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<p>Previously in [1], we constructed the <i>ℓ</i>-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 <i>ℓ</i>-loop integrand recursion clearer.</p>

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The bi-adjoint scalar ℓ-loop planar integrand recursion and graded inverse variables

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月18日

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  • Yi-Xiao Tao

摘要

Previously in [1], we constructed the ℓ-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 ℓ-loop integrand recursion clearer.