Complete algebraic reduction of one-loop tensor Feynman integrals

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 ¼ 5 are expressed by 4-point tensor integrals of rank R À 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 À 2 d 4 À 2 þ 2ðR À 1Þ, 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 higherdimensional 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 ¼ 4 À 2", 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.

RECOLA2: REcursive Computation of One-Loop Amplitudes 2

“…For the evaluation of the one-loop integrals, a task that demands high standards with respect to numerical stability and CPU performance, amplitude generators are either equipped with own internal implementations, or they rely on external libraries like FF [36], LoopTools [37], QCDLoop [38], OneLOop [39], Golem95C [40], PJFry [41], Package-X [42], and Collier [43,44]. In the case of Recola, the public Fortran95 library Collier is used which achieves a fast and stable calculation of tensor integrals via the strategies developed in Refs.…”

Reductions and Contractions of 1-loop Tensor Feynman Integrals

“…for lower rank results, see [15,16]. After contraction with chords q (differences of external momenta), we get…”

“…We are working on independent calculations for tensor contractions as an alternative to the PJFry reductions [11], based on work by DavydychevTarasov-Fleischer-Jegerlehner-Riemann-Yundin (DTFJRY) [12][13][14][15][16]. First numerical studies have been discussed in [17].…”

Introducing Loopedia

“…2a. [1,2], [1,4], [2,3], [3,4], [4,6], [3,5], [2,7], [6, −1], [6,5], [5,7], [ This index depends highly on the way we number the vertices. The crucial part is that if we write down all possible indices (for all numberings) according to the presented rules, there will be exactly one of them which is minimal in a specific sense.…”

“…For one-loop integrals there are already public tools and repositories such as: LoopTools [1], Golem95 [2], PJFry [3,4], QCDloop [5], ONELOOP [6], and Hepforge [7]. For two-loop and multi-loop integrals, however, there is no one-stop repository yet.…”