A Two-Loop Four-Point Form Factor at Function Level

TL;DR

This work elevates the two-loop four-point MHV form factor for ${\rm tr}\,\phi^2}$ in planar ${\mathcal N}=4$ SYM from symbol data to a full function-level description by constructing a coproduct-based function space up to weight four on a rational three-parameter surface. The authors fix beyond-the-symbol constants using integrability, first-entry and Steinmann constraints, dihedral symmetry, and boundary limits including soft, collinear, and FFOPE data, obtaining explicit $G$-function representations on the rational surface and in the bulk. They demonstrate numerical consistency through boundary slices, confirm antipodal self-duality at the function level (beyond the symbol), and provide extensive ancillary materials for the full coproducts and limits. The results enable detailed checks across kinematic regions (including 2D kinematics and OPE limits) and pave the way for extension to higher multiplicity and loop orders, with potential connections to MRK and self-crossing regimes.

Abstract

Recently, the maximally-helicity-violating four-point form factor for the chiral stress-energy tensor in planar $\mathcal{N}=4$ super Yang-Mills was computed to three loops at the level of the symbol associated with multiple polylogarithms. It exhibits {\it antipodal self-duality}, or invariance under the combined action of a kinematic map and reversing the ordering of letters in the symbol. Here we lift the two-loop form factor from symbol level to function level. We provide an iterated representation of the function's derivatives (coproducts). In order to do so, we find a three-parameter limit of the five-parameter phase space where the symbol's letters are all rational. We also use function-level information about dihedral symmetries and the soft, collinear, and factorization limits, as well as limits governed by the form-factor operator product expansion (FFOPE). We provide plots of the remainder function on several kinematic slices, and show that the result is compatible with the FFOPE data. We further verify that antipodal self-duality is valid at two loops beyond the level of the symbol.

Figures (4)

• Figure 1: The rational surface is parametrized by three kinematic variables $(u_3,v_1,v_2)$. It intersects a soft limit at the $v_2 = 1$ surface (red). The intersection with a triple collinear limit is at the $v_1=0$ surface (blue). The $u_3=0$ surface (yellow) maps into itself under two dihedral transformations ${\cal C}^2$ and ${\cal C}\cdot{\cal F}$, as indicated by "cycle/flip". The (green) square boundary of the $u_3=0$ surface intersects another two-parameter surface (\ref{['2Dkin']}) where the momenta all lie in two spacetime dimensions. The point $(0,1,1)$ makes contact with the OPE limit.
• Figure 2: The finite part $D_0(u_3,v_1,v_2)$ of the remainder function $\mathcal{R}_4^{(2)}$ on the ${v_1} = k({1-v_2})$ slice with the constant $k=$ 0.25, 0.5, 0.75, 1. The white curves highlight where $\mathcal{R}_4^{(2)}$ flips sign.
• Figure 3: The finite part of the remainder function $\mathcal{R}_4^{(2)}$ on the ${v_1} = k({1-v_2})$ slice when $k \ge 1$.
• Figure 4: Diagram showing paths along which we integrate to obtain the $u_2,v_1,v_2$ discontinuities of $\mathcal{R}_4^{(2)}$.