Bootstrapping the three-loop hexagon
aa r X i v : . [ h e p - t h ] N ov CERN–PH–TH/2011/189 SLAC–PUB–14528 LAPTH-029/11HU-EP-11-38 NSF-KITP-11-176
Bootstrapping the three-loop hexagon
Lance J. Dixon (1 , , James M. Drummond (1 , and Johannes M. Henn (4 , (1) PH-TH Division, CERN, Geneva, Switzerland (2)
SLAC National Accelerator Laboratory, Stanford University, Stanford, CA 94309, USA (3)
LAPTH, Universit´e de Savoie, CNRS, B.P. 110, F-74941 Annecy-le-Vieux Cedex, France (4)
Institut f¨ur Physik, Humboldt-Universit¨at zu Berlin, Newtonstraße 15, D-12489 Berlin, Germany (5)
Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA
Abstract
We consider the hexagonal Wilson loop dual to the six-point MHV amplitude inplanar N = 4 super Yang-Mills theory. We apply constraints from the operatorproduct expansion in the near-collinear limit to the symbol of the remainder func-tion at three loops. Using these constraints, and assuming a natural ansatz for thesymbol’s entries, we determine the symbol up to just two undetermined constants.In the multi-Regge limit, both constants drop out from the symbol, enabling us tomake a non-trivial confirmation of the BFKL prediction for the leading-log approx-imation. This result provides a strong consistency check of both our ansatz for thesymbol and the duality between Wilson loops and MHV amplitudes. Furthermore,we predict the form of the full three-loop remainder function in the multi-Reggelimit, beyond the leading-log approximation, up to a few constants representingterms not detected by the symbol. Our results confirm an all-loop prediction forthe real part of the remainder function in multi-Regge 3 → Introduction and outline
Scattering amplitudes in N = 4 super Yang-Mills theory (SYM) have fascinating properties,especially in the planar limit. One of their most surprising properties is an equivalence withcertain light-like Wilson loop configurations, for which there is strong empirical evidence at weakcoupling, as well as general arguments originating from strong coupling [1, 2, 3, 4, 5, 6, 7, 8].The equivalence relates the suitably-defined finite parts of maximally-helicity-violating (MHV)scattering amplitudes to the finite parts of Wilson loops evaluated on a null polygonal contourin dual (or region) space. The edges of the polygon are defined by the gluon momenta p µi via p µi = x µi − x µi +1 . (1)The contour has corners (or cusps) at the points x i . The equivalence between amplitudes andWilson loops implies that the analytic properties of Wilson loops in the dual space can beidentified with those of scattering amplitudes in momentum space.Wilson loops in a conformal field theory exhibit conformal symmetry. The null polygonalWilson loops related to scattering amplitudes are ultraviolet divergent due to the presence ofcusps on the contour. Nonetheless they still obey a conformal Ward identity [4, 9]. This identitycan be simply stated as follows. We write the logarithm of the Wilson loop with n cusps as asum of divergent and finite terms,log W n = [UV divergent] n + F WL n . (2)The Ward identity for the finite part is then K µ F WL n = − γ K n X i =1 (2 x µi − x µi − − x µi +1 ) log x i − ,i +1 , (3)where x i,j = x i − x j and x i + n ≡ x i , and K µ are the generators of (dual) special conformaltransformations, K µ = n X i =1 (cid:20) x µi x νi ∂∂x νi − x i ∂∂x i µ (cid:21) . (4)The cusp anomalous dimension [10] γ K is predicted to all orders in the coupling constant [11].The Ward identity (3) fixes F WL n , up to functions of conformally invariant cross ratios. Belowsix points there are no such cross-ratios and the solution is unique up to an additive constant.In fact this solution coincides precisely with the BDS ansatz [12] for the finite part of MHVscattering amplitudes. At six points and beyond there are cross ratios, so the solution is notunique. The BDS ansatz still provides a particular solution to the Ward identity for all n , butit does not give the complete answer. A convenient way of writing the solution to the Wardidentity is then F WL n = γ K F WL n , − loop + R n . (5)2ere F WL n , − loop is the one-loop result for F WL n , while R n is the ‘remainder function’, which is afunction only of conformal cross ratios and becomes non-vanishing at two loops [5]. In terms ofthe loop expansion parameter a ≡ g N c / (8 π ), the remainder function is expanded as R n = ∞ X l =2 a l R ( l ) n . (6)At six points, the remainder function depends on three dual conformal cross ratios, u = x x x x = s s s s , v = x x x x = s s s s , w = x x x x = s s s s , (7)which are in turn built from the Lorentz invariants s i,j = ( p i + p j ) and s i,j,k = ( p i + p j + p k ) . Thegluon momenta for the scattering process, p µi with i = 1 , , . . . ,
6, satisfy the on-shell conditions p i = 0.The conformal symmetry of the Wilson loop implies that the dual planar MHV amplitudes ex-hibit ‘dual conformal symmetry’. This symmetry has been observed in the form of the scatteringamplitudes in many guises: the form of the integrals in the perturbative expansion [13, 14, 15];the background isometry of the AdS sigma model after T-duality [1, 8, 16]; the structure of tree-level amplitudes, where it extends to dual superconformal symmetry [17], and combines with theoriginal Lagrangian superconformal symmetry to form a Yangian symmetry [18]; the structure ofthe scattering amplitudes on the Coulomb branch [19] and in higher dimensions [19, 20, 21, 22];and the form of the on-shell recursion relations for the four-dimensional planar integrand [23].Many review articles are available on different aspects of all of these developments, includingrefs. [24, 25, 26, 27, 28, 29, 30, 31]. For the purposes of this paper the important point is simplythat the Ward identity (3) requires the function R to depend only on the invariant cross ratios u , v and w .Much recent progress [32, 33, 34, 35, 36] has focused on understanding the structure of theremainder function, in part due to the fact that this same function governs the structure of scatter-ing amplitudes, both at strong coupling [1] and in the MHV sector in perturbation theory [5, 6, 7].Understanding its form then promises to greatly enhance our understanding of scattering am-plitudes in general. A very important result in this direction was the analytic calculation of theFeynman integrals appearing at two loops in the hexagonal Wilson loop [32, 33], which provideda closed-form expression for the remainder function in terms of (many) multi-dimensional poly-logarithms, or Goncharov polylogarithms. Remarkably, this seemingly complicated expressioncould be dramatically simplified into a few lines of classical polylogarithms [35]. An importanttool for finding such a compact form of the two-loop, six-point remainder function is the notionof the symbol of a transcendental function [37]. The symbol is a quantity which preserves the un-derlying algebraic nature of the function, while forgetting about certain analytic properties, suchas the particular branch cut on which the function should be evaluated. Complicated identitiesbetween polylogarithms reduce to simple algebraic relations at the level of symbols. The symbolcan therefore be a key step in discerning the analytic structure of amplitudes. For example,3 conjecture has been made recently for the symbol of the two-loop remainder function for anarbitrary number of points [38]. Of course, eventually one would like to reconstruct the actualfunction represented by the symbol.Another important property of polygonal Wilson loops is that they should respect a particularoperator product expansion (OPE) in the region where several consecutive edges are nearlycollinear [39, 40]. This idea has recently been used to argue that at two loops the hexagonremainder function can be uniquely fixed from the knowledge only of the leading corrections tothe energies of the exchanged states in the OPE [41]. The OPE has also recently been used toaddress the same problem for Wilson loops with more than six sides [42], and for super Wilsonloops associated with non-MHV amplitudes [43].An important kinematical limit of higher-point scattering amplitudes is the multi-Regge limit.This limit is a generalization of the high-energy limit of four-point scattering, but is one inwhich multiple parameters can survive, related to the ordering of the final-state particles inrapidity. For the MHV amplitudes in planar N = 4 super Yang-Mills theory, this structure hasbeen explored in several papers [44, 45, 46, 47, 48, 49, 50]. Indeed, this limit provided earlyevidence that the BDS ansatz needed to be corrected at two loops, starting with the six-pointamplitude [44]. While the remainder function R ( u, v, w ) vanishes in the Euclidean version ofmulti-Regge kinematics [51, 52, 53], in the physical region its discontinuity is nonzero and canbe analysed. When dual conformal invariance holds, this discontinuity depends nontrivially ontwo dimensionless variables, rather than the three variables u , v and w characterizing generickinematics.A consequence of the duality between MHV amplitudes and Wilson loops is that the multi-Regge behaviour of the amplitude should be consistent with the OPE behaviour of the Wilsonloop in the near-collinear limit. That is, there is a further limit one can take of the multi-Reggekinematics which is collinear in nature. This combined limit was studied recently [50], and itwas shown that constraints from the two limits pass a self-consistency test.In this paper, inspired by all these exciting recent developments, we will make an ansatz for thesymbol of the three-loop hexagon remainder function, R (3)6 ( u, v, w ), which is heavily constrainedby the structures described above. We are able to apply all of the physical requirements, suchas the correct collinear behaviour, OPE expansion, multi-Regge limits and so on, at the levelof the symbol. The correct near-collinear behaviour, governed by the OPE expansion, is one ofthe strongest constraints on our ansatz. It is quite non-trivial that there is a consistent solutionto the combined constraints. For general kinematics, the solution for the symbol is not unique,but contains 26 arbitrary constants. However, all but three of these parameters are irrelevantin the multi-Regge limit. Analysing the symbol in this limit, and imposing consistency with theleading logarithmic prediction [48], we find that two of the three parameters relevant in this limitcan be fixed. Hence the symbol is completely fixed, up to a single constant parameter, in thisregime. An additional constraint enables us to show that this extra constant parameter actuallyvanishes. The latter constraint is an all-loop-order prediction [49] concerning the behaviour ofthe real part of the remainder function in the multi-Regge limit, after analytic continuation