A complete algebraic reduction of one-loop tensor Feynman integrals

TL;DR

This work delivers a complete algebraic reduction for one-loop tensor Feynman integrals, specifically reducing 5-point tensors (rank up to 5) to 4-point tensor coefficients while avoiding inverse 5-point Gram determinants. It leverages Davydychev-style representations in shifted dimensions, symmetrized dimensional recurrences, and the algebra of signed minors to express higher-point tensors in terms of higher-dimensional 4-point functions and 3-point tensor coefficients. To address numerical instabilities from small Gram determinants, the authors develop analytic expansions in Gram determinant powers and apply Padé approximants to achieve stable, accurate results across phase space. The approach is further augmented by symmetrized recurrences and analytic contractions, enabling a compact, implementable framework with explicit formulas up to rank-5 5-point tensors and scalable generalization to higher ranks.

Abstract

We set up a new, flexible approach for the tensor reduction of one-loop Feynman integrals. The 5-point tensor integrals up to rank R=5 are expressed by 4-point tensor integrals of rank R-1, such that the appearance of the inverse 5-point Gram determinant is avoided. The 4-point tensor coefficients are represented in terms of 4-point integrals, defined in $d$ dimensions, $4-2ε\le d \le 4-2ε+2(R-1)$, with higher powers of the propagators. They can be further reduced to expressions which stay free of the inverse 4-point Gram determinants but contain higher-dimensional 4-point integrals with only the first power of scalar propagators, plus 3-point tensor coefficients. A direct evaluation of the higher dimensional 4-point functions would avoid the appearance of inverse powers of the Gram determinants completely. The simplest approach, however, is to apply here dimensional recurrence relations in order to reduce them to the familiar 2- to 4-point functions in generic dimension $d = 4-2\eps$, introducing thereby coefficients with inverse 4-point Gram determinants up to power $R$ for tensors of rank $R$. For small or vanishing Gram determinants - where this reduction is not applicable - we use analytic expansions in positive powers of the Gram determinants. Improving the convergence of the expansions substantially with Padé approximants we close up to the evaluation of the 4-point tensor coefficients for larger Gram determinants. Finally, some relations are discussed which may be useful for analytic simplifications of Feynman diagrams.

Figures (3)

• Figure 2.1: Momenta flow of the $n$-point function.
• Figure C.1: (a) A six-point topology; (b) a four-point topology derived from (a).
• Figure C.2: The tensor coefficient $D_{1111}(x)$; the data points for a least square fit are shown. Input: grey (red) dots originate from the expansion at small $x$, black dots from the values at larger $x$.