Contents

CERN–PH–TH/2011/251     SLAC–PUB–14632      LAPTH-042/11      HU-EP-11/44

Analytic result for the two-loop six-point NMHV amplitude in super Yang-Mills theory

Lance J. Dixon, James M. Drummond
and Johannes M. Henn
SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309, USA
PH-TH Division, CERN, Geneva, Switzerland
LAPTH, Université de Savoie, CNRS, B.P. 110, F-74941 Annecy-le-Vieux Cedex, France
Humboldt-Universität zu Berlin, Newtonstraße 15, 12489 Berlin, Germany
Institute for Advanced Study, Princeton, NJ 08540, USA

lance@slac.stanford.edu ,  drummond@lapp.in2p3.fr ,  jmhenn@ias.edu

Abstract

We provide a simple analytic formula for the two-loop six-point ratio function of planar super Yang-Mills theory. This result extends the analytic knowledge of multi-loop six-point amplitudes beyond those with maximal helicity violation. We make a natural ansatz for the symbols of the relevant functions appearing in the two-loop amplitude, and impose various consistency conditions, including symmetry, the absence of spurious poles, the correct collinear behaviour, and agreement with the operator product expansion for light-like (super) Wilson loops. This information reduces the ansatz to a small number of relatively simple functions. In order to fix these parameters uniquely, we utilize an explicit representation of the amplitude in terms of loop integrals that can be evaluated analytically in various kinematic limits. The final compact analytic result is expressed in terms of classical polylogarithms, whose arguments are rational functions of the dual conformal cross-ratios, plus precisely two functions that are not of this type. One of the functions, the loop integral , also plays a key role in a new representation of the remainder function in the maximally helicity violating sector. Another interesting feature at two loops is the appearance of a new (parity odd) (parity odd) sector of the amplitude, which is absent at one loop, and which is uniquely determined in a natural way in terms of the more familiar (parity even) (parity even) part. The second non-polylogarithmic function, the loop integral , characterizes this sector. Both and can be expressed as one-dimensional integrals over classical polylogarithms with rational arguments.

## 1 Introduction

Much progress has been achieved recently in the analytic understanding of seemingly complicated scattering processes. In particular, attention has been focused on the planar sector, or large limit, of maximally supersymmetric Yang-Mills theory. The scattering amplitudes in this sector of the theory obey many startling properties, which has led to the hope that the general scattering problem might be solvable, exactly in the coupling.

One of the major simplifications that the planar theory enjoys is dual conformal symmetry [1, 2, 3, 4, 5, 6, 7], which dictates how colour-ordered amplitudes behave under conformal transformations of the dual (or region) variables defined via . For the particular case of maximally helicity-violating (MHV) amplitudes, this symmetry is intimately connected to the relation between the amplitudes and Wilson loops evaluated on polygons with light-like edges, whose vertices are located at the  [6, 3, 8, 4, 5, 9, 10, 11]. The MHV amplitudes are infrared divergent, just as the corresponding light-like Wilson loops are ultraviolet divergent. The Wilson-loop divergence has the consequence that a suitably-defined finite part transforms anomalously under the dual conformal symmetry [4, 5]. The Ward identity describing this behaviour actually fixes the form of the four-point and five-point amplitudes to all orders in the coupling, to that given by the BDS ansatz [12]. From six points onwards, the existence of dual-conformal invariant cross-ratios means that the problem of determining the MHV amplitude reduces to finding a function that depends only on the cross-ratios — the so-called ‘remainder function’, which corrects the BDS ansatz.

Great advances have been made recently in understanding the form of the remainder function, which is non-trivial beginning at two loops. The need for a two-loop remainder function for Wilson loops was observed for a large number of points in ref. [7], and for six points in ref. [9]. The multi-Regge limit of the six-point scattering amplitude also implied a non-trivial remainder function [13]. At a few generic kinematic points, the Wilson loop [11] and amplitude [10] remainder functions were found to agree numerically. The six-point Wilson loop integrals entering the remainder function were computed analytically in terms of Goncharov polylogarithms [14, 15], and then simplified down to classical polylogarithms [16] using the notion of the symbol of a pure function [17, 18, 19, 16]. The integrals contributing to the six-point MHV scattering amplitude have also been evaluated analytically [20] in a certain kinematical regime using a mass regulator [21], and the remainder function has been found to agree with the Wilson-loop expression of ref. [16]. Very recently, the symbol for the three-loop six-point remainder function was determined up to two arbitrary parameters [22], by imposing a variety of constraints, in particular the operator product expansion (OPE) for Wilson loops developed in refs. [23, 24, 25].

