RResumming the POPE at One Loop
Ho Tat Lam (cid:68) and Matt von Hippel (cid:68)(cid:68)
Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
Abstract
The Pentagon Operator Product Expansion represents polygonal Wilson loops in planar N = 4 super Yang-Mills in terms of a series of flux tube excitations for finite coupling. Wedemonstrate how to re-sum this series at the one loop level for the hexagonal Wilson loopdual to the six-point MHV amplitude. By summing over a series of effective excitations wefind expressions which integrate to logarithms and polylogarithms, reproducing the knownone-loop result. a r X i v : . [ h e p - t h ] S e p ontents W MHV
Using the Symbol Map 9 Introduction
The Pentagon Operator Product Expansion (or POPE) has shown itself to be a powerfultool for the calculation of polygonal Wilson loops and their dual amplitudes in planar N = 4super Yang-Mills [1–3]. Making use of integrability, the POPE computes Wilson loops atfinite coupling, presented as an expansion in flux tube states propagating across the loop.Kinematically, this expansion corresponds to expanding around a particular collinear limit.For quite some time, it was unclear if this expansion could be re-summed to obtain thefull kinematic dependence of the amplitude. A partial resummation was achieved in [4], butit was only with recent work by Luc´ıa C´ordova that such a resummation was shown to bepossible in the limit of weak coupling for all flux tube states [7]. C´ordova shows that, forthe NMHV six-particle amplitude at tree level, it is possible to package all combinationsof states that can contribute into single effective excitations, creating a series which can bere-summed to match the full (tree-level) amplitude.In this work, we extend C´ordova’s calculation to one loop for the MHV case. While theexpressions that appear are of comparable complexity, computing a one-loop amplitude inthis way allows us to observe the appearance of transcendental functions from the POPE, ina way that should generalize to higher loop orders.We begin in section 2 by describing the effective one-particle excitations needed for MHV.In section 3 we show how to re-sum them into the polylogarithmic functions of the one-loophexagon Wilson loop. Finally, we conclude by discussing how this procedure might beextended to higher loops. We can start by considering the expression for the hexagon Wilson loop given by the POPEprogram [3], as a sum over all possible flux tube excitations: W = (cid:88) m S m (cid:90) du . . . du m (2 π ) m Π dyn Π mat . (1)Here S m are symmetry factors, u i are rapidities, and Π dyn and Π mat are referred to asthe dynamical part and matrix part respectively. The dynamical part contains all of thisexpression’s dependence on the coupling, while the matrix part takes care of R-symmetry.The excitations summed over here are combinations of fundamental excitations: gluons(and gluon bound states), fermions, and scalars. While gluons and scalars can be straightfor-wardly integrated in rapidity, fermions must be integrated over two different Riemann sheets.On one of these sheets the fermion momentum is large with respect to the coupling, whileon the other it is small. Hence we follow prior convention and divide fermion integrationsinto “large” and “small” fermions, which can be treated separately.Through one loop, only states with one fundamental excitation can contribute, with theexception of small fermions. In practice, then, we can sum effective excitations consisting of3igure 1: Table of effective weak coupling excitations including the first n descendants ofthe particles transforming in the vector representation of SU(4), from [7]. The plane in thebottom contains the primary excitations.one fundamental excitation and a string of small fermions. The small fermion contour allowsus to evaluate all small fermion rapidity integrations via residues, so