A Surprise in the Amplitude/Wilson Loop Duality
Andreas Brandhuber, Paul Heslop, Panagiotis Katsaroumpas, Dung Nguyen, Bill Spence, Marcus Spradlin, Gabriele Travaglini
aa r X i v : . [ h e p - t h ] A p r Brown-HET-1592, IPPP/10/30, DCPT/10/60, QMUL-PH-10–04
A Surprise in the Amplitude/Wilson Loop Duality
Andreas Brandhuber a , Paul Heslop b , Panagiotis Katsaroumpas a ,Dung Nguyen c , Bill Spence a , Marcus Spradlin c and Gabriele Travaglini aa Centre for Research in String TheoryDepartment of Physics, Queen Mary University of LondonLondon, E1 4NS, United Kingdom b Institute for Particle Physics PhenomenologyDepartment of Mathematical Sciences and Department of PhysicsDurham University, Durham, DH1 3LE, United Kingdom c Department of PhysicsBrown University, Providence, Rhode Island 02912, USA
Abstract
One of the many remarkable features of MHV scattering amplitudes is their conjec-tured equality to lightlike polygon Wilson loops, which apparently holds at all ordersin perturbation theory as well as non-perturbatively. This duality is usually expressedin terms of purely four-dimensional quantities obtained by appropriate subtraction ofthe IR and UV divergences from amplitudes and Wilson loops respectively. In thispaper we demonstrate, by explicit calculation, the completely unanticipated fact thatthe equality continues to hold at two loops through O ( ǫ ) in dimensional regulariza-tion for both the four-particle amplitude and the (parity-even part of the) five-particleamplitude. Introduction
Amongst the many remarkable features of the mathematical structure of scatteringamplitudes that have emerged in the past several years, one of the most mysteriousremains the apparent equality between planar maximally helicity violating (MHV)scattering amplitudes and lightlike Wilson loops in maximally supersymmetric Yang-Mills (SYM) theory. This new aspect of duality first emerged in [1], where Alday andMaldacena argued using AdS/CFT that the prescription for computing scatteringamplitudes at strong coupling was mechanically identical to that for computing theexpectation value of a Wilson loop over the closed contour obtained by gluing themomenta of the scattering particles back-to-back to form a polygon with lightlikeedges.Adhering to the principle that there is no such thing as a coincidence in SYMtheory, it was suggested in [2, 3] that MHV amplitudes and Wilson loops might beequal to each other not just at strong coupling, as the work of Alday and Maldacenaindicated, but perhaps even order by order in perturbation theory. This bold sugges-tion was confirmed by explicit calculations at one loop for four particles in [2] and forany number of particles in [3], and at two loops for four and five particles in [4, 5].Already for a few years prior to these developments planar MHV amplitudes inSYM had come under close scrutiny following the discovery of the ABDK relation [6],which expresses the four-point two-loop amplitude as a certain quadratic polynomialin the corresponding one-loop amplitude, a relation which was later checked to holdalso for the five-point two-loop amplitude [7, 8]. The all-loop generalization of theABDK relation, known as the BDS ansatz after the authors of [9], expresses anappropriately defined infrared finite part of the all-loop amplitude in terms of theexponential of the one-loop amplitude. This proposal has also been completely verifiedfor the three-loop four-point amplitude [9], and partially explored for the three-loopfive-point amplitude [10].However it was shown in [11] that the ABDK/BDS ansatz is incompatible withstrong coupling results in the limit of a very large number of particles, and indeedit was found in [12] that starting from six particles and two loops the ansatz isincomplete and the amplitude is given by the ABDK/BDS expression plus a nonzero‘remainder function’ (an analytic expression for which was obtained in [13–15]). Thebreakdown of the ABDK/BDS ansatz beginning at six particles can be understoodon the basis of dual conformal symmetry [16, 17], which completely determines theform of the four- and five-particle amplitudes but allows for an arbitrary function ofconformal cross-ratios beginning at n = 6 [5, 18]. While dual conformal invariance ofSYM scattering amplitudes remains a conjecture beyond one loop, it is necessary if theequality between amplitudes and Wilson loops is to hold in general since the symmetry1ranslates to the manifest ordinary conformal invariance of the corresponding Wilsonloops.Of course dual conformal symmetry alone does not imply the amplitude/Wilsonloop equality since they could differ by an arbitrary function of cross-ratios, butmiraculously precise agreement was found in [12, 18] between the two sides for n = 6particles at two loops. Evidently some magical aspect of SYM theory is at workbeyond the already remarkable dual conformal symmetry.This series of developments has opened up a number of interesting directionsfor further work. In this paper we turn our attention to a question which mighthave seemed unlikely to yield an interesting answer: does the amplitude/Wilson loopequality hold beyond O ( ǫ ) in the dimensional regularization parameter ǫ ? Thisquestion is motivated largely by the observation [3] that at one loop, the four-particleamplitude is actually equal to the lightlike four-edged Wilson loop to all orders in ǫ after absorbing an ǫ -dependent normalization factor. Furthermore, the parity-evenpart of the five-particle amplitude is equal to the corresponding Wilson loop to allorders in ǫ , again after absorbing the same normalization factor. To our pleasantsurprise we find a positive answer to this question at two loops: agreement between the n = 4 and the parity-even part of the n = 5 amplitude and the corresponding Wilsonloop continues to hold at O ( ǫ ) up to an additive constant which can be absorbed intovarious structure functions.Let us emphasize that this is a rather striking result which cannot reasonably becalled a coincidence: at this order in ǫ the amplitudes and Wilson loops we computedepend on all of the kinematic variables in a highly nontrivial way, involving poly-logarithmic functions of degree 5. It would be very interesting to continue exploringthis miraculous agreement and to understand the reason behind it. Dual conformalinvariance cannot help in this regard since the symmetry is explicitly broken in di-mensional regularization so it cannot say anything about terms of higher order in ǫ ,but of course as mentioned above already at O ( ǫ ) there must be some mechanismbeyond dual conformal invariance at work.The rest of the paper is organized as follows. In Section 2 we review key aspectsof one- and two-loop amplitudes and Wilson loops, in particular the ABDK/BDSansatz and the correspondence between MHV amplitudes and Wilson loops. We alsosummarize our main results on the equality of amplitudes and Wilson loops up to O ( ǫ )at four and five points in (2.17), (2.18), respectively. In Section 3 we present the four-and five-point amplitudes at one and two loops, and in Section 4 the correspondingWilson loops. Section 5 is devoted to discussing the numerical methods that have For n > ǫ two-mass easy boxfunctions, while the corresponding n -point amplitude contains additional parity-odd as well as parity-even terms which vanish as ǫ → O ( ǫ ) terms. Two appendices complete the paper. In the first one,we present results valid to all orders in ǫ for all Wilson loop diagrams contributingto the four-point case, with the exception of the so-called “hard” diagram topology,which is evaluated only up to and including O ( ǫ ) terms. In the second appendix,we present a novel expression for the all-orders in ǫ one-loop n -point Wilson loopdiagrams which have simple analytic