For more than six points, numerical results for the remainder function have been obtained via Wilson loop integrals [26, 27]. Integral representations for the MHV amplitudes have been presented at seven points [28] and for an arbitrary number of points using momentum twistors [29]. Recently, an expression for the symbol of the two-loop remainder function has been given for an arbitrary number of points [30], and the structure of the OPE for this case has been explored [31]. In special kinematics corresponding to scattering in two space-time dimensions, analytic results are available for a number of configurations at two loops [32, 33, 34] and conjecturally even at three loops [35].

When one considers amplitudes beyond the MHV sector, there is another finite dual conformally invariant quantity that one can consider, namely the ‘ratio function’ . This quantity is defined by factoring out the MHV superamplitude from the full superamplitude [36],

 A=AMHV×P. (1.1)

Infrared divergences are universal for all component amplitudes; hence the MHV factor contains all such divergences, leaving an infrared finite quantity . One of the central conjectures of ref. [36] is that is also dual conformally invariant. There is strong supporting evidence for this conjecture in the form of direct analytic one-loop results [37, 38, 39, 40, 41] and, in the six-point case, numerical evidence at two loops [42]. In this paper, we will construct the ratio function analytically at two loops for six external legs.

At tree level, the ratio function is given by a sum over dual superconformal ‘-invariants’ [36, 43, 44, 45, 46, 47]. These quantities are invariant under a much larger (infinite-dimensional) Yangian symmetry, obtained by combining invariance under both the original and dual copies of superconformal symmetry [48]. Beyond tree level one finds -invariants dressed by dual conformally invariant functions [36, 41, 42]. Since the -invariants individually exhibit spurious poles, which cannot appear in the final amplitude, they cannot appear in an arbitrary way. The particular linear combination appearing in the tree amplitude is free of spurious poles. At loop level, the absence of spurious poles implies restrictions on the dual conformally invariant functions that dress them [49]. Additional restrictions on these same functions come from the known behaviour of the amplitude when two of the external particles become collinear. These constraints will be important in our construction of at two loops.

A consequence of the duality between MHV amplitudes and light-like Wilson loops is that the remainder function can be analysed by conformal field theory methods, such as the operator product expansion (OPE) [23, 24, 25]. Various proposals have been put forward for extending the duality between amplitudes and Wilson loops beyond the MHV sector, either in terms of a supersymmetric version of the Wilson loop [50, 51, 30], or in terms of correlation functions [52, 53]. Although there may be various subtleties in realising such an object, compatible with the full supersymmetry in a Lagrangian formulation [54], one may instead justify the existence of such an object through the OPE. The framework for pursuing this approach was developed in ref. [55], and agreement was found with the known one-loop six-point next-to-MHV (NMHV) amplitude [37, 39, 40, 41]. This agreement provides non-trivial evidence that there does indeed exist a Wilson-loop quantity dual to all scattering amplitudes.

The aim of this paper is to combine various approaches in order to determine the six-point ratio function , or equivalently the NMHV amplitude, analytically at two loops. This quantity was expressed in terms of dual conformal integrals, and computed numerically, in ref. [42]. We proceed in a manner similar to our recent examination of the three-loop six-point remainder function [22]. In particular, we make an ansatz for the symbols of the various pure functions involved. (See appendix A for a brief introduction to pure functions and their symbols.) In other words, we assume that the functions that appear fall within a particular class of multi-dimensional iterated integrals, or generalized polylogarithms. We say functions rather than function because in general, beyond one loop, one can imagine that there are both (parity even)(parity even) and (parity odd)(parity odd) contributions, in a sense which we make specific in the next section. For convenience, we call these contributions ‘even’ and ‘odd’, respectively. At tree level and at one loop, the odd part vanishes.

After constructing an ansatz, the next step is to impose consistency conditions. We first impose the spurious pole and collinear conditions. Then we impose that a certain double discontinuity is compatible with the OPE [23, 24, 25, 55]. At this stage, we find that the symbols for the relevant functions contain nine unfixed parameters. We convert these symbols into explicit functions. In general, this step leads to ‘beyond-the-symbol ambiguities’. These ambiguities are associated with functions whose symbols vanish identically, namely transcendental constants, such as the Riemann values , multiplied by pure functions of lower degree. However, in the present case, after re-imposing the spurious and collinear restrictions at the level of functions, there is only one additional ambiguity, associated with adding the product of with the one-loop ratio function. This term obeys all constraints by itself and has vanishing symbol. We are thus left with a ten-dimensional space of functions. In particular, we find that the odd part is necessarily non-zero. Moreover, it is uniquely determined in terms of the even part.

