Hexagon Wilson loop OPE and harmonic polylogarithms

Abstract

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

A recent, integrability-based conjecture in the framework of the Wilson loop OPE for ... SYM theory, predicts the leading OPE contribution for the hexagon MHV remainder function and NMHV ratio function to all loops, in integral form. We prove that these integrals evaluate to a particular basis of harmonic polylogarithms, at any order in the weak coupling expansion. The proof constitutes an algorithm for the direct computation of the integrals, which we employ in order to obtain the full (N)MHV OPE contribution in question up to 6 loops, and certain parts of it up to 12 loops. We attach computer-readable files with our results, as well as an algorithm implementation which may be readily used to generate higher-loop corrections. The feasibility of obtaining the explicit kinematical dependence of the first term in the OPE in principle at arbitrary loop order, offers promise for the suitability of this approach as a non-perturbative description of Wilson loops/scattering amplitudes.

Published for SISSA by Springer

Received: October 30, 2013

Accepted: November 1, 2013 Published: November 19, 2013

Georgios Papathanasiou

LAPTh, CNRS, Universit de Savoie, Annecy-le-Vieux F-74941, France

Physics Department, Theory Division, CERN, CH-1211 Geneva 23, Switzerland

E-mail: mailto:[email protected]

Web End [email protected]

Abstract: A recent, integrability-based conjecture in the framework of the Wilson loop OPE for N = 4 SYM theory, predicts the leading OPE contribution for the hexagon

MHV remainder function and NMHV ratio function to all loops, in integral form. We prove that these integrals evaluate to a particular basis of harmonic polylogarithms, at any order in the weak coupling expansion. The proof constitutes an algorithm for the direct computation of the integrals, which we employ in order to obtain the full (N)MHV OPE contribution in question up to 6 loops, and certain parts of it up to 12 loops. We attach computer-readable les with our results, as well as an algorithm implementation which may be readily used to generate higher-loop corrections. The feasibility of obtaining the explicit kinematical dependence of the rst term in the OPE in principle at arbitrary loop order, o ers promise for the suitability of this approach as a non-perturbative description of Wilson loops/scattering amplitudes.

Keywords: Supersymmetric gauge theory, Wilson, t Hooft and Polyakov loops, Scattering Amplitudes, AdS-CFT Correspondence

ArXiv ePrint: 1310.5735

Open Access doi:http://dx.doi.org/10.1007/JHEP11(2013)150

Web End =10.1007/JHEP11(2013)150

Hexagon Wilson loop OPE and harmonic polylogarithms

JHEP11(2013)150

Contents

1 Introduction 1

2 The Wilson loop OPE 42.1 Kinematics and dynamics in the collinear limit 42.2 MHV hexagon 62.3 NMHV hexagon 9

3 General analysis of the integrals 103.1 Reduction to sum over residues 103.2 Z-sums and harmonic polylogarithms 14

4 Implementation and results 174.1 Algorithm 174.2 New predictions: MHV hexagon 184.3 New predictions: NMHV hexagon 23

5 Conclusions and outlook 24

A Review of useful functions 27A.1 The [notdef] functions 27

A.2 The fi functions 28A.3 Harmonic polylogarithms 28

1 Introduction

The discovery of the duality between Maximally Helicity Violating (MHV) amplitudes and null polygonal Wilson loops in planar N = 4 super-Yang Mills theory [13] has elucidated

remarkable features of its structure, such as dual conformal invariance [4] (see also [5] for a review). The controlled manner in which the latter symmetry is broken, implies that the all-loop behavior of the amplitudes is accurately captured by the BDS ansatz [6] for four and ve points, and needs to be corrected by a scalar function of conformally invariant cross-ratios ui, known as the remainder function Rn, for n = 6 points and beyond [7].

For the simplest nontrivial case of six points at two loops R(2)6(u1, u2, u3), a long expression involving transcendental functions of many variables, known as multiple (or

Goncharov) polylogarithms, was rst found on the Wilson loop side in [8]. This was then drastically simplied and reexpressed in terms of classical polylogarithms with the method of symbols in [9]. In extracting higher-loop information, it proves advantageous to consider kinematical limits where simplications occur, such as the multi-Regge [10, 11] and (near-) collinear [12] limit.

1

JHEP11(2013)150

The Operator Product Expansion (OPE) approach to null polygonal Wilson loops is precisely an expansion in terms approaching the collinear limit at di erent paces, each of which receives contributions at any loop order. For the hexagon, which will be the focus of this paper, it predicts that the leading term at weak t Hooft coupling = g2Y M N has the form

R6 = cos e

1

Xl=1 l

l1

Xn=0nf(l)n() + O(e2) , (1.1)

where [notdef], , [notdef] is a particular parametrization of [notdef]u1, u2, u3[notdef], in which ! 1 conve

niently describes the limit where two consecutive segments become collinear. As we review in the next section, each term in the expansion corresponds to a di erent excitation of a color-electric ux tube, created by the two segments adjacent to the ones becoming collinear, whose energy can be calculated exactly [13] with the help of AdS/CF T integrability (see [14] for a review).

The functions f(l)n() above are given in terms of a single Fourier integral, whose precise integrand was found in [12] only for n = l 1. Still, this information combined with other

reasonable assumptions was enough to x the 3-loop symbol of the hexagon remainder function up to two unknown parameters [15], which were later determined in [16]. Apart from the propagation of the ux tube excitation, what was further necessary for obtaining the integrals for any n, was knowledge of the transition amplitude, or form factor, describing how the excitation is emitted/absorbed at the two sides of the ux tube.

Great progress in this respect was recently made in [17, 18], where an all-loop expression for the aforementioned form factor, or pentagon transition was proposed. This formulation, which now holds for any n-gon, again crucially relies on integrability. In particular, it relates the pentagon transition to the S-matrix of excitations on top of the Gubser-Klebanov-Polyakov string [19, 20], which is the string dual to the ux tube vacuum. The integrals of the so called attened part f(l)0() were computed up to l = 4 loops by means of an ansatz, and perhaps more importantly, it is also possible to obtain more terms in the OPE expansion in this framework [21]. The data of the leading and subleading OPE contributions were su ciently strong constraints for determining the full 3-loop hexagon remainder function [22], with the authors of the latter paper stating that this procedure could be extended to four loops and higher.

A practical question that naturally arises in this context, is how one can e ciently evaluate the integrals dening f(l)n() at higher loops. A more conceptual question, is whether there exists a basis of functions large enough to describe the answer at any loop order, especially in light of the conjecture [23], that multiple polylogarithms should form such a basis for the hexagon Wilson loop in general kinematics.

In this work, we address the two aforementioned questions simultaneously. By identifying the building blocks of the leading OPE integrand, reducing the integral to a sum of residues, and employing the technology of Z-sums [24], we prove that at any loop order the -dependence of the contribution (1.1) to R6 is given by

f(l)n() =

Xs,r,mic[notdef]s,m1,...,mr e[notdef]sHm1,...,mr (e2) , (1.2)

2

JHEP11(2013)150

where the c[notdef] are numerical coe cients, and Hm1,...,mr (x) are a single-variable subset of multiple polylogarithms known as harmonic polylogarithms (HPLs) [25]. Our proof is algorithmic in nature, which allows us to perform the integrations in principle at any loop order. We implement it in order to determine f(l)n for any n up to l = 6 loops, and for n = l 1, l 2 up to l = 12, thereby providing new high-loop predictions for the MHV

hexagon remainder function.Finally, we also analyze a particular component of the NMHV ratio function R [26],

for which an OPE framework has also been developed [18, 27, 28]. This framework is based on e orts to extend the Wilson loop/amplitude duality beyond the MHV case [2933], and also on the establishment of an interesting triangle of relations of the latter two observables to correlation functions of operators in the stress tensor multiplet of N = 4 theory, in the

limit where their consecutive spacetime separations become lightlike [3438].1 We nd that at weak coupling

l

Xs,r,mics,m1,...,mr sHm1,...,mr (e2) , (1.4)

and the cs are numerical coe cients. We similarly employ our algorithm implementation in order to obtain explicit expressions for all f(l)n up to l = 6, and the n = l, l 1 terms up

to l = 12.

This paper is organized as follows. We begin by reviewing the basic ingredients of the OPE approach for the hexagon Wilson loop in section 2, and present the integral formulas describing the leading term for the MHV remainder function and for the component of the NMHV ratio function. In section 3 we prove that their weak coupling expansion at arbitrary loop order always evaluates to the basis of functions described above. Section 4 focuses on the utility of our proof as a direct evaluation method of these integrals. We rst summarize the steps of the algorithm, and then apply it in order to obtain new predictions for the remainder and ratio functions at high loop order. We conclude with an extensive discussion on the implications of our work, and possible directions of further inquiry.

The appendix contains additional information on several functions which were important in our treatment, and most notably harmonic polylogarithms. The results from all our high-loop calculations, as well as the Mathematica code used to generate them, are included in seven ancillary les accompanying the version of this paper on the arXiv.

1In particular, it was shown in [39] that the dimensional regularization of the initial super-Wilson loop proposal [29, 30] breaks superconformal symmetry, and hence cannot be in correspondence with scattering superamplitudes. More recent studies of the 1- and 2-loop super-Wilson loop suggest that anomalies appear only in certain components [32], for which the symmetry can be restored either by nite counterterms [31], or by explicitly rewritting them in an invariant form with the help of the superpropagator [33]. In any case the approach [18, 27, 28] may be thought of as a collinear-limit expansion of the null correlators [3438].

3

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R(6134)6 =

f(l)n() =

e

2 cosh

1

Xl=0 l

Xn=0nf(l)n() + O(e2) , (1.3)

where for any l,

JHEP11(2013)150

(a) (b)

Figure 1. In (a), the dashed null segments connect two non-intersecting edges of the hexagon, crossing from two of its cusps. They break it up into three squares, with every two adjacent ones forming a pentagon. In (b), using conformal transformations we place O at the origin, and P, F, S at null past, null future, and spacelike innity respectively. The conformal group element e(DM01) leaves the middle square invariant, and its action on A and B makes them parallel to x+ when ! 1.

2 The Wilson loop OPE

This section serves as a review of the OPE approach for the hexagon Wilson loop, and helps in establishing our notations. In subsection 2.1 we discuss how to take the collinear limit, and outline how at weak coupling the Wilson loop decomposes into terms approaching the limit at di erent paces, mostly based on [12, 13]. Subsections 2.2 and 2.3 focus on the extension and renement of this approach for the MHV and NMHV hexagon respectively, as presented in [17, 18], and also building on [2628].

For the knowledgeable reader, the equations which will form the basis of our subsequent analysis are the denition of the conformally invariant, nite Wilson loop observable (2.9), and its leading OPE contribution for the MHV case (2.12) and NMHV case (2.23), in terms of an all-loop integral.

2.1 Kinematics and dynamics in the collinear limit

In order to take the collinear limit of the hexagon, we start by picking two non-intersecting segments, and form a square by connecting them with another two lightlike segments (see gure 1(a)). We can then x all of its 16 coordinates, 4 of them from the lightlike constraints, and the rest by conformal transformations. Specically, the 4-dimensional conformal group SO(2, 4) will have 15 generators, which implies that not only all squares will be conformally equivalent, but also that each given square will be invariant under a subset of 3 transformations.

As we show in gure 1(b), a convenient choice will be to arrange the square to lie on the (x0, x1) plane, with its points located at the origin, past lightlike innity (x =

x0 x1 ! 1 with x+ = x0 + x1 xed), future lightlike innity (x+ ! 1 with x

4

xed), and spacelike innity (x1 ! 1 with x0 xed). In this case, the three symmetries of

the square will be dilatations, boosts in the (x0, x1) plane, and rotations in the transverse (x3, x4) plane, generated by D, M01 and M34 respectively.