continuation properties. The infinite sequence of n -point planar maximally helicity violating (MHV) ampli-tudes in N = 4 super-Yang-Mills theory (SYM) has a remarkably simple structure.Due to supersymmetric Ward identities [19–22], at any loop order L , the amplitudecan be expressed as the tree-level amplitude, times a scalar, helicity-blind function M ( L ) n : A ( L ) n = A tree n M ( L ) n . (2.1)In [6], ABDK discovered an intriguing iterative structure in the two-loop expansionof the MHV amplitudes at four points. This relation can be written as M (2)4 ( ǫ ) − (cid:0) M (1)4 ( ǫ ) (cid:1) = f (2) ( ǫ ) M (1)4 (2 ǫ ) + C (2) + O ( ǫ ) , (2.2)where IR divergences are regulated by working in D = 4 − ǫ dimensions (with ǫ < f (2) ( ǫ ) = − ζ − ζ ǫ − ζ ǫ , (2.3)and C (2) = − ζ . (2.4)The ABDK relation (2.2) is built upon the known exponentiation of infrared diver-gences [23, 24], which guarantees that the singular terms must agree on both sides of(2.2), as well as on the known behavior of amplitudes under collinear limits [25, 26].The (highly nontrivial) content of the ABDK relation is that (2.2) holds exactly aswritten at O ( ǫ ). However, ABDK observed that the O ( ǫ ) terms do not satisfy thesame iteration relation [6].In [6], it was further conjectured that (2.2) should hold for two-loop amplitudeswith an arbitrary number of legs, with the same quantities (2.3) and (2.4) for any n .In the five-point case, this conjecture was confirmed first in [7] for the parity-even partof the two-loop amplitude, and later in [8] for the complete amplitude. Notice thatfor the iteration to be satisfied parity-odd terms that enter on the left-hand side of3he relation must cancel up to and including O ( ǫ ) terms, since the right-hand side isparity even up this order in ǫ . So far this has been checked and confirmed at two-looporder for five and six particles [8, 27]. This is also crucial for the duality with Wilsonloops (discussed below) which by construction cannot produce parity-odd terms attwo loops.It has been found that starting from six particles and two loops, the ABDK/BDSansatz (2.2) needs to be modified by allowing the presence of a remainder function R n [12, 18], M (2) n ( ǫ ) − (cid:16) M (1) n ( ǫ ) (cid:17) = f (2) ( ǫ ) M (1) n (2 ǫ ) + C (2) + R n + E n ( ǫ ) , (2.5)where R n is ǫ -independent and E n vanishes as ǫ →
0. We parameterize the latter as E n ( ǫ ) = ǫ E n + O ( ǫ ) . (2.6)In this paper we will discuss in detail E n for n = 4 , R n .In a parallel development, Alday and Maldacena addressed the problem of calcu-lating scattering amplitudes at strong coupling in N = 4 SYM using the AdS/CFTcorrespondence. Their remarkable result showed that the planar amplitude at strongcoupling is calculated by a Wilson loop W [ C n ] := Tr P exp (cid:20) ig I C n dτ ˙ x µ ( τ ) A µ ( x ( τ )) (cid:21) , (2.7)whose contour C n is the n -edged polygon obtained by joining the lightlike momentaof the particles following the order induced by the colour structure of the planaramplitude. At strong coupling the calculation amounts to finding the minimal areaof a surface ending on the contour C n embedded at the boundary of a T-dual AdS space [1]. Shortly after, it was realised that the very same Wilson loop evaluated atweak coupling reproduces all one-loop MHV amplitudes in N = 4 SYM [2, 3]. Theconjectured relation between MHV amplitudes and Wilson loops found further strongsupport by explicit two loop calculations at four [4], five [5] and six points [12, 18, 27].In particular, the absence of a non-trivial remainder function in the four- and five-point case was later explained in [5] from the Wilson loop perspective, where it wasrealised that the BDS ansatz is a solution to the anomalous Ward identity for theWilson loop associated to the dual conformal symmetry [16].The Wilson loop in (2.7) can be expanded in powers of the ’t Hooft coupling4 := g N/ (8 π ) as h W [ C n ] i := 1 + ∞ X l =1 a l W ( l ) n := exp ∞ X l =1 a l w ( l ) n . (2.8)Note that the exponentiated form of the Wilson loop is guaranteed by the non-Abelianexponentiation theorem [29,30]. The w ( l ) n are obtained from “maximally non-Abelian”subsets of Feynman diagrams contributing to the W ( l ) n and in particular from (2.8)we find w (1) n = W (1) n , w (2) n = W (2) n −
12 ( W (1) n ) . (2.9)The UV divergences of the n -gon Wilson loop are regulated by working in D = 4 + 2 ǫ dimensions with ǫ <
0. The one-loop Wilson loop w (1) n times the tree-level MHVamplitude is equal to the one-loop MHV amplitude, first calculated in [31] using theunitarity-based approach [32], up to a regularization-dependent factor. This impliesthat non-trivial remainder functions can only appear at two and higher loops. At twoloops, which is the main focus of this paper, we define the remainder function R WL n for an n -sided Wilson loop as w (2) n ( ǫ ) = f (2)WL ( ǫ ) w (1) n (2 ǫ ) + C (2)WL + R WL n + E WL n ( ǫ ) , (2.10)where f (2)WL ( ǫ ) := f (2)0 + f (2)1 , WL ǫ + f (2)2 , WL ǫ . (2.11)Note that f (2)0 = − ζ , which is the same as on the amplitude side, while f (2)1 , WL = G (2)eik = 7 ζ [33]. In [28], the four- and five-edged Wilson loops were cast in the form(2.10) and by making the natural requirements R WL4 = R WL5 = 0 , (2.12)this allowed for a determination of the coefficients f (2)2 , WL and C (2)WL . The results foundin [28], are f (2)WL ( ǫ ) = − ζ + 7 ζ ǫ − ζ ǫ , (2.13)and C (2)WL = − ζ . (2.14)As noticed in [28], there is an intriguing agreement between the constant C (2)WL andthe corresponding value of the same quantity on the amplitude side. We follow the definitions and conventions of [28], to which we refer the reader for more details. We expect a remainder function at every loop order l and the corresponding equations wouldbe w ( l ) n ( ǫ ) = f ( l )WL ( ǫ ) w (1) n ( lǫ ) + C ( l )WL + R ( l ) n, WL + E ( l ) n, WL ( ǫ ). The O (1) and O ( ǫ ) coefficients of f (2)WL ( ǫ ) had been determined earlier in [4]. R n = R WL n . (2.15)A consequence of the precise determination of the constants f (2)2 , WL and C (2)WL is thatno additional constant term is allowed on the right hand side of (2.15). For the samereason, the Wilson loop remainder function must then have the same collinear limitsas its amplitude counterpart, i.e. R WL n → R WL n − , (2.16)with no extra constant on the right hand side of (2.16).The main result of the present paper is that for n = 4 , ǫ . In particular wefind E (2)4 = E (2)4 , WL − ζ , (2.17) E (2)5 = E (2)5 , WL − ζ . (2.18)Note that these results have been obtained (semi-)numerically with typical errors of10 − at n = 4 and 10 − for n = 5. Details of the calculations are presented in theremaining sections of this paper. More precisely E (2)4 is known analytically [9], whilethe analytic evaluation of E (2)4 , WL is discussed in appendix A. At five points all resultsare numerical and furthermore on the amplitude side we only considered the parity-even terms. It is an interesting open question whether the parity-odd terms cancelat O ( ǫ ) as they do at O ( ǫ ) [8]. In this section we review the ingredients necessary for our calculation of the O ( ǫ )terms in the ABDK relation for the n = 4 , We begin with the one-loop amplitudes, for which analytic results can be given to allorders in ǫ . An alternative interpretation of the duality in terms of certain ratios of amplitudes (Wilsonloops) has been given recently in [34]. M (1)4 = − stI (1)4 (3.1)where s = ( p + p ) , t = ( p + p ) are the usual Mandelstam variables and I (1)4 isthe massless scalar box integral I (1)4 = = e ǫγ iπ D/ Z d D p p ( p − p ) ( p − p − p ) ( p + p ) , (3.2)which we have written out in order to emphasize the normalization convention (fol-lowed throughout this section) that each loop momentum