Bootstrapping an NMHV amplitude through three loops

Abstract

Abstract:We extend the hexagon function bootstrap to the next-to-maximally-helicityviolating (NMHV) configuration for six-point scattering in planar N = 4 super-Yang-Mills theory at three loops. Constraints from theQ differential equation, from the operator product expansion (OPE) for Wilson loops with operator insertions, and from multi-Regge factorization, lead to a unique answer for the three-loop ratio function. The three-loop result also predicts additional terms in the OPE expansion, as well as the behavior of NMHV amplitudes in the multi-Regge limit at one higher logarithmic accuracy (NNLL) than was used as input. Both predictions are in agreement with recent results from the flux-tube approach. We also study the multi-particle factorization of multi-loop amplitudes for the first time. We find that the function controlling this factorization is purely logarithmic through three loops. We show that a function U , which is closely related to the parity-even part of the ratio function V , is remarkably simple; only five of the nine possible final entries in its symbol are non-vanishing. We study the analytic and numerical behavior of both the parity-even and parity-odd parts of the ratio function on simple lines traversing the space of cross ratios (u, v, w), as well as on a few two-dimensional planes. Finally, we present an empirical formula for V in terms of elements of the coproduct of the six-gluon MHV remainder function R 6 at one higher loop, which works through three loops for V (four loops for R 6 ).

The seven-gluon amplitude in multi-Regge kinematics beyond leading logarithmic accuracy

“…[14], dispersion integrals yielding the BDS-normalized amplitude for N = 6 gluons in MRK were derived in refs. [26][27][28][29][30]. In Fourier-Mellin space, they factorize into universal building blocks known as the adjoint BFKL eigenvalue and impact factor, which may be determined perturbatively from first principles, or extracted [31][32][33] from fixed-order expressions for the amplitude that have been obtained by other means [34][35][36].…”

“…A leading-order strong-coupling analysis is also possible [37,38], but even more remarkably, the building blocks in question can also be obtained to all loops [39] by means of analytic continuation from a collinear limit where the dynamics is governed by an integrable flux tube [40][41][42][43][44][45][46][47][48][49][50][51], see also [52][53][54]. These developments render the MRK as one of the best sources of 'boundary data' [54][55][56][57] for determining the six-gluon amplitude in general kinematics through five loops, by exploiting its analytic structure with the help of the bootstrap method [30,[58][59][60][61][62].…”

“…Focusing on the helicity configuration most commonly found in the literature, see e.g. [30], the analogue of (3.1)-(3.2) is 19) where the helicity flip kernel H will be defined below in eq. (3.43).…”

Positive amplitudes in the amplituhedron

“…For n = 6 the L = 2 and L = 3 cases will be discussed in [26], based on results obtained in [27,28].…”

Gravity on-shell diagrams

“…Taking the large N limit of the gauge group, the planar theory is even simpler and spawned most of the newly discovered structures, including dual conformal symmetry [9][10][11], Yangian symmetry [12], integrability [13,14], a dual interpretation of amplitudes in terms of Wilson loops [15][16][17][18][19][20], the expansion of amplitudes in special kinematic limits at finite coupling using OPE methods [21][22][23], the hexagon-function bootstrap [24][25][26] heavily using symbols and cluster polylogarithmics [27][28][29][30], as well as a variety of other structures. More recently, scattering amplitudes were expressed in terms of on-shell diagrams and the…”