual to the null-polygonal Wilson loops whose segments are light-like and proportional to external gluon momenta [15]. The duality implies a conformal symmetry in the dual space [1, 68]. This dual conformal symmetry strongly constrains the form of the amplitudes. In particular, the maximal helicity violating (MHV) amplitude is expressed as a sum of the Bern, Dixon and Smirnov (BDS) formula [9] and a nite remainder (remainder function), which is a function of the cross-ratios of the cusp coordinates for the null polygon.

The amplitudes have been studied intensively from both weak- and strong-coupling sides. At weak coupling, recent developments using the mixed motive theory have made it possible to evaluate the remainder function for the six-point amplitudes up to four-loop level [10]. Moreover a method based on the OPE and integrability has been proposed to calculate the scattering amplitudes in this theory, which is expected to be applicable to the intermediate coupling region [1114].

At strong coupling, the AdS/CFT correspondence asserts explicit relations between

N = 4 super Yang-Mills theory and the superstring theory in AdS5[notdef]S5. Alday and Mal

dacena have thereby proposed that the MHV amplitude can be evaluated by the area of the

1

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minimal surfaces in AdS with a null-polygonal boundary along the Wilson loop [1]. It turns out later that the remainder function for the null-polygonal minimal surfaces is calculated with the help of integrability [15]. Namely it is obtained by solving the Y-system or the thermodynamic Bethe ansatz (TBA) system related to certain two-dimensional quantum integrable systems [1618]. The cross-ratios are given by the Y-functions at special values of the spectral parameter and the remainder function is expressed by the free energy of the TBA system and the Y-functions.

For the null-polygonal minimal surfaces in AdS3 and AdS4 space-time, the relevant integrable systems are the homogeneous sine-Gordon models [19] with purely imaginary resonance parameters [18], which are the perturbed SU(N)k/U(1)N1 coset conformal eld theory (CFT) at level k = 2 and 4, respectively. Around the limit where the null boundary becomes regular polygonal, corresponding to the UV limit of the two-dimensional systems, the remainder functions are calculated analytically for lower point amplitudes [2022]. There, the free-energy part is evaluated by the standard bulk conformal perturbation theory (CPT). In order to evaluate the Y-functions, the g-function or the boundary entropy is utilized because the Y-function itself is not well incorporated in quantum eld theory. The boundary CPT then e ciently yields the analytic expansions of the Y-functions. The resultant remainder functions are observed to be close to the two-loop results after an appropriate normalization/rescaling. Numerically, one also nds that this similarity extends beyond the UV limit. The minimal surfaces in these cases, however, give the amplitudes with some specic kinematic congurations of gluon momenta.

The null-polygonal minimal surface in AdS5 with six cusps is the simplest non-trivial example that allows the most general kinematic conguration. At strong coupling, the relevant two-dimensional system is the Z4-symmetric integrable model [2325] with a boundary twist [26]. The remainder function around the UV limit in this case has been studied in detail in [27]. Although the free-energy part is analytically evaluated by the bulk CPT for the twisted Z4-parafermion, the expansion of the Y-functions there is determined by numerical tting. The di erence from the the AdS3 and AdS4 cases come from the fact that the TBA equations in the AdS5 case have a twist parameter, and it is unclear how to construct the g-function with this twist parameter.

Given the analytic results at weak coupling as well as the OPE method for nite coupling, the analytic data at strong coupling would provide pieces of the whole picture of the scattering amplitudes. They would also be useful for a check of the nite-coupling analysis. In this report, we thus decide to devote ourselves to the analytic expansions of the remainder function for the general kinematics.

In order to overcome the problem mentioned above, we take below another route for the UV expansion of the Y-functions, which does not rely on the g-function. Instead, our analysis is based on a seminal work by Bazhanov, Lukyanov and Zamolodchikov [28, 29], where the role of quantum monodromy matrix is claried in the minimal CFT, M2,2n+3, perturbed by the 1,3 operator. Most remarkably, a new object in eld theories, Baxters

Q operator, is introduced in this work. They noted the importance of the fundamental relations among the T- and Q-functions, the quantum Wronskian relation [29]. The T-and Y-systems can be regarded as colloraries of this. The Q operators and the quantum

2

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Wronskian relation have also played important roles in the non-equilibrium current problem [30], in the ODE/IM correspondence [31], in the spectral problem of the AdS5/CFT4 correspondence [32] and so on.

In this report, we provide a yet another application: the quantum Wronskian relation is very e cient in obtaining the analytic expansion of the Y-functions particularly in the CFT limit. Specically, we apply it, for the rst time, to the Z4-symmetric integrable model or its lattice regularization. The lattice regularization adopted here allows one to elucidate the analyticity of the T-/Y-functions numerically. The expansion of the Y-functions around the UV limit is then determined analytically up to and including the terms of order (mass)

43 .

A series of the higher-order coe cients is also determined recursively.

Combined with the free-energy part, the UV expansion of the Y-functions gives the analytic expansion of the six-point reminder function for the general kinematic conguration. We also compare the strong-coupling results with the perturbative ones, and nd that the rescaled remainder functions are close to each other for large ranges of the parameters. This is in accord with the previous observations in the AdS3 and AdS4 cases [2022, 33]

as well as in the perturbative cases [10, 34].

This paper is organized as follows: in section 2, we review the TBA-system for the six-point gluon scattering amplitudes at strong coupling and express the remainder functions using Y-functions. In section 3, we reconsider the Y-/T-functions based on the lattice model. Taking the scaling limit, we derive the TBA system for the amplitudes. In section 4, we study the CFT limit of the TBA system. Based on the T-Q relation and the quantum Wronskian relation, we calculate the analytic expansion of the Y-function in the CFT limit. The detailed analysis of the asymptotics of the related spectral determinant based on nonlinear integral equations is studied in appendix A. In section 5, we apply the analytic expansion of the Y-functions to determine the leading expansion of the remainder function for the six-point amplitudes and compare it with the perturbative calculations.

2 Y-system and TBA for scattering amplitudes at strong coupling

Let us begin with a review on the evaluation of the six-point MHV amplitudes at strong coupling using TBA of the twisted Z4-symmetric integrable model.

2.1 Hitchin system and Stokes data

Alday and Maldacena proposed a method of computing the gluon scattering amplitudes in N = 4 super Yang-Mills using the AdS/CFT correspondence [1, 15]. Consider the the

scalar part of the n-point gluon MHV scattering amplitudes in the strong coupling limit, and factor out the contribution of the tree amplitudes. According to [1], the result can be evaluated by computing the area A of the corresponding classical open string solutions in AdS5 spacetime:1

(amplitude)

(tree) e

1Recently, it was shown that there is another contribution from the S5 part of AdS5[notdef] S5 [35] in addition

to the area. However, this contribution is independent of the cross-ratios and hence does not a ect the discussions below.

3

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p

2 A

where denotes the t Hooft coupling. A string solution represents a minimal surface whose boundary is a polygon located on the boundary of AdS5. The polygon consists of n null edges given by the n momenta of incoming gluons. Note that the amplitudes are dened in space-time with signature (3,1), (2,2) or (1,3).