integral carries an overallfactor of e ǫγ /iπ D/ . The integral may be evaluated explicitly (see for example [36]) interms of the ordinary hypergeometric function F , leading to the exact expression M (1)4 = − e ǫγ ǫ Γ(1 + ǫ )Γ (1 − ǫ )Γ(1 − ǫ ) (cid:2) ( − s ) − ǫ F (1 , − ǫ, − ǫ, s/t ) + ( s ↔ t ) (cid:3) , (3.3)valid to all orders in ǫ . We will always be studying the amplitude/Wilson loopduality in the fully Euclidean regime where all momentum invariants such as s and t are negative. The formula (3.3) applies in this regime as long as we are careful tonavigate branch cuts according to the rule( − z ) − ǫ F ( − ǫ, − ǫ, − ǫ, z ) := lim ε → Re (cid:20) F ( − ǫ, − ǫ, − ǫ, z + iε )( − z + iε ) ǫ (cid:21) (3.4)when z > M ( L )5 contain both parity-even and parity-odd contri-butions after dividing by the tree amplitude as in (2.1). The parity-even part of theone-loop five-point amplitude is given by [37] M (1)5+ = − X cyclic s s I (1)5 , I (1)5 = , (3.5)where s i = ( p i + p i +1 ) and the sum runs over the five cyclic permutations of the exter-nal momenta p i . This integral can also be explicitly evaluated (see for example [36]),7igure 1: Integrals appearing in the amplitude M (2)5+ . Note that I (2) d contains theindicated scalar numerator factor involving q , one of the loop momenta. leading to the all-orders in ǫ result M (1)5+ = − e ǫγ ǫ Γ(1 + ǫ )Γ (1 − ǫ )Γ(1 − ǫ ) 12 X cyclic h (cid:18) − s − s s s (cid:19) ǫ F ( − ǫ, − ǫ, − ǫ, − s s − s )+ (cid:18) − s − s s s (cid:19) ǫ F ( − ǫ, − ǫ, − ǫ, − s s − s ) − (cid:18) − ( s − s )( s − s ) s s s (cid:19) ǫ F ( − ǫ, − ǫ, − ǫ, − s s ( s − s )( s − s ) ) i , (3.6)again keeping in mind (3.4). The two-loop four-point amplitude is expressed as [38] M (2)4 = 14 s tI (2)4 + ( s ↔ t ) , I (2)4 = , (3.7)which may be evaluated analytically through O ( ǫ ) using results from [9] (no all-ordersin ǫ expression for the double box integral is known), from which we find E = 5 Li ( − x ) − L Li ( − x ) + 12 (3 L + π ) Li ( − x ) − L L + π ) Li ( − x ) −
124 ( L + π ) log(1 + x ) + 245 π L − ζ + 2312 π ζ , (3.8)8here x = t/s and L = log x . A comment is in order here: In order to be able topresent the amplitude remainder (3.8) in this form, we have pulled out a factor of( st ) − Lǫ/ from each loop amplitude M ( L )4 . This renders the amplitudes, and hence theABDK remainder E ( ǫ ), dimensionless functions of the single variable x . We performthis step in the four-point case only, where we are able to present analytic results forthe amplitude and Wilson loop remainders.The parity-even part of the two-loop five-point amplitude involves the two integralsshown in Figure 1, in terms of which [7, 8, 39] M (2)5+ = 18 X cyclic (cid:0) s s I (2) a + ( p i → p − i ) (cid:1) + s s s I (2) d , (3.9)where s i = ( p i + p i +1 ) . To evaluate this amplitude to O ( ǫ ) we must resort to anumerical calculation using Mellin-Barnes parameterizations of the integrals (whichmay be found for example in [7]), which we then expand through O ( ǫ ), simplify,and numerically integrate with the help of the MB , MBresolve , and barnesroutines programs [40,41], In this manner we have determined the O ( ǫ ) contribution E (2)5 to thefive-point ABDK relation numerically at a variety of kinematic points. The resultsare displayed in Table 1. The one-loop Wilson loop was found in [3] for any number of edges and to all orders inthe dimensional regularization parameter ǫ . It is obtained by summing over diagramswith a single gluon propagator stretching between any two edges of the Wilson looppolygon. Diagrams with the propagator stretching between adjacent edges p i andcusp :=Γ(1 + ǫ ) e ǫγ × (cid:0) − ǫ ( − s i ) − ǫ (cid:1) finite :=Γ(1 + ǫ ) e ǫγ × F ǫ Figure 2:
One-loop Wilson loop diagrams. The expression of F ǫ is given in (B.12) ofequivalently in (B.14) . p i +1 are known as cusp diagrams, and give the infrared-divergent terms in the Wilsonloop, proportional to ( − p i · p i +1 ) − ǫ /ǫ = ( − s i ) − ǫ /ǫ .9n the other hand, diagrams for which the propagator stretches between two non-adjacent edges are finite. Their contribution to the Wilson loop can be found to allorders in ǫ and is (up to an ǫ -dependent factor) precisely equal to the finite part of atwo-mass easy or one-mass box function [3] (for details see appendix B). The general n -point one loop amplitude is given by the sum over precisely these two-mass easyand one-mass box functions [31] to O ( ǫ ). Thus we conclude that the Wilson loopis equal to the amplitude at one loop for any n up to finite order in ǫ only (and upto a kinematic independent factor).However at four and five points a much stronger statement can be made. Thefour-point amplitude and the parity-even part of the five-point amplitude are bothgiven by the sum over zero- and one-mass boxes to all orders in ǫ . Thus the Wilsonloop correctly reproduces these one-loop amplitudes to all orders in ǫ . Using theresults in appendix B we find that the four-point Wilson loop (in a form which ismanifestly real in the Euclidean regime s, t <
0) is given by W (1)4 = Γ(1 + ǫ ) e ǫγ n − ǫ (cid:2) ( − s ) − ǫ + ( − t ) − ǫ (cid:3) + F ǫ ( s, t, ,
0) + F ǫ ( t, s, , o = Γ(1 + ǫ ) e ǫγ ( − ǫ (cid:2) ( − s ) − ǫ + ( − t ) − ǫ (cid:3) + 1 ǫ (cid:16) ust (cid:17) ǫ (cid:20) (cid:18) ts (cid:19) ǫ F ( ǫ, ǫ ; 1 + ǫ ; − t/s ) + (cid:16) st (cid:17) ǫ F ( ǫ, ǫ ; 1 + ǫ ; − s/t ) − πǫ cot( ǫπ ) (cid:21)) . (4.1)Note in particular the additional cotangent term explained in detail at the end ofappendix B. The generic form of the function F ǫ is given in (B.12) of equivalently in(B.14).For the five-point amplitude we display a new form which has a simple analyticcontinuation in all kinematical regimes and also a very simple expansion in termsof Nielsen polylogarithms (see (B.11)). It is given in terms of F hypergeometric The all-orders in ǫ n -point amplitude contains new integrals contributing at O ( ǫ ). W (1)5 = X i =1 Γ(1 + ǫ ) e ǫγ h − ǫ ( − s i ) − ǫ + F ǫ ( s i , s i +1 , s i +3 , i = X i =1 Γ(1 + ǫ ) e ǫγ ( − ǫ ( − s i ) − ǫ − (cid:18) s i +3 − s i − s i +1 s i s i +1 (cid:19) ǫ (cid:20) s i +3 − s i s i +1 F (cid:0) , , ǫ ; 2 , s i +3 − s i s i +1 (cid:1) + s i +3 − s i +1 s i F (cid:0) , , ǫ ; 2 , s i +3 − s i +1 s i (cid:1) + H − ǫ ǫ − ( s i +3 − s i )( s i +3 − s i +1 ) s i s i +1 F (cid:0) , , ǫ ; 2 , ( s i +3 − s i )( s i +3 − s i +1 ) s i s i +1 (cid:1)(cid:21)) (4.2)where H n is the n th -harmonic number. Using hypergeometric identities one can showthat (up to the prefactor) the four- and five-sided Wilson loops (4.1), (4.2) are equalto the four-point and the (parity-even part of the) five-point amplitudes of (3.3)and (3.6).The precise relation between the Wilson loop and the amplitude is W (1)4 = Γ(1 − ǫ )Γ (1 − ǫ ) M (1)4 , W (1)5 = Γ(1 − ǫ )Γ (1 − ǫ ) M (1)5+ , (4.3)where M (1)4 is the one-loop four-point amplitude and M (1)5+ is the parity-even part ofthe five-point amplitude. At two-loop order, the n -point Wilson loop is given by a sum over six different typesof diagrams. These are described in general for polygons with any number of edgesin [28] and are displayed for illustration below.The computation of the four-point two-loop Wilson loop up to O ( ǫ ) was firstperformed in [4]. In appendix A we display all the contributing diagrams for thiscase and give expressions for these to all orders in ǫ in all cases except for the “hard”diagram, which we give up to and including terms of O ( ǫ ). Summing up the con-tributions from all these diagrams we obtain the result for the two-loop four-pointWilson loop to O ( ǫ ). This is displayed in (4.4) of the next subsection.11ross curtain factorised crossY hard self-energyFigure 3: The six different diagram topologies contributing to the two-loop Wilsonloop. For details see [28].
