IBP reduction coefficients made simple

Abstract

We present an efficient method to shorten the analytic integration-by-parts (IBP) reduction coefficients of multi-loop Feynman integrals. For our approach, we develop an improved version of Leinartas’ multivariate partial fraction algorithm, and provide a modern implementation based on the computer algebra system Singular. Furthermore, we observe that for an integral basis with uniform transcendental (UT) weights, the denominators of IBP reduction coefficients with respect to the UT basis are either symbol letters or polynomials purely in the spacetime dimension D. With a UT basis, the partial fraction algorithm is more efficient both with respect to its performance and the size reduction. We show that in complicated examples with existence of a UT basis, the IBP reduction coefficients size can be reduced by a factor of as large as ∼ 100. We observe that our algorithm also works well for settings without a UT basis.

Two-loop leading colour QCD corrections to $$ q\overline{q} $$ → γγg and qg → γγq

“…In contrast, the decomposition involving the kinematic denominators is computationally non-trivial. We For them, we employ the MultivariateApart package [69] to perform a multivariate partial fraction decomposition; see also [70][71][72] for related decomposition techniques. The decomposition of [69] is based on replacing all irreducible denominator factors d k ({s ij }) in a given expression according to…”

“…In particular, the reduction in complexity is particularly pronounced for the most complicated rank-5 identities, for which up to a factor of 40 reduction in size is seen. While it has been known for a long time that integration-by-parts identities can become substantially simpler even using naive variants of multivariate partial fraction decompositions, we would like to point out a systematic study of the impact of these new algorithms on the size of the reduction identities which has recently appeared in [72]. Starting from these substantially simpler identities, we perform the relevant crossings and then a second partial fraction decomposition.…”

Two-loop leading-color helicity amplitudes for three-photon production at the LHC

“…Within QCD, the current frontier for 2 → 1 and 2 → 2 processes is three loops [62][63][64][65][66][67][68], while for 2 → 3 scattering processes with massless partons it is two loops. A lot of work has already been carried out in this direction, mostly for planar amplitudes [69][70][71][72][73] but recently also for non-planar ones [49,57,59,[74][75][76][77]. All this progress has enabled the very recent first calculation of a 2 → 3 process at NNLO, namely, three-photon production at the LHC.…”

Loop-tree duality from vertices and edges

“…Let us also comment that for the topology with five vertices and eight edges, we have experimented an alternative approach based on the partial fractioning through the automated codes MultivariateApart[86] and the Singular[87] library pfd.lib[88], finding 102 causal terms instead of 98.…”