In order to fix the remaining free parameters, we turn to a representation of the even part of the two-loop six-point NMHV amplitude based on loop integrals [42]. We analyse this representation, appropriately rewritten with a mass regulator [21], in the symmetric regime with all three cross-ratios equal to . In this regime, the most cumbersome double-pentagon integrals can be traded for the MHV remainder function, plus simpler integrals. This observation allows us to perform an analytic expansion for small and large . Comparing these expansions with the ansatz, we are able to match them, precisely fixing all remaining free parameters. The fact that the ansatz agrees with the expansion of the loop-integral calculation in this regime is a highly non-trivial cross check, since an entire function is matched by an ansatz with just a few free parameters. Further confirmation that our result is correct comes from comparing with a numerical evaluation [42] at a particular asymmetric kinematical point. This latter check also confirms the expectation that is defined independently of any infrared regularization scheme. See also ref. [56] for a recent discussion of different infrared regularizations and regularization-scheme independence.

In contrast to the the two-loop six-point MHV amplitude [16], the two-loop six-point ratio function cannot quite be expressed in terms of classical polylogarithms. Two additional functions appear, one in the even part and one in the odd part. However, these functions have a very simple structure: we can write them as simple one-dimensional integrals over classical polylogarithmic functions of degree three. The even part of the ratio function can be written in terms of single-variable polylogarithmic functions whose arguments are rational in the three cross-ratios , plus one of the new functions, which coincides with the finite double-pentagon integral  [57]. We use the differential equations obeyed by this integral [58, 59] to derive various parametric integral representations for it. The odd part consists entirely of the second new function, , which also can be expressed as a single integral over classical polylogarithms of degree three. This function can also be identified as the odd part of another finite double-pentagon integral,  [57], which we compute using the differential equations derived in ref. [58].

This paper is organized as follows. In section 2 we review non-MHV amplitudes, and the definition of the ratio function in planar super Yang-Mills theory. We discuss the physical constraints satisfied by , namely in the collinear and spurious limits, and also those arising from the OPE expansion of super Wilson loops. We make an ansatz for the symbol of at two loops in section 3, and then apply the constraints. In order to promote the symbol to a function, we introduce in section 4 two new functions that are not expressible in terms of classical polylogarithms, but have simple parametric integral definitions. Next, in section 5, we parametrise the beyond-the-symbol ambiguities and apply the collinear and spurious constraints at the functional level, which leaves only ten unfixed parameters. In section 6 we determine these parameters by performing an analytic two-loop evaluation of the integrals contributing to the even part of the NMHV amplitude in a special kinematical regime. The final result for the full two-loop NMHV ratio function is presented in section 7. We conclude in section 8. Several appendices contain background material and technical details. We provide the symbols for several of the quantities appearing in this article as auxiliary material.

## 2 Non-MHV amplitudes and the ratio function

To describe the scattering amplitudes of super Yang-Mills theory, it is useful to introduce an on-shell superspace (see e.g. refs. [60, 61, 36, 62]). All the different on-shell states of the theory can be arranged into an on-shell superfield which depends on Grassmann variables transforming in the fundamental representation of ,

 Φ=G++ηAΓA+12!ηAηBSAB+13!ηAηBηCϵABCD¯¯¯¯ΓD+14!ηAηBηCηDϵABCDG−. (2.1)

Here , , , , and are the positive-helicity gluon, gluino, scalar, anti-gluino, and negative-helicity gluon states, respectively. These on-shell states carry a definite null momentum, which can be written in terms of two commuting spinors, . Note that the spinors and are not uniquely defined, given ; they can be rescaled by , . The transformation properties of the states and the variables are such that the full superfield has weight 1 under the following operator,

 h=−12[λα∂∂λα−~λ˙α∂∂~λ˙α−ηA∂∂ηA]. (2.2)

All the different (colour-ordered) scattering amplitudes of the theory are then combined into a single superamplitude , from which individual components can be extracted by expanding in the Grassmann variables associated to the different particles. The tree-level MHV superamplitude is the simplest cyclically invariant quantity with the correct scaling behaviour for each particle that manifests translation invariance and supersymmetry,

 A(0)MHV=iδ4(p)δ8(q)⟨12⟩⟨23⟩⋯⟨n1⟩. (2.3)

The arguments of the delta functions are the total momentum and total chiral supercharge , respectively. The full MHV superamplitude is the tree-level one multiplied by an infrared-divergent factor,

 AMHV=A(0)MHV×M. (2.4)

Moving beyond MHV amplitudes, we define the ratio function by factoring out the MHV superamplitude from the full superamplitude [36],