The five-point two-loop Wilson loop was calculated up to O ( ǫ ) in [5]. In order toobtain results at one order higher in ǫ we have proceeded by using numerical methods.In particular we have used Mellin-Barnes techniques to evaluate and expand all thetwo-loop integrals of Figure 3. This is described in more detail in Section 5. Here is our final result for the four-point Wilson loop at two loops expanded up toand including terms of O ( ǫ ): w (2)4 = C × h ( − s ) − ǫ + ( − t ) − ǫ i × h w ǫ + w ǫ + w + w − ǫ + O ( ǫ ) i , (4.4)where w = π , (4.5) w = − ζ , (4.6) w = − π (cid:0) log x + π (cid:1) + π
144 = − π (cid:18) log x + 23 π (cid:19) , (4.7) w − = − h − π log x − π log x + 75 π log(1 + x ) + 90 π log x log(1 + x )+15 log x log(1 + x ) + 240 π log x Li ( − x ) + 120 log x Li ( − x ) − π Li ( − x ) −
540 log x Li ( − x ) + 1440 log x Li ( − x ) − ( − x ) − π ζ − x ζ + 5940 ζ i , (4.8)12nd C := 2 h Γ(1 + ǫ ) e γǫ i = 2 (cid:16) ζ ǫ − ζ ǫ (cid:17) + O ( ǫ ) . (4.9)We recall that x = t/s .We would like to point out the simplicity of our result (4.4) – specifically, (4.5)–(4.8) are expressed only in terms of standard polylogarithms. Harmonic polyloga-rithms and Nielsen polylogarithms are present in the expressions of separate Wilsonloop diagrams, as can be seen in appendix A, but cancel after summing all contribu-tions. O ( ǫ ) Wilson Loop Remainder Function at Four Points
Using the result (4.4) and the one-loop expression for the Wilson loop, one can workout the expression for the remainder function at O ( ǫ ), as defined in (2.5) and (2.6).Our result is E , WL = 1360 (cid:20) π log x − π log(1 + x ) − π log x log(1 + x ) −
15 log x log(1 + x ) − π log x Li ( − x ) −
120 log ( x )Li ( − x )+180 π Li ( − x ) + 540 log x Li ( − x ) − x )Li ( − x )+1800Li ( − x ) + 690 π ζ − ζ (cid:21) , (4.10)where we recall that E n, WL is related to the quantity E n introduced in (2.5) and (2.6).Remarkably, (4.10) does not contain any harmonic polylogarithms. We will comparethe Wilson loop remainder (4.10) to the corresponding amplitude remainder (3.8) inSection 6.1. O ( ǫ ) Wilson Loop at Five Points and the Five-Point RemainderFunction
For the five-point amplitude and Wilson loop at two loops we resort to completelynumerical evaluation of the contributing integrals, and a comparison of the remainderfunctions is then performed. We postpone this discussion to section 6.2. Similarly to what was done for the amplitude remainder (3.8), in arriving at (4.10) we havepulled out a factor of ( st ) − ǫ/ per loop in order to obtain a result which depends only on the ratio x := t/s . Mellin-Barnes Integration
The two-loop five-point Wilson loop and amplitude have been numerically evalu-ated by means of the Mellin-Barnes (MB) method using the MB package [40] in MATHEMATICA . At the heart of the method lies the Mellin-Barnes representation1( X + Y ) λ = 12 πi λ ) Z + i ∞− i ∞ dz X z Y λ + z Γ( − z )Γ( λ + z ) . (5.1)We will use the integral representation for the hard diagram of the Wilson loop as anexample in order to describe the procedure we followed. The integral for the specificdiagram shown in Figure 4 has the expression f H ( p , p , p ; Q , Q , Q ) (5.2)= 18 Γ(2 + 2 ǫ )Γ(1 + ǫ ) Z ( Y i =1 dτ i ) Z ( Y i =1 dα i ) δ (1 − X i =1 α i )( α α α ) ǫ ND ǫ . We write the numerator and denominator as a function of the momentum invariants,i.e. squares of sums of consecutive momenta, D = − α α (cid:2) ( p + Q + p ) (1 − τ ) τ + ( p + Q ) (1 − τ )(1 − τ )+( Q + p ) τ τ + Q τ (1 − τ ) (cid:3) + cyclic(1 , , , (5.3) N = 2 [2( p p )( p Q ) − ( p p )( p Q ) − ( p p )( p Q )] α α + 2( p p )( p p ) [ α α (1 − τ ) + α α τ ] + cyclic(1 , , , (5.4)where 2 p i p i +1 = − ( p i + Q i +2 ) + Q i +2 − ( Q i +2 + p i +1 ) + ( Q i + p i +2 + Q i +1 ) , p i Q i = − ( p i + Q i +2 + p i +1 ) + ( Q i +2 + p i +1 ) − ( p i +2 + Q i +1 + p i ) + ( p i +2 + Q i +1 ) , p i Q j = ( p i + Q j ) − Q j . (5.5)By means of the substitution α → − τ , α → τ τ and α → τ (1 − τ ), we eliminateone integration and the delta function to get a five-fold integral over τ i ∈ [0 , P ms =1 X s ) λ = 1(2 πi ) m − λ ) m − Y s =1 Z + i ∞− i ∞ dz s ! Q m − s =1 X z s s Γ( − z s ) X λ + P m − s =1 z s m Γ( λ + P m − s =1 z s ) , (5.6)which introduces m − z s , where m is the number of terms inthe denominator. At this point, the integrations over the τ i ’s can be easily performedby means of the substitution Z dx x α (1 − x ) β = Γ( α + 1)Γ( β + 1)Γ( α + β + 2) . (5.7)14 p p Q Q Q Figure 4:
The hard diagram corresponding to (5.2) . We are now left with an integrand that is an analytic function containing powersof the momentum invariants ( − s ij ) f ( { z s } ,ǫ ) and Gamma functions Γ( g ( { z i } , ǫ )), where f and g are linear combinations of the z s ’s and ǫ . In order to perform the MBintegrations, one has to pick appropriate contours, so that for each z s the Γ( · · · + z s )poles are to the left of the contour and the Γ( · · · − z s ) poles are to the right.At this point we use various Mathematica packages to perform a series of op-erations in an automated way to finally obtain a numerical expression at specifickinematic points. We will briefly summarise the steps followed, while for more de-tails we refer the reader to the references documenting these packages and referencestherein. Using the
MBresolve package [41], we pick appropriate contours and resolvethe singularity structure of the integrand in ǫ . The latter involves taking residuesand shifting contours, and is essential in order to be able to Laurent expand the inte-grand in ǫ . Using the barnesroutines package [40, 41], we apply the Barnes lemmas,which in general generate more integrals but decrease their dimensionality, leading tohigher precision results. Finally, using the MB package [40] we numerically integrate atspecific Euclidean kinematic points to obtain a numerical expression. While all ma-nipulations of the integrals and the expansion in ǫ are performed in Mathematica , theactual numerical integration for each term is performed using the
CUBA routines [42]for multidimensional numerical integration in
FORTRAN . The high number of diagrams,and number of integrals for each diagram, makes the task of running the
FORTRAN integrations ideal for parallel computing. 15
Comparison of the Remainder Functions
The remainder functions for the four-point amplitude and Wilson loops are given in(3.8) and (4.10), respectively. From these relations, it follows that the difference ofremainders is a constant, x -independent term: E = E , WL − ζ , (6.1)as anticipated in (2.17).We would like to stress that this is a highly nontrivial result since there is noreason a priori to expect that the four-point remainder on the amplitude and Wilsonloop side, (3.8) and (4.10) respectively, agree (up to a constant shift). For example,anomalous dual conformal invariance is known to determine the form of the four- andfive-point Wilson loop only up to O ( ǫ ) terms [5], but does not constrain terms whichvanish as ǫ →
0. The expressions we derived for the amplitude and Wilson loopfour-point remainders at O ( ǫ ) are also pleasingly simple, in that they only containstandard polylogarithms. We have numerically evaluated both the five-point two-loop amplitude and Wilsonloop up to O ( ǫ ) at 25 Euclidean kinematic points, i.e. points in the subspace of thekinematic invariants with all s ij <
0. The choice of these points and the valuesof the remainder functions E (2)5 , E (2)5 , WL at O ( ǫ ) together with the errors reported bythe CUBA numerical integration library [42] appear in Table 1, while in Figures5 and 6 we plot both remainders for all kinematic points. We have calculated thedifference between the amplitude and Wilson loop remainders, see Table 2 and Figure7. Remarkably, this difference also appears to be constant (within our numericalprecision) as in the four-point case, and hence we conjecture that E (2)5 = E (2)5 , WL − ζ . (6.2)It is also intriguing that the constant difference is fit very well by a simple rationalmultiple of ζ , rather than a linear combination of ζ and ζ ζ as would have beenallowed more generally by transcendentality.In the last column of Table 2 we give the distance of our results from this conjecturein units of their standard deviation. 16 ( s , s , s , s , s ) E (2)5 E (2)5 , WL − , − , − , − , − − . ± . − . ± . − , − , − , − , − − . ± . − . ± . − , − , − , − , − − . ± . − . ± . − , − , − , − , − − . ± . − . ± . − , − , − , − , − − . ± . − . ± . − , − , − , − , − − . ± . − . ± . − , − , − , − , − − . ± . − . ± . − , − , − , − , − − . ± . − . ± . − , − , − , − , − − . ± . − . ± . − , − , − , − , − − . ± . − . ± . − , − , − , − , − − . ± . − . ± . − , − , − , − , − − . ± . − . ± . − , − , − , − , − − . ± . − . ± . − , − , − , − , − − . ± . − . ± . − , − , − , − , − − . ± . − . ± . − , − , − , − , − − . ± . − . ± . − , − , − , − , − − . ± . − . ± . − , − , − , − , − − . ± . − . ± . (cid:0) − , − , − , − , − (cid:1) − . ± . − . ± . (cid:0) − , − , − , − , − (cid:1) − . ± . − . ± . (cid:0) − , − , − , − , − (cid:1) − . ± . − . ± . (cid:0) − , − , − , − , − (cid:1) − . ± . − . ± . (cid:0) − , − , − , − , − (cid:1) − . ± . − . ± . (cid:0) − , − , − , − , − (cid:1) − . ± . − . ± . (cid:0) − , − , − , − , − (cid:1) − . ± . − . ± . Table 1: O ( ǫ ) five-point remainders for amplitudes ( E (2)5 ) and Wilson loops ( E (2)5 , WL ) . s , s , s , s , s ) E (2)5 − E (2)5 , WL |E (2)5 − E (2)5 , WL + ζ | /σ − , − , − , − , − − . ± . .