The equations of motion of the string under the Virasoro constraints are rephrased as the SU(4) Hitchin equations for the connections (Az, Az) and the adjoint scalar elds ( z, z) with the Z4 automorphism [16]. They are equivalently presented by the linear equations for a four component vector q(z, z; ),

Dz + 1 z

q(z, z; ) = 0,

Dz + z

q(z, z; ) = 0, (2.1)

with appropriate boundary conditions. Here Dz and Dz denote the covariant derivatives and stands for the spectral parameter. The explicit forms of Dz and Dz are given in [16]. The information of the null polygon is encoded in the asymptotic behavior of z, which is diagonalized at innity by an appropriate gauge transformation:

h1 zh !

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1p2diag P (z)1/4, iP (z)1/4, P (z)1/4, iP (z)1/4

. (2.2)

The polynomial degree of P (z) is n 4 and its coe cients parameterize the shape of the

polygon. When n > 4, the linear equation necessarily possesses irregular singularity at innity, which implies the Stokes phenomena. Customarily, the whole complex plane is divided into sectors,

Wk : (2k 3)

n +

4n arg < arg z <

4n arg . (2.3)

We denote by sk(z, z; ), the most recessive solution as [notdef]z[notdef] ! 1 in Wk. One can consistently

choose (sk, sk+1, sk+2, sk+3) as a linearly independent basis in Wk. This implies a linear

dependent relation among ve neighboring sjs,

sk + sk+4 = aksk+1 + bk+1sk+2 + ck+3sk+3, (2.4)

where bk+1 and ck+1 are some constants. Note the periodicity bi+3 = bi.

The normalization of sj(z, z; ) is xed such that

hsj, sj+1, sj+2, sj+3[angbracketright] = 1, (2.5)

where [angbracketleft]si, sj, sk, sl[angbracketright] det(sisjsksl). The Z4 automorphism results in the following relations

for the Stokes data,

hsk, sk+1, sj, sj+1[angbracketright]( ) = [angbracketleft]sk1, sk, sj1, sj[angbracketright](i ), (2.6)

hsj, sk, sk+1, sk+2[angbracketright]( ) = [angbracketleft]sj, sj1, sj2, sk[angbracketright](i ). (2.7)

Below, we conne our argument to the n = 6 case where

P (z) = z2 U. (2.8)

4

(2k 1)

n +

For Wj+6 = Wj, we shall impose the boundary condition,

sj+6 = [notdef](1)jsj. (2.9)

The multiplier [notdef] is parametrized as

[notdef] = ei

32 , (2.10)

where is real for solutions in the (1, 3) or in the (3, 1) signature of the four-dimensional space-time whereas it is purely imaginary for the (2, 2) signature. It also appears, e.g., in the relation among bi,

b1b2b3 = b1 + b2 + b3 + [notdef] + [notdef]1. (2.11)

2.2 Y-functions, TBA and evaluation of area

The key ingredients in the following discussion are the Y-functions, which are dened explicitly by2

Y1( ) = [angbracketleft]s2, s3, s5, s6[angbracketright](e ), (2.12)

Y2( ) = [angbracketleft]s1, s2, s3, s5[angbracketright][angbracketleft]s2, s4, s5, s6[angbracketright](e +i/4), (2.13)

Y3( ) = Y1( ). (2.14)

Then it was shown in [16] that the Hirota bilinear identities (or Plcker relations), as well as the relations (2.9), (2.6) and (2.7), lead to the following Y-system,

Y1

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+ i 4

Y1

i4[parenrightbigg]= 1 + Y2( ), (2.15)

Y2

+ i 4

Y2

i4[parenrightbigg]= 1 + [notdef]Y1( )

[parenrightbig][parenleftBigg]1

+ [notdef]1Y1( )

. (2.16)

The asymptotic behavior of the Y-functions is shown to be

log Y1( ) ! [notdef]Z[notdef]e[notdef]( i'), log Y2( ) ! p2[notdef]Z[notdef]e[notdef]( i') for Re ! [notdef]1, '

4 < Im < ' +

4 , (2.17)

where Z is a complex parameter with phase '. This is related to the moduli parameter U in (2.8) as

Z [notdef]Z[notdef]ei' = U

3

4

[integraldisplay]

1

1(1 t2)

14 dt = p (

14 )

U

3

4 . (2.18)

3 (34)

We further introduce

[epsilon1]( ) = log Y1( + i'), ~

[epsilon1]( ) = log Y2( + i'). (2.19)

2The Y-functions here are identied with those in [17, 22] as Y1( ) = [notdef]1[Y AMSV1,1(e )]1, Y2( ) = [Y AMSV2,1(e )]1, Y3( ) = [notdef][Y AMSV3,1(e )]1. The cusp coordinates which appear below are also related as

xa+2 = xAMSVa.

5

The asymptotic behavior (2.17), together with the assumption of the analyticity of [epsilon1], ~

[epsilon1] in

the strip Im 2 (4 , 4 ), leads to the integral equations,[epsilon1] = 2[notdef]Z[notdef] cosh + K2 log 1 + e~[epsilon1]

+ K1 log 1 + [notdef]e[epsilon1]

[parenrightbig][parenleftBigg]1

+ [notdef]1e[epsilon1]

, (2.20)

~[epsilon1] = 2p2[notdef]Z[notdef] cosh + 2K1 log 1 + e~[epsilon1]

+ K2 log 1 + [notdef]e[epsilon1]

[parenrightbig][parenleftBigg]1

+ [notdef]1e[epsilon1]

, (2.21)

where

K1( ) = 12 cosh , K2( ) =

p2 cosh

cosh 2 , (2.22)

and the symbol denotes the convolution, f g =

[integraltext]

1

1 d [prime]f( [prime])g( [prime]). These turn out

be identical to the TBA equations for the Z4-symmetric integrable model [2325] twisted by [notdef]. The equations (2.20) and (2.21) determine [epsilon1] and ~

[epsilon1] in the strip completely.

In the original setting, the geometric data such as cross-ratios are given rst, and then the area of the surfaces should be evaluated. Below, we slightly deform this logic: the TBA equations are given rst, then the cross-ratios and the area are evaluated second. Once the TBA equations are solved, the coe cient bk in eq. (2.4) and the cross-ratios of gluon momenta are given by

bk = Y1

(k 1)i 2

, Uk = 1 + Y2

(2k + 1)i 4

, (2.23)

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for k = 1, 2, 3 (mod 3), where

U1 = b2b3 = x214x236 x213x246

, U2 = b3b1 = x225x214 x224x215

, U3 = b1b2 = x236x225 x235x226

. (2.24)

The cusp coordinates xj are related to the external momenta through pj = xj xj+1. In

the literature, uk := 1/Uk are often used as a basis of independent cross-ratios for the six-point case. The number of the independent cross-ratios matches that of the parameters in the TBA system ([notdef]Z[notdef], ', [notdef]).

Naively, the values of Yj outside the analytic strip are necessary in order to evaluate Uk (1 k 3). Although this can be accomplished by the analytic continuation in

principle, we can avoid this by a clever choice of quantities. For example, suppose ' is negative and small. Two quantities, b1 and U2 = U1, are readily calculated by (2.23).

