Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space

Abstract

We introduce an algebro-geometrically motived integration-by-parts (IBP) reduction method for multi-loop and multi-scale Feynman integrals, using a framework for massively parallel computations in computer algebra. This framework combines the computer algebra system Singular with the workflow management system GPI-Space, which are being developed at the TU Kaiserslautern and the Fraunhofer Institute for Industrial Mathematics (ITWM), respectively. In our approach, the IBP relations are first trimmed by modern tools from computational algebraic geometry and then solved by sparse linear algebra and our new interpolation method. Modelled in terms of Petri nets, these steps are efficiently automatized and automatically parallelized by GPI-Space. We demonstrate the potential of our method at the nontrivial example of reducing two-loop five-point nonplanar double-pentagon integrals. We also use GPI-Space to convert the basis of IBP reductions, and discuss the possible simplification of IBP coefficients in a uniformly transcendental basis.

IBP reduction coefficients made simple

“…Relying on the module intersection IBP reduction method and its implementation in the Singular-GPI-Space framework for massively parallel computations, the analytic IBP reduction coefficients were calculated for the integrals with ISP up to the degree 4 in the sector (1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 0) in ref. [33]. 2 The size of the IBP reduction coefficients with respect to a Laporta basis is 2.4GB (with all parameters analytic).…”

“…They are listed in the following: As we have observed in ref. [33], if we reduce the target integrals to the UT basis, the size of coefficients is reduced to 712MB. More importantly, the IBP reduction coefficients with respect to UT basis have no "mixed" poles and all kinematic denominators are symbol letters.…”

“…Reduze and LiteRed [5][6][7][8][9][10][11][12][13][14][15][16], based on the Laporta algorithm [17] and/or the algebra structures of IBP relations [18][19][20]. In recent years, many new ideas and programs have appeared for use in the computation of complicated multi-loop IBP reductions, for example, syzygy approach [21][22][23][24][25][26], finite-field interpolation [27][28][29][30][31], module intersection [32,33], intersection theory [34][35][36][37], η expansion [38][39][40][41][42] and direct solution of IBP recursive relations [43].…”

“…In ref. [33], the master integral basis with uniform transcendental (UT) weights [46,47] was suggested to shorten the size of IBP reduction coefficients.…”

“…In particular, as mentioned in ref. [33], we suggest that when a UT master integral basis for the integral family under consideration exists, it is advantageous to first reduce Feynman integrals to JHEP12(2020)054 the UT basis, and then run our partial fraction algorithm to shorten the size of the IBP coefficients. The reason is that, in the examples we have tested, for Feynman integrals…”