82 ( − , − , − , − , − − . ± . .
23 ( − , − , − , − , − − . ± . .
324 ( − , − , − , − , − − . ± . − , − , − , − , − − . ± . .
356 ( − , − , − , − , − − . ± . − , − , − , − , − − . ± . .
828 ( − , − , − , − , − − . ± . .
29 ( − , − , − , − , − − . ± . . − , − , − , − , − − . ± . .
811 ( − , − , − , − , − − . ± . . − , − , − , − , − − . ± . .
313 ( − , − , − , − , − − . ± .
024 0 . − , − , − , − , − − . ± .
011 1 .
915 ( − , − , − , − , − − . ± .
017 0 . − , − , − , − , − − . ± .
010 0 . − , − , − , − , − − . ± .
11 0 . − , − , − , − , − − . ± . . (cid:0) − , − , − , − , − (cid:1) − . ± . . (cid:0) − , − , − , − , − (cid:1) − . ± . . (cid:0) − , − , − , − , − (cid:1) − . ± . . (cid:0) − , − , − , − , − (cid:1) − . ± . . (cid:0) − , − , − , − , − (cid:1) − . ± . . (cid:0) − , − , − , − , − (cid:1) − . ± . . (cid:0) − , − , − , − , − (cid:1) − . ± . . Table 2:
Difference of the five-point amplitude and Wilson loop two-loop remainderfunctions at O ( ǫ ) , and its distance from − ζ ∼ − . in units of σ , the standarddeviation reported by the CUBA numerical integration package [42]. æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à à à à à à à à à à à à à à à à - - - Kinematic point
Figure 5:
Remainder functions at O ( ǫ ) for the amplitude (circle) and the Wilson loop(square). E æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æà à à à à à à à à à à à à à à à à à à à à à à à - - - - Kinematic point
Figure 6:
Remainder functions at O ( ǫ ) for the amplitude (circle) and the Wilsonloop (square). In this Figure we have eliminated data point 17 and zoomed in on theothers. σ itdrifts away from it, hinting at a potential underestimate of the errors. To test ourerror estimates we used the remainder functions at O ( ǫ ), that are known to vanish.Our analysis confirmed that, as we increase the desired precision, the actual precisionof the mean value does increase, but on the other hand reported errors tend to becomeincreasingly underestimated. E - E æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ - - - - - - - - Kinematic point
Figure 7:
Difference of the remainder functions E (2)5 − E (2)5 , WL . Acknowledgements
We would like to thank M. Czakon and V. V. Khoze for many helpful discussions. MSis grateful to F. Cachazo for a comment which inspired his interest in this problem,and to L. Dixon and A. Volovich for useful correspondence and discussions. We wouldlike to thank the QMUL High Throughput Computing Facility, and especially AlexMartin and Christopher Walker, for providing us with the necessary computer powerfor the high precision numerical evaluations. We would also like to thank Terry Arterand Alex Owen for technical assistance. This work was supported by the STFC underthe Queen Mary Rolling Grant ST/G000565/1 and the IPPP Grant ST/G000905/1,by the US Department of Energy under contract DE-FG02-91ER40688, and by theUS National Science Foundation under grant PHY-0638520. GT is supported by anEPSRC Advanced Research Fellowship EP/C544242/1.19
Details of the Two-Loop Four-Point Wilson Loopto All Orders in ǫ In this appendix, we present the results for the separate classes of Wilson loop di-agrams contributing to a four-point loop. In all cases (with the exception of the“hard” diagram) our results are valid to all orders in the dimensional regularizationparameter ǫ . These expressions are given in terms of hypergeometric functions. Wealso expand these up to O ( ǫ ) with the help of the mathematica packages HPL and
HypExp [43, 44].
A.1 Two-loop Cusp Diagrams
Figure 8:
The two-loop cusp corrections. The second diagram appears with its mirrorimage where two of the gluon legs of the three-point vertex are attached to the otheredge; these two diagrams are equal. The blue bubble in the third diagram representsthe gluon self-energy correction calculated in dimensional reduction.
The total contributions of all diagrams that cross a single cusp (the crossed dia-gram across a cusp, the self-energy diagram across the cusp and the vertex across thecusp, as depicted in Figure 8) is easily seen to be ( − s ) − ǫ ǫ (cid:20) Γ(1 + 2 ǫ )Γ(1 − ǫ )Γ(1 + ǫ ) − (cid:21) . (A.1)Adding the contributions for the four cusps, we obtain W cusp = h ( − s ) − ǫ + ( − t ) − ǫ i ǫ (cid:20) Γ(1 + 2 ǫ )Γ(1 − ǫ )Γ(1 + ǫ ) − (cid:21) (A.2)= h ( − s ) − ǫ + ( − t ) − ǫ i × h ǫ π − ǫ ζ π − ǫ (cid:0) π ζ + 9 ζ (cid:1) + O ( ǫ ) i . (A.3) In this and the following formulae, a factor of C is suppressed in each diagram, where C is definedin (4.9). p p p Figure 9:
One of the four curtain diagrams. The remaining three are obtained bycyclic permutations of the momenta.
A.2 The Curtain Diagram
The contribution of all four curtain diagrams is W curtain = ( st ) − ǫ (cid:18) − ǫ (cid:19) (cid:20) − Γ(1 − ǫ ) Γ(1 − ǫ ) (cid:21) (A.4)= h ( − s ) − ǫ + ( − t ) − ǫ i (cid:20) − ǫ π − ǫ ζ − π
160 + π
48 log x − ǫ (cid:16) − ζ log x + 32 ζ − π ζ (cid:17) + O ( ǫ ) (cid:21) . A.3 The Factorised Cross Diagram
The factorised cross diagram is given by the product of two finite one-loop Wilsonloop diagrams, expressed each by (4.1)12 ǫ (cid:18) stu (cid:19) − ǫ h x − ǫ F ( ǫ, ǫ, ǫ, − /x ) + x ǫ F ( ǫ, ǫ, ǫ, − x ) − πǫ cot( ǫπ ) i . The result for the factorised cross is therefore − ǫ (cid:18) stu (cid:19) − ǫ (cid:20) F ( ǫ, ǫ ; 1 + ǫ ; − x ) x ǫ + F (cid:18) ǫ, ǫ ; 1 + ǫ ; − x (cid:19) x − ǫ − πǫ cot( ǫπ ) (cid:21) = h ( − s ) − ǫ + ( − t ) − ǫ i × (cid:16) g + g − ǫ + O ( ǫ ) (cid:17) , (A.5)21 p p p Figure 10:
One of the four factorised cross diagram. p p p p p p p p k k k k k k k k Figure 11:
The Y diagram together with the self-energy diagram. The sum of thesetwo topologies gives a maximally transcendental contribution. with g = − (cid:0) log x + π (cid:1) , (A.6) g − = 1192 (cid:16) log x + π (cid:17)h log x − x + 1) log x − ( − x ) log x , + 3 π log x − π log( x + 1) + 12 Li ( − x ) − ζ i . (A.7)22 .4 The Y Diagram The diagrams in Figure 11 correspond to the following contribution to the two-loopWilson loop: W Y = − ǫ Γ(1 + 2 ǫ )Γ(1 + ǫ ) u (A.8) × Z dτ Z dτ h I ( z ( τ ) , z ( τ ) , z ( τ )) − I ( z ( τ ) , z ( τ ) , k ) − I ( z ( τ ) , z ( τ ) , k ) i , where z ( τ ) = k − p τ , z ( τ ) = k − p τ , and I ( z , z , z ) = Z dσ ( σ (1 − σ )) ǫ h − ( − z + σz + (1 − σ ) z ) + i i − − ǫ , (A.9)where z , z must be lightlike. The evaluation of (A.8) gives W Y = (cid:18) stu (cid:19) − ǫ ǫ h − x + 1) − ǫ Γ(1 + 2 ǫ )Γ( − ǫ + 1)Γ(1 + ǫ )+ 4 x ǫ Γ( − ǫ + 1) Γ( − ǫ + 1) F ( ǫ, ǫ ; 1 + ǫ ; − x ) + x ǫ F (2 ǫ, ǫ ; 1 + 2 ǫ ; − x ) − πǫ cot(2 πǫ ) + Γ(1 + 2 ǫ )Γ( − ǫ + 1) + x ↔ x i We multiply by four to obtain the contribution of all such diagrams. Then theexpansion of this contribution in ǫ begins at O ( ǫ − ), h ( − s ) − ǫ + ( − t ) − ǫ i × h c ǫ + c + c − ǫ + O ( ǫ ) i , (A.10)where c = − h log x − x + 1) log x − ( − x ) log x + 3 π log x ) − π log( x + 1 + 12Li ( − x ) − ζ i , (A.11) c = 1960 h x −