For concreteness, we may choose the generators and one-parameter group elements to be

M01 =

0

B

B

B

@

0 1 0 0 1 0 0 0 0 0 0 0 0 0 0 0

1

C

C

C

A

, eM01 =

0

B

B

B

@

cosh sinh 0 0 sinh cosh 0 0 0 0 1 0 0 0 0 1

1

C

C

C

A

, (2.1)

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M34 =

0

B

B

B

@

0 0 0 0 0 0 0 0 0 0 0 i

0 0 i 0

1

C

C

C

A

, eiM34 =

0

B

B

B

@

1 0 0 00 1 0 00 0 cos sin

0 0 sin cos

1

C

C

C

A

, (2.2)

whereas D is just the identity matrix, e D = e . It is straightforward to show their action on the square leaves it invariant, also keeping in mind that in two dimensions spacelike innity is a single point.

Going back to the hexagon, we can parametrize all conformally inequivalent geometries by acting with the symmetries of the square on the cusps below the reference square. In fact, it will be advantageous to consider the group element exp [(D M01)], as the

collinear limit will now simply correspond to ! 1. In more detail,

[parenleftBig]

lim

!1

0

B

B

B

@

1

2

1

2 0 0

1

2

12 0 0 0 0 0 0 0 0 0 0

e(DM01)

[parenrightBig] [notdef]

x[notdef] =

[notdef]

x1 x2 x3 x4

1[parenleftBigg]0[parenleftBigg]1[parenleftBigg]= 12(x0 + x1) 0 B

B

B

@

1 1 0 0

1

C

C

C

A

. (2.3)

This implies that all segments below the middle square will atten out onto its lower edge, as we illustrate in gure 1(b), thus becoming collinear. Depending on the choice of initial hexagon, we will get a slightly di erent relation between the cross ratios ui and the group coordinates , , parametrizing the symmetries of the square. Following the conventions of [17, 18], we will be using2

u1 = x246x213 x236x214

= 12

e2+sech1 + e2 + 2 e cos + e2 ,

u2 = x215x224 x214x225

= 12esech ,

(2.4)

A great advantage of describing the collinear limit in a matter which takes into account the symmetries of the square, is that it also makes the description of the dynamics more transparent. In particular, we can think of the Wilson loop segments belonging to the

2Note that this parametrization di ers from the one used in the initial Wilson loop OPE approach [12].

5

u3 = x226x235 x225x236

= 1

1 + e2 + 2 e cos + e2 .

middle square as a ux tube sourced by two quarks moving at the speed of light, and decompose the Wilson loop with respect to all possible excitations of this ux tube. These excitations will be eigenstates of the symmetries of the square with eigenvalues E, p, , so that schematically we may write

W = [integraldisplay]

dneEn+ipn+imn CbotCtop , (2.5)

where the exponential part describes their propagation, n labels di erent excitations, and Cbot (Ctop) denotes the transition amplitude, or overlap, between the initial (nal) state of the bottom (top) part of the polygon and the intermediate eigenstate. This picture is reminiscent of expressing the product of two neighboring operators A and B as a sum of local operators, whose scaling dimensions control the dependence of the coe cients on the distance between A and B. Hence the decomposition (2.5) around the collinear limit has been coined Wilson loop OPE.

Since its derivation only relied on the symmetries of the problem, the above formula should hold for any conformal eld theory where the ux is conserved. The good news is that in N = 4 super Yang-Mills, the ux tube excitations are in 1-1 correspondence with

excitations of an integrable spin chain, with the collinear twist operator [40] D M01 as its hamiltonian. The states of the spin chain are single-trace operators, with the vacuum made of a sea of derivatives acting on the complex combination of two scalars of the theory Z,

vacuum = tr ZDS+Z

[parenrightbig]

, D+ = D0 + D1 (2.6)

and excited states built by inserting any fundamental eld of the theory on the vacuum, e.g.

single excitation = tr

Xl=1 lE(l)(p) . (2.8)

Due to integrability, the energy E can be calculated to all loops [13], and moreover for M elementary excitations we will have EM = M + O( ). This teaches us that at

weak coupling the Wilson loop OPE (2.5) will be a sum of terms with di erent integer exponential behaviors eM as ! 1, where M is the classical twist of the state, or

equivalently the number of elementary excitations it consists of. In this paper, we will

focus on the leading-twist, or single-particle term O(e), and we will frequently refer to

it as the leading OPE contribution.

2.2 MHV hexagon

Let us now become more specic, and dene the hexagon Wilson loop-related observable which will be most suited for analyzing its OPE. By construction, the middle square always

6

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ZDS1+ DS2+Z

[parenrightBig]

, (2.7)

where S S1+S2 0. In fact, minimizing the energy suggests that elementary excitations

of the spin chain can only be drawn from the components of which have minimal classical twist eigenvalue S = 1. At quantum level the (shifted with respect to the vacuum)

twist of the operators (2.7) receives anomalous contributions due to renormalization,

E(p) = ( S)1 ( S)vac = 1 +

1

has two of its segments coinciding with the cusps of the hexagon, and hence the bottom and top part of the hexagon which lie outside of it will also be squares. Then,

r log W log

, Wbot, Wtop are the Wilson loops dened on the contours

of the middle square, and of the pentagons created by joining the middle and lower, and middle and upper squares (see gure 1(a)). The particular ratio we are considering removes all (cusp-induced) ultraviolet divergences, leaving a nite function of conformal cross ratios. Aside this, it does not cause any loss of information, as the square and pentagon Wilson loops are given by the BDS ansatz [6].

Specializing on the MHV case, from symmetry arguments the single-particle contribution is expected to be bosonic and uncharged under the R-symmetry. This picks out only one out of the twist-1 excitations, the component F+i of the gauge eld [12], where the rst component is projected on x+ and the second component is on the (x3, x4) plane, according to the notations discussed in the beginning of section 2.1. As we also mentioned in that section, its all-loop dispersion relation has been found with the help of integrability [13], in parametric form with respect to the Bethe rapidity u. Following the notations of the latter paper, from this point on we will rescale the t Hooft coupling by

g2

1

(2.11)

where the functions ?[notdef](2gt) are independent of u, and as we review in appendix A.1, can

be obtained iteratively as a Taylor expansion in g 1.

In order to obtain information about the hexagon from the analogue of (2.5) for r, we need however to know the (rescaled) creation/absorption form factors Cbot, Ctop, which

also depend on g. Initially, these were determined to the rst few orders by comparing with the explicit computation of the hexagon up to two loops [8, 9]. Recently however, all-loop expressions for them were proposed, relying again on the integrability of the theory [17, 18]. In particular, these form factors, also dubbed as pentagon transitions, are related to the S-matrix of excitations on top of the GKP string [19, 20], which forms the vacuum of the ux tube.3

With this ingredient in place, and in the more natural rapidity parametrization, the nal formula for the leading (single-particle) OPE contribution for r reads

r = 2 cos e

[integraldisplay]

1 ?(2gt)1 et+ ?+(2gt)

et 1[bracketrightbigg]sin (ut)et/2 ,

du2 [notdef]1(u)e 1(u)+ip1(u) + O(e2), (2.12)

3The new formulation in the framework of the OPE approach also has the advantage of being applicable to Wilson loops with more cusps, and generalizable to multiparticle contributions [21].

7

W W[Box]

WbotWtop , (2.9)

where W is our hexagon, and W[Box]

JHEP11(2013)150

(4)2 , (2.10)

in terms of which the gauge eld dispersion relation reads

1(u) E1(u) 1 = [integraldisplay]

dt t

?+(2gt)1 et

?(2gt)

et 1[bracketrightbigg] [parenleftBig]cos (ut)et/2 1[parenrightBig]

,

p1(u) = 2u

[integraldisplay]

0

dt t

0

+1

1

where the measure [notdef]1(u) is given by

[notdef]1(u) =

g2 cosh (u)

u2 + 14

( x+x g2)

p(x+x+ g2)(xx g2)[notdef]

[notdef] exp

[bracketleftbigg] [integraldisplay]

1

0

dt

t (J0(2gt)1)

2et/2 cos(ut) J0(2gt) 1

et 1

+f3(u, u)f4(u, u)

.

(2.13)

[bracketrightbigg]

In the last formula, x[notdef] are the Zhukowski variables

x[notdef] = x

u [notdef]

i 2

, x(u) = u +

pu2 (2g)22 , (2.14)

Ji is the i-th Bessel function of the rst kind, and the form of the functions fi(u, v) is reviewed in appendix A.2. Once the observable (2.12) has been determined, the near collinear limit of the MHV hexagon remainder function R (we drop the index as we will only be dealing with n = 6),

W = W BDSeR(u1,u2,u3) (2.15)

parametrizing the part of the Wilson loop W which is not captured by the BDS ansatz W BDS [6], is given by

R = r rBDS , (2.16)

where [41]

rBDS log WBDS

= cusp 4

Li2 (u2) Li2 (1 u1) Li2 (1 u3) + log2 (1 u2)

log (u1) log (u3) log (u1/u3) log (1 u2) +

2 6

JHEP11(2013)150

(2.17)

1 + e2

[parenrightbig][bracketrightbig]+ O(e2) , (2.18)

is dened as in (2.9), but only including the BDS contribution for each of the polygons. The equality (2.16) is a consequence of the denition (2.15), given that the BDS ansatz accurately describes square and pentagon Wilson loops.

In more detail, the logarithm of the BDS part of any null polygonal Wilson loop in

N = 4 super Yang-Mills, is proportional to the logarithm of the same Wilson loop in

the U(1) theory, with the proportionality constant being a fourth of the cusp anomalous dimension cusp. The latter quantity can also be calculated to all loops (see [42] for a review), and is given to the rst few orders in the weak coupling by

cusp =

1

=

cusp

2 cos e

e log

1 + e2

[parenrightbig]+ e log

Xl=1g2l lcusp = 4g2 423 g4 +44445 g6 4 [parenleftbigg]736315 + 8 23[parenrightbigg]

g8 + O(g10) . (2.19)

8

2.3 NMHV hexagon

Scattering amplitudes involving gluons in other helicity congurations, or other particles of N = 4 super Yang-Mills, are most conveniently described by exploiting its (dual) super-

conformal symmetry [26], see also [5] for a review.

Very briey, one starts by packaging the particle content of the theory4 in a single supereld with the help of a Grassmann variable A, whose index transforms in the fundamental representation of the R-symmetry group SU(4). Namely, all external states of [notdef]1 helicity gluons G[notdef], [notdef]12 helicity Majorana fermions A,

A, and zero helicity real

3! A B C D[epsilon1]ABCDG , (2.20)

which in turn allows us to combine all n-point amplitudes in a superamplitude

An( 1, . . . , n).

All MHV amplitudes form the part of the superamplitude which has 8 powers of Grassmann variables, starting the the MHV gluon amplitude,

AMHVn = (2)4 (4)[parenleftbigg]

n

Xi=1pi

scalars SAB can be simultaneously described by

1 3! A B C[epsilon1]ABCD

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= G+ + A A + 1

2! A BSAB +

D + 1

[parenrightbigg] [summationdisplay]

( j)4( k)4AMHVn(1+ . . . j . . . k . . . n+) + . . . . (2.21)

On the basis of tree- and 1-loop level amplitude computations, it was argued in [26] that NMHV amplitudes will similarly organize in a homogeneous polynomial of degree 12 in Ai, and more importantly, that they have the same infrared divergence structure as the MHV amplitudes, so that the two superamplitudes are related by

ANMHVn = AMHVnRn , (2.22)

where Rn is the (dual conformal invariant) NMHV ratio function. Evidently, it will consist

of terms involving 4 powers of the Grassmann variables, whose components we can be denoted as R(ijkl)n.

Following attempts for generalizing the Wilson loop/scattering amplitude duality beyond the MHV case [29, 30], an analogous proposal for the OPE of certain components of the NMHV hexagon Wilson loop was put forth in [27, 28]. According to the latter, the dual of the R(i,i+1,j,j+1)6 component is given by a Wilson loop5 W(i,i+1,j,j+1)6 - normalized

by bosonic squares and pentagons as in (2.12) - which at tree-level has insertions of a complex scalar eld combination at the cusp between segments i, i + 1, and its complex conjugate eld at the cusp between segments j, j + 1. The two segments forming the ux tube reference square are chosen to lie between i, i + 1 and j, j + 1 without coinciding with any of them, and [notdef]i j[notdef] 3. Thus the top and the bottom part of the polygon have a

scalar insertion each, and it is also expected that only scalar excitations of the ux tube will propagate to leading order in the OPE expansion.