the only integration weneed to do explicitly is that of the fundamental excitation. The resulting effective excitationsare summarized in figure 1.While [7] had to consider general R-symmetry representations, here for the MHV casewe need only consider the singlets. These correspond to r = 0 and r = 4 in the notation ofthat paper. In particular, we do not need to re-derive the list of residues that must be takenin small fermion rapidity, as Figure 6 of that paper provides the needed information. Specif-ically, it instructs us to consider ten chains of fundamental excitations and correspondingdescendents: F b ( ψ S ¯ ψ S ) n , ψ ¯ ψ S ( ψ S ¯ ψ S ) n , φ ¯ ψ S ( ψ S ¯ ψ S ) n , ¯ ψ ¯ ψ S ( ψ S ¯ ψ S ) n , F − b ¯ ψ S ( ψ S ¯ ψ S ) n , and theirconjugates.The most straightforward procedure would then be to start with the effective measuresgiven in appendix B of [7], and find expressions that can be summed over helicity. Instead,we will take a shortcut, and begin with equation (17) of that paper for which this has alreadybeen done. For singlet excitations, we specialize to the case where r = 4 and r = 0. Thereare two cases: the positive helicity excitations (here, just F a ) and the rest, which we willrefer to as “gluonic” and “non-gluonic”. µ [4 , a,n ( u ) | gluonic = ( − a + n Γ (cid:16) | a | − iu (cid:17) Γ (cid:16) | a | + iu (cid:17) Γ( n + 1)Γ( | a | + n ) (cid:18) iu + | a | (cid:19) n (cid:18) iu + | a | (cid:19) n + O ( g ) (2) µ [4 , a,n ( u ) | non − gluonic = ( − a + n Γ (cid:16) | a | − iu − (cid:17) Γ (cid:16) | a | + iu + 3 (cid:17) Γ( n + 1)Γ( | a | + n + 2) (cid:18) iu + | a | (cid:19) n (cid:18) iu + | a | (cid:19) n + O ( g )(3)4or MHV, we have the same excitations, evaluated at the same residues with the samesymmetry factors. The only change is in the contributions referred to as NMHV form factors,presented in [8]. These are factors present in the NMHV amplitude that take into account thenontrivial R-symmetry of the external states. Since we are interested in the MHV amplitudehere, we need to divide the integrands in eq. 3 by these form factors in order to obtain ourdesired result.For the gluonic case, form factors contribute a factor of h − F a ( u ) from the fundamentalexcitation and a product of contributions from the descendants, where h F a ( u ) is the gluonicform factor. Expanded in g , h − F a ( u ) is h − F a ( u ) = 1 g (cid:16) a iu (cid:17) (cid:16) a − iu (cid:17) + O ( g ) . (4)This tells us two things. First, since the form factor for gluonic excitations starts at order g − , removing it means that what was previously a tree-level NMHV expression now gives usthe one-loop expression for MHV. Second, we must also remove a factor of (cid:0) a + iu (cid:1) (cid:0) a − iu (cid:1) .The contribution from the descendants is also simple to take into account. Expandingthe small fermion form-factors h ψ S ( v ) in g we find h − ψ S ( v ) = gv + O ( g ) , h − ψ S ( v ) = vg + O ( g ) (5)Since the descendants consist of pairs of ψ S and ¯ ψ S evaluated at different residues, the factorsof g cancel.For the gluonic case, the first pair of descendants of F a contains a ψ S at u − i a and a ¯ ψ S at u − i a − i . Subsequent descendants are at intervals of i . Then to leading order in thecoupling, the contribution from the descendant form factors is n (cid:89) k =1 u − i (cid:0) a + k + 1 (cid:1) u − i (cid:0) a + k − (cid:1) (6)for n descendants.Most of these factors cancel. We are left only with contributions from k = 1, k = 2, k = n − k = n . Together, these give an overall factor of (cid:0) u − i a − in (cid:1) (cid:0) u − i a − in − i (cid:1)(cid:0) u − i a (cid:1) (cid:0) u − i a − i (cid:1) , (7)which we must remove.Between eq. 4 and eq. 7 we have all that we need to convert the expressions in 3 tothe corresponding integrands for the MHV case in the gluonic sector. The calculation forthe non-gluonic