40 log( x + 1) log x + 120 log ( x + 1) log x + 10 π log x − π log( x + 1) log x + 480 log( x + 1)Li ( − x ) log x −
240 Li ( − x ) log x + 480 S , ( − x ) log x − ζ log x + 120 π log ( x + 1) −
480 log( x + 1)Li ( − x )+ 480Li ( − x ) − S , ( − x ) + 480 log( x + 1) ζ + π i , (A.12)23 − = 1240 h − x + 10 log( x + 1) log x + 10 log ( x + 1) log x + 20Li ( − x ) log x − π log x −
20 log ( x + 1) log x − ζ log x + 30 π log ( x + 1) log x −
120 log ( x + 1)Li ( − x ) log x + 120 log( x + 1)Li ( − x ) log x − ( − x ) log x + 120 S , ( − x ) log( x ) −
240 log( x + 1) S , ( − x ) log x − S , ( − x ) log x + 120 log( x + 1) ζ log x + 4 π log x − π log ( x + 1) − π log( x + 1) + 120 log ( x + 1)Li ( − x ) − ( − x )Li ( − x ) −
240 log( x + 1)Li ( − x ) + 240Li ( − x ) + 240 log( x + 1) S , ( − x )+ 40 H , ( − x ) + 120 H , ( − x ) + 40Li ( − x ) S , ( − x ) − H , , ( − x ) − H , , ( − x ) − ζ −
120 log ( x + 1) ζ − π ζ i . (A.13) A.5 The Half-Curtain Diagram
We now consider the “half-curtain” diagram, whose contribution to the Wilson loop p p k p k p k k Figure 12:
Diagram of the half-curtain topology. is W hc ( x ) = − Z dσ dρ dτ Z τ dτ s ( − sστ ) ǫ u ( − sτ − tρ − uρτ ) ǫ = 18 (cid:18) stu (cid:19) − ǫ (1 + x ) − ǫ Z dσ Z − /x da Z − x db Z − b x dτ στ ) ǫ − ab ) ǫ , (A.14)where we have changed variables in the second line to a = 1 + uρ/s , and b = 1 + uτ /t .24he evaluation of this diagram and of that with s ↔ t leads to W hc ( x ) + W hc (1 /x ) = 18 ǫ (cid:18) stu (cid:19) − ǫ × h πǫ cot(2 πǫ ) − x − ǫ F (2 ǫ, ǫ ; 1 + 2 ǫ ; − /x ) − x ǫ F (2 ǫ, ǫ ; 1 + 2 ǫ ; − x )+ h (1 + x ) − ǫ + (1 + 1 /x ) − ǫ i(cid:16) x − ǫ F ( ǫ, ǫ ; 1 + ǫ ; − /x )+ x ǫ F ( ǫ, ǫ ; 1 + ǫ ; − x ) − πǫ cot( πǫ ) (cid:17)i . (A.15)This can be expanded in ǫ using F ( ǫ, ǫ ; 1 + ǫ ; x ) = 1 + ǫ Li ( x ) − ǫ (cid:2) Li ( x ) − S ( x ) (cid:3) + ǫ (cid:2) Li ( x ) − S ( x ) + S ( x ) (cid:3) + O ( ǫ ) , (A.16)The contribution of all diagrams of the half-curtain type is obtained by multiplying(A.15) by a factor of four. One obtains thus h ( − s ) − ǫ + ( − t ) − ǫ i × h d ǫ + d + d − ǫ + O ( ǫ ) i , (A.17)where d = 2 c , (A.18) d = − c − (cid:0) π − log x (cid:1) (cid:0) log x + π (cid:1) , (A.19) d − = 72 c − (A.20) − h − log ( x ) + 6 log( x + 1) log x + 12Li ( − x ) log ( x ) − π log x + 6 π log( x + 1) log x − ( − x ) log x − ζ log x − π ζ i , and c j are the coefficients for the Y diagram, given in (A.11)–(A.13). A.6 The Cross Diagram
We now consider the cross diagram, whose expression is given by W cr ( x ) = − Z dτ dσ Z τ dτ Z σ τ u ( − sσ − tτ − uσ τ ) ǫ u ( − sσ − tτ − uσ τ ) ǫ = − (cid:18) stu (cid:19) − ǫ Z − /x da Z − x db Z a da Z b db − a b ) ǫ − a b ) ǫ , (A.21)25 p p p Figure 13:
One of the cross diagrams. As before, the remaining three can be generatedby cyclic permutations of the momentum labels. for which we find W cr ( x ) = − ǫ (cid:18) stu (cid:19) − ǫ n − F (2 ǫ, ǫ, ǫ ; 1 + 2 ǫ, ǫ ; − x ) x ǫ + Γ( − ǫ + 1) F (2 ǫ, ǫ, ǫ ; 1 + ǫ, ǫ ; − x ) x ǫ Γ( − ǫ + 1)+ 14 (cid:20) F ( ǫ, ǫ ; 1 + ǫ ; − x ) x ǫ + F (cid:16) ǫ, ǫ ; 1 + ǫ ; − x (cid:17) x − ǫ − πǫ cot( ǫπ ) (cid:21) − ǫ π cot(2 πǫ ) ( ψ (2 ǫ ) + γ ) − π ǫ cot( πǫ ) + (cid:16) x ↔ x (cid:17)o . (A.22)Notice the presence of the one-loop finite diagram squared. Expanding this andmultiplying by a factor of two to account for all diagrams leads to the followingresult, W cr ( x ) = h ( − s ) − ǫ + ( − t ) − ǫ i × h f + f − ǫ + O ( ǫ ) i , (A.23)26here f = − π + log x ) , (A.24) f − = − h − x + 30 log( x + 1) log x + 180Li ( − x ) log x − π log x + 60 π log( x + 1) log x − ( − x ) log x − ζ log x + 180 π Li ( − x ) log x + 2880Li ( − x ) log x − π log x + 30 π log( x + 1) − π Li ( − x ) − ( − x )+ 4320 ζ − π ζ i . (A.25) A.7 The Hard Diagram
The generic n -point hard diagram topology is depicted in Figure 4. In the four-pointcase, one diagram is obtained from Figure 4 by simply setting Q = Q = 0, and Q = p . There are four such diagrams, obtained by cyclic rearrangements of themomenta. We have evaluated the hard diagrams using Mellin-Barnes, arriving at thefollowing result: W hard = h ( − s ) − ǫ + ( − t ) − ǫ i × h h ǫ + h ǫ + h + h − ǫ + O ( ǫ ) i , (A.26)where h = π , (A.27) h = − (cid:16) c + ζ (cid:17) , (A.28) h = − c − (cid:0) log x + π (cid:1) − π (cid:0) log x + π (cid:1) + 311440 π (A.29)27 − = − h H , ( − x ) + 600 H , ( − x ) − H , , ( − x ) − H , , ( − x )+ 200 ζ Li ( − x ) − ( − x )Li ( − x ) − ( − x )Li ( x + 1) + 1320Li ( − x )+ 20Li ( − x ) log ( x ) + 60Li ( − x ) log ( x ) − ( − x ) log ( x + 1) log x − ( x + 1) log ( x + 1) log x + 100Li ( − x ) log( − x ) log ( x + 1)+ 600Li ( − x ) log ( x + 1) + 20 π Li ( − x ) log( x ) + 600Li ( − x ) log( x + 1) log( x ) − ( − x ) log( x ) + 1200Li ( x + 1) log x + 200Li ( − x )Li ( x + 1) log( x + 1) − ( − x ) log( x + 1) + 600 S , ( − x ) log x + 1200 S , ( − x ) log( x + 1) − ζ log x − ζ log ( x + 1) − ζ log( x + 1) log x − x + 5 log( x + 1) log x −
400 log( − x ) log ( x + 1) log x − π log ( x + 1) − π log( x + 1) log x + 150 π log ( x + 1) log x + 50 log ( x + 1) log x −
100 log ( x + 1) log x + 7 π log x − π log( x + 1) − ζ − π ζ i . (A.30)The analytical evaluation of this diagram up to O ( ǫ ) was obtained in [4]. Ourevaluation of the O ( ǫ ) terms agrees precisely with that of [28] (and with [4] up toa constant term). The evaluation of the O ( ǫ ) term is new and has been performednumerically. We have then compared the entire Wilson loop expansion to the analyticexpression for the amplitude remainder given in (3.8), finding the relation (2.17). B The One-Loop Wilson Loop Reloaded