4The on-shell elds of N = 4 super Yang-Mills form the CPT self-conjugate, doubleton representation

of the superconformal group P SU(2, 2|4) [43].

5More precisely, R6 and W6 will only di er by O(e2) terms [18].

9

1 j<k n

For a particular component of the NMHV ratio function, this leading OPE contribution was also predicted in [18], by a similar analysis of the scalar pentagon transitions. It is the component with the scalar insertions between edges 6-1 and 3-4, and edges 2 and 5 forming the reference frame, for which the OPE reads (we drop the lower index)

W(6134) =

1g2 e [integraldisplay]

du2 [notdef]0(u) e 0(u)+ip0(u) + O(e2) , (2.23)

where 0(u) and p0(u) are the anomalous part of the energy and the momentum of a scalar excitation propagating between the bottom and top part of the polygon [13],

0(u) =

[integraldisplay]

JHEP11(2013)150

dt t

1 ?+(2gt) ?(2gt)

et 1

et/2 cos (ut) 1[parenrightBig] [integraldisplay]

10dt

t ?+(2gt) ,

0

(2.24)

p0(u) = 2u

[integraldisplay]

dt t

1 ?+(2gt) + ?(2gt)

et 1

et/2 sin (ut) ,

0

and the scalar integration measure [notdef]0 is now given by

[notdef]0(u) = g2 cosh (u)[notdef]

[notdef] exp

[bracketleftbigg] [integraldisplay]

1

0

dt

t (J0(2gt) 1)

2et/2 cos(ut) J0(2gt) 1

et 1

+ f3(u, u) f4(u, u)

[bracketrightbigg]

.

(2.25)

Here as well Ji is the i-th Bessel function of the rst kind, ?[notdef] are the same functions which

appeared in the MHV hexagon subsection, and which are reviewed in appendix A.1, and fi(u, v) are functions which are in turn reviewed in appendix A.2. Well also have the same parametrization (2.4) of the conformally invariant cross ratios u1, u2, u3 in terms of , , .

3 General analysis of the integrals

In section 2, we recalled that the Wilson loop OPE approach yields the leading, O(e) term in the ! 1 collinear limit of the (N)MHV hexagon as a 1-dimensional Fourier integral,

to all orders in g. Here we will focus on the weak coupling expansion of the integral, and prove that it can be evaluated at any loop order, in terms of harmonic polylogarithms.

3.1 Reduction to sum over residues

Let us start by combining the descriptions of the MHV and NMHV integrals (2.12), (2.23) simultaneously,6

I =

[integraldisplay]

du2 [notdef](u) e (u)+ip(u) , (3.1)

6We drop the indices in , , p in order to avoid clutter, but we should keep in mind they correspond to di erent functions for = 0, 1.

10

with the help of the denition

[notdef](u) = g2

cosh (u)

x+x

(x+x g2)

p(x+x+ g2)(xx g2)

!

[notdef]

1

0

[notdef] exp

[bracketleftbigg] [integraldisplay]

dt

t (J0(2gt) 1)

2et/2 cos(ut) J0(2gt) 1

et 1

+ f3(u, u) f4(u, u)

[bracketrightbigg]

,

(3.2)

where = 1 corresponds to the propagation of a gauge eld excitation and describes the MHV case, and = 0 corresponds to a scalar excitation and describes the NMHV case. Similarly , p and the functions f3, f4 will implicitly depend on , as reviewed in appendix A.2, equations (A.4), (A.5) and (A.7). In obtaining (3.2), we made use of the identity

[integraldisplay]

1

0

JHEP11(2013)150

dt

t 2et/2 [1 J0(2gt)] cos(tu) = log

x+x

u2 + 14

, (3.3)

which may be proven as was done in [13, 44] for similar integrals involving Bessel functions,7 in order to replace et ! et in the second line of (2.13).

We will proceed to evaluate the integral (3.1) order by order at weak coupling g by turning it into a sum over residues. To this aim, we will need to analyze the general dependence of [notdef], and p on the integration parameter u.

Focusing rst on the exponential part of [notdef], and in particular the fi functions contained therein, and dened in (A.4), it is evident that the dependence on u enters only through the , ~

vectors (A.5). As a result of the regular Taylor expansion of the Bessel functions involved (A.3), at weak coupling these can always be expressed in terms of the following integrals which can be calculated exactly,8

[integraldisplay]

1

0 dt [parenleftbigg]

tmezt

1 et

et

t m,0

[parenrightbigg]

= (1)m+1 (m)(z) , m 0 , (3.4)

where (m)(z) is the polygamma function. It is easy to see that in both NMHV ( = 0) and MHV ( = 1) case the possible arguments include z = 12 [notdef] iu and z = 1, and for the

second argument we further reduce to Riemann zeta functions,

(m)(1) = (1)m+1m! m+1 . (3.5)

For the MHV case, there exists one additional possibility for the argument, z = 32 [notdef] iu, and by denition the weak coupling expansion of the fi functions will contain bilinear combinations of these -functions.

A similar analysis can be performed for the remaining exponential part of [notdef](u) (3.2), and also for (u) and p(u), which shows that we now obtain monomials of the aforementioned polygamma functions with the specic arguments. For (u), p(u) this is particularly easy to see due to the alternative expression (A.7) mentioned in appendix A.2, which relates them to the same building blocks of the fi functions.

7We thank Benjamin Basso for pointing this out to us.

8In particular the m = 0 case only appears in 1, and as the di erence (z) (1), so that the second

term in the integral cancels out.

11

Finally, it is straightforward to show that at any loop order in g 1, the x[notdef]-dependent

factor of the measure (3.2) will be a sum of products of inverse powers of u [notdef] i2. In more detail, the factor in question is equal to (u2 + 14)1 at g = 0, has a regular Taylor expansion around that point, and due to9

x[notdef] = x(u [notdef] i2) , x(u) =

u +

pu2 (2g)22 =

u 2

"1 +1

Xk=0

1/2k

[parenrightbigg] [parenleftbigg]

4g2 u2

[bracketrightBigg]

k

, (3.6)

it has e ective expansion parameters g/(u [notdef] i2).

Gathering the information we obtained, we deduce that the weak coupling expansion of the integral (3.1) will be a sum of the general form

I =

[integraldisplay]

JHEP11(2013)150

du2 [notdef](u) e (u)+ip(u)

l1

Xn=0 n

l1

=

1

Xl=1g2l

[integraldisplay]

du(l)n(u, )

1

Xl=1g2l

Xn=0nh(l)n() , (3.7)

(l)n(u, ) =

Xc e2iusech(u)

Yi

(mi)

12 [notdef] iu

[parenrightbigg][parenleftBigg][parenleftBigg][producttext]

j (mj)(32 [notdef] iu) (u + i2)r1 (u i2)r 2

! (3.8)

where we have absorbed all factors that dont depend on u in the coe cients c (which obviously also depend on the di erent indices li, mi, ri etc).

From the last formula, we arrive at the following important conclusion: The only possible locations where the integrand may have poles are for u = (k + 12)i, k 2 Z. In particular these poles may come from the denominator (if any), the hyperbolic cosecant, or the polygamma functions at negative integer arguments.

In what follows we will restrict to > 0, in which case we can close the contour with a semicircle on the u > 0 plane, whose integral at innite radius will go to zero due to the decaying exponential. Therefore by Cauchys residue theorem we will have

h(l)n() [integraldisplay]

1

k + 12[parenrightbigg]

i

[parenrightbigg]

, (3.9)

and in order to proceed we will need to nd an analytic expression for the residues as a function of k. This cannot be achieved by directly Taylor expanding the expressions for the integrands around u = (k + 12)i, as (n)

du(l)n(u, ) = 2i

Xk=0 Res

(l)n, u =

32 + iu

[parenrightbig]

and (n) 12 + iu

[parenrightbig]

develop poles there.

Instead, we rst employ the reection formula

(n)(z) = (1)n (n)(1 z)

@n

@zn cot(z) , (3.10)

9We have chosen the branch pu2 = u, which is equivalent to assuming x(u) starts at O(g0). The opposite

branch corresponds to x(u) starting at O(g2), and the two solutions are related by x(u) ! g2/x(u), as can

be seen by the dening relation u = x(u) + g2/x(u). From this it follows that the expansions in the two branches of the factor in question only di er by an overall sign, but as pointed out in [18], there already exists a sign ambiguity in (u), that needs to be xed by physical input.

12

which allows us isolate the singular terms into elementary functions with known expansions. In more detail, we rst use the recurrence relation

(n)(z + 1) = (n)(z) + (1)nn!zn1 , (3.11)

with z = 12 [notdef] iu, in order to eliminate the (m)(32 [notdef] iu) factors in (3.8).10 Then, we

apply (3.10) with z = 12 + iu, also noting that the cotangent derivative will give a polynomial in cotangents and cosecants, which in terms of u become i tanh u and sechu respectively.11

In this manner, we have achieved to reexpress (3.8) as a sum of products of a restricted set of functions, having known expansions for u = (k + 12)i + [epsilon1], [epsilon1] small, which we also present below:

1

e2iu = e(2k+1)+i[epsilon1] = e(2k+1)

Xn=0(i[epsilon1])n

n! , (3.12)

(m)

tanh u = coth [epsilon1]

JHEP11(2013)150

12 iu

[parenrightbigg]= (m)

k + 1

i[epsilon1]2[parenrightbigg]=

1

Xn=1 (m+n)(k + 1) n!

i[epsilon1] 2

n

, (3.13)

1

2 =

2[epsilon1] +

Xn=1 22nB2n (2n)!

[epsilon1] 2

2n1 , (3.14)

sechu = i(1)kcsch

[epsilon1]

2

= i(1)k [bracketleftBigg]

2[epsilon1]

1

Xn=12 22n1 1

[parenrightbig]

[epsilon1] 2

B2n

2n1

[bracketrightBigg]

, (3.15)

(2n)!

1u i2

= i

= i

[parenleftbigg]

i[epsilon1] 2(k + 1)

n

, (3.16)

k + 1 i[epsilon1]2

k + 1

1

Xn=0

1u + i2

= i

8

>

>

>

<

>

>

>

:

2 [epsilon1] k = 0 ,

i k

(3.17)

k i[epsilon1]2

=

1

Xn=0

[parenleftbigg]

i[epsilon1] 2k

n

k 1 ,

where B2n are the Bernoulli numbers.

A few remarks are in order. First, since the hyperbolic functions become periodic in the imaginary axis, their k-dependence reduces to an overall sign at most. Furthermore, because of the last equation, we will have to determine the residue for k = 0 separately. Aside this value, we may use the above expansions in order to determine the residues in (3.9) for

10Notice that after expanding, we will not get any new rational factors apart from the already existing u i2 in (3.8).

11In more detail, it can be shown that

@n@zn cot(z) =

[m12]

Xk=0ck cos(z)m12k sin(z)m1 ,

which implies we will always have even powers of csc(z) ! sechu.

13

general positive integer k. Finally, the fact that the residues will only contain -functions with integer arguments, allows us to replace them with generalized harmonic numbers,

(k + 1) (0)(k + 1) = E + S1(k)

(m1)(k + 1) = (1)m(m 1)!( m Sm(k)) ,

(3.18)

dened as

Xn=11nm , (3.19)

where E = (1) =[similarequal] 0.577 is the Euler-Mascheroni constant.3.2 Z-sums and harmonic polylogarithms

Let us now focus on the structure of the residue of(l)n at u = (k+ 12)i, as a function of k. In the previous section, we demonstrated that(l)n will be a sum of products of the restricted set of functions (3.13)(3.18), which we expanded around the positions of all poles. From this analysis, it immediately follows that

Res (l)n, u = k + 12[parenrightbigg]

i

[parenrightbigg]=

Xc e(e2)ks [parenleftbigg]1kl1 (k + 1)l2

r

Sm(k) =

k

JHEP11(2013)150

Yi=1Smi(k) (3.20)

for di erent values of the indices s, r, l1, l2, m1, . . . mr and numerical constants c.

