New features of the Mathematica code FIRE are presented. In particular, it can be applied together with the recently developed code LiteRed by Lee in order to provide an integration by parts reduction to master integrals for quite complicated familie
s of Feynman integrals. As as an example, we consider four-loop massless propagator integrals for which LiteRed provides reduction rules and FIRE assists to apply these rules. So, as a by-product one obtains a four-loop variant of the well-known three-loop computer code MINCER. We also describe various ways to find additional relations between master integrals for several families of Feynman integrals.
We evaluate a Laurent expansion in dimensional regularization parameter $epsilon=(4-d)/2$ of all the master integrals for four-loop massless propagators up to transcendentality weight twelve, using a recently developed method of one of the present co
authors (R.L.) and extending thereby results by Baikov and Chetyrkin obtained at transcendentality weight seven. We observe only multiple zeta values in our results. Therefore, we conclude that all the four-loop massless propagator integrals, with any integer powers of numerators and propagators, have only multiple zeta values in their epsilon expansions up to transcendentality weight twelve.
We evaluate three typical four-loop non-planar massless propagator diagrams in a Taylor expansion in dimensional regularization parameter $epsilon=(4-d)/2$ up to transcendentality weight twelve, using a recently developed method of one of the present
coauthors (R.L.). We observe only multiple zeta values in our results.
We evaluate analytically higher terms of the epsilon-expansion of the three-loop master integrals corresponding to three-loop quark and gluon form factors and to the three-loop master integrals contributing to the electron g-2 in QED up to the transc
endentality weight typical to four-loop calculations, i.e. eight and seven, respectively. The calculation is based on a combination of a method recently suggested by one of the authors (R.L.) with other techniques: sector decomposition implemented in FIESTA, the method of Mellin--Barnes representation, and the PSLQ algorithm.
We evaluate, exactly in d, the master integrals contributing to massless three-loop QCD form factors. The calculation is based on a combination of a method recently suggested by one of the authors (R.L.) with other techniques: sector decomposition im
plemented in FIESTA, the method of Mellin--Barnes representation, and the PSLQ algorithm. Using our results for the master integrals we obtain analytical expressions for two missing constants in the ep-expansion of the two most complicated master integrals and present the form factors in a completely analytic form.
The program FIESTA has been completely rewritten. Now it can be used not only as a tool to evaluate Feynman integrals numerically, but also to expand Feynman integrals automatically in limits of momenta and masses with the use of sector decomposition
s and Mellin-Barnes representations. Other important improvements to the code are complete parallelization (even to multiple computers), high-precision arithmetics (allowing to calculate integrals which were undoable before), new integrators and Speer sectors as a strategy, the possibility to evaluate more general parametric integrals.
One of the two existing strategies of resolving singularities of multifold Mellin-Barnes integrals in the dimensional regularization parameter, or a parameter of the analytic regularization, is formulated in a modified form. The corresponding algorit
hm is implemented as a Mathematica code MBresolve.m