In this section we derive a new expression for the all-orders in ǫ one-loop finite Wilsonloop diagrams, and hence also a new expression for the all orders in ǫ (finite part of the)2 me box function. We also improve a previous expression for use in all kinematicalregimes.A general one-loop Wilson loop diagram is given by the following integral QpqP := Γ(1 + ǫ ) e ǫγ × F ǫ ( s, t, P , Q )= Γ(1 + ǫ ) e ǫγ × R dτ R dσ u/ {− [ P + σ ( s − P )+ τ ( t − Q + στu )] − iε } ǫ . (B.1)Here we have defined s = ( p + P ) , t = ( p + Q ) and u = P + Q − s − t . The relation28etween this Wilson loop diagram and the corresponding 2 me box function is [3]finite part of QPp q ! = e ǫγ Γ(1 + ǫ )Γ (1 − ǫ )Γ(1 − ǫ ) × F ǫ ( s, t, P , Q ) . (B.2)Notice that we have included an infinitesimal negative imaginary part − iε in the de-nominator which dictates the analytic properties of the integral. This has the oppositesign to the one expected from a propagator term in a Wilson loop in configurationspace. On the other hand it has the correct sign for the present application, namelyfor the duality with amplitudes [45]. One simple way to deal with this is simply toadd an identical positive imaginary part to all kinematical invariants s → s + iε, t → t + iε, P → P + iε, Q → Q + iε . (B.3)We will assume this in the following.Changing variables to σ ′ = σ − ( P − t ) /u and τ ′ = τ − ( P − s ) /u and thendropping the primes, this becomes F ǫ ( s, t, P , Q ) = Z Q − sut − P u dσ Z Q − tus − P u dτ u/ − ( a − + στ u )] ǫ (B.4)= − ǫ Z Q − sut − P u dσσ − ( a − + στ u )] ǫ (cid:12)(cid:12)(cid:12)(cid:12) τ =( Q − t ) /uτ =( s − P ) /u , (B.5)where a = u/ ( P Q − st ). Note that the analytic continuation of a implied by (B.3)is a → a − iε . Now we split the integration into two parts, Z Q − sut − P u = − Z t − P u + Z Q − su , (B.6)and rescale the integration variable so that it runs between 0 and 1 in each case. It isimportant to split the integral in this way since there is a singularity at σ = 0 whichone must be very careful when integrating over. We obtain in this way F ǫ ( s, t, P , Q ) = ( − a ) ǫ ǫ Z dσσ (cid:26) − (1 − aP ) σ ] ǫ + 1[1 − (1 − aQ ) σ ] ǫ − − (1 − as ) σ ] ǫ − − (1 − at ) σ ] ǫ (cid:27) , (B.7)where a = u/ ( P Q − st ). Now each of the four terms by itself is divergent (even for ǫ = 0), only the sum gives a finite integral. A straightforward way of regulating eachterm individually is to simply subtract 1 /σ from each term in the integrand, thusremoving the divergence at σ = 0. We thus have F ǫ ( s, t, P , Q ) = ( − a ) ǫ h f (1 − aP ) + f (1 − aQ ) − f (1 − as ) − f (1 − at ) i , (B.8)29here f ( x ) = 1 ǫ Z dσ (1 − xσ ) − ǫ − σ = 1 ǫ Z x dσ (1 − σ ) − ǫ − σ . (B.9)The problem becomes that of finding the integral f ( x ). It has two equivalent forms,both given in terms of hypergeometric functions. The first form is given by f ( x ) = x × F (1 , , ǫ ; 2 , x ) = ∞ X n =1 ǫ n S n +1 ( x ) . (B.10)Notice the very simple expansion in terms of Nielsen polylogarithms. The secondform is f ( x ) = − ǫ (cid:2) ( − x ) − ǫ F ( ǫ, ǫ ; 1 + ǫ ; 1 /x ) + ǫ log x (cid:3) + constant , (B.11)where the constant is there to make f (0) = 0, and is not important since it will cancelin F ǫ .We thus arrive at two different forms for the Wilson loop diagram. The first formis F ǫ ( s, t, P , Q )= ( − a ) ǫ h (1 − aP ) F (1 , , ǫ ; 2 ,
2; 1 − aP ) + (1 − aQ ) F (1 , , ǫ ; 2 ,
2; 1 − aQ ) − (1 − as ) F (1 , , ǫ ; 2 ,
2; 1 − as ) − (1 − at ) F (1 , , ǫ ; 2 ,
2; 1 − at ) i , (B.12)and it is manifestly finite. Furthermore, since F (1 , , ǫ ; 2 , x ) = Li ( x ) /x , thisform directly leads to the expression derived in [46,47] for the finite 2 me box function, F ǫ =0 ( s, t, P , Q ) = 12 (cid:2) Li (1 − aP ) + Li (1 − aQ ) − Li (1 − as ) − Li (1 − at ) (cid:3) . (B.13)We also notice that the simple expansion of (B.10) gives a correspondingly simpleexpansion for the Wilson loop diagram in terms of Nielsen polylogarithms.The more familiar looking second form for the two-mass easy box function is (see(A.13) of [48]) F ǫ ( s, t, P , Q ) = − ǫ × (cid:20) (cid:18) a − aP (cid:19) ǫ F ( ǫ, ǫ ; 1 + ǫ ; 1 / (1 − aP )) + (cid:18) a − aQ (cid:19) ǫ F ( ǫ, ǫ ; 1 + ǫ ; 1 / (1 − aQ )) − (cid:18) a − as (cid:19) ǫ F ( ǫ, ǫ ; 1 + ǫ ; 1 / (1 − as )) − (cid:18) a − at (cid:19) ǫ F ( ǫ, ǫ ; 1 + ǫ ; 1 / (1 − at ))+ ǫ ( − a ) ǫ (cid:16) log(1 − aP ) + log(1 − aQ ) − log(1 − as ) − log(1 − at ) (cid:17)(cid:21) . (B.14)30his second form was derived in [3, 48] except for the last line which is an additionalcorrection term needed to obtain the correct analytic continuation in all regimes. Theidentity (1 − aP )(1 − aQ )(1 − as )(1 − at ) = 1 , (B.15)implies that if all the arguments of the logs are positive then this additional termvanishes, but for example if we have 1 − aP , − aQ > − as, − at < a )2 πi ( − a ) ǫ /ǫ . This becomes important when considering this expressionat four and five points in the Euclidean regime.For applications in this paper we are interested in taking either one massive legmassless (for the five-point case) or both massive legs massless (for the four-pointcase). Using the first expression for the finite Wilson loop diagram in terms of F functions and using that F (1 , , ǫ ; 2 ,
2; 1) = − ψ (1 − ǫ ) − γǫ = − H − ǫ ǫ (where ψ ( x ) is thedigamma function, γ is Euler’s constant and H n is the harmonic number of n ), weobtain the four- and five-point one-loop Wilson loop expressions of (4.1) and (4.2).For completeness we also consider the limit with P = Q = 0 using the secondexpression for the finite diagram (B.14), since this and similar expressions have beenused throughout appendix A. When P = Q = 0, we have a = 1 /s + 1 /t , 1 − as = − s/t and 1 − at = − t/s and using F ( ǫ, ǫ ; 1 + ǫ ; 1) = ǫπ csc( ǫπ ) , (B.16)we get F ǫ ( s, t, ,
0) = − ǫ × (cid:20) − (cid:16) ust (cid:17) ǫ (cid:18) ts (cid:19) ǫ F ( ǫ, ǫ ; 1 + ǫ ; − t/s ) − (cid:16) ust (cid:17) ǫ (cid:16) st (cid:17) ǫ F ( ǫ, ǫ ; 1 + ǫ ; − s/t )+ 2( a ) ǫ ǫπ csc( ǫπ ) + ǫ ( − a ) ǫ (cid:16) − log(1 − as ) − log(1 − at ) (cid:17)(cid:21) . (B.17)We wish to know this in the Euclidean regime in which s, t, a <
0. The first line isthen manifestly real, whereas the second gives2 πǫ ( − a ) ǫ e − iǫπ csc( ǫπ ) + 2 πǫi ( − a ) ǫ = 2 πǫ ( − a ) ǫ cot( ǫπ ) . (B.18)This is the form used for the one-loop Wilson loop throughout the paper, for examplein (4.1) and (A.5). 31 eferences [1] L. F. Alday and J. Maldacena, Gluon scattering amplitudes at strong coupling,
JHEP (2007) 064, .[2] J. M. Drummond, G. P. Korchemsky and E. Sokatchev,
Conformal properties offour-gluon planar amplitudes and Wilson loops,