Then one evaluates b3 by the second equation in (2.24). The nal piece, b2, is obtained from (2.11). Given bi, other cross ratios are now accessible via (2.24). Alternatively, one may also use the Y-system (2.15), (2.16) as recurrence relations to generate Yj(ki/4) for any k 2 Z from a set of Yj(k[prime]i/4) in the analytic strip. As discussed shortly, the Y-

functions have the periodicity Yj( + 3i/2) = Yj( ), and thus the procedure terminates after a few steps.

We are now in position to write down the area A of the 6-cusp solutions or the scalar magnitude of the gluon scattering amplitudes in the strong coupling limit. Instead of A itself, we deal with the nite remainder dened by

R = ABDS A, (2.25)

6

where ABDS is the all-order ansatz for the MHV amplitude proposed by Bern, Dixon and Smirnov [9], including the divergent part. The present formulation then yields

R = ABDS Aperiods Afree, (2.26)

and each part reads

ABDS =

. (2.29)

The minus of the last term F = Afree coincides with the free energy whereas 2[notdef]Z[notdef] is

identied with the mass/scale parameter of the Z4-symmetric integrable model. The overall coupling dependence p has been omitted above.

Within the framework described above, the numerical solutions of (2.20) and (2.21) yield explicit evaluation of the gluon scattering amplitudes [27] . Although the analytic solution to the TBA equations for generic Z and [notdef] is beyond our reach, two limiting cases are accessible [36]. One is the limit where [notdef]Z[notdef] ! 1. In this case, the integrable model

reduces to a free massive theory, and the free-energy part Afree and the Y-functions Yj( ) are expanded by multiple integrals. Via analytic continuation, this limit is also relevant for the amplitudes in the Regge limit [37, 38]. Another limit is [notdef]Z[notdef] ! 0, which we are

interested in here.

When [notdef]Z[notdef] is strictly zero, Afree and Yj are obtained as the central charge of the Z4-

parafermion theory and a solution to the constant Y-system, respectively. Moreover, for small [notdef]Z[notdef] the free-energy part is expanded by the bulk conformal perturbation theory

(CPT). The Y-functions are expanded by the boundary CPT through the relation to the g-function for [notdef] = 1 [22], corresponding to the minimal surfaces in AdS4. For generic [notdef], however, it is still unclear how to incorporate [notdef] in the framework of the boundary CPT.

In the following, we take an approach to the problem, which is di erent from any of the above, and is based on the integrable eld theoretical structure proposed in [29]. This allows us to analytically evaluate the Y-functions for small [notdef]Z[notdef], as shown in section 4.

3 T-functions from the lattice regularization and their scaling limit

In this section we embed the Y-system into another tractable object in integrable systems, the T-system. This enable us to apply the machinery of the latter to evaluate the Y-functions for small [notdef]Z[notdef] in the next section.

There are several ways to introduce the T-system which is equivalent to the Y-system in (2.15) and (2.16). Here we start with the lattice regularization [39]. One advantage of

7

3

Xk=1

Li2 (1 Uk) , (2.27)

Aperiods = [notdef]Z[notdef]2, (2.28)

Afree = 1 2

[integraldisplay]

1 4

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d

1 2[notdef]Z[notdef] cosh log 1 + [notdef]e[epsilon1]( )

[parenrightbig][parenleftBigg]1

+ [notdef]1e[epsilon1]( )

[parenrightbig]

+ 2p2[notdef]Z[notdef] cosh log 1 + e~[epsilon1]( )

[parenrightbig][parenrightbigg]

this choice is that analyticity assumptions, necessary to derive the TBA equations, can be checked numerically.

As is well known, the Z4 parafermion model is related to the spin-12 XXZ model. Its Hamiltonian and spectrum can be studied from the transfer matrix. In order to dene the latter, we introduce R(v), the Uq(

csl2) R matrix of spin 12 representation:

R(v) =

0

B

B

B

@

1

C

C

C

A

a(v)

b(v) c(v)

c1(v) b(v)

a(v)

,

sin(v)sin , c(v) = ev,

where q = ei . Let V (m) be the m + 1 dimensional Uq(sl2) module and V (m)(v) be the corresponding Uq(

csl2) module. By V (m)i(v) we mean its i-th copy. The R matrix acting on V (1)i(vi) V (1)j(vj) is denoted by Ri,j(vi vj).

Then one can construct an inhomogeneous transfer matrix T1(x) acting on 2N sites by,

T1(x) = Tr0DR0,2N(ix + i )R0,2N1(ix i ) [notdef] [notdef] [notdef] R0,2(ix + i )R0,1(ix i ), (3.1) where su x 0 denotes the auxiliary space. The spectral parameter x is set via v = ix for later convenience. We have also introduced the diagonal twist matrix D = [e

a(v) = sin(v + )

sin , b(v) =

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2 i]

under the trace. The quantity [notdef] = ei

32 is to be identied with the multiplier in (2.10).

To diagonalize T1, Baxter [40] ingeniously introduced an operator Q which commutes with T1(x). They satisfy Baxters TQ relation,

T1(x)Q(x) = (x + i 2 )Q(x i ) + (x i

2 )Q(x + i ), (3.2)

where

N .

Below we shall consider T1 and Q on their common eigenspace, thus we do not distinguish operators from their eigenvalues. The eigenvalue of Q is explicitly written with a set of Bethe roots [notdef]xj()[notdef] with twist ,

Q(x) = e

x 2

m

(x) = 4 sinh(x ) sinh(x + )

Yj=12 sinh(x xj()).

This, together with (3.2), parameterizes the eigenvalue of the transfer matrix.

The fusion method generates a series of vertex models such that the auxiliary space is V (j)(v). Let Tj(x) be the corresponding inhomogeneous transfer matrix, with a suitable normalization.3 By construction [notdef]Tj(x)[notdef] constitute a commutative family and they satisfy

the T-system of SU(2) type,

Tj(x + 2 i)Tj(x

2 i) = fj(x) + Tj+1(x)Tj1(x), j 2 N (3.3)

3Note su x j is twice of that in [29].

8

where

T0(x) := (x), fj(x) = T0(x + j + 1

2 i)T0(x

j + 1

2 i).

Note the periodicity of Tj(x) in the present normalization is

Tj(x + i) = Tj(x). (3.4)

There is an additional relation when q is at a root of unity, which is crucial in obtaining a closed set of functional relations. From now on we x

= 2

3 . (3.5)

Then the desired relation is

T3(x) = T1(x) + T0(x) [notdef] + [notdef]1). (3.6)

The equation for j = 2 in (3.3) can be thus rewritten as

T2(x + 3 i)T2(x

3 i) = (T1(x) + [notdef]T0(x))(T1(x) + [notdef]1T0(x)), (3.7) thereby yielding a closed functional relations among T1 and T2.

