Bootstrapping an NMHV amplitude through three loops

Abstract

We extend the hexagon function bootstrap to the next-to-maximally-helicity-violating (NMHV) configuration for six-point scattering in planar $\mathcal{N}=4$ super-Yang-Mills theory at three loops. Constraints from the $\bar{Q}$ 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 $R_6$).

69 pages, 12 figures, 1 table, 3 ancillary files; v2, minor typo's corrected

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• Multi-Regge kinematics and the moduli space of Riemann spheres with marked points
• Elliptic polylogarithms and Feynman parameter integrals
• Adjoint BFKL at finite coupling: a short-cut from the collinear limit
• A Cluster Bootstrap for Two-Loop MHV Amplitudes
• Multi-Loop Positivity of the Planar $\mathcal{N}=4$ SYM Six-Point Amplitude
• Stratifying On-Shell Cluster Varieties: the Geometry of Non-Planar On-Shell Diagrams
• Manifestly Dual-Conformal Loop Integration
• The Sklyanin Bracket and Cluster Adjacency at All Multiplicity
• Cross-Order Integral Relations from Maximal Cuts
• Analytic structure of the $n=7$ scattering amplitude in $\mathcal{N}=4$ SYM theory in multi-Regge kinematics: Conformal Regge cut contribution
• Regge meets collinear in strongly-coupled $\mathcal{N}=4$ super Yang-Mills