 A=AMHV×P. (2.5)

Here has an expansion in terms of increasing Grassmann degree, corresponding to the type of amplitudes (MHV, NMHV, NMHV, etc.),

 P=1+PNMHV+PN2MHV+…+P¯¯¯¯¯¯¯¯¯¯¯MHV. (2.6)

The number of terms in the above expansion of is , where is the number of external legs. The Grassmann degrees of the terms are . At six points, which is the case of interest for this paper, there are just three terms, corresponding to MHV, NMHV and NMHV. The NMHV amplitudes for are equivalent to amplitudes, which are simply related to the MHV amplitudes by parity. Thus the non-trivial content of the ratio function at six points is in the NMHV term.

At tree level, is given by a sum over dual superconformal ‘-invariants’ [36]. In particular, for six points we have

 P(0)NMHV=R1;35+R1;36+R1;46. (2.7)

The -invariants can be described using dual coordinates , defined by

 pα˙αi=λαi~λ˙αi=xα˙αi−xα˙αi+1,qαAi=λαiηAi=θαAi−θαAi+1. (2.8)

Then we have [36, 43]

 (2.9)

The -invariants take an even simpler form in terms of momentum twistors [63, 45]. These variables are (super)twistors associated to the dual space with coordinates . They are defined by

 Zi=(Zi|χi),ZR=α,˙αi=(λαi,xβ˙αiλiβ),χAi=θαAiλiα. (2.10)

The momentum (super)twistors transform linearly under dual (super) conformal symmetry, so that is a dual conformal invariant. The -invariants can then be written in terms of the following structures:

 [abcde]=δ4(χa(bcde)+cyclic)(abcd)(bcde)(cdea)(deab)(eabc), (2.11)

which contain five terms in the sum over cyclic permutations of in the delta function. The bracket notation serves to make clear the totally anti-symmetrised dependence on five momentum supertwistors. The quantity is a special case of this general invariant,

 Rr;ab=[r,a−1,a,b−1,b]. (2.12)

At the six-point level it is clear that there are six different such invariants. We label them compactly by , using the momentum twistor that is absent from the five arguments in the brackets:

 (1)≡[23456], (2.13)

and so on.

In general -invariants obey many identities; see for example refs. [36, 41]. These identities can be organised as residue theorems in the Grassmannian interpretation [44]. At six points, the only identity we need is [36]

 (1)−(2)+(3)−(4)+(5)−(6)=0. (2.14)

Using eqs. (2.12), (2.13) and (2.14), we can rewrite the NMHV tree amplitude (2.7) as

 P(0)NMHV=[12345]+[12356]+[13456]=(6)+(4)+(2)=(1)+(3)+(5). (2.15)

Beyond tree level, the -invariants in the ratio function are dressed by non-trivial functions of the dual conformal invariants [36]. In the six-point case, there are three independent invariants. We may parametrise the invariants by the cross-ratios,

 u=x213x246x214x236,v=x224x251x225x241,w=x235x262x236x225. (2.16)

Often it will also be useful to use the variables defined by,

 yu=u−z+u−z−,yv=v−z+v−z−,yw=w−z+w−z−, (2.17)

where

 z±=12[−1+u+v+w±√Δ],Δ=(1−u−v−w)2−4uvw. (2.18)

In terms of momentum twistors, the cross-ratios are expressed as

 u=(6123)(3456)(6134)(2356),v=(1234)(4561)(1245)(3461),w=(2345)(5612)(2356)(4512), (2.19)

while the variables simplify to

 yu=(1345)(2456)(1236)(1235)(3456)(1246),yv=(1235)(2346)(1456)(1234)(2456)(1356),yw=(2345)(1356)(1246)(1345)(2346)(1256). (2.20)

In this form, it is clear that a cyclic rotation by one unit maps the variables as follows,

 yu⟶1yv,yv⟶1yw,yw⟶1yu, (2.21)

while the cross-ratios behave in the following way,

 u⟶v,v⟶w,w⟶u. (2.22)

The parity operation which swaps the sign of the square root of (i.e. inverts the variables) is equivalent to a rotation by three units in momentum twistor language. Indeed one can think of the cross-ratios as independent, parity-invariant combinations of the variables. Specifically we have

 u=yu(1−yv)(1−yw)(1−yuyv)(1−yuyw),1−u=(1−yu)(1−yuyvyw)(1−yuyv)(1−yuyw), (2.23)

and similar relations obtained by cyclic rotation. Because of the ambiguity associated with the sign of the square root of in eq. (2.17), the primary definition of the variables is through the momentum twistors and eq. (2.20). Further relations between these variables are provided in appendix F.

