Bootstrapping an NMHV amplitude through three loops

TL;DR

The authors extend the hexagon function bootstrap to the six-point NMHV amplitude at $3$-loops in planar ${\cal N}=4$ SYM, achieving a unique determination of the NMHV ratio function through a synthesis of the $\bar{Q}$ differential equation, Wilson-loop OPE data, and multi-Regge factorization. They show that the three-loop functions $V$ and $\tilde{V}$ are highly constrained, study their behavior in collinear, near-collinear, MRK, and multi-particle factorization limits, and reveal surprisingly simple structures such as a purely logarithmic multi-particle factorization function and an especially simple function $U$ with only five final entries. An empirical relation ties $V$ to coproduct components of the six-gluon MHV remainder $R_6$ at one higher loop, linking NMHV and MHV sectors in a predictive way. The results validate the hexagon bootstrap framework at three loops and point toward a coherent, cross-validated path to higher-loop NMHV amplitudes and to a deeper understanding of the interplay between OPE, MRK, and coproduct structures in ${\cal N}=4$ SYM.

Abstract

We extend the hexagon function bootstrap to the next-to-maximally-helicity-violating (NMHV) configuration for six-point scattering in planar ${\cal N}=4$ 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 $R_6$).

Figures (12)

• Figure 1: (a) Multi-particle factorization of a six-point amplitude into two four-point amplitudes, in the limit $s_{345}\rightarrow0$. (b) The most general multi-particle factorization of an $n$-point amplitude into a $(j-i+1)$-point amplitude and an $(n-j+i+1)$-point amplitude, in the limit that $K^2 = s_{i,i+1,\ldots,j-1}\rightarrow0$.
• Figure 2: $V^{(1)}(u,u,1)$, $V^{(2)}(u,u,1)$ and $V^{(3)}(u,u,1)$, normalized to unity at $(1,1,1)$. One loop is in green, two loops is in purple, and three loops is in blue.
• Figure 3: $V^{(1)}(u,1,u)$, $V^{(2)}(u,1,u)$ and $V^{(3)}(u,1,u)$, normalized to unity at $(1,1,1)$. One loop is in green, two loops is in purple, and three loops is in blue.
• Figure 4: $V^{(1)}(u,1,1)$, $V^{(2)}(u,1,1)$ and $V^{(3)}(u,1,1)$, normalized to unity at $(1,1,1)$. One loop is in green, two loops is in purple, and three loops is in blue.
• Figure 5: $V^{(1)}(1,v,1)$, $V^{(2)}(1,v,1)$ and $V^{(3)}(1,v,1)$, normalized to unity at $(1,1,1)$. One loop is in green, two loops is in purple, and three loops is in blue.