In more detail, (3.14) will only contribute numerical factors independent of k. The same will apply for (3.15), except for an overall (1)k factor, as the discussion in footnote 11

implies that we will always have odd powers of sechu. The latter k-dependent factor combines with the exponential coming from (3.12), which will also contribute the powers of . Finally the inverse powers of k, k + 1 come from (3.16), (3.17), and the products of harmonic numbers from (3.13) due to (3.18).

At this point, we will choose to concentrate on the MHV case ( = 1), and come back to examine what changes for the simpler NMHV case at the end of this subsection. There exist three additional steps we need in order to bring these terms in a form where we will be able to perform the summation over all poles k. First of all, we can always partial fraction in order to get only inverse powers of either k or k + 1, since the degree of the numerator is always smaller than the degree of the denominator, and in particular zero. Secondly, we can perform certain manipulations in order to reduce the number of di erent arguments in the sums. This procedure, generally known as synchronization [45], is necessary for the e cient evaluation of sums of this type with the help of the computer.

In our case, achieving this goal involves treating the terms with powers of k and k + 1 di erently. In particular, we can eliminate the latter in terms of the former by redening the summation index k[prime] = k +1, and extending the summation range to also include k[prime] = 1 by adding and subtracting the term in question. More concretely,

1

Xk=1e(e2)k(k + 1)r . . . =

1 e(e2)k[prime]

k[prime]r . . . =[parenleftBigg][parenleftBigg]

1 e(e2)k

kr . . .

[parenrightBigg] [parenleftBigg][parenleftBigg]e

[parenrightbig]

(3.21)

. . .

14

and notice that this redenition of the summation index will turn e ! e in (3.20). In addition, the argument of the harmonic numbers included in these terms will now be k 1, and we can turn this into the argument of the harmonic numbers multiplying inverse

powers of k, by replacing

Sm(k) = Sm(k 1) +

1km , (3.22)

The formula above follows immediately from the denition (3.19), and is the analogue of (3.11) for harmonic numbers. Evidently we do not need to partial fraction again, as we only obtain additional powers of the same monomial in k.

Finally, we will make use of the fact that harmonic numbers are the simplest case of a more general set of nested sums, known as Z-sums [24]. These are dened as

Z(n; m1, . . . , mj; x1, . . . , xj) =

Xn i1>i2>...>ij>0

xi11 im11

. . . xijj imjj

, (3.23)

or recursively by

Z(n; m1, . . . , mj; x1, . . . , xj) =

n

Xi1=1xi11im11Z(i1 1; m2, . . . , mj; x2, . . . , xj) , (3.24)

JHEP11(2013)150

where Z(n) is equal to the unit step function. The sum of mi is known as the weight, or transcendentality, and the number of summations j as the depth. Clearly, harmonic numbers are depth-1 Z-sums, Sm(k) = Z(k; m; 1).

An important property of these objects, also known as the quasi-shu e algebra,12 is

that a product of two Z-sums with the same outer summation index can be reexpressed as a linear combination of single Z-sums. This easily follows by splitting the square double summation range into regions with denite index ordering, namely

n

Xi=1

n

Xi=1

i1

n

Xj=1aij =

j1

Xj=1aij +

n

Xj=1

Xi=1aij +

n

Xi=1aii . (3.25)

For our purposes, rather than the general recursive procedure for decomposing a product of Z-sums, we will just need the particular case (we set all xi = 1, and for compactness drop both them and the outer summation index n from our notation),

Z(l)Z(m1, . . . , mj) = Z(l, m1, . . . , mj) + Z(m1, l, . . . , mj) + . . . + Z(m1, . . . , mj, l)

+ Z(m1 + l, . . . , mj) + . . . + Z(m1, . . . , mj + l) .(3.26)

In other words we take all permutations that preserve the order of indices of the two Z-rums on the left hand side, and also add the l index to all mi. The above formula can then be used recursively in order to decompose the product of harmonic numbers/depth-1 Z-sums into single harmonic sums in (3.20).

12In fact, the quasi-shu e algebra forms part of a larger Hopf algebra structure [24].

15

After these three steps, the k-dependent part of all terms in the sum over residues (3.20) will be itself proportional to a Z-sum,

1

Xk=1(e2)k

km1 Z(k 1; m2, . . . , mj; 1, . . . , 1) = Z(1; m1, m2, . . . , mj; e2, 1, . . . , 1) .

(3.27)

The critical observation, based on [24], is that this particular Z-sum precisely coincides with the series representations of the harmonic polylogarithm Hm1,m2,...,mj (e2) of Remiddi and Vermaseren [25]!13

We have therefore rigorously proven that the integral yielding the single-particle contribution to the Wilson loop OPE of the MHV hexagon (2.12) will have a weak coupling expansion of the form (3.7), with its dependence always given by

h(l)n() =

Xs,r,mic[notdef]s,m1,...,mr e[notdef]sHm1,...,mr (e2) , mi 1 , (3.28)

where c[notdef] are numerical coe cients.

Finally, let us go back and extend the last part of our analysis to the case of the NMHV hexagon component integral, namely (3.7), (3.8) with = 0. We recall that since the latter formula will not contain any (m)(32 [notdef] iu) or inverse powers of u [notdef] i2, the sum

over residues (3.20) will not contain any inverse powers of k, k + 1. Thus we may directly reexpress the harmonic numbers as a linear combination of Z-sums with the help of the the quasi-algebra (3.26) in (3.20), and obtain

1

JHEP11(2013)150

Xk=0 Res

(l)n, u = 2k + 12 i

[parenrightbigg]

se

1

Xk=0(e2)kZ(m1, . . . , mr)

d dy

1

se

Xk=0yk+1k + 1Z(m1, . . . , mr)

[vextendsingle][vextendsingle][vextendsingle]y=e2

(3.29)

se

d

dy H1,m1,...,mr (y)[notdef]y=e2

,...,mr (e2) .

The equality of the last line follows from the denition of HPLs (A.9), also taking into account the discussion about their a- and m-notation before equation (A.12). We similarly conclude that the weak coupling expansion of the single-particle contribution to the Wilson loop OPE of the NMHV hexagon component (2.23) will be proportional to (3.7) with

h(l)n() = 1

2 cosh

Xs,r,mics,m1,...,mr sHm1,...,mr (e2) , mi 1 . (3.30)

13For an earlier, implicit denition of harmonic polylogarithms in terms of the inverse Mellin transform of nested sums, see also [46].

16

= se

1 + e2 Hm1

The general, all-loop structure of the hexagon Wilson loop OPE contributions for the MHV (3.28) and NMHV (3.30) case agree with the ansatze, based on empirical evidence,14

used in [18].15

Before concluding, we should note that by rening our analysis so as to keep track of the maximum powers of g that can multiply the terms of the integrand (3.8), it is possible to show that the indices of the terms summing up to hln() (3.28), (3.30) are constrained as follows,

s +

Xi

mi 2(l 1) + n , (3.31)

where again = 0, 1 corresponds to the NMHV, MHV case respectively. This of course is in agreement with the fact that all N = 4 amplitudes computed to date have maximal

transcendentality 2L at L loops, given also that the Taylor expansion of transcendental functions yields terms with the same or smaller transcendentality.

4 Implementation and results

4.1 Algorithm

In the previous section, we presented a general proof for the exact basis of harmonic poly-logarithms, including the dependence of their coe cients and arguments on the kinematical data, which is suitable for describing the leading OPE contribution of the hexagon Wilson loop at any order in the weak coupling expansion g 1.

Furthermore, it is perhaps evident that our proof forms an algorithmic process which allows us to directly compute the Fourier integrals (3.7) and obtain explicit expressions for the relevant part of the hexagon in terms of the kinematical variables (2.4). We stress again that this method of computation can be applied at arbitrary loop order l, subject to restrictions in computational power.

Let us now summarize the steps of the algorithm, which facilitate its implementation in any computer algebra system. We start with the weak coupling expansion of the integrand (3.1), which as we reviewed in section 2, can be performed with the help of the denitions of [notdef], , p, and the integral (3.4). Then, we

1. Replace all -functions appearing with (n) 12 iu [parenrightbig]

and (3.11).

2. Replace u = (k + 12)i + [epsilon1] and nd the residue for general k 1, with the help

of (3.12)(3.17).

3. Replace (m1)(k + 1) ! Hm(k) = Z(k, m, 1) by means of (3.18).4. Partial fraction the inverse powers of k, k + 1, if any.

14The fact that harmonic polylogarithms is a suitable basis for describing the hexagon Wilson loop OPE was rst noted in [47].

15Note however that this paper uses the a-notation for harmonic polylogarithms, whereas we are using the m-notation. The relation between the two notations is discussed in appendix A.3.

17

JHEP11(2013)150

with the help of (3.10)

5. For the k 1 sum over residues, synchronize the arguments by applying (3.21) to

the terms with k + 1 powers, and (3.22) to terms with k powers.

6. Reduce products of Z-sums to linear combinations of single Z-sums by recursively applying (3.26).

7. Replace

1

result of the previous step.

4.2 New predictions: MHV hexagon

With the steps of the algorithm set in place, we can now proceed with its implementation, in order to determine the leading OPE contribution of the hexagon remainder function to high loop order. We remind the reader that this contribution will be given in terms of the h(l)n() functions as

R = 2 cos e

+ O(e2) , (4.1)

by virtue of equations (2.12), (2.16), (2.18), (2.19) and (3.7).

We have included the results, together with a Mathematica code that generates them, in ancillary text les accompanying the version of this paper on the arXiv. The code is fully general, and can be used in principle for any given value of l. In particular, we have used it to obtain h(l)n() for all allowed values of l 1 n 0, up to l = 6, thereby

providing information about the 5- and 6-loop hexagon for the rst time.

Generating results at higher loops from this code is only a matter of computational power and optimization, and to illustrate this we also calculated the n = l 1, l 2

components up to l = 12. As far as the e ciency of the code is concerned, we should stress that even without particular attention to optimization, the computation of all 4-loop leading-twist terms takes about 10 seconds on a portable computer, and all 5-loop terms about a minute!

Let us start by briey mentioning what is already known about the hexagon in the near-collinear limit, and in the rst few orders at weak coupling. In [18], the n = 0, also dubbed attened part of the hexagon was computed up to 4 loops under the assumption of an ansatz for the general structure of the expression. More specically, the free parameters

18

Xk=1(e2)k

km1 Z(k 1; m2, . . . , mj; 1, . . . , 1) ! Hm1,m2,...,mj (e2) ,

1

Xk=0(e2)kZ(k; m2, . . . , mj; 1, . . . , 1) !11 + e2 Hm2,...,mj (e2) .

8. Obtain nal result for the integral by evaluating and adding Res(h(l)n, u = i

2 ) to the

JHEP11(2013)150

1

Xl=1g2l

[bracketleftBigg]

[bracketrightBigg]

Xn=0nh(l)n() + lcusp 4

e log(1 + e2) + e log(1 + e2)

[bracketrightbig]

l1

of the ansatz were determined by comparing its Taylor expansion with a nite number of terms in the sum over residues, which the Fourier integral (3.7) reduces to.16

In the previous section, we proved that the structure of this ansatz is correct at any loop order, which thus places the aforementioned calculations on a rmer setting. As a further consistency check, we rst aimed to reproduce the results reported in [18] for the h(l)0() functions,17 n = 1, . . . , 4 with the help of our direct computation method.

More specically, we compared with the expressions contained in the Mathematica le Functionshf.nb accompanying the latter paper, and found indeed agreement. Due to the many functional identities between harmonic polylogarithms, the two expressions are not identical, but they can be brought in the same form with the help of the HPL package [49, 50]. To this end, we use property (A.16) in order to change the argument of the HPLs to e2, and also replace all powers of ! 12H0(e2) multiplying them.

Then, we employ the command HPLProductExpand in both expressions, which eliminates any products of HPLs, in favor of their linear combinations.