At six points it can also be convenient to simplify the momentum-twistor four-brackets by introducing [16] antisymmetric two-brackets of variables via [53]

 (ij)=14!ϵijklmn(klmn), (2.24)

so that we have

 u=(12)(45)(14)(25),1−u=(24)(15)(14)(25),yu=(26)(13)(45)(46)(12)(35), (2.25)

plus six more relations obtained by cyclic permutations.111In comparison with ref. [22], the indexing of the variables differs by one unit, and a square-root ambiguity in defining the variables was resolved in the opposite way.

Having specified our notation for the invariants we need, we now parametrise the six-point NMHV ratio function in the following way,

 PNMHV=12[ [(1)+(4)]V3+[(2)+(5)]V1+[(3)+(6)]V2 + [(1)−(4)]~V3−[(2)−(5)]~V1+[(3)−(6)]~V2]. (2.26)

The and are functions of the conformal invariants and of the coupling, with the even under parity while the are odd (recall that parity is equivalent to a rotation by three units). The cyclic and reflection symmetries of the amplitude (and hence the ratio function ) mean that the and are not all independent. Indeed, choosing and , we can write

 PNMHV=12[ [(1)+(4)]V(u,v,w)+[(2)+(5)]V(v,w,u)+[(3)+(6)]V(w,u,v) + [(1)−(4)]~V(yu,yv,yw)−[(2)−(5)]~V(yv,yw,yu)+[(3)−(6)]~V(yw,yu,yv)]. (2.27)

The functions and obey the symmetry properties,

 V(w,v,u)=V(u,v,w),~V(yw,yv,yu)=−~V(yu,yv,yw). (2.28)

Note that we have written the parity-odd function as a function of the variables, while , being parity even, can be written as a function of the cross-ratios. The functions and depend on the coupling. We expand them perturbatively as follows,

 V(a)=∞∑l=0alV(l),~V(a)=∞∑l=0al~V(l). (2.29)

Here

 a≡g2N8π2, (2.30)

where is the Yang-Mills coupling constant for gauge group ; the planar limit is with held fixed.

At tree level, we have while vanishes. One can see that the expression (2.27) with agrees with eq. (2.15). At one loop, still vanishes, while is a non-trivial function involving logarithms and dilogarithms,

 V(1) = 12[−logulogw+log(uw)logv+Li2(1−u)+Li2(1−v)+Li2(1−w)−2ζ2], (2.31) ~V(1) = 0. (2.32)

The main results of this paper are analytical two-loop expressions for and , both of which are non-vanishing.

Let us discuss some general constraints that the functions and obey.

Physical poles in amplitudes are associated with singular factors in the denominator involving sums of color-adjacent momenta, of the form . In the -invariants, in the notation of eq. (2.11), such poles appear as four brackets of the form

 (i−1,i,j−1,j)=⟨i−1,i⟩⟨j−1,j⟩x2ij. (2.33)

However, the -invariants also contain spurious poles, which arise from the four brackets that are not of this form. The full amplitude must not have such poles. Therefore the functions and must conspire to cancel the pole with a zero in the corresponding kinematical configuration.

In the dual-coordinate notation (2.9), the -invariants contain poles from denominator factors of the form . For special values of , such factors can simplify into physical singularities, but for generic values they correspond to spurious poles. In the six-point case, for example, contains a factor of

 ⟨1|x14x46|5⟩=⟨1|x14|4]⟨45⟩ (2.34)

in the denominator. While the pole at is a physical (collinear) singularity, the pole at is spurious. In momentum-twistor notation, the spurious pole comes from any four-bracket in the denominator which is not of the form .

For example, in the six-point case the -invariants and both contain the spurious factor in the denominator. (In the dual-coordinate notation, this particular pole is proportional to rather than .) In the tree-level amplitude (2.15) there is a cancellation between the two terms, so we see that

 (1)≈−(3) as (2456)→0. (2.35)

At loop level, using this relation, we find that the absence of the spurious pole implies the following condition on and ,

 [V(u,v,w)−V(w,u,v)+~V(yu,yv,yw)−~V(yw,yu,yv)](2456)=0=0. (2.36)

As the spurious bracket vanishes, we find the following limiting behaviour,

 w→1,yu→(1−w)u(1−v)(u−v)2,yv→1(1−w)(u−v)2v(1−u),yw→1−u1−v. (2.37)

This is easiest to see in the two-bracket notation of eq. (2.24), in which corresponds to and hence to . The above condition reduces to the one of ref. [49] on the assumption that .222This is true after correcting a typo in eq. (3.63) of that reference. The one-loop expression for , eq. (2.31), satisfies the above constraint with , since in the limit.