to4 → N = 4 super-Yang-Mills theory. Thesame restriction has also been identified within a supersymmetric formulation of the Wilsonloop [38]. We find that imposing this final-entry condition fixes the symbol completely up tojust two free parameters, and furthermore it determines the symbol uniquely in the multi-Reggelimit, and consistently with the all-loop-order prediction for 3 → u , v and w , the three-loopremainder function cannot be described in terms of classical polylogarithms, in contrast to whathappened at two loops. In section 7 we present our conclusions and give a brief outlook. Threeappendices give some useful relations between different sets of kinematic variables, as well as analternate representation of the logarithmic coefficients in the multi-Regge limit.In additional files accompanying this article, as both Mathematica notebooks and plain textfiles, we provide the symbol for the three-loop remainder function, after imposing the final-entryconstraint. We also provide the symbols associated with the remainder function in the multi-Regge limit. The remainder function of N = 4 SYM is expected to be described in terms of pure functions.We define a pure function of degree (or weight) k recursively, by demanding that its differential5atisfies d f ( k ) = X r f ( k − r d log φ r . (8)Here the sum over r is finite and φ r are algebraic functions. This recursive definition is forall positive k ; the only degree zero pure functions are constants. The definition (8) includeslogarithms and classical polylogarithms, as well as other iterated integrals, such as harmonicpolylogarithms of one [58] or more [59, 60, 61, 62] variables.The symbol [37] S ( f ) of a pure function f is defined recursively with respect to eq. (8), S ( f ( k ) ) = X r S ( f ( k − r ) ⊗ φ r . (9)If we continue this process until we reach degree 0, we find that S ( f ( k ) ) is an element of the k -fold tensor product of the space of algebraic functions, S ( f ( k ) ) = X ~α φ α ⊗ . . . ⊗ φ α k , (10)where ~α ≡ { α , . . . , α k } . The symbol of a function does not contain all the information aboutthe function. In particular, it loses information about which logarithmic branch the integrandof an iterated integral is on, at each stage of integration. It also does not detect functions thatare transcendental constants multiplied by pure functions of lower degree. (That is, such func-tions have zero symbol.) The symbol therefore corresponds to an equivalence class of functionsthat differ in these aspects. Nevertheless, the symbol is extremely useful, because complicatedidentities between transcendental functions defined by iterated integrals become simple algebraicidentities.If a symbol can be expressed as a sum of terms, with all entries in each term belonging toa given set of variables, then we say that the symbol can be factorised in terms of that set ofvariables. From the definition of the symbol, a term containing an entry which is a product canbe split into the sum of two terms, according to . . . ⊗ φ φ ⊗ . . . = . . . ⊗ φ ⊗ . . . + . . . ⊗ φ ⊗ . . . . (11)Performing this factorisation is usually necessary to identify all algebraic relations between terms.It is often necessary to perform the step again after taking a kinematic limit, because the algebraicrelations in the limit are different than for generic kinematics.The elements of the symbol are not all independent. In particular the integrability condition d f ( k ) = 0 for any function implies relations among the different elements. These relations canbe described simply: One picks two adjacent slots in the symbol φ α i ⊗ φ α i +1 and replaces thecorresponding elements by the wedge product d log φ α i ∧ d log φ α i +1 in every term. The resultingexpression must vanish. 6t is very helpful in our analysis to consider the discontinuities of the functions involved. Thesymbol makes clear the locations of the discontinuities of the function. If we have S ( f ( k ) ) = X ~α φ α ⊗ . . . ⊗ φ α k , (12)then the degree k function f ( k ) will have a branch cut starting at φ α = 0. The discontinuityacross this branch cut, denoted by ∆ φ α f ( k ) , will also be a pure function, of degree k −
1. Itssymbol is found by clipping the first element off the symbol for f ( k ) : S (∆ φ α f ( k ) ) = X ~α φ α ⊗ . . . ⊗ φ α k . (13)In general, taking discontinuities commutes with taking derivatives. We will now describe a procedure for constraining the form of the remainder function based ona plausible ansatz for its symbol. Our experience with six-point integrals in both four and sixdimensions [54, 55, 63] is that their symbols are always formed of terms with entries drawn fromthe following set of nine elements, { u, v, w, − u, − v, − w, y u , y v , y w } . (14)Here we use the notation y u = u − z + u − z − , y v = v − z + v − z − , y w = w − z + w − z − , (15)where z ± = 12 h − u + v + w ± √ ∆ i , ∆ = (1 − u − v − w ) − uvw . (16)Thus our ansatz for the remainder function at l loops will be the most general symbol of degree2 l that we can make from the above set of nine elements. That is, we assume that the symbolfor the remainder function can be factorised in terms of the set (14).We can also write the cross ratios in terms of ratios of two-brackets of CP variables w i , u = (23)(56)(25)(36) , v = (34)(61)(36)(41) , w = (45)(12)(41)(52) , (17)where ( ij ) = − ( ji ) = ǫ ab w ai w bj . In these variables, ∆ is a perfect square, √ ∆ = ± (12)(34)(56) + (23)(45)(61)(14)(25)(36) . (18)7aking the positive branch of the square root, and using the Schouten identity for the two-brackets, we have1 − u = (35)(26)(25)(36) , − v = (46)(31)(36)(41) , − w = (51)(42)(41)(52) , (19) y u = (23)(46)(15)(56)(13)(24) , y v = (61)(24)(35)(34)(51)(26) , y w = (45)(62)(31)(12)(35)(46) . (20)Note that under a cyclic permutation, w i → w i +1 , with indices modulo 6, the sign of √ ∆ flips, √ ∆ → −√ ∆. So the y variables permute as y u → /y v → y w → /y u . This inversion willnot affect the symmetry properties of the parity-even functions and symbols in which we areinterested, which involve even numbers of y variables.From eqs. (17), (19) and (20) we see that our ansatz is equivalent to saying that the symbolcan be factorised in terms of two-brackets ( ij ) (or equivalently momentum-twistor four-brackets)at the six-point level. (There are 15 two-brackets ( ij ), but only combinations that are invariantunder rescaling of individual w i coordinates are allowed, which reduces the number of independentcombinations to the nine exhibited in eqs. (17), (19) and (20).) Note that we can fix a coordinatechoice w i = (1 , z i ), where these variables coincide with the z i variables of ref. [35], so that ouransatz is also equivalent to assuming that the symbol can be factorised in terms of differences ofthe z i . The form of our ansatz is certainly sufficient at the two-loop level, because the remainderfunction is explicitly known [6, 7, 32, 33, 35], and its symbol is indeed of this form [35]. In theabove variables, it is given by S ( R (2)6 )= − nh u ⊗ (1 − u ) ⊗ u (1 − u ) + 2 (cid:0) u ⊗ v + v ⊗ u ) ⊗ w − v + 2 v ⊗ w − v ⊗ u i ⊗ u − u + h u ⊗ (1 − u ) ⊗ y u y v y w − u ⊗ v ⊗ y w i ⊗ y u y v y w o + permutations , (21)where the sum is over the 6 permutations of u , v and w , which correspondingly permute y u , y v and y w .What constraints should the symbol of the remainder function obey? • It should be integrable, i.e. it should actually be the symbol of a function. • The first entry in any term of the symbol should be a cross ratio u , v or w . The leadingentries describe the locations of the discontinuities of the function, which can only originateat x ij = 0, as can be seen by considering the unitarity cuts of the amplitude [41]. Thesepoints correspond to cuts in u , v or w originating at either 0 or ∞ . A first entry containing1 − u , y u , etc. , would lead to a discontinuity starting at an unphysical point.Within our ansatz for the symbol of the three-loop remainder function, these two constraintsare sufficient to show (by explicit enumeration) that the second entry of the symbol can only8e drawn from the set { u, v, w, − u, − v, − w } . This result is consistent with a conjectureof some of the authors of ref. [41]. The second-entry property is of course true for the knowntwo-loop remainder function. We also have the following further conditions on the symbol of theremainder function: • It should be completely symmetric in the cross ratios u, v, w . • It should be parity even. Because the y variables of eq. (15) invert under parity (theexchange of z + and z − ), there should be an even number of y entries in any given term inthe symbol. • It should vanish in the collinear limit. This constraint can be implemented at the level ofthe symbol as follows. In the limit w →
0, we find that the y variables behave as y u −→ u − v , y v −→ v − u , y w −→ w (1 − u )(1 − v )(1 − u − v ) . (22)The collinear limit can be obtained by first taking the w → v → − u . The symbol of the remainder function should thenvanish. (A term in the symbol vanishes if at least one of its entries goes smoothly to 1.)We have analysed the implications of the above constraints up to three loops ( i.e. up tosymbols of degree six). At one loop we find that there are no symbols obeying all of the aboveproperties. This result is expected, since the remainder function, which vanishes in the collinearlimit, starts appearing only at two loops and beyond. At two loops there is a four-parameterfamily of symbols obeying the constraints that we have outlined. Not surprisingly, it containsthe symbol of the two-loop remainder function which is explicitly known [35] and satisfies theabove conditions. At three loops we find a 59-parameter space of symbols obeying the constraintsoutlined above. We would like to impose more constraints to see if we can further restrict thespace of possible solutions. We have the following two classes of additional constraints: • As well as vanishing in the strict collinear limit, the Wilson loop in the near-collinear regimeshould have an OPE expansion as described in refs. [39, 40, 41]. Roughly speaking, thisexpansion comes about because a Wilson loop can be expanded around the limit wherea set of adjacent sides becomes collinear. A scaling parameter τ measures how close theWilson loop is to the collinear configuration ( τ → ∞ corresponds to the strict collinearlimit). In terms of this parameter the Wilson loop should have an expansion of the form W = Z dn C n e − E n τ . (23)Here n is shorthand for the set of labels corresponding to the state being exchanged, E n isthe ‘energy’ of the state ( i.e. its eigenvalue under the τ scaling), and C n corresponds roughly More accurately, one considers the logarithm of a particular finite, conformally invariant ratio of Wilson loops.