Next, we moved on to determine all non-attened leading-twist contribitions h(l)n(),

n [negationslash]= 0 for l 4. As all HPLs in the expressions have argument e2, we will omit it for

compactness. At two loops we have

h(2)1 = e [(4 4)H1 4H1,1] + ( ! ) , (4.2)

at three loops18

h(3)2 =e[bracketleftbigg]

JHEP11(2013)150

16( 1)H1,1 + 16H1,1,1 + H1

4 2 4 + 3

[parenrightbig]

+ 2

3

[parenrightbigg]

2H2(2 + 1) 4H3

[bracketrightbigg]

+ ( ! ) , (4.3)

4 3 + 82 823 40 + 22 + 36[parenrightbigg] 2H2(2 + 1)

+ H3(2 4) 8H1,2 + 2 20 + 2 24 + 42 [parenrightbig]

h(3)1 =e

H1

H1,1

4(2 + 1)H2,1 + 48(1 )H1,1,1 + 48H1,1,1,1

[bracketrightbigg]

+ ( ! ) , (4.4)

whereas at four loops

h(4)3 =e

8 3 + 1639 162 +429 + 48 423 803[parenrightbigg]

+ H2 1623 + 8 429 +323[parenrightbigg]+ H3

H1

323 163[parenrightbigg]

16See also [48] for an application of the same method in the multi-Regge limit.

17In the notations of [18], h(l)0() ! fl().

18Recently [22] appeared, which calculates the 3-loop hexagon in general kinematics, and also specializes to the near-collinear kinematics considered here. We have checked that the results of the two calculations agree if we take into account that the expansion parameter in the latter reference is 2g2. See also [51] for a calculation of the analogue of h(3)2 in the earlier Wilson loop OPE approach [12].

19

+ 162 + 64 423 48

H1,1 + 16H1,2 + (16 + 8)H2,1 (4.5)

40H1,3

+(64 64)H1,1,1 +

3 + 8H2,2 + 16H3,1 64H1,1,1,1

[bracketrightbigg]

+ ( ! ) ,

H1[parenleftbigg] 8 3 + 40 3 + 1633 16223 642 +5223 + 224

74

15

h(4)2 =e

402

3 160

[parenrightbigg]

+ H2

8 3 82 + 42 + 24 +223 + 32 [parenrightbigg]

+ H3

8 + 823 + 16[parenrightbigg]+ H4(16 + 8) + 24H5 + H1,1

40 3 + 1633 802

+ 522

522

3 + 304

3 224

[parenrightbigg]

+ (8 + 40)H1,3

+ H1,2

162 + 80 423[parenrightbigg]+ H2,1

162 + 40 423 + 40 [parenrightbigg]

802 + 384 5223 304 [parenrightbigg]

+ 80(H1,1,2 + H1,2,1) + (80 + 40)H2,1,1 + 384( 1)H1,1,1,1 + 8H1,4 + 16(H2,3 + H3,2) + 24H3,1 + 40(H1,1,3 + H1,3,1)

+ 24(H1,2,2 + H2,1,2 + H2,2,1) + 48H3,1,1 384H1,1,1,1,1 [bracketrightbigg]

+ ( ! ) , (4.6)

JHEP11(2013)150

+ (24 + 24)H2,2 + H1,1,1

h(4)1 =e

[bracketleftbigg][parenleftbigg]

423

9 + 83 8 32 1222 1122 +

1134

45

+1222

82 3

914

3 + 464 +

3 + 76 3 + 32 5

45 282 400[parenrightbigg]

H1

+

423 + 20 + 22[parenrightbigg]

H3 +

82 + 4 + 223 8

H4

4322 82 +1423 + 48 + 12 3

49 +823 + 32 [parenrightbigg]

+ (24 24)H5 +

+ H2

3233 1222 1522 +15223 + 592 + 96 3

914

45 382 464[parenrightbigg]

H1,1 + 8H1,4 +

833 402 + 102 + 152

+ 20 3

102 3

H1,2 +

16 + 1023 + 60

H1,3 20H1,5

+ 162 +2823 + 64 + 24 3 + 22 + 64[parenrightbigg] H2,1

+ 82 + 36 +423 + 32

H2,2 + (8 + 12)(H2,3 + H3,2)

12(H2,4 + H4,2 + H3,3) + 24 + 42 + 20

[parenrightbig]

H3,1 24H5,1 + (8 + 4)H4,1

20

JHEP11(2013)150

P5n=0 nh(6)n(), as a function of , . Colors of the visible spectrum denote di erent values of h(6), increasing from blue to red. The function is always positive, and monotonically increasing and decreasing in and respectively.

Figure 2. Plot of the MHV hexagon leading OPE contribution at 6 loops, h(6)

3233 1922 +15223 + 768 + 96 3 482 592[parenrightbigg] H1,1,1

(H1,1,2 + H1,2,1) + (16 + 80)(H1,1,3 + H1,3,1)

+ 96H3,1,1,1 + (56 + 40)(H1,2,2 + H2,1,2) + 20(H1,2,3 + H1,3,2)

+ 322 + 96 823 + 88

H1,1,1,1

+ 192(H1,1,1,2 + H1,1,2,1 + H1,2,1,1) + 80(H1,1,1,3 + H1,1,3,1 + H1,3,1,1)

+ 40(H1,1,2,2 + H1,2,1,2 + H1,2,2,1) + (16 + 40)H3,1,1

+ 48(H2,1,1,2 + H2,1,2,1 + H2,2,1,1) + 960( 1)H1,1,1,1,1 960H1,1,1,1,1,1

+ ( ! ) . (4.7)

Evidently the length of the expressions grows quite fast with the loop order, due to the increase not only in the number of terms in the integrand, but also in the quasi-shu e algebra decomposition (3.26) for higher powers of harmonic numbers.

Finally, we computed all 5- and 6-loop leading-twist contributions to the OPE of the MHV hexagon, as well as the h(l)l1() and h(l)l2() terms up to l = 12 loops. For

21

+

+ 402 + 192 102 3

H2,1,1

+ 24(H2,1,3 + H2,3,1H2,2,2 + H3,1,2 + H3,2,1) + (56 + 44)H2,2,1

+ 96(1 + 2)H2,1,1,1 + 1922 + 960 482 768

[parenrightbig]

[bracketrightbigg]

MHV hexagon leading OPE term, t[Equal]0

105

Loop order

l[Equal]6

l[Equal]5

l[Equal]4

l[Equal]3

l[Equal]2

104

1000

l[RParen1]

l h 0[LParen1]

[LParen2][Minus]1[RParen2]

100

10

1

0 1 2 3 4

s

Figure 3. Log-linear plot of the h(l)0 component, yielding the leading OPE contribution at = 0, as a function of at di erent loop orders l. Its sign is given by (1)l, as it varies continuously without

vanishing. Increasing l by one increases the magnitude by roughly a factor of 10, maintaining similar shape.

compactness, we will content ourselves with writing down only the simplest 5-loop term,

h(5)4 =e

JHEP11(2013)150

32 33 32 3 +449 6439 +2229 + 482 1629 3203 +74 540

+ 42 + 140 3

[parenrightbigg]

H1

+ H2

32 33 833 [phone] +1029 30 [parenrightbigg]

+ H3

423 + 16

29 583[parenrightbigg]+ H4(4 + 2) + 4H5

+ H1,1

32 3 6439 + 642 1629 192 +1623 +3203[parenrightbigg]

+

6423 64 + 162 9

H1,2 +

6423 32 [phone]

H2,1

+

323 160 3

H1,3 +

642 256 +1623 + 192

H1,1,1

+

323 32

H2,2 +

643 128 3

H3,1 323 H1,4 64(H1,1,2 + H1,2,1)

32(2 + 1)H2,1,1 + 256(1 )H1,1,1,1

64

3 (H2,3 + H3,2)

160

3 (H1,1,3 + H1,3,1) 32(H2,1,2 + H2,2,1 + H1,2,2)

64H3,1,1 + 256H1,1,1,1,1

[bracketrightbigg]

+ ( ! ) .

(4.8)

We also present a plot of the full 6-loop O(e) term for the MHV hexagon in gure 2, as

well as a comparison of the h(l)0() component at di erent loop orders in gure 3.

22

As mentioned in the introductory remarks of this subsection, the rest of our results at 5 loops and beyond are included in the ancillary les accompanying this article. In more detail, the Mathematica implementation of the algorithm is included in algorithm.m, the full single-particle OPE contribution up to 6 loops in MHV full.m, and the part of the latter with leading and subleading powers of up to 12 loops in MHV leadtau.m and MHV subleadtau.m respectively.

4.3 New predictions: NMHV hexagon

Similarly, we employed our algorithm implementation in order to compute all O(e) contributions in the OPE of the NMHV hexagon component (2.23) up to 6 loops, and all of its leading and subleading powers of up to 12 loops. Also in this case, our results agreed with the previous indirect computation of [18], contained in the le Fnl.nb attached to the latter paper, up to 3 loops.

Of our new results, for reasons of space we will only mention here three out of the ve single-particle contributions at 4 loops, saving the rest for the attached ancillary les. After we rescale the overall -dependent factor in (3.30), and also shift the loop index so as to measure the total powers of g2 in (2.23),

h(l+1)n() = 1

2 cosh F (l)n() , (4.9)

the aforementioned component of the NMHV ratio function will be given by

R(6134) =

JHEP11(2013)150

e

2 cosh

1

Xl=0g2l

l

Xn=0nF (l)n() + O(e2) , (4.10)

where

F (4)4 =

16

9 H1 18 3 + 43 + 2

[parenrightbig]

+ 16

9 H2 122 + 2

[parenrightbig]

+ 32

3 H3

32

3 H4

64(H1,2 + H2,1) +

16

3 122 + 2

[parenrightbig]

H1,1 256H1,1,1 32H2,2

160

3 (H1,3 + H3,1) + 256H1,1,1,1 +

49 24 3 + 4

[parenrightbig]

+ 222

9 +

74540 , (4.11)

F (4)3 = H1

224 33 +1649 +248229 +3194135[parenrightbigg]+ H2

224 33 3233 242[parenrightbigg]

64

3 H4 56H5 + H3

3223 11229[parenrightbigg]+ H1,1

288 3 44839 88029[parenrightbigg]

+

44823 + 1122 9

(H1,2 + H2,1) + 5762 + 1122

[parenrightbig]

H1,1,1 + 32

3 (H1,3 + H3,1)

576(H1,1,2 + H1,2,1 + H2,1,1)

416

3 H2,2 2560H1,1,1,1

448

3 (H2,3 + H3,2)

224

3 (H1,4 + H4,1)

1120

3 H1,1,3 224(H1,2,2 + H2,1,2 + H2,2,1)

1120

3 (H1,3,1 + H3,1,1) + 2560H1,1,1,1,1

16 32

3

762 3

9

248 5

444

45 , (4.12)

23

3

32

9 23

16 32 1242 33 376 5 1129 23 18449[parenrightbigg]

+ H2 48 3 + 443 +98223 +5034180[parenrightbigg]+ H3 16 3 823[parenrightbigg]

64H5

F (4)2 =H1

322H4

3 42H6 + 1682 182

[parenrightbig]

H2,2 136H3,3

304 3 12833 36823[parenrightbigg]

+ H1,1,1 1248 3 1923 5602 [parenrightbig]

+ H1,1 224 3 + 1643 +424223 +979445[parenrightbigg]+

322 1522 3

+ (H1,2 + H2,1)

JHEP11(2013)150

(H1,3 + H3,1)

+

6082 + 1522 3

(H1,1,2 + H1,2,1 + H2,1,1) + 24962 + 5922

[parenrightbig]

H1,1,1,1

96(H1,4 + H4,1) 64(H2,3 + H3,2) 32(H1,1,3 + H1,3,1 + H3,1,1)

640(H1,2,2 + H2,1,2 + H2,2,1) 2496(H1,1,1,2 + H1,1,2,1 + H1,2,1,1 + H2,1,1,1)

11520H1,1,1,1,1 88(H1,5 + H5,1)

124(H2,4 + H4,2) 224(H1,1,4 + H1,4,1 + H4,1,1)

528(H1,2,3 + H1,3,2 + H2,1,3 + H2,3,1 + H3,1,2 + H3,2,1)

1440(H1,1,1,3 + H1,1,3,1 + H1,3,1,1 + H3,1,1,1) 528H2,2,2

832(H1,1,2,2+H1,2,1,2+H1,2,2,1+H2,1,1,2 + H2,1,2,1 + H2,2,1,1) + 11520H1,1,1,1,1,1

16 33

224

30142

1016

168 . (4.13)

Notice in particular the symmetry under permutations of all HPL integer indices, as a result of ordering the independent summation ranges of the harmonic numbers in (3.20) in all possible ways, so at to express them in terms of HPLs. In the MHV case the rst HPL index is special due to the presence of inverse powers of k, k 1, however the permutation

symmetry still holds for the remaining indices.