There is also a constraint from the collinear behaviour. There are two types of collinear limits, a ‘-preserving’ one where NMHV superamplitudes are related to NMHV superamplitudes with one fewer leg, and a ‘-decreasing’ one which relates NMHV superamplitudes to NMHV superamplitudes with one fewer leg. These two operations are related to each other by parity and correspond to a supersymmetrisation of the two splitting functions found when analysing pure gluon amplitudes [64, 65, 37, 66, 42]. For the six-point NMHV case, we only need to examine one of the collinear limits; the other will follow automatically by parity.

Under the collinear limit, the -point amplitude should reduce to the -point one multiplied by certain splitting functions. The splitting functions are automatically taken care of by the MHV prefactor in eq. (2.5). The -point ratio function should then be smoothly related to the -point one. Consequently, in the collinear limit the loop corrections to the six-point ratio function should vanish, because the five-point ratio function (containing only MHV and components) is exactly equal to its tree-level value. The -invariants behave smoothly in the limit, either vanishing or reducing to lower-point invariants. In the case at hand we can consider the limit , which also corresponds to , or , or with . In this limit, all -invariants vanish except for and , which become equal. Beyond tree level, the sum of their coefficients must therefore vanish in the collinear regime. This implies the constraint,

 [V(u,v,w)+V(w,u,v)+~V(yu,yv,yw)−~V(yw,yu,yv)]w→0, v→1−u=0. (2.38)

In fact the parity-odd function drops out of this constraint. The reason is that the collinear regime can be approached from the surface (see eq. (2.18)), and all parity-odd functions should vanish on this surface.

The final constraint we will need comes from the predicted OPE behaviour of the ratio function [55]. The general philosophy that an operator product expansion governs the form of the amplitudes comes from the relation of amplitudes to light-like Wilson loops. Light-like Wilson loops can be expanded around a collinear limit and the fluctuations can be described by operator insertions inside the Wilson loop [23, 24, 25]. By extending this philosophy [55] to supersymmetrised Wilson loops [51, 50] (or equivalently correlation functions [52]) one can avoid questions about giving a precise Lagrangian description of the object under study. In this sense the OPE can be used to justify the existence of a supersymmetrised object dual to non-MHV amplitudes.

The analysis of ref. [55] allows one to choose various components of the ratio function. Let us consider the component proportional to . The only term in eq. (2.27) that contributes to this component is the first one,

 P(2356)NMHV=1(2356)V(u,v,w). (2.39)

In order to examine the OPE, we follow ref. [55] and choose coordinates by fixing a conformal frame where

 1(2356)=14(coshσcoshτ+cosϕ)=√uvw2(1−v), (2.40)

and the three cross-ratios are given by

 u=eσsinhτtanhτ2(coshσcoshτ+cosϕ),v=1cosh2τ,w=e−σsinhτtanhτ2(coshσcoshτ+cosϕ). (2.41)

Extrapolating the results of ref. [55] to two loops, the OPE predicts the leading (double) discontinuity of the component of the ratio function to be,

 ΔvΔvP(2356)NMHV ∝ ∞∑m=−∞∫∞−∞dp2πeimϕ−ipσC(2356)m(p)F(2356)|m|+1,p(τ)[γ1+|m|(p)]2, (2.42)

where

 F(2356)E,p(τ) =sechEτ2F1[12(E−ip),12(E+ip);E;sech2τ], (2.43) C(2356)m(p) =14(−1)mB[12(|m|+1+ip),12(|m|+1−ip)], (2.44) γ1+|m|(p) (2.45)

Here is the hypergeometric function, is the Euler beta function, and is the logarithmic derivative of the function.

## 3 Ansatz for the symbol of the two-loop ratio function

In order to make a plausible ansatz for the ratio function at two loops we assume that the functions and are pure functions of , and , i.e. iterated integrals or multi-dimensional polylogarithms of degree four. Moreover, we make an ansatz for the symbols of and , requiring that their entries are drawn from the following set of nine elements,

 {u,v,w,1−u,1−v,1−w,yu,yv,yw}. (3.1)

We summarise some background material on pure functions and symbols in appendix A. We recall that the variables invert under parity. The parity-even function should have a symbol which contains only terms with an even number of entries. Likewise the symbol of the parity-odd function should contain only terms with an odd number of entries. The ansatz for the symbol entries is the same as the one used recently for the three-loop remainder function [22] (after omitting restrictions on the final entry of the symbol). It is consistent with every known function appearing in the six-point amplitudes of planar super Yang-Mills theory, in particular the simple analytic form of the two-loop remainder function found in ref. [16]. It is also consistent with the results for explicitly known loop integrals appearing in such amplitudes, see refs. [58, 59, 67]. In the ensuing analysis we will find many strong consistency checks on our ansatz.

