A systematic approach to $\ell$-loop planar integrands from the classical equation of motion

TL;DR

The paper presents a recursive, off-shell framework to construct $\ell$-loop planar integrands in colored quantum field theories starting from the classical equation of motion. Central to the method are the comb component and the $\ell$-loop kernel, which together with generalized Berends-Giele currents enable a controlled recursion that builds both irreducible and reducible planar contributions. The authors formulate explicit prescriptions for bi-adjoint scalar theory and Yang-Mills theory, including graph-factor corrections to avoid overcounting, and provide nontrivial two-loop examples that agree with standard Feynman-rule results. The approach is designed to be broadly applicable, potentially extendable to non-Lagrangian theories and non-planar or gravitational settings, highlighting a unifying, theory-agnostic pathway to loop integrands via planar recursions.

Abstract

In this paper, we present a recursive method for $\ell$-loop planar integrands in colored quantum field theories. We start with the classical equation of motion and then pick out the comb component, which will help us to define the loop kernels. Then we construct the $\ell$-loop integrands based on some recursion rules for the $\ell$-loop kernels. Finally, we reach a recursion formula for the $\ell$-loop planar integrands. Our method can be easily generalized to general quantum field theories, even non-Lagrangian theories, to obtain the planar part of the whole $\ell$-loop integrands.

Figures (3)

• Figure 1: All diagrams in the 2-way 2-loop kernel
• Figure 2: 2-point 2-loop diagram 1
• Figure 3: 2-point 2-loop diagram 2 ($l_2$ here is the shifted one)