Plots of the full 5 and 6loop O(e) OPE contribution to the NMHV ratio function

component are presented in gure 4. Analytic expressions for all F (l)n terms up to l = 6 loops may be found in the le NMHV full.m, and the F (l)l and F (l)l1 for 7 l 12 may be

found in NMHV leadtau.m and NMHV subleadtau.m respectively.

5 Conclusions and outlook

In this paper, we explored the implications of the recent conjecture [17, 18], which adds signicant new ingredients to the OPE approach to null polygonal Wilson loops, for the case of the MHV and NMHV hexagon. We found that the integrability underlying this

24

3 + 122 3 + 64 5 4 23 +

9 +

45 +

JHEP11(2013)150

(a) (b)

Figure 4. Plots of the leading OPE contribution F (l)

Pln=0 nF (l)n() to the NMHV ratio function component (4.10) at l = 5, 6 loops, as a function of , . As the functions change sign, we have also included the F (l) = 0 plane for comparison.

proposal, and manifesting itself in the presence of polygamma functions in the energy and higher conserved charges of the ux tube,19 heavily constrains the conformally invariant functions parametrizing the hexagon: It implies that the leading OPE contribution for these functions is expressed in terms of harmonic polylogarithms with specic kinematical dependence, at any loop order in the weak coupling expansion!

This general result is consistent with the evidence based on a particular d log form for the all-loop integrand, and presented in [23], that a basis of multiple (or Goncharov) polylogarithms is su cient for describing the MHV and NMHV hexagon. In particular, harmonic polylogarithms are a single-variable subset of multiple polylogarithms. The basis of harmonic polylogarithms we found also agrees with the ansatze assumed in [18] in order to compute (parts of) the leading OPE contribution up to 3 loops for the NMHV case, and up to 4 for the MHV, thus further supporting their validity.

Moreover, our analysis has important consequences in the practical front of extracting data for the hexagon at higher orders in the weak coupling expansion. Starting from the Fourier integral form of the aforementioned OPE contribution as predicted by [18], our proof of general structure forms an algorithm for the direct computation of these integrals, in principle at any loop order. By implementing this algorithm in Mathematica, we were able to obtain new results for the full O(e) term in the near-collinear limit of the MHV

remainder function and NMHV ratio function up to 6 loops, and the part of this term with the leading and subleading powers of up to 12 loops. We include these results in the

19In [52], the hamiltonian density of the Heisenberg XXX, or SL(2), spin chain of spin s was found to be a digamma function of the operator measuring the total spin of the adjacent sites. A similar expression for the hamiltonian density of the P SU(2, 2|4) spin chain, giving rise to the complete 1-loop dilatation operator

of N = 4 SYM, was found in [53] by appropriate identication of an SL(2) subsector. The generalization to

the all-loop dilatation operator relies on a deformation of the spectral parameter, which loosely speaking is responsible for the appearance of polygamma functions.

25

attached ancillary les, together with an implementation of the algorithm, which can be readily used to evaluate higher-order contributions as well.

Input from the Wilson loop OPE has been crucial, together with other reasonable assumptions, for the recent determination of the full 3-loop MHV remainder function in general kinematics [22]. The authors of the latter paper report that the same methodology may be used for obtaining the remainder function at 4 loops and higher, and we hope that the data presented in this paper will again prove useful in that respect. The same applies for the NMHV ratio function, which is currently known to 2 loops [54]. Furthermore, it has been argued in [22], that the hexagon MHV remainder and NMHV ratio functions are described by hexagon functions, a more restricted class of combinations of multiple polylogarithms with proper branch cuts. It would be an interesting consistency check to investigate whether their near-collinear limit reduces to the basis (3.28) at arbitrary loop order.

There is a number of exciting open questions which require further inquiry. Reconstruction of the full hexagon Wilson loop from its OPE would require obtaining all terms with higher powers of e in the near-collinear limit, which are contributions of multi-particle states propagating in the ux tube. Indeed, it is possible to extend the current framework in order to incorporate 2-particle excitations [21] and higher, and one would ideally like to nd a general basis/method to calculate the resulting integrals for these terms as well. Note that even though the dimensionality of the integrals will be equal to the excitation number, data from the 3-loop hexagon remainder function show that the e2 terms are described by harmonic polylogarithms as well [22]. Navely similar multidimensional integrals, which are however not expected to only yield HPLs, appear in the near-collinear limit of the heptagon. For the latter, very little is known beyond the 2-loop total di erential [55, 56] and motivic avatar structure [57] for the MHV case, and symbol for the NMHV case [16].

Aside the near-collinear limit, a similar all-loop integral formula for the hexagon remainder function also exists in the multi-Regge limit [10, 11]. Based on evidence that the relevant class of functions for describing this limit are certain combinations of HPLs that are single-valued on the complex plane, the authors of [48] were able to extract the (next-to-)leading-logarithmic part of the aforementioned formula to high loop order. Given the resemblance of the integral formulas in the near-collinear and multi-Regge limit, it should be possible to extend our method in order to rigorously prove the appropriateness of the single-valued HPLs, and possibly the conjecture [58] generalizing the results of [48] for the leading-lorarithmic part to all loops. More broadly, the similarities between the two limits seem to be quite extensive, and suggest that a physical picture based on integrability should exist for the multi-Regge limit as well. For example, the main quantities entering the master formula of [10, 11] only depend on polygamma functions up to the currently known order.

Finally, our work may prove useful for the computation of other observables of N = 4

super Yang-Mills theory. In [59, 60], a very interesting class of infrared-nite weighted cross-sections was introduced and extensively studied. These measure the total ow of charge registered by detectors positioned at spatial innity, for an initial state generated by the scalar half-BPS operator acting on the vacuum. For the choice of charges considered

26

JHEP11(2013)150

in the latter papers, and for two detectors measuring them at di erent directions simultaneously, these were shown to be related to 4-point correlation functions of components of the stress-tensor multiplet. In particular, they are given by the convolution of the Mellin transform of the Euclidean correlators with a coupling-independent detector kernel. The resulting inverse Mellin integral was computed from its known ingredients at one loop, and a generalization of our method may be applicable for performing the computation at higher loops, given the relation between Fourier and Mellin transforms,

1

2

[integraldisplay]

+1

1

ds(ex)sf(is) . (5.1)

We expect that again harmonic polylogarithms will play a role in describing the dependence of these double ow correlations on the single variable parametrizing them, the angle between the two detectors.

Acknowledgments

Wed like to thank Marcus Spradlin, Christian Vergu and Anastasia Volovich for collaboration at the early stages of this project. We are also grateful to Benjamin Basso, James Drummond and Emery Sokatchev for enlightening discussions and comments on the manuscript. This work was supported in part by the US Department of Energy under Grant No. DE-FG02-97ER-41029 at the University of Florida, and the French National Agency for Research (ANR) under contract StrongInt (BLANC-SIMI-4-2011) at LAPTh.

A Review of useful functions

A.1 The functionsFor completeness, we review the functions ?[notdef] [13] which enter in the expressions (2.11)

and (2.24), for the anomalous dimensions and momenta of the ux tube excitations propagating between the two parts of the polygon.

These are dened as

?(t) = 2

1

where the coe cients ?n depend on g and obey

?n + [integraldisplay]

= 2g n,1 . (A.2)

The last two equations can be solved perturbatively in g 1, with the help of the following

Taylor explansion of the i-th Bessel function of the rst kind Ji(z) around z = 0,

Ji(z) =

In any case, from the latter formula it is evident that for small t, ?[notdef] have a regular

Taylor expansion.

27

dxeipxf(p) = 1

[integraldisplay]

2i

+i1

i1

JHEP11(2013)150

1

Xn=1(2n 1) ?2n1J2n1(t) ,

?+(t) = 2

(A.1)

Xn=1 (2n) ?2nJ2n1(t) ,

dt

t Jn(2gt)

?+(2gt) (1)n ?(2gt)

et 1

1

0

Xn=0(1)n k! (i + n + 1)

z 2

1

i+2n

, (A.3)

A.2 The fi functions

In this appendix, we review the computation of the fi functions [18] appearing in the integration measure (2.13), (2.25) of the leading (N)MHV hexagon OPE contribution.

The functions in question are dened as

f1(u, v) = 2 ~

(u) [notdef] Q [notdef] M [notdef] (v) , f2(u, v) = 2 ~

(v) [notdef] Q [notdef] M [notdef] (v) ,

f3(u, v) = 2 ~

(u) [notdef] Q [notdef] M [notdef] ~

(v) , f4(u, v) = 2 (v) [notdef] Q [notdef] M [notdef] (v) ,

(A.4)

where the , ~

are vectors with elements

j(u) [integraldisplay]

JHEP11(2013)150

dt t

1 Jj(2gt)(J0(2gt) cos(ut)

et/2 (1) [notdef]j ) et 1

~

j(u)

(A.5)

with = 0 for the NMHV case and = 1 for MHV case, and Q, M are matrices independent of u. In particular, Q has matrix elements, Qij = ij(1)i+1i, and M is related to another matrix K,

M (1 + K)1 =

1

0

[integraldisplay]

dt

t (1)j+1

1

0

Jj(2gt) sin(ut)

(1) [notdef](j+1) et 1

et/2

Xn=0 (K)n ,

(A.6)

The Taylor expansion of the Bessel functions Ji (A.3) implies that Kij starts at order O(gi+j). Hence even though the vectors and matrices entering (A.4) are innite-

dimensional (equivalently they are an innite triple sum), if we wish to obtain fi up to

O(gn), we can truncate them to their rst i, j = 1, . . . , n 1 components.

The building blocks of the fi functions can also be used in order to calculate all remaining ingredients of the leading OPE contribution for the (N)MHV hexagon. Namely the anomalous dimension and momentum of the single-particle ux tube excitations can be written as

(u) = 4g (Q [notdef] M [notdef] (u))1 , p(u) = 2u 4g (Q [notdef] M [notdef] ~

(u))1 , (A.7)

Kij = 2j(1)j(i+1) [integraldisplay]

1

0

dt t

Ji(2gt)Jj(2gt) et 1

.

and the cusp anomalous dimension as

cusp = 4g(Q [notdef] M)11 . (A.8)

The expressions (A.7), which also implicitly depend on by virtue of (A.5), are equivalent to the ones used in the main text (2.11), (2.24), and can be derived from them.

A.3 Harmonic polylogarithms

In this appendix, we review the denition and basic properties of harmonic polylogarithms (HPL) [25], also based on [50, 61].

28

For x 2 (0, 1) and ai = [notdef]1, 0, 1[notdef], harmonic polylogarithms are dened asH(x) = 1 ,

H(a1, . . . , an; x) =

(A.9)

where the auxiliary functions fa entering the case of the recursive denition are given by

f1(x) =

11 + x ,

f0(x) = 1 x ,

f1(x) = 1

8

>

>

<

>

>

:

1n! logn x if a1 = . . . an = 0 ,

[integraldisplay]

x

0 dyfa1(t)H(a2, . . . , an; t) otherwise,

(A.10)

The number of ai indices, or the number of integrations when the recursive denition applies, is called the weight or transcendentality of the harmonic polylogarithm. Obviously at weight n we have 3n distinct (but not all functionally independent) HPLs. For example for weight zero we just have H(x) = 1, and for weight 1,

H(1, x) =

[integraldisplay]

x

JHEP11(2013)150

.

1 x

0 f1(t) = [integraldisplay]

x

11 + x = log(1 + x) ,

H(0, x) = log x ,

H(1, x) = [integraldisplay]

x

0

(A.11)

Up to weight 3, HPLs can be expressed in terms of classical polylogarithms of more general arguments [25]. However for weight 4 and higher, the latter only form a subset of the wider basis of HPL functions.