9o the probability of emission and absorption of a given state. In principle, a completeknowledge of the set of states labeled by n , and expressions for the energies E n and theoverlap functions C n entirely fix the remainder function. In fact, armed with a knowledgeof only the leading corrections to the energies of the simplest single-particle states, wecan predict the leading discontinuity at any loop order. At two loops this information issufficient to determine the entire symbol [41], because the leading discontinuity is just asingle discontinuity, ∆ v R (2)6 . The discontinuities in the other two cross ratios, ∆ u R (2)6 and∆ w R (2)6 , are related by symmetry. Using the fact that the first entry of the symbol is either u , v or w , and eq. (13) for the symbol of the discontinuity, we see that knowing ∆ v R (2)6 allows the full two-loop symbol to be reconstructed by appending a v to the front andsumming over cyclic permutations. At three loops, the leading corrections to the E n sufficeto constrain the double discontinuity, ∆ v ∆ v R (3)6 . This is a powerful constraint, although itdoes not uniquely determine the remainder function on its own. • The remainder function should also obey particular constraints in multi-Regge kinemat-ics [47, 48, 49, 50]. In this limit, u →
1, while v and w vanish in a particular way, u −→ , v − u −→ x , w − u −→ y . (24)Here x and y are free parameters. One must be careful about the branch on whichthe limit is taken. In fact, the functions we are interested in vanish in this limit in theEuclidean region [51, 52, 53] (when all separations x ij are taken to be spacelike) but arenon-vanishing and even logarithmically divergent in physical regions for 2 → → unique solution to theOPE constraints (thus adding support to the correctness of the ansatz we have adopted). Thusthe OPE fixes 33 of the 59 free parameters of our symbol.Analysing the multi-Regge limits we find that, of the 26 functions without any double discon-tinuity, only three are non-vanishing in the multi-Regge kinematics. One has beyond-leading-logbehaviour (it is proportional to log (1 − u ) in the limit (24)), and is therefore ruled out. Anotherparameter is fixed by the known leading-log behaviour, proportional to log (1 − u ) [48, 49]. Thusa single parameter is left undetermined in the multi-Regge limit. This free parameter appears inthe next-to-leading log behaviour, but not at the next-to-next-to-leading log level. We will see The variable y introduced in eq. (24) should not be confused with the variables y u , y v and y w used for generickinematics. → S ( R (3)6 ) = S ( X ) + X i =1 α i S ( f i ) . (25)The first term, S ( X ), is the piece that is fixed by the OPE constraints. The remaining freeparameters α i accompany symbols of functions f i which have no double discontinuity. Examiningthe form of S ( X ) we find it can be written in such a way that its final entries are always of theform, n u − u , v − v , w − w , y u , y v , y w o . (26)Note that this is not in contradiction with the ansatz (14), since the entries can always befactorised. Instead it is a more restrictive statement, because only 6 out of the 9 potentialvariables actually appear in the final entry. This result concerning the restricted structure of thefinal entries of S ( X ) is closely connected to the observations of ref. [38], which suggests that thisfact may be related to a supersymmetric formulation of the Wilson loop. Similar restrictionsappear [57] in differential equations obeyed [54, 55, 56] by integrals related to planar N = 4super-Yang-Mills scattering amplitudes [23]. These observations suggest that the full symbol S ( R (3)6 ), not just S ( X ), should be of a form in which its final entries are drawn from the list (26).Imposing this condition on the final entries of S ( R (3)6 ) reduces the number of free parametersto just two. The fact that it is possible to impose this restriction, consistently with the knownmulti-Regge behaviour, is highly non-trivial.Finally, let us note that even if we were able to fix the entire symbol and find a functionwith all the desired analytic properties, vanishing in the collinear limit with the correct OPEbehaviour, etc. , there would always remain the possibility of adding some amount of the two-loopremainder function multiplied by ζ , that is, R (3)6 −→ R (3)6 + γ ζ R (2)6 , (27)for some constant γ . We will see such ‘beyond-the-symbol’ ambiguities appearing in a particularway in our predictions for the multi-Regge behaviour of the three-loop remainder function.We will now discuss the OPE analysis in further detail, and then describe the predictionsfor the three-loop remainder function in the multi-Regge kinematics. We will conclude with adiscussion of the conditions on the final entries, and the remaining ambiguities after imposing allour constraints. In order to describe the OPE expansion for a light-like Wilson loop, the authors of ref. [39]introduced a reference square with null sides, denoted by W square in Fig. 1. Two of the sides of11 orig W top W bottom W square Figure 1:
The four different Wilson loops entering the definition of the ratio (28). The reference squareis shown by the dashed line. The top and bottom Wilson loops are obtained by replacing a sequence ofedges by the corresponding part of the square. the square coincide with two of the sides of the Wilson loop, while the other two sides are formedby finding other null lines that intersect the two previous ones as well as two of the corners ofthe original loop. One can then consider the finite, conformally invariant quantity made from aratio of Wilson loops, r = log W orig W square W top W bottom . (28)The four different Wilson loops appearing in the ratio are depicted in Fig. 1.Note that at six points, the top and bottom loops are five-sided. The four-sided and five-sidedloops appearing in the ratio r are entirely determined by the conformal Ward identity (3). Thusknowledge of the quantity r is equivalent to knowing the six-point remainder function.As described in ref. [41], the Wilson loop, or more precisely the ratio r , is expected to havean OPE expansion of the form r = Z dn C n e − E n τ . (29)At one loop, the states labelled by n are free single-particle exchanges between the bottom halfof the the loop and the top half. Beyond one loop there can be interactions between the particlesand the vertical Wilson lines in Fig. 1, as well as multi-particle exchanges, and so forth. Thequantities C n and E n entering the OPE should be expanded in the coupling constant. In principle,to determine r (and hence the remainder function) one needs to know the space of states andthe dependence of C n and E n on the coupling.There is, however, a piece of the remainder function that is completely constrained at l loops,just from knowing the one-loop anomalous dimensions [64] of the single-particle states beingexchanged [41]. In the near-collinear limit, one of the cross ratios vanishes, say v →
0. Itvanishes exponentially quickly as τ → ∞ ; that is, τ is proportional to log v in this limit. Thespecial piece of the remainder function (or r ) is the leading discontinuity in v , which is therepeated ( l − l − v r . This discontinuity can be extracted from the OPE by12rst Taylor expanding the energies of the intermediate states in the coupling constant, E n = E (0) n + g E (1) n + g E (2) n + . . . . (30)After Taylor expanding the exponential in eq. (23) in g we find r = Z dn C n e − E (0) n τ h − g τ E (1) n + g (cid:0) τ ( E (1) n ) − τ E (2) n (cid:1) + . . . i . (31)Because τ is proportional to log v as τ → ∞ , the leading discontinuity in v at any loop orderis given by the term involving the highest power of τ . This term is always obtained from theone-loop corrections E (1) n to the energies of the simplest single-particle states — those stateswhose overlap functions C n are non-vanishing at order g .The exchanged states carry other quantum numbers in addition to the energy E n . There is a‘momentum’ p conjugate to the other scaling ( σ -scaling) invariance of the square and a discretelabel m , conjugate to the rotational invariance ( φ -rotation) in the two directions orthogonal tothe square. These three invariances of the square can be used to completely parametrise thethree variables u, v and w on which r (or the six-point remainder function) depends. Explicitly,the variables σ, τ and φ are related to u, v and w via u = e σ sinh τ tanh τ σ cosh τ + cos φ ) , v = 1cosh τ , w = e − σ sinh τ tanh τ σ cosh τ + cos φ ) . (32)A more detailed description of the leading discontinuity of r at l loops is then∆ l − v r ( l ) ∝ ( − l ( l − Z dp π e − ipσ (cid:18) ∞ X m =1 [ γ m +2 ( p )] l − cos( mφ ) p + m + ∞ X m =2 [ γ m − ( p )] l − cos(( m − φ ) p + ( m − (cid:19) C m ( p ) F m/ ,p ( τ ) . (33)The one-loop anomalous dimensions γ m ( p ) are the energies E (1) n of conformal primary states, andthey are given by [64], γ m ( p ) = ψ (cid:0) m + ip (cid:1) + ψ (cid:0) m − ip (cid:1) − ψ (1) . (34)The explicit formulae for the overlap functions C m ( p ) and the conformal blocks F m/ ,p ( τ ), whichaccount for the exchange of conformal descendant states, are given in ref. [41]. The formula (33)has been slightly adapted from the corresponding one for two loops [41] by raising the anomalousdimensions γ m ( p ) to the power l − τ at each loop order in eq. (31), as in this term the anomalous dimension appears inthe exponent accompanied by a factor of τ . In summary, the leading discontinuity in any of thecross ratios (which are all related by the permutation symmetry) is completely predicted by theOPE, in a formula very similar to the two-loop case.Evaluating the expression (33) is quite involved. However, following ref. [41] we can saythat it must obey certain differential equations. The differential operators D ± of ref. [41] should13nnihilate any function given by a sum of two towers of conformal blocks. Using results fromAppendix A, one can work out the form of these operators in terms of the cross ratios u , v and w : D ± = 41 − v h − z ± u∂ u − (1 − v ) v∂ v − z ± w∂ w + (1 − u ) vu∂ u u∂ u + (1 − v ) v∂ v v∂ v + (1 − w ) vw∂ w w∂ w + ( − u − v + w ) (cid:0) (1 − v ) u∂ u v∂ v − vu∂ u w∂ w + (1 − v ) v∂ v w∂ w (cid:1)i . (35)At any given loop order beyond one loop, the symbol of the remainder function R ( l )6 is actuallyequal to the symbol of the Wilson loop ratio r ( l ) . The difference between the two functions comesfrom additional terms in the expansion of eq. (28) in the coupling. For example, in eq. (5), R n is a constant for the four- and five-point contributions to r , but there are degree two functions(at most) related to F WL n , − loop that will contribute to the difference between r and the remainderfunction, when they are multiplied by transcendental constants from higher-order terms in γ K .These terms drop out of the symbol.For our three-loop analysis we require that the symbol of the leading (double) discontinuity(∆ v ) R (3)6 is annihilated by the product of D + and D − , S (cid:0) D + D − ∆ v ∆ v R (3)6 ( u, v, w ) (cid:1) = 0 . (36)The above is a very general constraint, which should apply to all expressions admitting an OPEexpansion of the form described in ref. [39]. Within our specific ansatz it becomes extremelypowerful. We find that it fixes 33 out of the 59 coefficients that were undetermined after im-posing integrability, symmetry and the collinear limit. The remaining 26 terms have no doublediscontinuity in any single channel, so they cannot be fixed without supplying additional infor-mation.In ref. [41] the sum (33) was performed for the single discontinuity ∆ v R (2)6 at two loops, forwhich only a single power of the anomalous dimensions γ m ( p ) appears. One method used is tocompute the discontinuities of the discontinuity. We can perform a similar analysis for the doublediscontinuities of the double discontinuity at three loops.At two loops the discontinuity ∆ v R (2)6 has further discontinuities of the type ∆ u , ∆ w and ∆ − v .The double discontinuity ∆ w ∆ v R (2)6 is a degree two function. When computing the integral (33)over p as a sum over residues, it arises from double poles in the p plane for l = 2. The formula (34)for γ m ( p ) contains single poles, with constant residues, at p = i ( m + 2 k ) for non-negative integers k . They can combine with single poles at the same locations in the overlap functions C m ( p ). For p = im they can also combine with poles from the p + m denominator factor. Double polesgive rise to derivatives with respect to p , which can hit the exponential e − ipσ (the only place σ appears) and bring down a factor of σ . Because log w is proportional to σ in eq. (32) as σ → + ∞ ,the coefficient of the term linear in σ yields the discontinuity with respect to w .14imilarly, at three loops the double discontinuity of the double discontinuity ∆ w ∆ w ∆ v ∆ v R (3)6 arises from triple poles in the p plane in the expression (33) for l = 3, which generate twoderivatives with respect to p acting on e − ipσ . The analysis of appendix B.1 of ref. [41] is almostdirectly applicable to ∆ w ∆ w ∆ v ∆ v R (3)6 . However, there is a small mismatch due to the factor of p + m in the denominator of the terms in the first sum in eq. (33). This factor contributesa pole at p = im , which combines with the pole coming from C m ( p ) to produce a double-polecontribution to the two-loop expression ∆ w ∆ v R (2)6 , without requiring a pole from γ m +2 ( p ). Thereare no such contributions for the three-loop expression ∆ w ∆ w ∆ v ∆ v R (3)6 , because the only triplepoles come from combining [ γ m ( p )] with C m ( p ).On the other hand, if we could remove the p + m factor in the denominator of eq. (33),then the same analysis for the two-loop problem would also apply directly at three loops. Itis important for this conclusion that the residues of γ m ( p ) are constants, independent of m and p . Removing the denominator amounts to acting with the particular second-order operator (cid:3) = − ( ∂ σ + ∂ φ ) described in ref. [41]. In terms of the cross ratios, using results from AppendixA, the operator (cid:3) is given by (cid:3) = 4 uw − v h u∂ u + w∂ w − (1 − u ) ∂ u u∂ u − (1 − w ) ∂ w w∂ w + (1 − u − v − w + 2 uw ) ∂ u ∂ w i . (37)We therefore conclude that (cid:3) ∆ w ∆ w ∆ v ∆ v R (3)6 ∝ (cid:3) ∆ w ∆ v R (2)6 = w (1 − u + v − w )(1 − v )(1 − w ) . (38)The second equation can be found by acting with (cid:3) on the symbol for the discontinuity of R (2)6 , S (∆ w ∆ v R (2)6 ) = − n u ⊗ uvw (1 − u )(1 − v )(1 − w ) − (1 − w ) ⊗ v − v − (1 − v ) ⊗ w − w − y u ⊗ y u y v y w o , (39)which is easily extracted from the symbol (21) for R (2)6 . It can also be found by applying (cid:3) tothe explicit representation for the discontinuity X found in ref. [41]. (We have not yet fixedthe overall normalization of R (3)6 ; we will do this subsequently when we match to the leading-logarithmic behaviour in the multi-Regge limit.) Remarkably, the symbol obtained after imposingthe condition (36) is perfectly consistent with the condition (38), which is a non-trivial check ofour analysis.In conclusion, after imposing the leading OPE constraints we find a solution consistent withour ansatz containing 26 unfixed parameters α i , S ( R (3)6 ) = S ( X ) + X i =1 α i S ( f i ) . (40)15ach of the symbols appearing in the above expression is required to be integrable, and so theredo exist functions X, f i with those symbols. The double discontinuities of X and the f i obey∆ v ∆ v X = 0 , D + D − ∆ v ∆ v X = 0 , S (∆ v ∆ v f i ) = 0 . (41)Although the symbol for X is one of the central results of this article, it is also rather lengthy.Therefore we do not present it directly in the text. Instead we give it in accompanying Mathe-matica and plain text files. In these files, a term a ⊗ b ⊗ . . . ⊗ f is written as SB[ a, b, . . . , f ]. Usingsymbol(ic) manipulation programs, it is straightforward to extract information about various lim-its and discontinuities from the symbol. The next section describes one such limit, multi-Reggekinematics. We now analyse our symbol in the multi-Regge limit (24), in which u → v and w vanish.First we find that in the Euclidean version of this limit, the symbol we have found vanishes, inagreement with observations [51, 52, 53] about the consistency of the BDS ansatz in this typeof limit. Next we analytically continue to a physical branch, defined by letting u → e − πi u . Forphysical 2 → v and w remain at their Euclidean values. The imaginary terms onthe physical branch that are generated by this transformation of u come from the discontinuityof the function in the u channel in the multi-Regge limit. As mentioned in section 2, the symbolof the discontinuity of a function f in a given channel ( u say) can be found by taking the termsin the original symbol S ( f ) with initial entry u and stripping off that entry. The result, aftermultiplying by ( − πi ), is the symbol of the discontinuity S (∆ u f ). The real terms for 2 → u channel. They are found from S (∆ u ∆ u f ),after multiplying by (2 πi ) . (In principle, there can be contributions to the imaginary and realparts from triple and higher order discontinuities in u as well. However, through three loopsthere are no such terms.)The behaviour we expect for the l -loop remainder function in the multi-Regge limit in thephysical region is R ( l )6 −→ (2 πi ) l − X r =0 log r (1 − u ) h g ( l ) r ( x, y ) + 2 πi h ( l ) r ( x, y ) i , (42)where the logarithmic expansion coefficients g ( l ) r and h ( l ) r are functions that depend only on thefinite ratios x and y defined in eq. (24). It is convenient to change variables to describe thesefunctions. Following ref. [48], we introduce the variables w, w ∗ defined by x = 1(1 + w )(1 + w ∗ ) , y = ww ∗ (1 + w )(1 + w ∗ ) . (43) The new variable w in the multi-Regge limit (which is always accompanied by a w ∗ ) should not be confusedwith the original cross ratio w .