Below we will show that (3.3) for j = 1 and (3.7) can be transformed into TBA equations. Before doing this, we elucidate the analytic properties of Tj deduced from numerics, as a merit in the lattice regularization. By denition, Tj has 2N zeros and has no poles in complex x plane. Led by numerical observations we conjecture that all zeros of T1(x) are on the Im x = 2 line while those of T2(x) are on the real axis in the ground state. Below, quantities which have no zeros and poles in the strip including the real axis will play an important role. We thus dene

T_j(x) = Tj(x + (j 1)

2 i) j = 1, 2. (3.8)

Then the closed functional relations now read

T_1(x +

6 i)T_1(x

6 i) = f1(x +

2 i) + T0(x +

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2 i)T_2(x),

6 i) = (T_1(x) + [notdef]T0(x))(T_1(x) + [notdef]1T0(x)).

By adopting the change of variables,

Y1(x) = T_1(x)

T0(x) , Y2(x) =

T0(x + 2 i)T_2(x) f1(x + 2 i)

T_2(x +

6 i)T_2(x

,

it is easily checked that Y1 and Y2 satisfy the same Y-system as (2.15) and (2.16) if = 3x2.

Thanks to the knowledge on the zeros of the T-functions, one concludes that Y1(x)

possesses poles of order N at x = [notdef] while Y2(x) possesses poles of order N at x = [notdef] [notdef]6 i.

There are no other poles or zeros of Y1(x), Y2(x) in the strip Im x 2 [6 , 6 ]. This

motivates us to dene the pole-free functions,

eY1(x) = Y1(x)D1(x), eY2(x) = Y2(x)D2(x),

9

where

D1(x) = tanh

N , D2(x) = D1(x + i6 )D1(x i

6 ).

One then derives the functional equations,

eY1(x + 6 i) eY1(x 6 i)

eY2(x)

=

3

4(x ) tanh

3

4(x + )

1 + Y2(x)

1

,

eY2(x + 6 i)

eY2(x 6 i)

eY1(x)

2 = (1 + [notdef]Y11(x))(1 + [notdef]1Y11(x)),

where both sides do not have any zeros or poles in the strip. Thanks to the analyticity, one arrives at

log Y1(x) = log D1(x) + K1 log

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1 + [notdef]1Y1[parenrightBig][parenleftBig]1 + [notdef]Y1

(x) + K2 log

1 + 1 Y2

(x),

log Y2(x) = log D2(x) + K2 log

1 + [notdef]1Y1[parenrightBig][parenleftBig]1 + [notdef]Y1

(x) + 2K1 log

1 + 1 Y2

(x),

where

K1(x) = 3

, K2(x) = 3 cosh 32x

p2 cosh 3x.

4 cosh 32x

Now consider the following scaling limit,

lim

N!1

4Ne

32 = 2[notdef]Z[notdef] = [lscript]. (3.9)

In this limit, the driving terms become

lim

N!1

log D1(x) = [lscript] cosh

3x 2

, limN!1log D2(x) = p2[lscript] cosh [parenleftbigg] 3x 2

. (3.10)

We denote the Y-functions in the scaling limit by Yscj(x). If changing the variables as = 32x, we recover (2.20) and (2.21) by the identication,

log Ysc1(x) = [epsilon1]( ), log Ysc2(x) = ~

[epsilon1]( ).

We also dene the T-functions in the scaling limit by

Tscj(x) = lim

N!1

e2 NTj(x), (3.11)

where especially Tsc0(x) = 1. In the scaling limit, the relations between the Y-functions and the T-functions are drastically simplied,

Ysc1(x) = Tsc1(x), Ysc2(x) = Tsc2(x + i

2 ). (3.12)

For later use, we shall also discuss the scaling limit of Q. The Bethe ansatz roots are roughly classied into two clusters, x[lscript]j and xrj . We thus adopt parameterizations,

~xrj() = xrj() , ~x[lscript]j() = x[lscript]j() + ,

10

and

= ex = e

2

3 , rj() = e~x

rj()(2N)

2

3 , [lscript]j() = e~x

[lscript]j ()(2N)

2

3 .

Then the scaling limit of Qsc reads

Qsc( ) = limN!1,[lscript]=xed

eN Q(x)

2

3

rj()

2

= C()

3

4

Yj

1

( [lscript]2)

[parenrightBig] [productdisplay]

2 , (3.13)

where we assumed the numbers of roots in the left and the right clusters are identically equal to N2 . The prefactor stands for

C() =

Yje~x

rj()~x[lscript]j()+i. (3.14)

1

( [lscript]2)

23 1 [lscript]j()

j

This Q-function plays an important role for studying analytical properties of the Y-functions.

4 Expansions of T-/Y-functions around CFT limit

In this section, we consider the expansions of the T- and Y-functions around [lscript] = 2[notdef]Z[notdef] = 0.

As mentioned in the previous section, such expansions for the minimal surface in AdS3 or

AdS4 were studied in detail in [2022] through the bulk and boundary conformal perturbation theory. Also, in [27], the expansions of the Y-functions for the six-point case in AdS5 were considered, but there remained an unknown function Y (1,0)j() at order [lscript]4/3. Our goal here is to determine the analytic form of this unknown function. The key idea is to use the quantum Wronskian, which is naturally derived from the discretized lattice regularization reviewed in the previous section. The quantum Wronskian determines not only Y (1,0)j()

but also the expansion of Yj in the CFT limit dened below. For the time being, we set Z to be real, i.e., ' = 0.

4.1 General argument

Let us start with a general argument on the expansions of the T- and Y-functions. Solutions to the Y-system have a periodicity as conjectured rst in [41], and it played an important role in the analysis of the perturbed CFT (see, e.g. [36]). Here we have an apparent periodicity4 inherited from the underlying lattice model (see (3.4)), and it motivates the expansions of Yj( ) and Tj( ),

Yj( ) =

1

JHEP08(2014)162

Xp=02T (p)j[lscript]43 p cosh 4p3 , (4.1)

4Note that the periodicity can be proved even without information on its lattice origin. The cluster algebraic structure is shown to be essential [4245]. The argument here is meant to explain the periodicity in a simple manner.

Xp=02Y (p)j[lscript]43 p cosh 4p3 , Tj( ) =

1

11

where Yj( ) (Tj( )) is Yscj(x) (Tscj(x)) as a function of . The conformal perturbative argument suggests that the coe cients are also expanded around [lscript] = 0,5

Y (p)j =

Xq

Y (p,q)j[lscript]43 q T (p)j = Xq

T (p,q)j[lscript]43 q. (4.2)

The central issue here is to determine these coe cients.

4.2 Quantum Wronskian and CFT limit

We remind that Baxters TQ-relation (3.2) is a second order di erence equation, and there are two linearly independent solutions. Let [notdef]xj()[notdef] be a set of BAE roots for negative

twist , then the second solution reads

Q(x) = e

x

Qj 2 sinh(x xj()). Its scaling

limit is obtained from (3.13) by ! .

There exist remarkable relations (the quantum Wronskian relations) between such two independent solutions of Baxters TQ relation and the fusion transfer matrices [29],

2i sin

2 Tj(x) =Q(x i

j + 1

3 )

Q(x + ij + 1

3 )

j + 1

3 )

j + 1

Q(x + i

Q(x i

3 ), j = 0, 1, 2, [notdef] [notdef] [notdef] . (4.3)

The same relation naturally arises in the context of the massive generalization of the ODE/IM correspondence [46]. A little bit di erent view from the lattice model is remarked in [47].