Let us pause to note that our assumption that the relevant functions are pure functions of a particular degree equal to twice the loop order is by no means an innocent one. Although it is true that such general polylogarithmic functions generically show up in amplitudes in four-dimensional quantum field theories, it is certainly not true in general that they always appear with a uniform degree dependent on the loop order. In QCD, for example, the degrees appearing range from twice the loop order to zero, and the transcendental functions typically appear with non-trivial algebraic prefactors. In fact the observed behaviour of having maximal degree only is limited to super Yang-Mills theory, and the most evidence is for the planar sector. This behavior is the generalization, to non-trivial functions of the kinematics, of the maximal degree of transcendentality for harmonic sums that has been observed in the anomalous dimensions of gauge-invariant local operators [68].

The symbols we construct from the set of letters (3.1) should obey certain restrictions. They should be integrable; that is, they should actually be symbols of functions. The initial entries of the symbol should be drawn only from the set , because the leading entry determines the locations of branch points of the function in question, and branch integrable symbols of degree 4 for , and 2 for , obeying the initial entry condition as well as the symmetry conditions (2.28). The spurious pole conditions (2.36) provide 14 constraints and the collinear conditions (2.38) provide 14 more, leaving 15 free parameters at this stage. In order to impose the constraints from the leading discontinuity predicted by the OPE, we use the fact that the sum (2.42) is annihilated by the following differential operator [55],

 D = ∂2τ+2coth(2τ)∂τ+sech2τ∂2σ+∂2ϕ+1. (3.2)

In the variables this differential operator is given by

 D = 12(D++D−)+1, (3.3)

where

 D±=41−v[ −z±u∂u−(1−v)v∂v−z±w∂w +(1−u)vu∂uu∂u+(1−v)2v∂vv∂v+(1−w)vw∂ww∂w +(−1+u−v+w)((1−v)u∂uv∂v−vu∂uw∂w+(1−v)v∂vw∂w)]. (3.4)

Imposing that the double discontinuity (2.39) is annihilated by the operator gives 5 further conditions, leaving 10 free parameters in the symbol. One of these parameters, denoted by below, is just the overall normalisation of the symbol of the double discontinuity, which is non-zero and convention-dependent. In the following we provide functions with physical branch cuts which represent the symbol. We find that the solution has the form,

 S(V) = αXS(VX)+9∑i=1αiS(fi),S(~V) = αXS(~VX)+α8S(~f), (3.5)

where and through are the constant free parameters, and the quantities , , and will be defined below.333In section 6 we will fix the ten parameters using an analytical computation for particular kinematics. We have compared the symbols (3.5) for and with all parameters fixed to an independent computation of these symbols from a formulation of the super Wilson loop [69]; the results agree precisely.

The double discontinuities of the functions appearing in eq. (3.5) obey

 S(ΔvΔvVX) = 2αX[u⊗(1−u)+u⊗u+w⊗(1−w)+w⊗w (3.6) +2(u⊗w+w⊗u−uw⊗(1−v)−(1−v)⊗uw +(1−v)⊗(1−v))], S(ΔvΔvfi) = 0, (3.7) S(ΔvΔv~VX) = S(ΔvΔv~f) = 0. (3.8)

Consistency with the spurious pole condition (2.36) forces the odd part to be non-zero, given that is non-zero. The odd part contains no ambiguity at the level of the symbol (or beyond it), once we fix the even part, particularly the two parameters and .

The symbol of the double discontinuity of and is entirely controlled by , through eq. (3.6). We can find a function compatible with this symbol, and compare it to eq. (2.42) to fix the terms. We find that

 ΔvΔvP(2356)NMHV ∝ 1(2356)[log2u+log2w+4logulogw+2log2(1−v) (3.9) −4log(uw)log(1−v)−2(Li2(1−u)+% Li2(1−w)−2ζ2)].

In order to present the symbols appearing in eq. (3.5) explicitly and compactly, it is very useful to employ harmonic polylogarithms [70, 71, 72]. This presentation simultaneously accomplishes the following step, of turning the symbols into functions, up to certain beyond-the-symbol ambiguities. The functions we will present are of degree at most four, and almost all of them can be represented in terms of classical functions. Thus the use of harmonic polylogarithms may seem unnecessarily complicated. However, it is a very useful way to represent, at any degree, a symbol only involving the letters , whereas functions are often insufficient beyond degree four.

