The bi-adjoint scalar $\ell$-loop planar integrand recursion and graded inverse variables

TL;DR

This paper introduces a new formalism using graded inverse variables to clarify the recursion process for constructing $ ext{l}$-loop planar integrands in bi-adjoint scalar theories.

Contribution

It proposes a novel formalism with graded inverse variables that simplifies and clarifies the recursion rules for $ ext{l}$-loop planar integrand construction.

Findings

The new formalism makes the loop kernel recursion more elegant.

Graph factors and symmetry factors can be derived from monomials of variables.

The approach enhances understanding of $ ext{l}$-loop integrand recursion.

Abstract

Previously in \cite{Tao:2025fch}, we constructed the $\ell$-loop planar integrands using loop components and loop kernels by some recursion rules. In this paper, we propose a new formalism to express the loop kernel recursion. We define ``graded inverse variables" to make the loop kernel recursion more elegant. And the graph factor, including the symmetry factor, can be figured out from each monomial of some variables. This new formalism makes the previous $\ell$-loop integrand recursion clearer.