Below we will show that coe cients Y (p,0)j (j = 1, 2) are determined by applying the above relations. We start with the scaling limit of (4.3),

2i sin

2 Tscj( ) = Qsc( q

j+1

2 ) Qsc( q

JHEP08(2014)162

j+1

2 ) Qsc( q

j+1

2 ) Qsc( q

j+1

2 ). (4.4)

Note q = e

2

3 i. We further take the CFT limit [lscript] ! 0 with the shift !

[lscript] 2

23 . Let us

dene scaled functions,

TCFTj( 2) = lim

[lscript]!0

Tscj( [lscript]

2

23 ), (4.5)

A( ) = lim

[lscript]!0

rj()

2 ,( ) = lim [lscript]!0

Yj

1

rj()[parenrightbig]2[parenrightBig]

. (4.6)

Yj

1

Then the quantum Wronskian relation takes the form,

2i sin

2 TCFTj( 2) = ei

j+1

2 ) ei

j+1

j+1

2 ), (4.7)

5When [notdef] = 1, the Y-functions for the n-point amplitudes, Ya,s( ) (a = 1, 2; s = 1, . . . , n 5),

have the quasi-periodicity Ya,s( + ni/4) = Ya,n4s( ). They are also expanded [22] as Ya,s( ) =

Pp,q y(p,2q)a,s[lscript](p+2q)(1 ) cosh(4p /n) with = (n4)/n. For n = 6, the above quasi-periodicity is promoted to the periodicity Ya( + ni/4) = Ya( ), where Ya := Ya,1, and thus only p even is allowed. This gives the expansion of the form as in (4.1) and (4.2).

2 (j+1)A( q

j+1

2 )( q

2 (j+1)A( q

2 )( q

12

where we used C()C() = 1, as [notdef]xj()[notdef] = [notdef]xj()[notdef] resulting form the Bethe ansatz

equations. The same limit of the Y-functions is denoted by Y CFTj( 2) = lim[lscript]!0 Yj( log [lscript]2), which has an obvious expansion from (4.1) and (4.2),

Y CFTj( 2) = 2Y (0,0)j +

1

Xp=1Y (p,0)j24p3 2p. (4.8)

In the literature, the limit, [lscript] ! 0 without rescaling in the spectral parameter is also

referred to as the CFT limit. In this paper, we call it the UV limit in order to avoid confusion. In the CFT limit, the mass terms in the TBA equations become chiral, and we are left with the massless TBA system.

Our strategy here is to use the quantum Wronskian relation to evaluate Y CFTj( 2) and

then read o the coe cients Y (p,0)j. Let the n-th (inverse) moment of BAE roots be an andn,

an = lim

[lscript]!0

1 n

Xj

[parenleftBig]

1 ( rj())2

n

,n = lim [lscript]!01 n

[parenleftBig]

1 ( rj())2 [parenrightBig]n

. (4.9)

JHEP08(2014)162

Xj

Then the following expansion is valid for 0 2( ),

ln A( ) =

1

Xn=1an 2n, ln( ) =

1

Xn=1n 2n. (4.10)

In the CFT limit, the relations between the Y-functions and the T-functions are simple,

Y CFT1( 2) = TCFT1( 2), Y CFT2( 2) = TCFT2( 2). (4.11)

Using the quantum Wronskian (4.7), one can relate the coe cients Y (p,0)j to an. To do so, let us introduce polynomials pn(y) in [notdef]y1, y2, [notdef] [notdef] [notdef] [notdef] by

exp(

1

X =0p (y)x . (4.12)

Here p0(y) = 1, p1(y) = y1, p2(y) = 12y21 + y2 etc. Then from (4.7), one obtains

Y (0,0)j =

sin

Xn=1ynxn) =

1

j+12 [parenrightBig]

2 sin 2

, (4.13)

Y (n,0)j = (1)n(j1)2

43 n

n

X =0p (a)pn ()sin j+12

4

3 (n 2 ) +

sin 2

, (n 1).

13

Some examples are listed as follows,

Y (0,0)1 = cos

2

,

1

2

43 Y (1,0)1 =

a1 cos

+ 6[parenrightBig]

1 cos

6

[parenrightBig][parenrightBig]

, (4.14)

sin

2[parenrightBig]

2

83 Y (2,0)1 = 1 2 sin

2[parenrightBig]

2a11 sin() +21 22

[parenrightbig]

cos

+ 6[parenrightBig] [parenleftBigg][parenleftBigg]a21 2a2

[parenrightbig]

cos

6

[parenrightBig][parenrightBig]

,

24Y (3,0)1 = 1 6 sin

2[parenrightBig]

[parenleftBig][parenleftBigg]

31 + 612 a31 + 6a2a1 6(a3 +3) [parenrightbig]

sin()

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3 a21 2a2

[parenrightbig]

1 cos

+ 6[parenrightBig]+ 3a121 22

[parenrightbig]

cos

6

[parenrightBig][parenrightBig] ,

and

Y (0,0)2 = cos() + 12, 2

43 Y (1,0)2 = (1 + a1) (2 cos() + 1), (4.15)

2

83 Y (2,0)2 = 12 2a11 +21 22 + a21 2a2

[parenrightbig]

(2 cos() + 1),

24Y (3,0)2 = 16 (1 + a1) (1 + a1) 2 6 (2 + a2)

[parenrightbig]

+ 6 (3 + a3)

[parenrightbig]

(2 cos() + 1).

Thus once the moments an andn are known, Y (p,0)j are easily evaluated. These moments satisfy the discrete Wiener-Hopf equations [31]. They are obtained from j = 0 case in (4.7) by expanding the both sides order by order in 2. Explicitly they are of the form,

n

X =0pn(a)pn () sin1 2

43 (n 2 ) +

[parenrightbigg]= 0. (4.16)

Since pn(a) is of the form an + [notdef] [notdef] [notdef] with the ellipses being a polynomial in a1, [notdef] [notdef] [notdef] , an1,

the above relation reduces to

sin(n

2 )an sin(n +

2 )n = rn(), (4.17)

where is dened in (3.5). The explicit forms of the rst few rn() read

r1() = 0,

r2() = sin 2

2

a21 +21 + 2 cos 2 a11

,

r3() = 1631 + a31

[parenrightbig]

sin

2[parenrightbigg] (12 + a1a2) sin

2[parenrightbigg]

+ a21 cos

16( 3)

[parenrightbigg]

a12 cos

16(3 + )

.

In general rn is a polynomial of aj andj for 1 j n 1.

14

Remarkably, they are su cient to determine an() [30, 31]. To be precise, we need two assumptions to accomplish this. First, we presume that an (n) is analytic for > 23

( < 23).This is consistent with the observation from BAE: as ! 23 one of the roots r ! 0

thus an diverges. This implies the existence of overlapping of the analytic strips near the origin of for both an andn. Second, the following asymptotic behavior of an as ! 1

is postulated,

an n

2

1 4n3 , (4.18)

where the coe cient reads explicitly,

n = (n3) (2n3 12)

4

1

2 n!

