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

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.