Nucl. Phys. B (2008) 385, .[3] A. Brandhuber, P. Heslop and G. Travaglini,
MHV Amplitudes in N=4 Su-per Yang-Mills and Wilson Loops,
Nucl. Phys. B (2008) 231, .[4] J. M. Drummond, J. Henn, G. P. Korchemsky and E. Sokatchev,
On planargluon amplitudes/Wilson loops duality,
Nucl. Phys. B (2008) 52, .[5] J. M. Drummond, J. Henn, G. P. Korchemsky and E. Sokatchev,
ConformalWard identities for Wilson loops and a test of the duality with gluon amplitudes, .[6] C. Anastasiou, Z. Bern, L. J. Dixon and D. A. Kosower,
Planar amplitudes inmaximally supersymmetric Yang-Mills theory,
Phys. Rev. Lett. (2003) 251602, hep-th/0309040 .[7] F. Cachazo, M. Spradlin and A. Volovich, Iterative structure within the five-particle two-loop amplitude,
Phys. Rev. D (2006) 045020, hep-th/0602228 .[8] Z. Bern, M. Czakon, D. A. Kosower, R. Roiban and V. A. Smirnov, Two-loopiteration of five-point N = 4 super-Yang-Mills amplitudes,
Phys. Rev. Lett. (2006) 181601, hep-th/0604074 .[9] Z. Bern, L. J. Dixon and V. A. Smirnov, Iteration of planar amplitudes in max-imally supersymmetric Yang-Mills theory at three loops and beyond,
Phys. Rev.D (2005) 085001, hep-th/0505205 .[10] M. Spradlin, A. Volovich and C. Wen, Three-Loop Leading Singularities andBDS Ansatz for Five Particles,
Phys. Rev. D , 085025 (2008), .[11] L. F. Alday and J. Maldacena, Comments on gluon scattering amplitudes viaAdS/CFT,
JHEP (2007) 068, .[12] Z. Bern, L. J. Dixon, D. A. Kosower, R. Roiban, M. Spradlin, C. Vergu andA. Volovich,
The Two-Loop Six-Gluon MHV Amplitude in Maximally Super-symmetric Yang-Mills Theory,
Phys. Rev. D (2008) 045007, . 3213] V. Del Duca, C. Duhr and V. A. Smirnov, An Analytic Result for the Two-Loop Hexagon Wilson Loop in N = 4 SYM,
JHEP (2010) 099 .[14] V. Del Duca, C. Duhr and V. A. Smirnov,
The Two-Loop Hexagon Wilson Loopin N = 4 SYM, .[15] J. H. Zhang,
On the two-loop hexagon Wilson loop remainder function in N=4SYM, [16] J. M. Drummond, J. Henn, V. A. Smirnov and E. Sokatchev,
Magic identitiesfor conformal four-point integrals,
JHEP (2007) 064, hep-th/0607160 .[17] J. M. Drummond, J. Henn, G. P. Korchemsky and E. Sokatchev,
Dual super-conformal symmetry of scattering amplitudes in N=4 super-Yang-Mills theory,
Nucl. Phys. B (2010) 317, .[18] J. M. Drummond, J. Henn, G. P. Korchemsky and E. Sokatchev,
Hexagon Wilsonloop = six-gluon MHV amplitude,
Nucl. Phys. B , 142 (2009), [19] M. T. Grisaru, H. N. Pendleton and P. van Nieuwenhuizen,
Supergravity AndThe S Matrix,
Phys. Rev. D (1977) 996.[20] M. T. Grisaru and H. N. Pendleton, Some Properties Of Scattering AmplitudesIn Supersymmetric Theories,
Nucl. Phys. B (1977) 81.[21] M. L. Mangano and S. J. Parke,
Multi-Parton Amplitudes in Gauge Theories,
Phys. Rept. (1991) 30, hep-th/0509223 .[22] L. J. Dixon,
Calculating scattering amplitudes efficiently, hep-ph/9601359 .[23] S. Catani,
The singular behaviour of QCD amplitudes at two-loop order,
Phys.Lett. B (1998) 161, hep-ph/9802439 .[24] G. Sterman and M. E. Tejeda-Yeomans,
Multi-loop amplitudes and resummation,
Phys. Lett. B (2003) 48, hep-ph/0210130 .[25] D. A. Kosower and P. Uwer,
One-loop splitting amplitudes in gauge theory,
Nucl.Phys. B (1999) 477, hep-ph/9903515 .[26] Z. Bern, V. Del Duca, W. B. Kilgore and C. R. Schmidt,
The infrared behaviorof one-loop QCD amplitudes at next-to-next-to-leading order,
Phys. Rev. D (1999) 116001, hep-ph/9903516 .[27] F. Cachazo, M. Spradlin and A. Volovich, Leading Singularities of the Two-Loop Six-Particle MHV Amplitude,
Phys. Rev. D (2008) 105022, . 3328] C. Anastasiou, A. Brandhuber, P. Heslop, V. V. Khoze, B. Spence andG. Travaglini, Two-Loop Polygon Wilson Loops in N=4 SYM,
JHEP (2009)115, .[29] J. G. M. Gatheral,
Exponentiation Of Eikonal Cross-Sections In NonabelianGauge Theories,
Phys. Lett. B (1983) 90.[30] J. Frenkel and J. C. Taylor,
Nonabelian Eikonal Exponentiation,
Nucl. Phys. B (1984) 231.[31] Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower,
One Loop N Point GaugeTheory Amplitudes, Unitarity And Collinear Limits,
Nucl. Phys. B (1994)217, hep-ph/9403226 .[32] Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower,
Fusing gauge theory treeamplitudes into loop amplitudes,
Nucl. Phys. B , 59 (1995), hep-ph/9409265 .[33] I. A. Korchemskaya and G. P. Korchemsky,
On lightlike Wilson loops,
Phys. Lett.B (1992) 169.[34] P. Heslop and V. V. Khoze,
Regular Wilson loops and MHV amplitudes at weakand strong coupling, [35] M. B. Green, J. H. Schwarz and L. Brink,
N=4 Yang-Mills And N=8 SupergravityAs Limits Of String Theories,
Nucl. Phys. B (1982) 474.[36] Z. Bern, L. J. Dixon and D. A. Kosower,
Dimensionally regulated pentagon in-tegrals,
Nucl. Phys. B (1994) 751, hep-ph/9306240 .[37] Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower,
One-loop self-dual andN = 4 superYang-Mills,
Phys. Lett. B (1997) 105, hep-th/9611127 .[38] Z. Bern, J. S. Rozowsky and B. Yan,
Two-loop four-gluon amplitudes in N = 4super-Yang-Mills,
Phys. Lett. B , 273 (1997), hep-ph/9702424 .[39] Z. Bern, J. Rozowsky and B. Yan,
Two-loop N = 4 supersymmetric amplitudesand QCD, hep-ph/9706392 .[40] M. Czakon,
Automatized analytic continuation of Mellin-Barnes integrals,
Com-put. Phys. Commun. (2006) 559, hep-ph/0511200 .[41] A. V. Smirnov and V. A. Smirnov,
On the Resolution of Singularities of MultipleMellin-Barnes Integrals,
Eur. Phys. J. C (2009) 445, .[42] T. Hahn, CUBA: A library for multidimensional numerical integration,
Comput.Phys. Commun. , 78 (2005), hep-ph/0404043 .3443] D. Maitre,
HPL, a Mathematica implementation of the harmonic polylogarithms,
Comput. Phys. Commun. (2006) 222, hep-ph/0507152 .[44] T. Huber and D. Maitre,
HypExp, a Mathematica package for expanding hyper-geometric functions around integer-valued parameters,
Comput. Phys. Commun. (2006) 122, hep-ph/0507094 .[45] G. Georgiou,
Null Wilson loops with a self-crossing and the Wilsonloop/amplitude conjecture,
JHEP (2009) 021, .[46] G. Duplancic and B. Nizic,
Dimensionally regulated one-loop box scalar integralswith massless internal lines,
Eur. Phys. J. C (2001) 357 hep-ph/0006249 .[47] A. Brandhuber, B. Spence and G. Travaglini, One-loop gauge theory amplitudesin N = 4 super Yang-Mills from MHV vertices,
Nucl. Phys. B (2005) 150, hep-th/0407214 .[48] A. Brandhuber, B. Spence and G. Travaglini,
From trees to loops and back,
JHEP (2006) 142, hep-th/0510253hep-th/0510253