4n3 . (4.19)

This is deduced from the analysis on the Bethe ansatz equations in the large limit, as shortly discussed in appendix A. The rst member, 1 can also be xed by the expansion of the Y-functions for the AdS4 case corresponding to = 0 [22].

Once these assumptions are taken for granted, we can successively determine an. The

rst order equation is simply solvable and one nds

a1() = (13 +

2 )

JHEP08(2014)162

12 (14) (34)

1. (4.20)

For Re > 0, the second moment is given explicitly,

a2() = 3 21 83

(23 +

(23 +

2 )

13

ix2

[parenrightbigg][parenrightBigg]3. (4.21)

For Re < 0, a2() is evaluated by the analytic continuation of the above expression. A double integral formula for the third moment a3 is derived similarly. For the evaluation up to Y (3,0)j, however, the explicit forms of a3 and3 are dispensable. This is due to the fact that only the sum, a3 +3, appears in Y (3,0)j and this sum also appears in the condition (4.17) for n = 3, in the special case = 23 ,

a3 +3 =

r3()

sin 2

2 )

[integraldisplay]

dx 2

sinh x2x + i

13 +ix 2

(13 +

2 )

1

1

. (4.22)

In this way, one can determine Y (p,0)j successively. In particular, the rst non-trivial coe cients are especially simple, to be given explicitly by

2

43 Y (1,0)1 = 2

p3 1

(23 +

2 ) (23

2 )

p3 12

2

43 Y (1,0)2 = 4

,

(23 +

2 ) (23

2 ) (12 +

2 ) (12

2 )

15

with

As a check, one can compare the above result with the relations of Y (p,q)j which follow from the Y-system (2.15), (2.16). For example, the Y-system requires Y (1,0)2/Y (1,0)1 =

2 cos(/2), which indeed agrees with (4.14), (4.15) and (4.20). At the next order, the Y-system gives a linear relation among Y (2,0)j and (Y (1,0)j)2. This is also conrmed from the above result. Moreover, one nds that Y (0,1)j = 0 (j = 1, 2). This means that, up to and including O([lscript]4/3), the Y-functions in the original TBA are xed only by the information

from the CFT limit.

By recovering the phase ' by Yj( ) ! Yj( i'), we nally obtain the expansion of

the massive Y-functions,

Yj( ) = 2Y (0,0)j + 2Y (1,0)j[lscript]

P5k=0 y(k)2l4k/3 for O(105) < [lscript] < O(101) can also reproduce Y (0,0)2 and Y (1,0)2 with 12- and 8-digit accuracy, respectively. At O(l8/3), the

coe cient Y (2,0)2 explains about 44 per cent of y(2)2, whereas about 51 per cent are from Y (0,2)2, which is determined by and proportional to (Y (1,0)2)2. The rest is carried by the undetermined Y (1,1)2. At O(l12/3), Y (3,0)2 explains about 16 per cent of y(3)2. Figure 1 (b)

shows a plot of the expansions of Y2(0) obtained from the analytic data in the CFT limit,i.e., Y (p,0)2, up to p = 1, 2, 3, respectively. ' and [notdef] are the same as in (a). Up to [lscript] 1,

the CFT data approximate Y2 relatively well. Combining them with the Y-system yields better approximation.

5 Application to six-point amplitudes at strong coupling

Now, we apply the expansion of the Y-functions for small [lscript] to the six-point amplitudes or the null-polygonal Wilson loops dual to the amplitudes. Our evaluation based on the quantum Wronskian relation allows us to analyze the amplitudes/Wilson loops corresponding to the minimal surfaces in AdS5 or [notdef] [negationslash]= 1. In the small-[lscript] or the UV limit, the Wilson loops

16

1 = 1 4

7 6

1 6

13[parenrightbigg][parenleftBigg] (14) (34)

! 43.

4( i')3[parenrightbigg]+ O([lscript]83 ) (j = 1, 2). (4.23)

For [notdef] = 1, corresponding to the minimal surfaces in AdS4, this reduces to the expansion in [22]. It is also in agreement with the expansion in [27] with [notdef] [negationslash]= 1 determined numerically.

Once the expansion of the Y-functions is found, one can immediately know the expansion of the T-functions through the relation (3.12). The quantum Wronskian relation thus provides a systematic and simple way to determine the coe cients Y (p,0)j and T (p,0)j .

Figure 1 (a) shows a plot of Y2(0) as [lscript] varies. As an example, the phase is xed to be ' = /20 and the chemical potential to be [notdef] = 10. We have chosen a real [notdef]

(imaginary ) so that we can compare our data against the three-loop result in term of the multiple polylogarithms [34] , which is discussed in the next section. Our expansions are valid also for real [notdef]. We nd a good agreement between the numerical results and our analytic expansion. A t by a function

JHEP08(2014)162

43 cosh

JHEP08(2014)162

(a) (b)

Figure 1. Plot of Y2(0) for ' = /20 and [notdef] = 10 as [lscript] varies. In both (a) and (b), the points

represent numerical results. In (a), the solid line represents the leading expansion (4.23) evaluated at = 0. (b) shows the expansions from the data in the CFT limit. The broken (), solid ()

and dotted ([notdef] [notdef] [notdef] ) lines represent

Pkp=0 2Y (p,0)2 [notdef] [lscript]4p3 cos 4p3' with k = 1, 2, 3, respectively. The case

with k = 1 is equivalent to (4.23).

become regular polygonal. The expansion thus gives the amplitudes/Wilson loops under small deformations around the regular polygonal contour.

To evaluate the amplitudes, we rst recall that the remainder function in (2.26) consists of three terms. One of the terms denoted by Afree is nothing but the free energy of the Z4-symmetric integrable model twisted by . Since this Z4-symmetric integrable model reduces, in the UV limit, to the twisted Z4-parafermion CFT, the free energy Afree is expanded by the bulk conformal perturbation theory [27]:

Afree = 6

1 9222[parenrightBig] [notdef]Z[notdef]2

+ 21/3 24

1 3

13 + 2

13

2

[parenrightBig] [notdef]

Z[notdef]83 + O([notdef]Z[notdef]163 ) , (5.1)

where

4 = 1

2

1

2

16[parenrightBig][bracketleftBig]p

34[parenrightBig][bracketrightBig]43, (5.2)

and (z) = (z)/ (1 z). The second term cancels Aperiod.

The expansion of the remaining term ABDS is derived by utilizing the results in (2.23), (2.24) and (2.26),

ABDS =

3

4 Li2 1 U0

[parenrightbig]

+ 3 [notdef] 2

2

3 (U0 1 + log U0)

U0(U0 1)2

(Y (1,0)2)2[notdef]Z[notdef]

83 + O([notdef]Z[notdef]

16

3 ) . (5.3)

, (5.4)

common to all k = 1, 2, 3. We note that ABDS is expanded in powers of Z4/3 = [notdef]Z[notdef]4/3ei

43 '

Here, U0 is the value of Uk in the UV limit,

U0 = 1 + 2Y (0,0)2 = 4 cos2 /2

[parenrightbig]

and its complex conjugate, but the '-dependence remains only at O([notdef]Z[notdef]8n/3) with n 2 3Z.