A di erent, more compact notation for the indices of HPLs may be obtained by replacing any string of m 1 subsequent zeros followed by a [notdef]1 index as follows,

m1

[bracehtipdownleft] [bracehtipupright][bracehtipupleft] [bracehtipdownright]

0 f1(t) = [integraldisplay]

x

11 x

= log(1 x) .

0

0, 0, . . . 0, [notdef]1 ! [notdef]m . (A.12)

For example in the new notation, which is sometimes is refered to as m-notation as opposed to the initial a-notation,

H(1, 0, 0, 1, 0, 1, 0; x) = H1,3,2,0(x) , (A.13)

In the main text, we will be exclusively using the m-notation.

Similarly to Z-sums, a product of two HPLs with the same argument can be expressed as a linear combination single HPLs. In a-notation, if we denote the vector of indices (a1, . . . , an) a, we have in particular

H(a; x)H(b; x) =

Xc=a

9

bH(c, x) (A.14)

29

9 b denotes all possible permutations of the elements of a and b combined, such that the internal order of the elements in a and b is preserved.20 For example

H(a1, a2; x)H(b1, b2; x) = H(a1, a2, b1, b2; x) + H(a1, b1, a2, b2; x) + H(b1, a1, a2, b2; x)

+ H(b1, a1, b2, a2; x)+H(b1, b2, a1, a2; x) + H(a1, b1, b2, a2; x) ,(A.15)

and more generally it is not hard to see that the product of two HPLs with weights (number of indices) w1 and w2 will decompose into a linear combination of (w1 + w2)!/(w1!w2!)

single HPLs.

Precisely because of the shu e algebra, not all HPLs of a given weight will be algebraically independent, as they can be expressed in terms of other HPLs of the same weight, and products of lower weights. In any case it is always possible to construct a basis of algebraically independent HPLs, whose indices form Lyndon words [62], although we will refrain from exploiting this property here.

Finally, an argument transformation property for HPLs with nonzero indices in the m-notation, which we will be making use of, is that

Hm1,...,mk (x) = (1)kHm1,...,mk (x) , for mk [negationslash]= 0 . (A.16)

This follows simply from the denition (A.9), after we change the sign of the integration variables.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

[1] L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, http://dx.doi.org/10.1088/1126-6708/2007/06/064

Web End =JHEP 06 http://dx.doi.org/10.1088/1126-6708/2007/06/064

Web End =(2007) 064 [http://arxiv.org/abs/0705.0303

Web End =arXiv:0705.0303 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0705.0303

Web End =INSPIRE ].

[2] J. Drummond, G. Korchemsky and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops, http://dx.doi.org/10.1016/j.nuclphysb.2007.11.041

Web End =Nucl. Phys. B 795 (2008) 385 [http://arxiv.org/abs/0707.0243

Web End =arXiv:0707.0243 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0707.0243

Web End =INSPIRE ].

[3] A. Brandhuber, P. Heslop and G. Travaglini, MHV amplitudes in N = 4 super Yang-Mills and Wilson loops, http://dx.doi.org/10.1016/j.nuclphysb.2007.11.002

Web End =Nucl. Phys. B 794 (2008) 231 [http://arxiv.org/abs/0707.1153

Web End =arXiv:0707.1153 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0707.1153

Web End =INSPIRE ].

[4] J. Drummond, J. Henn, V. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, http://dx.doi.org/10.1088/1126-6708/2007/01/064

Web End =JHEP 01 (2007) 064 [http://arxiv.org/abs/hep-th/0607160

Web End =hep-th/0607160 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0607160

Web End =INSPIRE ].

[5] J. Drummond, Review of AdS/CFT integrability. Chapter V.2: dual superconformal symmetry, http://dx.doi.org/10.1007/s11005-011-0519-4

Web End =Lett. Math. Phys. 99 (2012) 481 [http://arxiv.org/abs/1012.4002

Web End =arXiv:1012.4002 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.4002

Web End =INSPIRE ].

[6] Z. Bern, L.J. Dixon and V.A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, http://dx.doi.org/10.1103/PhysRevD.72.085001

Web End =Phys. Rev. D 72 (2005) 085001 [http://arxiv.org/abs/hep-th/0505205

Web End =hep-th/0505205 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0505205

Web End =INSPIRE ].

20This operation of element mixing is called a shu e, and the corresponding product algebra the shu e algebra, since it is equivalent to all possible ways to ri e shu e two decks of cards.

30

where a

JHEP11(2013)150

[7] J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Conformal Ward identities for Wilson loops and a test of the duality with gluon amplitudes, http://dx.doi.org/10.1016/j.nuclphysb.2009.10.013

Web End =Nucl. Phys. B 826 (2010) 337 [http://arxiv.org/abs/0712.1223

Web End =arXiv:0712.1223 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0712.1223

Web End =INSPIRE ].

[8] V. Del Duca, C. Duhr and V.A. Smirnov, The two-loop hexagon Wilson loop in N = 4 SYM, http://dx.doi.org/10.1007/JHEP05(2010)084

Web End =JHEP 05 (2010) 084 [http://arxiv.org/abs/1003.1702

Web End =arXiv:1003.1702 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1003.1702

Web End =INSPIRE ].

[9] A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical polylogarithms for amplitudes and Wilson loops, http://dx.doi.org/10.1103/PhysRevLett.105.151605

Web End =Phys. Rev. Lett. 105 (2010) 151605 [http://arxiv.org/abs/1006.5703

Web End =arXiv:1006.5703 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1006.5703

Web End =INSPIRE ].

[10] L. Lipatov and A. Prygarin, BFKL approach and six-particle MHV amplitude in N = 4 super Yang-Mills, http://dx.doi.org/10.1103/PhysRevD.83.125001

Web End =Phys. Rev. D 83 (2011) 125001 [http://arxiv.org/abs/1011.2673

Web End =arXiv:1011.2673 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1011.2673

Web End =INSPIRE ].

[11] V. Fadin and L. Lipatov, BFKL equation for the adjoint representation of the gauge group in the next-to-leading approximation at N = 4 SUSY, http://dx.doi.org/10.1016/j.physletb.2011.11.048

Web End =Phys. Lett. B 706 (2012) 470 [http://arxiv.org/abs/1111.0782

Web End =arXiv:1111.0782 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1111.0782

Web End =INSPIRE ].

[12] L.F. Alday, D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, An operator product expansion for polygonal null Wilson loops, http://dx.doi.org/10.1007/JHEP04(2011)088

Web End =JHEP 04 (2011) 088 [http://arxiv.org/abs/1006.2788

Web End =arXiv:1006.2788 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1006.2788

Web End =INSPIRE ].

[13] B. Basso, Exciting the GKP string at any coupling, http://dx.doi.org/10.1016/j.nuclphysb.2011.12.010

Web End =Nucl. Phys. B 857 (2012) 254 [http://arxiv.org/abs/1010.5237

Web End =arXiv:1010.5237 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1010.5237

Web End =INSPIRE ].

[14] N. Beisert et al., Review of AdS/CFT integrability: an overview, http://dx.doi.org/10.1007/s11005-011-0529-2

Web End =Lett. Math. Phys. 99 (2012) http://dx.doi.org/10.1007/s11005-011-0529-2

Web End =3 [http://arxiv.org/abs/1012.3982

Web End =arXiv:1012.3982 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.3982

Web End =INSPIRE ].

[15] L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, http://dx.doi.org/10.1007/JHEP11(2011)023

Web End =JHEP 11 http://dx.doi.org/10.1007/JHEP11(2011)023

Web End =(2011) 023 [http://arxiv.org/abs/1108.4461

Web End =arXiv:1108.4461 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1108.4461

Web End =INSPIRE ].

[16] S. Caron-Huot and S. He, Jumpstarting the all-loop S-matrix of planar N = 4 super Yang-Mills, http://dx.doi.org/10.1007/JHEP07(2012)174

Web End =JHEP 07 (2012) 174 [http://arxiv.org/abs/1112.1060

Web End =arXiv:1112.1060 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1112.1060

Web End =INSPIRE ].

[17] B. Basso, A. Sever and P. Vieira, Space-time S-matrix and ux-tube S-matrix at nite coupling, http://dx.doi.org/10.1103/PhysRevLett.111.091602

Web End =Phys. Rev. Lett. 111 (2013) 091602 [http://arxiv.org/abs/1303.1396

Web End =arXiv:1303.1396 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1303.1396

Web End =INSPIRE ].

[18] B. Basso, A. Sever and P. Vieira, Space-time S-matrix and ux tube S-matrix II. Extracting and matching data, http://arxiv.org/abs/1306.2058

Web End =arXiv:1306.2058 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1306.2058

Web End =INSPIRE ].

[19] B. Basso and A. Rej, Bethe ansaetze for GKP strings, http://arxiv.org/abs/1306.1741

Web End =arXiv:1306.1741 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1306.1741

Web End =INSPIRE ].

[20] D. Fioravanti, S. Piscaglia and M. Rossi, On the scattering over the GKP vacuum, http://arxiv.org/abs/1306.2292

Web End =arXiv:1306.2292 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1306.2292

Web End =INSPIRE ].

[21] B. Basso, A. Sever and and P. Vieira, Space-time S-matrix and ux-tube S-matrix III. The two-particle contributions, to appear.

[22] L.J. Dixon, J.M. Drummond, M. von Hippel and J. Pennington, Hexagon functions and the three-loop remainder function, http://arxiv.org/abs/1308.2276

Web End =arXiv:1308.2276 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1308.2276

Web End =INSPIRE ].

[23] N. Arkani-Hamed et al., Scattering amplitudes and the positive grassmannian, http://arxiv.org/abs/1212.5605

Web End =arXiv:1212.5605 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1212.5605

Web End =INSPIRE ].

[24] S. Moch, P. Uwer and S. Weinzierl, Nested sums, expansion of transcendental functions and multiscale multiloop integrals, http://dx.doi.org/10.1063/1.1471366

Web End =J. Math. Phys. 43 (2002) 3363 [http://arxiv.org/abs/hep-ph/0110083

Web End =hep-ph/0110083 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0110083

Web End =INSPIRE ].

[25] E. Remiddi and J. Vermaseren, Harmonic polylogarithms, http://dx.doi.org/10.1142/S0217751X00000367

Web End =Int. J. Mod. Phys. A 15 (2000) http://dx.doi.org/10.1142/S0217751X00000367

Web End =725 [http://arxiv.org/abs/hep-ph/9905237

Web End =hep-ph/9905237 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/9905237

Web End =INSPIRE ].

[26] J. Drummond, J. Henn, G. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in N = 4 super-Yang-Mills theory, http://dx.doi.org/10.1016/j.nuclphysb.2009.11.022

Web End =Nucl. Phys. B 828 (2010) 317 [http://arxiv.org/abs/0807.1095

Web End =arXiv:0807.1095 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0807.1095

Web End =INSPIRE ].

31

JHEP11(2013)150

[27] A. Sever, P. Vieira and T. Wang, OPE for super loops, http://dx.doi.org/10.1007/JHEP11(2011)051

Web End =JHEP 11 (2011) 051 [http://arxiv.org/abs/1108.1575

Web End =arXiv:1108.1575 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1108.1575

Web End =INSPIRE ].

[28] A. Sever, P. Vieira and T. Wang, From polygon Wilson loops to spin chains and back, http://dx.doi.org/10.1007/JHEP12(2012)065

Web End =JHEP http://dx.doi.org/10.1007/JHEP12(2012)065

Web End =12 (2012) 065 [http://arxiv.org/abs/1208.0841

Web End =arXiv:1208.0841 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1208.0841

Web End =INSPIRE ].

[29] L. Mason and D. Skinner, The complete planar S-matrix of N = 4 SYM as a Wilson loop in twistor space, http://dx.doi.org/10.1007/JHEP12(2010)018

Web End =JHEP 12 (2010) 018 [http://arxiv.org/abs/1009.2225

Web End =arXiv:1009.2225 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1009.2225

Web End =INSPIRE ].