states is similar, and is omitted for brevity. Removing these factors, andsimplifying using the definition of the Pochhammer symbol a n , we are left with the followingexpressions: µ MHV , gluonic a,n ( u ) = g ( − a + n Γ (cid:16) | a | − iu (cid:17) Γ (cid:16) | a | + iu (cid:17)(cid:16) | a | − iu (cid:17)(cid:16) | a | + iu (cid:17) Γ( n + 1)Γ( | a | + n ) (cid:18) iu + | a | (cid:19) n + O ( g ) (8)5 MHV , non − gluonic a,n ( u ) = g ( − a + n Γ (cid:16) | a | − iu + 1 (cid:17) Γ (cid:16) | a | + iu + 1 (cid:17)(cid:16) | a | − iu (cid:17)(cid:16) | a | + iu (cid:17) Γ( n + 1)Γ( | a | + n + 2) (cid:18) iu + | a | (cid:19) n + O ( g )(9)To sum up, one loop MHV, W MHV = 1 + g W MHV,(1) + O ( g ), is given by the followingPOPE series, W MHV = 1 + 2 ∞ (cid:88) n =0 (cid:90) d u π e − (2 n +2) τ +2 iuσ µ MHV , non − gluonic0 ,n ( u )+ (cid:88) a (cid:54) =0 ∞ (cid:88) n =0 (cid:90) d u π e − ( | a | +2 n ) τ + iaφ +2 iuσ (cid:2) µ MHV , gluonic a,n ( u ) + e − τ µ MHV , non − gluonic a,n ( u ) (cid:3) . (10) The resummation of the MHV POPE series can be carried out following a similar strategyto that employed in [7]. The resummation is performed beginning with an expansion in thecollinear limit, exp( τ ) → ∞ , which is then analytically continued to arbitrary kinematics.The key step is to replace the summation over n with integrations over t using the series andintegral representations of hypergeometric functions, F ( a, b, c ; z ) = ∞ (cid:88) n =0 Γ( a + n )Γ( b + n )Γ( c )Γ( a )Γ( b )Γ( c + n ) z n n ! = Γ( c )Γ( b )Γ( c − b ) (cid:90) d t t b − (1 − t ) c − b − (1 − tz ) − a (11)where this integral representation is valid only when Re( c ) > Re( b ) >
0. After this replace-ment, the POPE series is converted into the following expression, W MHV,(1) = (cid:88) a (cid:54) =0 (cid:90) d t (cid:90) ∞ + i(cid:15) −∞ + i(cid:15) d u π (cid:0) I gluonic a + I non − gluonic a (cid:1) +2 (cid:90) d t (cid:90) ∞ + i(cid:15) −∞ + i(cid:15) d u π I non − gluonic0 (12)with the following integrands, I gluonic a = ( − a t | a | + iu − (1 − t ) | a | − iu − ( e − τ t + 1) − | a | − iu e − τ | a | + iaφ +2 iσu u + | a | / I non − gluonic a = ( − a t | a | + iu (1 − t ) | a | − iu ( e − τ t + 1) − | a | − iu − e − τ ( | a | +2)+ iaφ +2 iσu u + | a | / u can be evaluated by taking residues at u = ± i | a | /
2. Only one ofthe residues will be picked up depending on how the contour is closed. All of the integrandsare of the form exp [ if ( t ) u ] /u in the limit of large u , with f ( t ) = 2 σ − log (cid:20) (1 + e − τ t )(1 − t ) t (cid:21) t ∗ = 12 (cid:20) − e σ +2 τ − e τ + (cid:113) (1 − e σ +2 τ − e τ ) + 4 e τ (cid:21) . When t ∈ ( t ∗ , f ( t ) >
0, the contour closes in the upper half complex plane and picksup a pole at u = i | a | / t ∈ (0 , t ∗ ), f ( t ) <
0, the contour closes in the lower half-plane and picks up a poleat u = − i | a | /
2. The prescription is however different when a = 0. There, the integrationcontour is shifted upwards on the complex plane as suggested by the POPE proposal (thisallows us to reproduce the correct Riemann sheet for the large fermions, which give the a = 0contribution). When t ∈ (0 , t ∗ ), a double pole at u = 0 is selected and when t ∈ ( t ∗ , u , the integration domain of t breaks into two pieces, W MHV,(1) = (cid:90) t ∗ d t (cid:88) a (cid:54) =0 ( − a e iaφ −| a | σ −| a | τ | a | (1 − t ) | a | (cid:18) t − t − t + e τ (cid:19) + (cid:90) t ∗ d t (cid:34) f ( t ) t + e τ + (cid:88) a (cid:54) =0 ( − a e iaφ + | a | σ −| a | τ | a | (cid:18) e τ tt + e τ (cid:19) | a | (cid:18) t − t − t + e τ (cid:19)(cid:35) . (14)We move the summation over helicity inside the integration. This summation converges inthe collinear limit, and it has a closed form, found from the following simple relation, ∞ (cid:88) a =1 x a a = − log(1 − x ) . (15)We are left with the following integrations to be performed, W