17

This is a consequence of the Z6-symmetry of the six-point amplitudes ! + i/4 or

' ! ' + /4, which corresponds to cyclically renaming the cusp points xa ! xa+1. This symmetry also explains the absence of the O([notdef]Z[notdef]4/3) term.

Combining the above results, we obtain the UV expansion of the remainder function,

R =

1

Xk=0r(k)(', ) [lscript]83 k , (5.5)

where

r(0) =

6 +

3

4 2

3

4 Li2 1 U0

[parenrightbig]

,

JHEP08(2014)162

r(1) = 3 24

[bracketleftBig][parenleftBig]

13 +2 ,13

2

[parenrightBig]

, (5.6)

and B(x, y) is the beta function, (x) (y)/ (x+y). The expansion of the Y-functions (4.23) also yields the cross-ratios for small [lscript],

Uk = U0 + 2Y (1,0)2[lscript]

43 cos

1

8p3

9

(1 U0) log U0[bracketrightBig] [notdef]B2

32(2)

2

3

2k + 14 '

[parenrightBig][bracketrightbigg]+ O([lscript]83 ) . (5.7)

Inverting this relation, one can express the parameters in the TBA equations by the cross-ratios [27],

cos2 2 =

112(U1 + U2 + U3),

tan 43' =

4 3

p3(U2 U3) 2U1 U2 U3

, (5.8)

[lscript]

43 = 2U1 + U2 + U3

6Y (1,0)2 cos 43'

,

which is valid up to O([lscript]

83 ).

Our formulas can be checked numerically. Figure 2 (a) shows the trajectories of uk =

1/Uk as [lscript] varies with ' and [notdef] = e

32 i xed to be /20 and 10, respectively. They are

obtained by solving the TBA equations and evaluating Afree numerically. Figure 2 (b) is a comparison of the same trajectories for small [lscript] and the expansion of uk obtained from (5.7). Figure 3 (a) shows the remainder function at strong coupling R for the cross-ratios uk given in gure 2. The points represent the numerical results, whereas the solid line represents our expansion R = r(0) + r(1)[lscript]8/3. Figure 3 (b) is the same plot for small [lscript]. From gure 2 and 3, we nd again a good agreement between the numerical results and our analytic expansions for small [lscript].

6 Comparison with perturbative results

In [2022, 33], the remainder functions of the strong-coupling amplitudes corresponding to the minimal surfaces in AdS3 and AdS4 were compared with the two-loop results. It was found there that after an appropriate normalization/rescaling they are close to each

18

(a) (b)

Figure 2. (a) Trajectories of uk = 1/Uk as [lscript] varies with ' = /20 and [notdef] = 10. Points denoted

by , +, [notdef] represent u1, u2, u3, respectively. (b) The same trajectories for small [lscript] (points) and

the expansion from (5.7). The broken (), solid () and dotted ([notdef] [notdef] [notdef] ) lines represent u1, u2, u3, respectively.

(a) (b)

Figure 3. (a) Six-point reminder function at strong coupling as [lscript] varies with ' = /20 and [notdef] =

10. The points represent the numerical results, whereas the solid line represents our expansion (5.5) with (5.6). (b) The same plot for small [lscript].

other. In this section, we compare the six-point remainder function at strong with those at two [48, 49], three [34] and four [10] loops.

For this purpose, we normalize/rescale the remainder function [33] so that it vanishes at [lscript] = 0 and approaches 1 for large [lscript]:

Rstrong = Rstrong RstrongUV

RstrongUV RstrongIR

, (6.1)

where we have introduced the notation Rstrong6 := R. RstrongUV (RstrongIR) is the value at [lscript] = 0 ([lscript] = 1) along the trajectory in the space of the cross-ratios parametrized by [lscript] with

', [notdef] xed. For the remainder function at L loops appearing in the perturbative expansion R6 =

P LR(L), one can also dened the rescaled reminder functions R(L) similarly.

At strong coupling, the UV value is read o from the expansion in (5.5), RstrongUV =

r(0). To nd the IR value RstrongIR, we use the large-[lscript] values of the cross-ratios Uk ! (1, ep2[lscript] cos( 4 +'), ep2[lscript] cos( 4 ')), which are found from the Y-system (2.15), (2.16) and the

19

JHEP08(2014)162

[lscript] 1/5 1 3 5 34/5 9 10 R(2) 3.202 [notdef] 104 -0.02351 -0.3618 -0.7814 -0.9388 -0.9890 -0.9951

R(3) 2.626 [notdef] 104 -0.01953 -0.3194 -0.7411 -0.9204

R(4) 2.214 [notdef] 104 -0.01643 -0.2827 -0.7014 -0.9004

Rstrong 4.145 [notdef] 104 -0.03007 -0.4226 -0.8304 -0.9584 -0.9935 -0.9973

Table 1. Samples of the numerical values of the rescaled remainder functions plotted in gure 4.

asymptotic behavior Y2( ) ! 2p2(

Ze + Ze ) for large [lscript] with [notdef] Im '[notdef] < /2. Since

Afree ! 0 and Aperiod cancels the leading term from ABDS we are left with RstrongIR =

2/12. On the perturbative side, R(L)UV are obtained from the cross-ratios U0 in the UV limit, whereas the IR values just vanish R(L)IR = 0.

We evaluate these rescaled remainder functions for the cross-ratios uk(= 1/Uk) given in gure 2, which are parametrized by [lscript] with ' = /20, [notdef] = 10. At strong coupling, it is

readily read o from the results in the previous section. At two loops, we use the simple analytic expression given in [49], whereas at three loops we use the expression given in [34] and evaluate it by using the C++ library GiNaC [50, 51]. The direct three-loop expression in terms of the multiple polylogarithms is given for the parameter region with real [notdef], which corresponds to the (2,2) signature of the four-dimensional space-time. In order to use this expression, we have chosen a real [notdef]. At four loops, it can be evaluated from the analytic expression in [10].6

Figure 4 (a) is a plot of the rescaled remainder functions at two, three and four loops and at strong coupling. Figure 4 (b) is a plot of the ratios of these rescaled remainder functions; R(3)/ R(2), R(4)/ R(3) and R(2)/ Rstrong. As [lscript] increases, it becomes harder to

evaluate R(3) and gure 4 includes the data for [lscript] 7. It also includes the four-loop

data for [lscript] 34/5. Some of the numerical values of the rescaled remainder functions

plotted in gure 4 are listed in table 1. From these gures and the table, we nd that the rescaled remainder functions stay close to each other as [lscript] varies, but the perturbative results gradually move away from those at strong coupling as the number of loops increases. Their ratios changes slowly along [lscript], and accumulate to 1 for large [lscript] as assured by denition. The ratios of the perturbative results, R(3)/ R(2), R(4)/ R(3), are very similar. These are in

accord with the observations in [2022, 33] mentioned above and those in [10, 34] that the ratios of the remainder functions at two, three and four loops are relatively constant for large ranges of the cross-ratios.

As [notdef] increases, we have observed that R(2)6, R(3)6 and Rstrong6 change in a similar manner: they tend to start decreasing for larger [lscript]. Although their di erences increase, they are still kept relatively close to each other. The dependence on [notdef] seems very weak. For example, even for [notdef] = 106, the ratios are still of O(1). At the order of the expansion in (5.5)

and (5.6), the '-dependence does not appear. Although it does appear at higher orders, the behavior of the rescaled remainder functions is still qualitatively similar.