16n terms of these variables, the symbols of the functions g ( l ) r and h ( l ) r have as their only entries w, w ∗ , (1 + w ), and (1 + w ∗ ).Both g ( l ) r and h ( l ) r are invariant under two Z symmetries:conjugation : w ←→ w ∗ , (44)which is a reality condition for the case that w ∗ is the complex conjugate of w , andinversion : w ←→ /w, w ∗ ←→ /w ∗ . (45)The combined operation of inversion and conjugation is the reflection symmetry x ↔ y , whichis inherited from the permutation symmetry v ↔ w for generic kinematics. We also expect thefunctions g ( l ) r and h ( l ) r to be single-valued as w is rotated around the origin of the complex plane.Finally, the functions should vanish for | w | →
0, which is the collinear limit on top of the Reggelimit.In taking the multi-Regge limit (24) of symbols, we note that any symbol containing a u ora y u entry can be discarded, because u → y u → x and y in eq. (24). The variables y v and y w go to finite values, ˜ y v and ˜ y w , in the limit: y v −→ ˜ y v = − − x + y + p ˜∆ − − x + y − p ˜∆ = 1 + w ∗ w , (46) y w −→ ˜ y w = − x − y + p ˜∆ − x − y − p ˜∆ = (1 + w ) w ∗ w (1 + w ∗ ) , (47)where ˜∆ = (1 − x − y ) − xy is the limit of ∆ / (1 − u ) . The relation of ˜ y v and ˜ y w to the ( w, w ∗ )variables can be found with the aid of formulae in Appendix B.The symbols S ( g ( l ) r ) and S ( h ( l ) r ) do not fix the functions g ( l ) r and h ( l ) r uniquely. One can alwaysadd transcendental constants such as ζ , multiplied by lower transcendentality functions whichvanish in the symbol. However, the above symmetries, eqs. (44) and (45), and analytic propertiesaround w = 0, greatly restrict the form of such potential ambiguities. In particular there are nosuch functions of degree 0 or 1 obeying these constraints.Before describing the three-loop predictions, we recall [47, 48] the corresponding expan-sion (42) at two loops, as obtained from the formula of ref. [35], g (2)1 ( w, w ∗ ) = 14 log | w | log | w | | w | , (48) g (2)0 ( w, w ∗ ) = 14 log | w | log | w | −
16 log | w | + 12 log | w | h Li ( − w ) + Li ( − w ∗ ) i − Li ( − w ) − Li ( − w ∗ ) . (49)It is not always the case that w ∗ is the complex conjugate of w . (That only happens if p ˜∆ isimaginary.) In the general case, | w | is just a shorthand for ww ∗ , and | w | is a shorthand for(1 + w )(1 + w ∗ ). 17he functions controlling the real parts depend on whether the scattering is 2 → → → h (2)1 ( w, w ∗ ) = 0 , (50) h (2)0 ( w, w ∗ ) = 0 . (51)In the case of 3 → v and w have to be analytically continued to the opposite signfrom their Euclidean values [49]; that is, u → | u | e πi , v → | v | e πi , w → | w | e πi . (52)In fact, the remainder function for 3 → → − u ) −→ log( u − − iπ , (53)followed by complex conjugation [49].Whereas the function g (2)1 in eq. (48) is manifestly invariant under both conjugation andinversion symmetries, g (2)0 in eq. (49) only has manifest invariance under w ↔ w ∗ . On the otherhand, this form makes clear that g (2)0 vanishes as | w | →
0, and also that it acquires no phase as w isrotated around the origin of the complex plane. The latter property is obvious for | w | < | w | >
1. Simple polylogarithm identities can be used to demonstratethe w inversion symmetry. In fact, assuming maximal transcendentality, the functions g (2)1 and g (2)0 , of degree 2 and 3 respectively, can be fixed uniquely, just by knowing the symbol of thetwo-loop remainder function and imposing these requirements. The uniqueness holds because thesymbol fixes the functions up to constants like ζ or ζ , multiplied by functions of correspondinglower degree, and there are no functions with degree 0 or 1 obeying the constraints.At three loops we find that in the multi-Regge limit, the symbol S ( X ) has the form of thesymbol of the right-hand side of eq. (42) for l = 3, with the leading divergence being a doublelogarithmic one. We also find that in this limit, all but three of the S ( f i ) vanish. We will call thefunctions with non-vanishing symbols in the limit f , f , f . We find that one symbol, S ( f ),has a triple logarithmic divergence in the multi-Regge limit, which is one logarithm beyondthe known degree of divergence. Therefore the coefficient α must vanish. The symbol S ( X )contributes to the double logarithmic divergence exactly what is required to match the symbolof the leading-log prediction [48]. We find that S ( f ) also contributes a double logarithmicdivergence (different in form from that of S ( X )). Hence we deduce that its coefficient α mustvanish, so that it does not spoil the agreement with the leading-log prediction. The final symbol S ( f ) then contributes to the next-to-leading-log term ( i.e. to S ( g (3)1 )) but not to the next-to-next-to-leading one ( i.e. not to S ( g (3)0 )). Because it is the only arbitrary coefficient from theexpression (40) that survives in the multi-Regge limit (after imposing the constraints we havejust discussed), we give it a new name, α = c .Now we describe our predictions for the multi-Regge limit, after imposing the conditions, α = c, α = 0 , α = 0 . (54)18e find (as described above) that the symbol of g (3)2 agrees precisely with the symbol of thecoefficient of the log (1 − u ) term predicted in ref. [48], namely S ( g (3)2 ) = 132 (cid:16) x ⊗ x ⊗ y + 3 x ⊗ y ⊗ xy − x ⊗ ˜ y w ⊗ ˜ y v ˜ y w (cid:17) + ( x ←→ y ) . (55)We have adjusted the overall normalization of X so that this term in the multi-Regge limit agreeswith ref. [48]. This normalization is based on the loop expansion parameter a = g N c / (8 π ) andeq. (6).When written in terms of the w, w ∗ variables, the symbol (55) can be seen to be the symbolof the following function, g (3)2 ( w, w ∗ ) = 18 g (2)0 ( w, w ∗ ) −
132 log | w | log | w | | w | log | w | | w | , (56)exactly as predicted in ref. [48]. Just as in the two-loop case, this degree 3 function is uniquelydetermined by its symbol, because there are no suitable degree 1 or 0 functions one could add.Also, we find from the double u discontinuity that the real part at leading-log level vanishes, h (3)2 ( w, w ∗ ) = 0 , (57)as expected.We also have predictions for the symbols of g (3)1 , g (3)0 and h (3)0 (and their corresponding func-tions) which are new. The function h (3)1 for 2 → w , and manifestly has good behaviouras | w | →
0. We choose to express the functions in a form where the | w | → g (3)1 ( w, w ∗ ) = 18 ( log | w | (cid:20) Li (cid:18) w w (cid:19) + Li (cid:18) w ∗ w ∗ (cid:19)(cid:21) + (5 log | w | − | w | ) h Li ( − w ) + Li ( − w ∗ ) i −
32 log | w | log | w | | w | h Li ( − w ) + Li ( − w ∗ ) i −
112 log | w | (cid:20) log | w | (log | w | + 2 log | w | ) −
10 log | w | | w | (cid:21) + 12 log | w | log | w | | w | log(1 + w ) log(1 + w ∗ ) − ζ log | w | ) + (cid:18)
52 + γ ′ (cid:19) ζ g (2)1 ( w, w ∗ ) + c g a . (58)19or this degree-four function there are only two constants to determine. The first one, γ ′ ,corresponds to the freedom to add the two-loop remainder function, multiplied by ζ , to thethree-loop remainder function, as in eq. (27). The second constant, c , is the remaining ambiguityat the level of the symbol. It multiplies the function, g a ( w, w ∗ ) = 4 log | w | | w | h Li ( − w ) + Li ( − w ∗ ) i − | w | (cid:20) Li (cid:18) w w (cid:19) + Li (cid:18) w ∗ w ∗ (cid:19)(cid:21) + 2 h Li ( − w ) − Li ( − w ∗ ) + log | w | log 1 + w w ∗ ih Li ( − w ) − Li ( − w ∗ ) i + 16 log | w | (log | w | + 2 log | w | ) − | w | log | w | log(1 + w ) log(1 + w ∗ )+ 8 ζ log | w | . (59)We will see later that this function does not enter, i.e. that c = 0, if we impose consistencywith the all-loop-order prediction for 3 → R (3)6 .We rule out additional constants multiplying lower-degree transcendental functions in eq. (58)by first assuming that potential functions must be built from logarithms and (at high enoughdegree) polylogarithms containing the same arguments found in the leading-transcendentality(symbol-level) terms, namely log w , log(1 + w ), Li m ( − w ), Li m ( w/ (1 + w )) and Li m (1 / (1 + w ))(for m = 2 , | w | → w is rotatedaround the origin of the complex plane. These constraints rule out functions of degree 0 or 1.The unique function of degree 2 obeying these conditions is g (2)1 ( w, w ∗ ) . If we had omitted thefinal-entry condition, for example, we could have added a term proportional to ζ log (cid:18) w w ∗ (cid:19) log (cid:18) (1 + w ) w ∗ (1 + w ∗ ) w (cid:19) . (60)This term has both symmetries and vanishes as | w | →