MHV,(1) = (cid:90) t ∗ d t log (cid:2) e iφ − σ − τ (1 − t ) (cid:3) (cid:18) t − − t − t + e τ (cid:19) + (cid:90) t ∗ d t (cid:20) f ( t ) t + e τ + log (cid:18) e iφ + σ + τ tt + e τ (cid:19) (cid:18) t − − t − t + e τ (cid:19)(cid:21) + h . c . (16)We organize the integrands as follows, W MHV,(1) = (cid:90) t ∗ d t log (cid:2) t + e τ + e iφ + τ + σ t (cid:3) (cid:18) t − − t − t + e τ (cid:19) + dd t log (cid:2) t + e τ (cid:3) log (cid:20) e τ + σ ) t − t (cid:21) + (cid:90) t ∗ d t log (cid:2) e iφ − σ − τ − e iφ − σ − τ t (cid:3) (cid:18) t − − t − t + e τ (cid:19) + h . c . (17)where some of the integrands are combined into total derivatives and the rest are of thefollowing type, (cid:90) d t log( b + at ) t + c = log( b + at ) log (cid:20) a ( t + c ) ac − b (cid:21) + Li (cid:18) b + atb − ac (cid:19) . t ∗ and the other one with nodependence on t ∗ , which we refer to as the middle term and the boundary term, W boundary = π − T ] log (cid:20) − T ( F S + T )(1 + F ST + T ) F S (cid:21) + Li (cid:20) − F TS (cid:21) − Li (cid:20) F SF S + T (cid:21) − Li (cid:20) F ST F ST + T (cid:21) − Li (cid:20)
11 + F ST + T (cid:21) + h . c . W middle = π (cid:20) t + 1 T (cid:21) log (cid:20) − S T (1 + tT ) (cid:21) + log (cid:20) vw (1 + tT ) (cid:21) log (cid:20) t − t (1 + tT ) (cid:21) + log (cid:20) vw ∗ T (cid:21) log (cid:20) F S ( S + F T )( ST + F + F T ) (cid:21) − Li [ v ] − Li (cid:20) − F TS v (cid:21) + Li (cid:20) F ST + F + F T v (cid:21) − Li [ w ] + Li (cid:20) F SF S + T w (cid:21) + Li (cid:20) F ST F ST + T w (cid:21) + h . c . (18)where S = e σ , T = e − τ , F = e − iφ/ , v = ( F + StT + F tT ) /F and w = ( F S + T − tT ) /F S . (Note that v and w are not the dual conformal cross-ratios used in sources like [9], which werefer to here as u , u , and u .) Both terms are symmetric under complex conjugation so wecan replace some of the terms by their complex conjugates ( w to w ∗ and so on). The newvariables v and w satisfy a few relations when t = t ∗ , vw (1 + tT ) = 1 , vw ∗ = v ∗ w = ST + F + F S + F T + F STF . (19)Using these relations the one-loop MHV expression W MHV,(1) = W boundary + W middle canbe further simplified to reach the known expression, W MHV,(1) = π S ] log[1 + T ] − log [ u ] log [ u ] + Li [ u ] − Li [1 − u ] − Li [1 − u ] , (20)where u = F S (1 + T ) ( ST + F + F S + F T + F ST ) u = T T u = F ST + F + F S + F T + F ST . (21)A particularly straightforward way to see this simplification is to use symbol methods,as we illustrate in Appendix A. 8
Conclusions and Outlook
Extending the results of [7], we have demonstrated how to re-sum the Pentagon OperatorProduct Expansion at one loop to obtain an MHV amplitude. In particular, we have shownhow logarithms and polylogarithms emerge in two ways: from the sum over helicity, and viaintegral representations of hypergeometric functions.At higher loop orders (and for one loop NMHV) the integrands we found here multiplysums of polygamma functions. Above one loop, we also need to consider multiple effectiveexcitations. Either will make this procedure more complex, but neither should compromisethe core of our program. Going forward, it should be possible either to find appropriatechoices of integral representations of these functions (similar to that used for the hypergeo-metric function) or to take their residues in an explicit infinite sum, in either case makingthe transcendentality properties of the resummation manifest.Looking farther afield, we anticipate that it may be possible to re-sum the POPE forfinite coupling. Doing so will likely involve an as-yet unknown basis of functions. Neverthe-less, hints at this stage indicate that this may be more feasible than one would assume. Inparticular, summing over descendants reduces the complexity of the needed sums over statesdramatically, leaving a much simpler sum over effective excitations.
