Analytic two-loop form factors in N=4 SYM

TL;DR

The paper computes the three-point, two-loop MHV form factor of half-BPS operators in ${\cal N}=4$ SYM using generalized unitarity, yielding a compact analytic remainder after subtracting the ABDK/BDK-like infrared ansatz. It demonstrates exponentiation of infrared divergences, establishes universal collinear factorisation, and derives a unique symbol that fixes the remainder function, which is subsequently expressed in terms of classical polylogarithms. A striking result is that the remainder is maximally transcendental and matches the corresponding part of a two-loop QCD amplitude in the large-top-mass limit, revealing a deep connection between ${\cal N}=4$ SYM and QCD. The work also develops a thorough numerical and symbolic program (via Mellin-Barnes techniques and symbol calculus) to validate and illuminate the analytic remainder, advancing the understanding of form factors in gauge theories.

Abstract

We derive a compact expression for the three-point MHV form factors of half-BPS operators in N=4 super Yang-Mills at two loops. The main tools of our calculation are generalised unitarity applied at the form factor level, and the compact expressions for supersymmetric tree-level form factors and amplitudes entering the cuts. We confirm that infrared divergences exponentiate as expected, and that collinear factorisation is entirely captured by an ABDK/BDS ansatz. Next, we construct the two-loop remainder function obtained by subtracting this ansatz from the full two-loop form factor and compute it numerically. Using symbology, combined with various physical constraints and symmetries, we find a unique solution for its symbol. With this input we construct a remarkably compact analytic expression for the remainder function, which contains only classical polylogarithms, and compare it to our numerical results. Furthermore, we make the surprising observation that our remainder is equal to the maximally transcendental piece of a closely related two-loop amplitude in QCD.

Figures (6)

• Figure 1: The two two-particle cuts contributing to the Sudakov form factor.
• Figure 2: Double two-particle cuts and three-particle cuts of three-point form factor.
• Figure 3: A particular three-particle cut of the three-point form factor.
• Figure 4: There are twenty integrals contributing to the three-particle cut example discussed in the text.
• Figure 5: The two appearances of the integral TriPent in the three-particle cut. They give different contributions to the cut integrand.