0; in fact, it is the unique term at degreetwo that satisfies the other three constraints but violates the phase condition.The degree-three function controlling the real part at next-to-leading-log level, h (3)1 , can befound from the multi-Regge limit of the double u discontinuity. (There is an overall factor of 1 / u is 2 u ⊗ u .) We find S ( h (3)1 ) = S ( g (3)2 ) − h x ⊗ y ⊗ y + y ⊗ x ⊗ y + y ⊗ y ⊗ x + ( x ←→ y ) i , (61)which integrates to h (3)1 ( w, w ∗ ) = g (3)2 ( w, w ∗ ) + 116 log | w | log | w | | w | log | w | | w | . (62)20his result agrees with that found in ref. [48].Moving on to next-to-next-to-leading-log level, we find the degree-five function controllingthe imaginary part, g (3)0 ( w, w ∗ ) = − ( − h (cid:16) Li ( − w ) + Li ( − w ∗ ) (cid:17) − log | w | (cid:16) Li ( − w ) + Li ( − w ∗ ) (cid:17)i + 12 (cid:20) (cid:18) Li (cid:18) w w (cid:19) + Li (cid:18)
11 + w (cid:19) + 124 log w log (1 + w )+ Li (cid:18) w ∗ w ∗ (cid:19) + Li (cid:18)
11 + w ∗ (cid:19) + 124 log w ∗ log (1 + w ∗ ) (cid:19) + log | w | | w | (cid:18) Li (cid:18) w w (cid:19) + Li (cid:18) w ∗ w ∗ (cid:19)(cid:19) + log | w | (cid:18) Li (cid:18)
11 + w (cid:19) −
16 log w log (1 + w )+ Li (cid:18)
11 + w ∗ (cid:19) −
16 log w ∗ log (1 + w ∗ ) (cid:19)(cid:21) − (cid:16) | w | − log | w | ) + 6 log | w | log | w | (cid:17)(cid:16) Li ( − w ) + Li ( − w ∗ ) (cid:17) − | w | log | w | | w | (cid:18) Li (cid:18) w w (cid:19) + Li (cid:18) w ∗ w ∗ (cid:19)(cid:19) − | w | log | w | log | w | | w | (cid:16) Li ( − w ) + Li ( − w ∗ ) (cid:17) + 53 log | w | −
52 log | w | log | w | + 43 log | w | log | w | − log | w | log (1 + w ) log (1 + w ∗ ) − | w | log(1 + w ) log(1 + w ∗ )+ ζ log | w | log | w | (log | w | − | w | ) + 4 ζ log | w | log | w | − ζ ) + ζ d g (2)1 ( w, w ∗ ) + ζ γ ′′ g (2)0 ( w, w ∗ ) + ζ d log | w | log | w | | w | log | w | | w | . (63)Note that although Li m (1 / (1 + w )) has logarithmic branch-cut behaviour near w = 0, the com-bination Li m (cid:18)
11 + w (cid:19) − ( − m ( m − w log m − (1 + w ) (64)is well-behaved. This property can be verified inductively by differentiating with respect to w and using ddw Li m (cid:18)
11 + w (cid:19) = −
11 + w Li m − (cid:18)
11 + w (cid:19) . (65)After using the combination (64) in g (3)0 , there are no other bare log w terms; they all come21long with a log w ∗ to form log | w | . Note that for m = 3 one can use an identity to eliminateLi (1 / (1 + w )) in favor of Li m ( − w ) and Li ( w/ (1 + w )), but there is no such identity for m > g (3)1 , all possible constraints will be satisfied by a function proportionalto the two-loop remainder function, multiplied by ζ . This accounts for the term proportional to g (2)0 ( w, w ∗ ). In addition, we can multiply the two-loop leading-log multi-Regge coefficient g (2)1 by ζ , to get something with the right transcendental degree and satisfying the above constraints.Presumably its coefficient, d , can be fixed by additional beyond-the-symbol information. Finally,there is another degree-three function satisfying all the constraints we imposed, with a coefficient d which we expect to be fixed in a similar fashion. This purely-logarithmic degree-three functionis a linear combination of the next-to-leading-log two-loop function g (2)0 and the leading-log three-loop function g (3)2 , as in eq. (56).The real part at next-to-next-to-leading-log level is given by the degree-four function, h (3)0 ( w, w ∗ ) = 116 ( − (cid:16) | w | − | w | (cid:17)h Li ( − w ) + Li ( − w ∗ ) i + log | w | (cid:20) Li (cid:18) w w (cid:19) + Li (cid:18) w ∗ w ∗ (cid:19)(cid:21) + 12 log | w | log | w | | w | h Li ( − w ) + Li ( − w ∗ ) i −
12 log | w | + 56 log | w | log | w | −
14 log | w | log | w | + 12 log | w | log | w | | w | log(1 + w ) log(1 + w ∗ ) − ζ log | w | ) + ζ γ ′′′ g (2)1 . (66)As was the case for eq. (58), the term containing an explicit ζ in eq. (66) is fixed using thesymmetries and the vanishing of h (3)0 as | w | →
0. There is an arbitrary constant γ ′′′ multiplying g (2)1 , but we will see shortly how to fix it.In ref. [46], the scattering amplitude in the multi-Regge limit was expressed as a sum ofRegge pole and Mandelstam cut contributions. By using this representation, general formulaewere obtained for the multi-Regge limit of the remainder function in both 2 → → f ( ω ; x, y ) characterizing the partial waves entering theMandelstam cut,exp[ R + iπδ ] = cos πω ab + i Z i ∞− i ∞ dω πi f ( ω ; x, y ) e − iπω | − u | − ω (2 → , (67)exp[ R − iπδ ] = cos πω ab − i Z i ∞− i ∞ dω πi f ( ω ; x, y ) | − u | − ω (3 → . (68)22ere exp[ R ] = 1 + a R (2)6 + a R (3)6 + . . . , (69) δ = − γ K | w | | w | , (70) ω ab = γ K | w | , (71)and the cusp anomalous dimension γ K is given by γ K = 4 a − ζ a + 22 ζ a + . . . , (72)in terms of the coupling constant a = g N c / (8 π ). Note that the quantity appearing in eqs. (67)and (68) is the ratio of the full amplitude (or Wilson loop) to the BDS ansatz, which accordingto our conventions (see eqs. (5) and (6)) is the exponential of the remainder function. The phase δ comes from the behavior of the BDS ansatz in the multi-Regge limit, while ω ab is derived fromthe Regge-pole contribution.Remarkably, the second term in eq. (68), containing f ( ω ; x, y ), drops out when we take thereal part of the equation, leading to the all-loop-order relation for 3 → n exp[ R − iπδ ] o = cos πω ab (3 → . (73)The factor of e − iπω inside the integral in eq. (67) prevents an analogously simple relation fromholding for 2 → g ( l ) r and h ( l ) r through l = 3, we can easily testeq. (73) at the three-loop level. We assemble the exponential of the remainder function, exp[ R ]in eq. (69), using eq. (42) for 2 → → e − iπδ and taking the real part, we find that eq. (73) is satisfied precisely, throughthree loops — but only if we set c = 0 in eq. (58) for g (3)1 . In addition we must fix the constant γ ′′′ in eq. (66) for h (3)0 to the value, γ ′′′ = 94 + γ ′ . (74)The imaginary part g (3)1 for 2 → R (3)6 for 3 → g (3)1 is multiplied by log(1 − u ) ineq. (42). In fact, the only function that does not enter eq. (73) is the degree-five function g (3)0 ,because it is from the imaginary part and has no log(1 − u ) multiplying it. Hence eq. (73) is apowerful check on our results.The c = 0 constraint imposed by eq. (73) also arises from considering restrictions on the finalentry of the symbol, as we shall do in the next section.23 Constraints on the final entry of the symbol