Harmonic polylogarithms are single-variable functions defined by iterated integration. It is very simple to write down their symbols. We use harmonic polylogarithms with labels (weight-vector entries) “0” and “1” only. The symbol of a harmonic polylogarithm of argument is obtained by reversing the list of labels and replacing all “0” entries by and all “1” entries by . Finally, one multiplies by where is the number of “1” entries. For example, the symbol of is , and the symbol of is  .

We also use the common convention of shortening the label list by deleting each “0” entry, while increasing by one the value of the first non-zero entry to its right, so that, for example, and . Apart from the logarithm function, we take all arguments of the harmonic polylogarithms to be , or . This representation guarantees that the functions we are using to represent the symbol do not have any branch cut originating from an unphysical point. We then compactify the notation further by writing , and so on. Finally, we recall that the symbol of a product of two functions is given by the shuffle product of the two symbols.

With this notation we can immediately write down a function which has the symbol, , of the part of with non-zero double discontinuity, i.e. the part fixed by the OPE,

 VX={ 4Hu3,1+logu(Hv3+2Hu2,1−5Hv2,1+6Hw2,1+32Hu2logw)+log2u(Hu2−3Hw2) +logv[Hu3−3Hu2,1−12logu(Hu2+Hv2)−32log2ulogw]+log2v(−Hu2+12log2u) +(u↔w)}+4Hv3,1+2logulog2vlogw−12log2ulog2w. (3.10)

We can similarly write down functions with the correct symbols for the first seven ambiguities in the even part (the double discontinuity of each function vanishes):

 f1= Hu2Hw2, f2=[ −logu(Hw3+Hw2,1+Hu2logw)−log2u(Hw2+12logvlogw)+(u↔w)], f3=[ −Hw2logulogv+(u↔w)]+Hv2logulogw, f4=[ −Hu2logulogw−log2u(2Hw2+logvlogw)+(u↔w)]−Hv2logulogw, f5=[ Hu2Hv2+Hu2,2+logu(2Hv2,1−2Hw2,1−Hu2logw)+logv(2Hu2,1+logu(Hu2+Hv2)) +(u↔w)]+Hv2,2, f6=[ −2Hu2Hv2−2Hu2,2−4Hu3,1+logu(−2Hu3−2Hv2,1+2Hw2,1+Hu2logw) −logv(2Hu2,1+logu(Hu2+Hv2))+(u↔w)]−2Hv2,2−4Hv3,1−2Hv3logv, f7=[ −3Hu4−3Hu2,1,1+logu(Hv3−2Hu2,1+Hw2,1+Hu2logw)+log2u(−12Hu2+Hw2) +logv(Hu3+12log2ulogw)+(u↔w)]−3Hv4−3Hv2,1,1−2Hv2,1logv−12Hv2log2v.

For the even part there remain two more ambiguities whose symbols cannot be expressed in terms of those of the single-variable harmonic polylogarithms with the arguments we have been using,

 f8=[ −Hu2Hv2−2Hu4+Hu2,2−4Hu3,1+6Hu2,1,1+logu(−Hu3−Hv3+Hw3+2Hu2,1) +Hw2log2u+logv(Hu3−Hw2logu)+(u↔w)] − Hv2,2−2Hv3,1+Hv2logulogw+12log2ulog2w−Hv3logv−2Ω(2)(w,u,v), f9= R(2)6(u,v,w).

Here stands for the two-loop remainder function, whose symbol is known [16]. The appearance of the two-loop remainder function as an ambiguity should not be surprising. It is a function with physical branch cuts, which vanishes in the collinear limit. Also, it is totally cyclic and hence automatically satisfies the spurious pole condition on its own. Furthermore, it has vanishing double discontinuities and hence drops out from the leading-discontinuity OPE criterion (2.42). It is known [16] that can in fact be expressed in terms of single-variable classical polylogarithms. However, to do so one must use arguments involving square roots of polynomials of the cross-ratios.

The other quantity not given in terms of the harmonic polylogarithms, which enters , is the integral . In fact its symbol can also be recognised from other considerations [58, 59], as we will discuss in the next section. The symbol of is,

 S(Ω(2)(u,v,w))=−12[S(qϕ)⊗ϕ+S(qr)⊗r+S(~Φ6)⊗yuyv], (3.11)

where

 ϕ=uv(1−u)(1−v),r=u(1−v)v(1−u). (3.12)

Here is the one-loop six-dimensional hexagon function [59, 67], whose symbol is given explicitly in terms of the letters of our ansatz [59],

 S(