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

Yi-Xiao Tao

TL;DR

This work addresses the computation of $\ell$-loop planar integrands in the bi-adjoint scalar theory by improving the recursion framework. The authors introduce graded inverse variables to reformulate the loop-kernel recursion, and they develop a systematic procedure to extract graph factors from monomials without drawing diagrams. They define a graded sewing operation and a map from graded-monomials to loop kernels, enabling compact, algorithmic construction of $\ell$-loop integrands and explicit kernel expressions. Potential extensions include applications to Yang-Mills theory, differential equations for these variables, and connections to perturbiner expansions.

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.

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

TL;DR

This work addresses the computation of -loop planar integrands in the bi-adjoint scalar theory by improving the recursion framework. The authors introduce graded inverse variables to reformulate the loop-kernel recursion, and they develop a systematic procedure to extract graph factors from monomials without drawing diagrams. They define a graded sewing operation and a map from graded-monomials to loop kernels, enabling compact, algorithmic construction of -loop integrands and explicit kernel expressions. Potential extensions include applications to Yang-Mills theory, differential equations for these variables, and connections to perturbiner expansions.

Abstract

Previously in \cite{Tao:2025fch}, we constructed the -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 -loop integrand recursion clearer.
Paper Structure (11 sections, 35 equations)

This paper contains 11 sections, 35 equations.

Table of Contents

  1. Introduction
  2. A review of the off-shell engineering
  3. Comb components and 1-loop kernels
  4. Graph factors: avoid overcounting
  5. Towards $\ell$-loop integrands
  6. A new formalism of the recursion
  7. Graded inverse variables
  8. Graph factors
  9. Recursion using new variables
  10. The map from monomials to loop kernels
  11. Conclusion