Acknowledgments
We owe special thanks to Luc´ıa C´ordova, who shared an early manuscript of her NMHVresummation which allowed us to attempt this work. We would also like to thank PedroVieira and the POPE team at the PSI Winter School for helpful discussions during earlystages of this project. This research was supported in part by Perimeter Institute for Theo-retical Physics. Research at Perimeter Institute is supported by the Government of Canadathrough the Department of Innovation, Science and Economic Development Canada and bythe Province of Ontario through the Ministry of Research, Innovation and Science.
A Simplifying W MHV
Using the Symbol Map
In this appendix we will show the equivalence of our expression for the one-loop MHV Wilsonloop in eq. 18 to the known expression in eq. 20 using symbols [10–12]. The symbol mapspolylogarithmic functions to tensor products of rational functions. For our purposes we needonly the action of the symbol map on the dilogarithm and on products of two logarithms:Li [ z ] ∼ (1 − z ) ⊗ /z log[ x ] log[ y ] ∼ x ⊗ y + y ⊗ x (22)The symbol map is not one-to-one. In particular, constants vanish under the symbol map,so any two functions that differ by a constant are mapped to the same symbol. Symbols9bey the following relations: φ ⊗ . . . ⊗ φ i φ j ⊗ . . . ⊗ φ n = φ ⊗ . . . ⊗ φ i ⊗ . . . ⊗ φ n + φ ⊗ . . . ⊗ φ j ⊗ . . . ⊗ φ n ,φ ⊗ . . . ⊗ φ ai ⊗ . . . ⊗ φ n = a ( φ ⊗ . . . ⊗ φ i ⊗ . . . ⊗ φ n )The symbols of the middle and boundary terms can be simplified by using the above relationsalong with those in eq. 19. In the end, we collect symbols with the same first entry and get, W boundary ∼ a ⊗ a F ST + b ⊗ b F ST + c ⊗ − caT + d ⊗ − dbT + F ⊗ − ST bdF + S ⊗ T S + T ⊗ F S a b c d T + (cid:0) T (cid:1) ⊗ F S T ab W middle ∼ a ⊗ F ST a + b ⊗ F ST b + c ⊗ − aT c + d ⊗ − bT d + F ⊗ − bdF ST + S ⊗ abSF T + T ⊗ abcdT F S + ST + F (1 + S + T ) + F STF T ⊗ F S abcd (23)where a = 1 + F ST + T , b = ST + F + F T , c = F S + T and d = S + F T . Combiningthem, the symbol of the full expression is, W MHV , (1) ∼ S ⊗ F S cd + (1 + T ) ⊗ F S T ab + ST + F (1 + S + T ) + F STF S ⊗ F S abcd (24)which is exactly the expected symbol for one-loop MHV, as can straightforwardly be obtainedfrom the full expression: W MHV , (1) = π S ] log[1 + T ] − log [ u ] log [ u ] + Li [ u ] − Li [1 − u ] − Li [1 − u ] . (25)The symbol maps transcendental constants to zero, so in principle our expression maydiffer from the known result by a term proportional to π . However, we can check thisconstant by taking the collinear limit ( T → References [1] B. Basso, A. Sever and P. Vieira, Phys. Rev. Lett. , 091602 (2013) [arXiv:1303.1396[hep-th]].[2] B. Basso, A. Sever and P. Vieira, JHEP , 008 (2014) [arXiv:1306.2058 [hep-th]].[3] B. Basso, A. Sever and P. Vieira, arXiv:1508.03045 [hep-th].[4] J. M. Drummond and G. Papathanasiou, JHEP , 185 (2016)doi:10.1007/JHEP02(2016)185 [arXiv:1507.08982 [hep-th]].105] G. Papathanasiou, Int. J. Mod. Phys. A , no. 27, 1450154 (2014)doi:10.1142/S0217751X14501541 [arXiv:1406.1123 [hep-th]].[6] G. Papathanasiou, JHEP , 150 (2013) doi:10.1007/JHEP11(2013)150[arXiv:1310.5735 [hep-th]].[7] L. C´ordova, arXiv:1606.00423 [hep-th].[8] B. Basso, J. Caetano, L. Cordova, A. Sever and P. Vieira, “OPE for all He-licity Amplitudes II. Form Factors and Data analysis,” JHEP (2015) 088doi:10.1007/JHEP12(2015)088 [arXiv:1508.02987 [hep-th]].[9] L. J. Dixon, M. von Hippel and A. J. McLeod, JHEP1601