Color-kinematic Duality for Form Factors

TL;DR

The paper extends color-kinematic duality from scattering amplitudes to form factors in gauge theories, validating the approach in ${ m \,N=4}$ SYM up to three loops and producing a four-loop Sudakov form factor integrand that passes extensive unitarity checks. It develops a general formalism and calculational strategy based on generating trivalent topologies, enforcing Jacobi-like relations for numerators, and verifying with unitarity cuts, yielding results that reproduce known literature in a unified framework. The key contributions include a demonstration of color-dual numerators for form factors, explicit master-integral-based constructions at four loops, and detailed consistency checks that support the conjectured duality for gauge-theory observables beyond amplitudes. The work suggests broad implications for extending duality to less supersymmetric theories and exploring a Lagrangian origin, as well as potential gravity duals via the double-copy idea, marking a significant step toward a universal perturbative framework for gauge observables.

Abstract

Recently a powerful duality between color and kinematics has been proposed for integrands of scattering amplitudes in quite general gauge theories. In this paper the duality proposal is extended to the more general class of gauge theory observables formed by form factors. After a discussion of the general setup the existence of the duality is verified in two- and three-loop examples in four dimensional maximally supersymmetric Yang-Mills theory which involve the stress energy tensor multiplet. In these cases the duality reproduces known results in a particularly transparent and uniform way. As a non-trivial application we obtain a very simple form of the integrand of the four-loop two-point (Sudakov) form factor which passes a large set of unitarity cut checks.

Figures (12)

• Figure 1: Trivalent diagrams for four-point tree amplitudes.
• Figure 2: Color-kinematics duality at loop level.
• Figure 3: Diagrams for four-point tree form factor.
• Figure 4: The integrals for the two-point two-loop form factor.
• Figure 5: The integrals for the two-point three-loop form factor.