We have shown that within the specific ansatz (14) we were able to write the symbol of thethree-loop remainder function in the form S ( R (3)6 ) = S ( X ) + X i =1 α i S ( f i ) . (75)There are 24 unfixed parameters α i , after imposing all of the constraints we have outlined,including the constraints coming from the multi-Regge limit (54). Moreover, by examining thesymbol S ( X ) we find that it is possible to write it so that the final entries are drawn from thefollowing set, n u − u , v − v , w − w , y u , y v , y w o . (76)The same restriction is true for the symbol of the full remainder function R (2)6 at two loops,given in eq. (21). As mentioned above, it has been suggested [38] that this fact is related to asupersymmetric formulation of the Wilson loop; and similar restrictions appear [57] in differentialequations [54, 55, 56] for integrals related to scattering amplitudes [23]. It is reasonable to thinkthat the full symbol S ( R (3)6 ) should obey this condition, including the ambiguities S ( f i ). Infact, it is possible to impose this condition on the remaining ambiguities, leaving just two freeparameters, S ( R (3)6 ) = S ( X ) + α S ( f ) + α S ( f ) . (77)The fact that this form for the symbol is consistent with the known Regge behaviour is highlynon-trivial. Indeed, one can adopt the constraint on the final entries from the beginning. Inthis case, after imposing the OPE constraints, the triple-log in the multi-Regge limit vanishesautomatically, and the leading-log contribution g (3)2 is uniquely fixed to agree with the predictionof refs. [49, 50]. Finally, the single remaining free parameter in the multi-Regge limit (whichappears in the function g (3)1 in eq. (58)) is fixed, c = 0 , (78)leaving a completely unambiguous prediction for the symbol of g (3)1 in this limit (the symbol for g (3)0 was already fixed unambiguously). It is reassuring that the same vanishing value for c is alsodictated by the relation (73) for the multi-Regge limit for 3 → S ( f ) is extremely simple: It is entirely composed from the entries { u, v, w, − u, − v, − w } ; the square-root containing y variables in eq. (14) do not appear in S ( f ). Thisproperty makes it straightforward to find an explicit function f , which has the symbol S ( f ).The function can be written in the form, f ( u, v, w ) = h ( u ) h ( v ) + h ( u ) h ( w ) + h ( v ) h ( w ) + k ( u ) + k ( v ) + k ( w ) . (79)24ere the single-variable functions h and k are given by h ( u ) = log u + log u Li (1 − u ) − Li (1 − u ) − (1 − /u ) , (80) k ( u ) = − log u H + log u ( H − H , − H , ) − log u ( H , − H , + H , , + 6 H , , + 18 H , , )+ 3 H , + 4 H , + 3 H , + H , , − H , , − H , , − H , , + 9 H , , − H , , , − H , , , − H , , , . (81)The arguments of the harmonic polylogarithms appearing in k ( u ) are all (1 − u ) and have beensuppressed to save space. A subscript m stands for m − H , , = H , , , , , (1 − u ).The function f above has been chosen so that it obeys ∂ u f = 1 u (1 − u ) (pure function) . (82)The fact that taking the derivative yields a pure function with the particular 1 / ( u (1 − u )) prefactoris the functional consequence of the final-entry condition on the symbol. The function f is real-valued in the Euclidean region but does not vanish in the collinear limit. It only vanishes up toterms involving explicit appearances of ζ ( π ) and ζ , which is what is guaranteed by the formof its symbol. In fact, already at the ζ level we find that f is divergent in this limit,lim w → f = ζ h log w (cid:16) log u log (1 − u ) + log u Li ( u ) + 2 log(1 − u )Li ( u ) − ( u ) + 3 H , ( u ) (cid:17) + finite i + ζ h . . . i . (83)In fact there exists no degree 4 function with a symbol within our ansatz, and also obeying theproperty (82), which could be used to remove this divergence in the collinear limit. This factsuggests that if we insist on preserving the functional consequence of the final entry condition (82),beyond the level of the symbol, then there will be additional constraints on the parameter α when completing the symbol S ( R (3)6 ) to a genuine function.The function f is intermediate in complexity between f and X . Its symbol contains termswith up to two y -variable entries, while the symbol for X has terms with four y -variable entries.Files containing the symbols for f , f and X , as well as the symbols of the functions charac-terizing the multi-Regge limit (which for c = 0 come entirely from X ), are provided as auxiliarymaterial for this paper.We leave to later work an explicit construction of functions associated with the other symbols,particularly S ( X ) and S ( f ). However, we can already say some things about the full three-loopremainder function. In particular, for any values of α and α , it is impossible to represent itssymbol by a function given in terms of (products of) only single-variable harmonic polylogarithms H ~w ( x ), whose weight vectors ~w contain only the entries 0 and 1. As a corollary, it is not possibleto represent the symbol by a function given purely in terms of the classical polylog functions25i n ( x ), for any choices of x . This result can be obtained by performing symmetry operationssimilar to those described in ref. [35]. It is sufficient, and a bit simpler, to test not the full symbol,but a particular piece of it. We take the double discontinuity in w , and then set w →
0, usingthe relations (22). This symbol is given by S (∆ w ∆ w X ) | w → = 18 ( u ⊗ u ⊗ (cid:20) − (1 − u ) ⊗ uv (1 − u )(1 − u − v ) + v ⊗ − v − u − v (cid:21) + u ⊗ (1 − u ) ⊗ (cid:20) − u (1 − u − v ) ⊗ uv (1 − u )(1 − u − v ) + u (1 − u − v ) ⊗ (1 − u ) (1 − v )(1 − u − v ) + v (1 − u − v ) ⊗ (1 − u )(1 − v )(1 − u − v ) (cid:21) + u ⊗ v ⊗ (cid:20) − − u )(1 − v )1 − u − v ⊗ uv (1 − u − v ) + u (1 − u − v ) ⊗ − v − u − v + v (1 − u − v ) ⊗ − u − u − v (cid:21)) + ( u ←→ v ) . (84)We replace each term of the form a ⊗ b ⊗ c ⊗ d in this expression with the following antisym-metrisation [35]: h ( a ⊗ b ⊗ c ⊗ d − ( c ↔ d )) − ( a ↔ b ) i − h ( a, b ) ↔ ( c, d ) i . (85)We find that eq. (84) is nonvanishing under this operation. The symbol of a degree four functionconstructed solely from products of single-variable harmonic polylogarithms with labels 0 and 1(which includes all Li n functions) vanishes under this operation. Hence (∆ w ∆ w X ) w → , and also X itself, must include functions beyond this class.We have also performed a similar test on the full degree six function. Given a degree sixsymbol which is a sum of terms of the form a ⊗ b ⊗ c ⊗ d ⊗ e ⊗ f , we replace each term with thefollowing antisymmetrisation, h(cid:0) ( a ⊗ b ⊗ c ⊗ d ⊗ e ⊗ f − ( e ↔ f )) − ( c ↔ d ) (cid:1) − ( a ↔ b ) i − h ( a, b ) ↔ ( e, f ) i . (86)The symbol of a degree six function constructed solely from products of single-variable harmonicpolylogarithms with labels 0 and 1 vanishes under this operation. We find that S ( X ) does notvanish under this operation, so again we conclude that X must include functions beyond thisclass. In this paper we determined the symbol of the remainder function for the three-loop hexagonWilson loop, or six-point MHV scattering amplitude, in planar N = 4 super-Yang-Mills theory,26p to a few undetermined constants. There are 26 such constants in a more general ansatz,but this number drops to just two if a final-entry restriction is imposed on the symbol. TheOPE expansion, as analysed in refs. [39, 40, 41], provides a powerful constraint for this problem,which is straightforward to implement with the aid of symbols. In particular, we uniquelydetermined the symbol S ( X ) for the part of the three-loop remainder function that has a leadingdiscontinuity.In the multi-Regge limit, all but one of the symbol-level constants drop out (all of themdrop out when we impose the final-entry restriction). In this limit, we are able to complete thesymbols for the coefficients in the logarithmic expansion into full analytic functions of degree3, 4 and 5. These functions depend