A simple method for multi-leg loop calculations 2: a general algorithm

TL;DR

The paper tackles the tensorial complexity of multi-leg, one-loop diagrams with arbitrary internal masses and external momenta. It extends a previous reduction technique by extracting the loop-momentum dependence through a massless spinor basis, employing the identity $/q=\frac{1}{2\, (\ell_1\cdot\ell_2)}\left[ 2\,(q\cdot\ell_2)\,/\ell_1 + 2\,(q\cdot\ell_1)\,/\ell_2 -\,/\ell_1 \, / q \, /\ell_2 -/\ell_2 \, / q \, /\ell_1 \right]$, and reconstructing the denominators to reduce tensor integrals to scalar and rank-1 forms. The approach is generalized to $n$ dimensions by using $\underline{q}=q+\tilde{q}$ with adjusted relations like $q^2 = D_1 - \tilde{q}^{\,2} + m_1^2$, and it is shown to handle collinear and Gram-determinant-related singularities. The contributions include a general, iteratable algorithm applicable to generic one-loop integrals and an explicit example illustrating the reduction, yielding compact expressions and broader applicability for high-leg processes in particle physics. Overall, the work provides a robust framework to simplify and accelerate practical one-loop calculations across a wide range of masses and kinematics.

Abstract

The method introduced in a previous paper to simplify the tensorial reduction in multi-leg loop calculations is extended to generic one-loop integrals, with arbitrary internal masses and external momenta.