[30] S. Caron-Huot, Notes on the scattering amplitude/Wilson loop duality, http://dx.doi.org/10.1007/JHEP07(2011)058

Web End =JHEP 07 (2011) 058 [http://arxiv.org/abs/1010.1167

Web End =arXiv:1010.1167 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1010.1167

Web End =INSPIRE ].

[31] A. Belitsky, Conformal anomaly of super Wilson loop, http://dx.doi.org/10.1016/j.nuclphysb.2012.04.022

Web End =Nucl. Phys. B 862 (2012) 430 [http://arxiv.org/abs/1201.6073

Web End =arXiv:1201.6073 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1201.6073

Web End =INSPIRE ].

[32] A. Belitsky, A note on two-loop superloop, http://dx.doi.org/10.1016/j.physletb.2012.10.018

Web End =Phys. Lett. B 718 (2012) 205 [http://arxiv.org/abs/1207.1924

Web End =arXiv:1207.1924 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1207.1924

Web End =INSPIRE ].

[33] A. Belitsky and S. Caron-Huot, Superpropagator and superconformal invariants, http://dx.doi.org/10.1016/j.physletb.2012.11.018

Web End =Phys. Lett. http://dx.doi.org/10.1016/j.physletb.2012.11.018

Web End =B 718 (2013) 1083 [http://arxiv.org/abs/1209.0224

Web End =arXiv:1209.0224 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1209.0224

Web End =INSPIRE ].

[34] L.F. Alday, B. Eden, G.P. Korchemsky, J. Maldacena and E. Sokatchev, From correlation functions to Wilson loops, http://dx.doi.org/10.1007/JHEP09(2011)123

Web End =JHEP 09 (2011) 123 [http://arxiv.org/abs/1007.3243

Web End =arXiv:1007.3243 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1007.3243

Web End =INSPIRE ].

[35] B. Eden, G.P. Korchemsky and E. Sokatchev, From correlation functions to scattering amplitudes, http://dx.doi.org/10.1007/JHEP12(2011)002

Web End =JHEP 12 (2011) 002 [http://arxiv.org/abs/1007.3246

Web End =arXiv:1007.3246 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1007.3246

Web End =INSPIRE ].

[36] B. Eden, G.P. Korchemsky and E. Sokatchev, More on the duality correlators/amplitudes, http://dx.doi.org/10.1016/j.physletb.2012.02.014

Web End =Phys. Lett. B 709 (2012) 247 [http://arxiv.org/abs/1009.2488

Web End =arXiv:1009.2488 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1009.2488

Web End =INSPIRE ].

[37] B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, The super-correlator/super-amplitude duality: part I, http://dx.doi.org/10.1016/j.nuclphysb.2012.12.015

Web End =Nucl. Phys. B 869 (2013) 329 [http://arxiv.org/abs/1103.3714

Web End =arXiv:1103.3714 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1103.3714

Web End =INSPIRE ].

[38] B. Eden, P. Heslop, G.P. Korchemsky and E. Sokatchev, The super-correlator/super-amplitude duality: part II, http://dx.doi.org/10.1016/j.nuclphysb.2012.12.014

Web End =Nucl. Phys. B 869 (2013) 378 [http://arxiv.org/abs/1103.4353

Web End =arXiv:1103.4353 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1103.4353

Web End =INSPIRE ].

[39] A. Belitsky, G. Korchemsky and E. Sokatchev, Are scattering amplitudes dual to super Wilson loops?, http://dx.doi.org/10.1016/j.nuclphysb.2011.10.014

Web End =Nucl. Phys. B 855 (2012) 333 [http://arxiv.org/abs/1103.3008

Web End =arXiv:1103.3008 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1103.3008

Web End =INSPIRE ].

[40] V. Braun, G. Korchemsky and D. Mueller, The uses of conformal symmetry in QCD, http://dx.doi.org/10.1016/S0146-6410(03)90004-4

Web End =Prog. http://dx.doi.org/10.1016/S0146-6410(03)90004-4

Web End =Part. Nucl. Phys. 51 (2003) 311 [http://arxiv.org/abs/hep-ph/0306057

Web End =hep-ph/0306057 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0306057

Web End =INSPIRE ].

[41] D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, http://dx.doi.org/10.1007/JHEP12(2011)011

Web End =JHEP 12 http://dx.doi.org/10.1007/JHEP12(2011)011

Web End =(2011) 011 [http://arxiv.org/abs/1102.0062

Web End =arXiv:1102.0062 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1102.0062

Web End =INSPIRE ].

[42] L. Freyhult, Review of AdS/CFT integrability. Chapter III.4: twist states and the cusp anomalous dimension, http://dx.doi.org/10.1007/s11005-011-0483-z

Web End =Lett. Math. Phys. 99 (2012) 255 [http://arxiv.org/abs/1012.3993

Web End =arXiv:1012.3993 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1012.3993

Web End =INSPIRE ].

[43] M. Gnaydin and N. Marcus, The spectrum of the S5 compactication of the chiral N = 2, D = 10 supergravity and the unitary supermultiplets of U(2, 2/4), http://dx.doi.org/10.1088/0264-9381/2/2/001

Web End =Class. Quant. Grav. 2 http://dx.doi.org/10.1088/0264-9381/2/2/001

Web End =(1985) L11 [http://inspirehep.net/search?p=find+J+Class.Quant.Grav.,2,L11

Web End =INSPIRE ].

[44] B. Eden and M. Staudacher, Integrability and transcendentality, http://dx.doi.org/10.1088/1742-5468/2006/11/P11014

Web End =J. Stat. Mech. 0611 (2006) http://dx.doi.org/10.1088/1742-5468/2006/11/P11014

Web End =P11014 [http://arxiv.org/abs/hep-th/0603157

Web End =hep-th/0603157 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0603157

Web End =INSPIRE ].

[45] J. Vermaseren, Harmonic sums, Mellin transforms and integrals, http://dx.doi.org/10.1142/S0217751X99001032

Web End =Int. J. Mod. Phys. A 14 http://dx.doi.org/10.1142/S0217751X99001032

Web End =(1999) 2037 [http://arxiv.org/abs/hep-ph/9806280

Web End =hep-ph/9806280 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/9806280

Web End =INSPIRE ].

[46] J. Blumlein and S. Kurth, Harmonic sums and Mellin transforms up to two loop order, http://dx.doi.org/10.1103/PhysRevD.60.014018

Web End =Phys. http://dx.doi.org/10.1103/PhysRevD.60.014018

Web End =Rev. D 60 (1999) 014018 [http://arxiv.org/abs/hep-ph/9810241

Web End =hep-ph/9810241 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/9810241

Web End =INSPIRE ].

32

JHEP11(2013)150

[47] G. Papathanasiou, Evaluating the Hexagon Wilson loop OPE, presentation at https://dl.dropbox.com/u/4015110/Papathanasiou_WilsonLoopOPE.nb

Web End =Symbolic https://dl.dropbox.com/u/4015110/Papathanasiou_WilsonLoopOPE.nb

Web End =Computation in Theoretical Physics School (2012).

[48] L.J. Dixon, C. Duhr and J. Pennington, Single-valued harmonic polylogarithms and the multi-Regge limit, http://dx.doi.org/10.1007/JHEP10(2012)074

Web End =JHEP 10 (2012) 074 [http://arxiv.org/abs/1207.0186

Web End =arXiv:1207.0186 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1207.0186

Web End =INSPIRE ].

[49] D. Matre, HPL, a Mathematica implementation of the harmonic polylogarithms, http://dx.doi.org/10.1016/j.cpc.2005.10.008

Web End =Comput. http://dx.doi.org/10.1016/j.cpc.2005.10.008

Web End =Phys. Commun. 174 (2006) 222 [http://arxiv.org/abs/hep-ph/0507152

Web End =hep-ph/0507152 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0507152

Web End =INSPIRE ].

[50] D. Matre, Extension of HPL to complex arguments, http://dx.doi.org/10.1016/j.cpc.2011.11.015

Web End =Comput. Phys. Commun. 183 (2012) http://dx.doi.org/10.1016/j.cpc.2011.11.015

Web End =846 [http://arxiv.org/abs/hep-ph/0703052

Web End =hep-ph/0703052 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0703052

Web End =INSPIRE ].

[51] J. Bartels, L. Lipatov and A. Prygarin, Collinear and Regge behavior of 2 ! 4 MHV

amplitude in N = 4 super Yang-Mills theory, http://arxiv.org/abs/1104.4709

Web End =arXiv:1104.4709 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1104.4709

Web End =INSPIRE ].

[52] V. Tarasov, L. Takhtajan and L. Faddeev, Local hamiltonians for integrable quantum models on a lattice, http://dx.doi.org/10.1007/BF01018648

Web End =Theor. Math. Phys. 57 (1983) 1059 [http://inspirehep.net/search?p=find+J+Theor.Math.Phys.,57,1059

Web End =INSPIRE ].

[53] N. Beisert, The complete one loop dilatation operator of N = 4 super Yang-Mills theory, http://dx.doi.org/10.1016/j.nuclphysb.2003.10.019

Web End =Nucl. Phys. B 676 (2004) 3 [http://arxiv.org/abs/hep-th/0307015

Web End =hep-th/0307015 ] [http://inspirehep.net/search?p=find+EPRINT+hep-th/0307015

Web End =INSPIRE ].

[54] L.J. Dixon, J.M. Drummond and J.M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N = 4 super Yang-Mills theory, http://dx.doi.org/10.1007/JHEP01(2012)024

Web End =JHEP 01 (2012) 024 [http://arxiv.org/abs/1111.1704

Web End =arXiv:1111.1704 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1111.1704

Web End =INSPIRE ].

[55] S. Caron-Huot, Superconformal symmetry and two-loop amplitudes in planar N = 4 super Yang-Mills, http://dx.doi.org/10.1007/JHEP12(2011)066

Web End =JHEP 12 (2011) 066 [http://arxiv.org/abs/1105.5606

Web End =arXiv:1105.5606 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1105.5606

Web End =INSPIRE ].

[56] J. Golden and M. Spradlin, The di erential of all two-loop MHV amplitudes in N = 4

Yang-Mills theory, http://dx.doi.org/10.1007/JHEP09(2013)111

Web End =JHEP 09 (2013) 111 [http://arxiv.org/abs/1306.1833

Web End =arXiv:1306.1833 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1306.1833

Web End =INSPIRE ].

[57] J. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic amplitudes and cluster coordinates, http://arxiv.org/abs/1305.1617

Web End =arXiv:1305.1617 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1305.1617

Web End =INSPIRE ].

[58] J. Pennington, The six-point remainder function to all loop orders in the multi-Regge limit, http://dx.doi.org/10.1007/JHEP01(2013)059

Web End =JHEP 01 (2013) 059 [http://arxiv.org/abs/1209.5357

Web End =arXiv:1209.5357 ] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1209.5357

Web End =INSPIRE ].

[59] A. Belitsky, S. Hohenegger, G. Korchemsky, E. Sokatchev and A. Zhiboedov, From correlation functions to event shapes, http://arxiv.org/abs/1309.0769

Web End =arXiv:1309.0769 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1309.0769

Web End =INSPIRE ].

[60] A. Belitsky, S. Hohenegger, G. Korchemsky, E. Sokatchev and A. Zhiboedov, Event shapes in N = 4 super-Yang-Mills theory, http://arxiv.org/abs/1309.1424

Web End =arXiv:1309.1424 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1309.1424

Web End =INSPIRE ].

[61] J. Ablinger, Computer algebra algorithms for special functions in particle physics, http://arxiv.org/abs/1305.0687

Web End =arXiv:1305.0687 [http://inspirehep.net/search?p=find+EPRINT+arXiv:1305.0687

Web End =INSPIRE ].

[62] J. Blumlein, Algebraic relations between harmonic sums and associated quantities, http://dx.doi.org/10.1016/j.cpc.2003.12.004

Web End =Comput. http://dx.doi.org/10.1016/j.cpc.2003.12.004

Web End =Phys. Commun. 159 (2004) 19 [http://arxiv.org/abs/hep-ph/0311046

Web End =hep-ph/0311046 ] [http://inspirehep.net/search?p=find+EPRINT+hep-ph/0311046

Web End =INSPIRE ].

33

JHEP11(2013)150