on two variables, yet they can all be expressed in terms ofclassical polylogarithms. Three of these functions represent new predictions for the behaviourof the amplitude in the multi-Regge limit. We found confirmation of the final-entry restrictionby testing an all-order relation for the remainder function in multi-Regge kinematics for 3 → f , one of the two terms that we could not fixusing the leading discontinuity (assuming the final-entry restriction), we were able to accomplishthis task. This function is particularly simple due to the fact that its symbol does not dependon the y variables, but only on { u, v, w, − u, − v, − w } . It factorises into single-variablefunctions constructed out of harmonic polylogarithms.The next simplest component is f . It is only quadratic in the y variables, so in some sense itis not much more complicated than the two-loop remainder function, although it is of degree sixinstead of four. The most complicated term is X , which is quartic in the y variables. However,we are optimistic that a relatively compact representation for it, as well as f , may be possibleto find. We are also encouraged by the fact that the collinear limits of f , which diverge beyondthe symbol level, cannot be repaired within functions obeying the final-entry restriction. Thisfact suggests that the repair may come instead through the collinear behaviour of X and f ,which could in turn fix one or both of the arbitrary constants α and α . It would be remarkableif the three-loop remainder function could be completely determined, or perhaps up to a singleambiguity associated with the two-loop remainder function, without ever directly evaluating asingle loop integral, for either a Wilson loop or a scattering amplitude. Acknowledgements
We would like to thank Harald Dorn and Thomas Gehrmann for useful discussions. J.M.H. isgrateful to the KITP, Santa Barbara, for hospitality while this work was carried out. This workwas supported in part by the National Science Foundation under Grant No. PHY05-51164, andby the US Department of Energy under contract DE–AC02–76SF00515.27 ppendix A
In this appendix we provide handy equations for relating various differential operators in term ofthe variables τ , σ and φ to those in terms of the cross ratio variables u , v and w . From eq. (32)we have the auxiliary relations1 − u − v − w − v = cos φ cosh σ cosh τ + cos φ , uvw (1 − v ) = 1(cosh σ cosh τ + cos φ ) , (87) √ ∆1 − v = i sin φ cosh σ cosh τ + cos φ , tanh τ = √ − v . (88)Using these relations, it is simple to show that1 u ∂u∂τ = 1 w ∂w∂τ = 1 − u + v − w √ − v , v ∂v∂τ = − √ − v , (89)1 u ∂u∂σ = 1 − u − v + w − v , w ∂w∂σ = − u − v − w − v , ∂v∂σ = 0 , (90)1 u ∂u∂φ = 1 w ∂w∂φ = 1 i √ ∆1 − v , ∂v∂φ = 0 . (91)Then the operators differentiating with respect to τ , σ and φ are ∂ τ = 1 √ − v h − − v ) v∂ v + (1 − u + v − w )( u∂ u + w∂ w ) i , (92) ∂ σ = 11 − v h (1 − u − v + w ) u∂ u − (1 + u − v − w ) w∂ w i , (93) ∂ φ = √ ∆ i (1 − v ) ( u∂ u + w∂ w ) . (94)Inserting these expressions into the form for D ± given in ref. [41], D ± = ∂ τ + 2 coth(2 τ ) ∂ τ + sech τ ∂ σ + ∂ φ ( ∂ φ ∓ i ) , (95)it is straightforward to obtain the form in terms of u , v and w given in eq. (35). Similarly, theoperator (cid:3) = − ( ∂ σ + ∂ φ ) is found to have the form given in eq. (37). Appendix B
Although the y variables are constructed using square roots of the original cross ratios u , v and w , the cross ratios themselves are rational combinations of the variables y u , y v and y w . The28xplicit relations are, u = y u (1 − y v )(1 − y w )(1 − y w y u )(1 − y u y v ) , v = y v (1 − y w )(1 − y u )(1 − y u y v )(1 − y v y w ) , w = y w (1 − y u )(1 − y v )(1 − y v y w )(1 − y w y u ) , (96)1 − u = (1 − y u )(1 − y u y v y w )(1 − y w y u )(1 − y u y v ) , − v = (1 − y v )(1 − y u y v y w )(1 − y u y v )(1 − y v y w ) , (97)1 − w = (1 − y w )(1 − y u y v y w )(1 − y v y w )(1 − y w y u ) , √ ∆ = (1 − y u )(1 − y v )(1 − y w )(1 − y u y v y w )(1 − y u y v )(1 − y v y w )(1 − y w y u ) , (98)where we have picked a particular branch of √ ∆. These formulas are also useful in the multi-Regge limit. The limit (24) corresponds to taking y u → y v → ˜ y v , y w → ˜ y w . We find in thelimit, x = ˜ y v (1 − ˜ y w ) (1 − ˜ y v ˜ y w ) , y = ˜ y w (1 − ˜ y v ) (1 − ˜ y v ˜ y w ) , p ˜∆ = (1 − ˜ y v )(1 − ˜ y w )1 − ˜ y v ˜ y w . (99)The variables w and w ∗ used in the multi-Regge limit, defined in eq. (43), are also rationalcombinations of ˜ y v and ˜ y w : w = 1 − ˜ y v ˜ y v (1 − ˜ y w ) , w ∗ = ˜ y w (1 − ˜ y v )1 − ˜ y w . (100)Inverting these equations gives the expressions for ˜ y v and ˜ y w in terms of w and w ∗ given ineqs. (46) and (47). Appendix C
Here we write the three-loop functions g (3) r and h (3) r in a form that makes the w inversion and w ↔ w ∗ symmetries manifest. To do so, we introduce functions ˆ g ( l ) r ( w, w ∗ ) and ˆ h ( l ) r ( w, w ∗ ) suchthat the sum over images under the two symmetries yields the full functions: g ( l ) r ( w, w ∗ ) = ˆ g ( l ) r ( w, w ∗ ) + ˆ g ( l ) r ( w ∗ , w ) + ˆ g ( l ) r (1 /w, /w ∗ ) + ˆ g ( l ) r (1 /w ∗ , /w ) , (101) h ( l ) r ( w, w ∗ ) = ˆ h ( l ) r ( w, w ∗ ) + ˆ h ( l ) r ( w ∗ , w ) + ˆ h ( l ) r (1 /w, /w ∗ ) + ˆ h ( l ) r (1 /w ∗ , /w ) . (102)We find thatˆ g (3)2 ( w, w ∗ ) = − (cid:20) ( − w ) − log | w | Li ( − w ) −
112 log | w | (cid:18) log | w | − | w | | w | (cid:19)(cid:21) , (103)which agrees with eq. (56) and with ref. [48] after the use of a few polylogarithm identities.Similarly, h (3)1 can be written symmetrically usingˆ h (3)1 ( w, w ∗ ) = − (cid:20) ( − w ) − log | w | Li ( − w ) −
112 log | w | (cid:18) log | w | + 3 log | w | | w | (cid:19)(cid:21) . (104)29he new functions found in this paper are g (3)1 , g (3)0 and h (3)0 . For g (3)1 the symmetric formusesˆ g (3)1 ( w, w ∗ ) = − (cid:26) (cid:16) | w | − | w | (cid:17) Li (cid:18)
11 + w (cid:19) + 3 log | w | log | w | | w | Li ( − w )+ 316 (cid:20) log | w | + 8 log | w | log(1 + w ∗ ) log 1 + ww + 2 log | w | log ww ∗ log (1 + w ) w (cid:21) − | w | | w | log | w | log(1 + w ) log 1 + ww + 32 ζ log | w | − ζ log | w | (cid:27) + ζ γ ′ g (2)1 ( w, w ∗ ) + c ˆ g a , (105)The constant c multiplies the function,ˆ g a = − | w | Li (cid:18) w w (cid:19) + Li ( − w ) h Li ( − w ) − Li ( − w ∗ ) + log | w | log 1 + w w ∗ i + 124 log | w | −
14 log | w | log | w | log | w | | w | + 12 log | w | log w log(1 + w ) log 1 + ww ∗ + 13 log | w | log (1 + w ) (2 log(1 + w ) − w ) − ζ log | w | log | w | | w | + 4 ζ log | w | . (106)Recall that c = 0 if we impose either the all-loop-order prediction for 3 → R (3)6 , as described insection 6. 30he function entering the symmetric form for g (3)0 isˆ g (3)0 ( w, w ∗ ) = − (cid:26) − (cid:16) ( − w ) − log | w | Li ( − w ) (cid:17) + 12 (cid:18) (cid:18)
11 + w (cid:19) + log | w | Li (cid:18)
11 + w (cid:19)(cid:19) + (cid:16) − | w | − | w | log | w | + 5 log | w | (cid:17) Li ( − w )+ log | w | log | w | | w | Li (cid:18)
11 + w (cid:19) − | w | log | w | log | w | | w | Li ( − w ) −
148 log | w | | w | (cid:18) log | w | + log | w | | w | − | w | log | w | | w | (cid:19) + 132 log ww ∗ log | w | log | w | −
132 log 1 + w w ∗ (cid:18) log | w | + 2 log | w | log | w | | w | (cid:19) × (cid:18) ww ∗ log | w | − log 1 + w w ∗ log | w | | w | (cid:19) + 12 ζ log | w | − ζ (cid:27) + ζ d g (2)1 ( w, w ∗ ) + ζ γ ′′ g (2)0 ( w, w ∗ ) + ζ d log | w | log | w | | w | log | w | | w | . (107)Finally, the function needed to write h (3)0 in a symmetric form isˆ h (3)0 ( w, w ∗ ) = 1128 (cid:26) (cid:16) | w | − | w | (cid:17) Li (cid:18)
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