6The four-loop data used here were provided to us by Lance Dixon, James Drummond, Claude Duhr and Je rey Pennington. We would like to thank them for providing these data.

20

JHEP08(2014)162

JHEP08(2014)162

(a) (b)

Figure 4. (a) Rescaled remainder functions at two loops (+), three loops ([notdef]), four loops ([diamondmath]) and

strong coupling ( ) for the cross-ratios given in gure 2 with ' = /20 and [notdef] = 10. (b) Ratios

of the rescaled remainder functions. Points denoted by +, [notdef], represent

R(3)/ R(2), R(4)/ R(3) and

7 Conclusions

In this paper we studied six-point gluon scattering amplitudes in N = 4 super Yang-Mills

theory at strong coupling by using the AdS/CFT correspondence. The area of the corresponding null-polygonal minimal surface in AdS5 is evaluated by solving the T-/Y-system for the Z4-symmetric integrable model with a twist parameter. The leading expansion to the remainder function is determined around the UV limit explicitly. We compared this result with the recent perturbative calculations, to nd that the rescaled remainder functions are close to each other along the trajectories parametrized by the scale parameter.

Our results at the leading order relied on the quantum Wronskian relation, which determines the expansion of the T-/Y-functions around the CFT (massless) limit. For higher order terms, one needs to study the massive TBA system intrinsically. As in [2022], the boundary CPT would be a way toward this direction based on the relation between the T-/Y-functions and the g-function [28, 52, 53]. Through the T-/Y-functions, one could also study the quantum Wronskian relation from a di erent perspective by using the boundary CPT. However, it is yet to be gured out how to incorporate the twist of the perturbed

Z4-parafermion theory in this framework.

Alternatively, the quantum Wronskian relation exists even in the massive case [46]. The more involved analyticity, however, dees the analytical determination of the moments of A,. Along [28, 29], the massive TBA systems have also been analyzed in [54, 55]. Hopefully one can detour the di culties, and obtain the systematic higher expansions.

Finally, given the recent developments on the nite-coupling amplitudes around the collinear limit [1214], it would be worthwhile to explore the extrapolation to the nite coupling also around the regular-polygonal limit. We hope to come back to these issues in near future.

21

R(2)/ Rstrong, respectively.

Acknowledgments

We would like to thank Lance Dixon, James Drummond, Claude Duhr and Je rey Pennington for providing to us the four-loop data of the remainder function. We would also like to thank Lance Dixon and Takahiro Ueda for useful correspondences. The work ofK. I., Y. S. and J. S. is supported in part by JSPS Grant-in-Aid for Scientic Research (C) No. 23540290, 24540248 and 24540399. The work of K. I. and Y. S. is also supported in part by JSPS Japan-Hungary Research Cooperative Program.

A Non-linear integral equation

We supplement a treatment on the Bethe ansatz equations based on suitably chosen auxiliary functions. The approach utilizes Non Linear Integral Equations satised by them [26, 56], and thus is referred to as the NLIE approach.7 Here we choose auxiliary functions as

a(x) = (x i 2 )Q(x + i ) (x + i 2 )Q(x i )

, A(x) = 1 + a(x),

(x) = (x + i 2 )Q(x i ) (x i 2 )Q(x + i )

JHEP08(2014)162

,(x) = 1 +(x).

The Bethe ansatz equations are equivalent to

a(xj) = 1.

Numerically, one observes the following properties of the nite size system in the ground state:

1. [notdef]a(x)[notdef] < 1 ([notdef](x)[notdef] < 1) in the upper (lower) half plane.2. In a narrow strip including the real axis, the zeros of A(x) ((x)) are located only on the real axis and they coincide with the Bethe ansatz roots [notdef]xj[notdef].

3. The extended branch cut function, 1i ln a(x), is an increasing function of real x.

These properties are enough in deriving the NLIE. After the scaling and the conformal limit, for Im positive small, it is explicitly given by

ln a( ) = da( ) +

[integraldisplay]

1+i0

1+i0

F ( [prime]) ln A( [prime])

d [prime]

2

[integraldisplay]

1i0

1i0

F ( [prime]) ln

( [prime])d [prime]

2 ,

da( ) = ie i

[integraldisplay]

3

2 , F ( ) =

eik

1 2 cosh 2 k

dk, (A.1)

where we use the same symbols a etc for functions of (= 3x2) by abuse of notation.

7It is also referred to as the DDV approach in the context of integrable eld theories.

22

We are interested in 1 behavior of ln A( ). The driving term da in the above

suggests that there exists B of order ln such that the smallest root 1 is greater than B. We assume that [notdef] ln A( )[notdef] 1 and [notdef] ln

( )[notdef] 1 if > 1. Introduce

g( ) = ln A( + B + i[epsilon1])

( + B i[epsilon1])

.

Then for 0, the NLIE is approximated by the following Wiener-Hopf type equation [29],

g( ) = da( + B) +

[integraldisplay]

d [prime]

0 F ( [prime])g( [prime])

2 . (A.2)

It is convenient to deal with the equation in the Fourier space,

bg(!) = [integraldisplay]

1 g( )ei! d

2 , g( ) =

[integraldisplay]

lim

JHEP08(2014)162

1

1

bg(!)ei! d!.

The standard recipe is to introduce the factorized Kernel,

G+(!)G(!) = 1

bF (!)

1, G(!) = G+(!)

such that

!!1

G[notdef](!) ! 1.

Explicitly we choose

r23 (1 3!4 i) (12 !2 i) (1 !4 i) ei !,

= 12

ln

G+(!) =

1 2

3 2

32 ln

1 2

.

Then the solution in the Fourier space is given as follows,

bg(!) = G+(!)Q+(!),

Q+(!) = 1 2

32G(i[epsilon1])

! + i[epsilon1]

G(i)

! + i eB

.

The comparison of the asymptotic behavior ! ! 1 of the Fourier transformation of (A.2)

concludes Q+(1) = 0. This determines the relation between parameters B and ,

eB =

2p

(14)

(34)

e . (A.3)

This makes the expression for

bg(!) simpler,

bg(!) = i (1 3!4 i)2p(! + i[epsilon1])(! + i) (12 !2 i) (1 !4 i)ei !. (A.4)

We introduce A+( ) := ln A( +B) for < 0. A simple manipulation leads to the following expression of its Fourier transformation

bA+(!) in terms of

bg(!),

bA+(!) =

bg(!)4 sinh 4 ! cosh 2 !e34 !. (A.5)

23

By substituting (A.4) into (A.5), one nds that ! = 43(n + 1)i, n 2 Z 0 are the only

relevant poles in the inverse Fourier transformation of

bA+(!). We thus nds for 1,

ln A( ) =

Xn 1e2

n( + B) (n(1 )) (12 + n) (2 n )n!

.

By comparing the above expansion with (4.10), making use of the explicit form of B in (A.3), we have the following asymptotic behavior of an(),

an()

The explicit form of the coe cient n in (4.18) is then determined as in (4.19).

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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