A recursive reduction of tensor Feynman integrals

TL;DR

The paper develops a fully recursive method to reduce one-loop tensor Feynman integrals with up to six external legs ($n\le 6$) and rank up to $R\le n$ by expressing $(n,R)$-integrals in terms of $(n,R-1)$- and $(n-1,R-1)$-integrals using higher-dimensional scalar representations and a novel master formula for five-point functions. The core advance is the master relation $I_5^{\mu_1\dots\mu_{R-1}\mu}=I_5^{\mu_1\dots\mu_{R-1}} Q_0^{\mu}-\sum_{s=1}^5 I_4^{\mu_1\dots\mu_{R-1},s} Q_s^{\mu}$, which enables systematic, sequential reductions across ranks and point-counts, complemented by explicit recursions for the $(5,3)$, $(5,4)$, and $(5,5)$ families and an analogous $(6,R)$ extension. The approach relies on auxiliary vectors $Q_s^{\mu}$ and a careful treatment of $g^{\mu\nu}$-terms and Gram-determinant factors, delivering a compact framework suitable for massless and massive propagators in dimensional regularization with numerical cross-checks against established tools. Overall, the work provides a practical, algorithmic tensor-reduction scheme applicable to NLO computations in LHC/ILC physics and meson factories. The method balances mathematical rigor with numerical implementability, enabling stable evaluation of pentagon and hexagon tensor integrals within a purely recursive pipeline.

Abstract

We perform a recursive reduction of one-loop $n$-point rank $R$ tensor Feynman integrals [in short: $(n,R)$-integrals] for $n\leq 6$ with $R\leq n$ by representing $(n,R)$-integrals in terms of $(n,R-1)$- and $(n-1,R-1)$-integrals. We use the known representation of tensor integrals in terms of scalar integrals in higher dimension, which are then reduced by recurrence relations to integrals in generic dimension. With a systematic application of metric tensor representations in terms of chords, and by decomposing and recombining these representations, we find the recursive reduction for the tensors. The procedure represents a compact, sequential algorithm for numerical evaluations of tensor Feynman integrals appearing in next-to-leading order contributions to massless and massive three- and four- particle production at LHC and ILC, as well as at meson factories.