The leading power Regge asymptotic behaviour of dimensionally regularized massless on-shell planar triple box

Abstract

The leading power asymptotic behaviour of the dimensionally regularized massless on-shell planar triple box diagram in the Regge limit t/s → 0 is analytically evaluated.1 E-mail: smirnov@theory.sinp.msu.ru Systematical analytical evaluation of two-loop Feynman diagrams with four external lines within dimensional regularization [1] began three years ago. In the pure massless case with all end-points on-shell, i.e. p 2 i = 0, i = 1, 2, 3, 4, the problem of analytical evaluation of two-loop four-point diagrams in expansion in ǫ = (4 − d)/2, where d is the space-time dimension, has been completely solved in [2,3,4,5,6,7]. The corresponding analytical algorithms have been successfully applied to the evaluation of two-loop virtual corrections to various scattering processes [8] in the zero-mass approximation.In the case of massless two-loop four-point diagrams with one leg off-shell the problem of the evaluation has been solved in [9,10], with subsequent applications [11] to the process e + e − → 3jets. (See [12] for recent reviews of the present status of NNLO calculations. See [13] for a brief review of results on the analytical evaluation of various double-box Feynman integrals and the corresponding methods of evaluation.) For another three-scale calculational problem, where all four legs are on-shell and there is a non-zero internal mass, a first analytical result was obtained in [14] for the scalar master double box.The purpose of this paper is to turn attention to three-loop on-shell massless four-point diagrams. As a first step, the leading power asymptotic behaviour of the dimensionally regularized massless on-shell planar triple box diagram shown in Fig. 1 in the Regge limit t/s → 0 will be analytically evaluated. This calculation will demonstrate that a three-loop BFKL analysis [15] (at least its virtual part which can be reduced to the evaluation of Regge asymptotics) is possible 2 . The calculation will be based on the technique of alpha parameters and MellinBarnes (MB) representation which was successfully used in [2,4,9,14] and reduces, due to taking residues and shifting contours, to a decomposition of a given MB integral into pieces where a Laurent expansion of the integrand in ǫ becomes possible. At a final stage, summation formulae for series of S 1 (n)S 3 (n)/n 2 , ψ ′′ (n + 1)S 2 (n)/n etc., where S k (n) = n j=1 j −k , are used. A table of such formulae is presented in Appendix.2 In three loops, non-planar diagrams as well as higher terms of expansion of double boxes in ǫ are also needed. 1The general planar triple box Feynman integral without numerator takes the formwhere s = (p 1 + p 2 ) 2 and t = (p 2 + p 3 ) 2 are Mandelstam variables, and k, l and r are loop momenta. Usual prescriptions k 2 = k 2 + i0, s = s + i0, etc. are implied.To evaluate the leading power asymptotic behaviour of the master triple box (1), i.e. for all a i = 1, in the limit t/s → 0 one can use the strategy of expansion by regions [16,17,18]. It shows that in the leading power only (1c-1c-1c) and (2c-2c-2c) region...

Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond

“…In the case of the ladder triple box (3.1), in the Regge limit t/s → 0, only the (1c-1c -1c) and (2c-2c-2c) regions participate in the leading power-law behavior [55].…”

“…[24] (see the discussion near eqs. (55) of that reference), we may construct the required set of identities by induction on the weight of the harmonic polylogarithms. For the first few weights, in the region −1 ≥ x ≥ 0, and letting L = ln(s/t) = ln(−1/x), we have, for example,…”

Duality between Wilson loops and gluon amplitudes

“…In a similar manner, the residue of w (2) at the single pole in is proportional to ζ 3 and this comes about as the result of a cancelation between various terms in (213) containing rational numbers, π 2 −terms as well as the integrals M 1 and M 2 . The most striking simplifications occur in the sum of finite O( 0 ) terms (214). We find that the integrals M 2 , M 3 , M 2 1 as well as the rational corrections and the terms proportional to π 2 and ζ 3 cancel in the sum of all diagrams leading to − 7 720 π 4 + 1 12 π 2 M 1 .…”

Applying Mellin-Barnes technique and Groebner bases to the three-loop static potential

“…1 which we denote by F(a 1 ,... ,a 11 ). A straightforward implementation of the loop-by-loop strategy leads to a six-fold MB representation which reads 3,4,6,7,8,9,11 …”