Tropical Feynman integration in the Minkowski regime

Cosmology meets cohomology

“…While it is possible to integrate the above two-site/four-point integral, direct integration becomes intractable at higher points. Similar integrals often appear in the study of dimensionally regulated Feynman integrals where there is a rich literature describing their evaluation and understanding their analytic structure (see [16][17][18][38][39][40][41][42][43][44][45][46][47][48][49] for a very incomplete list). In…”

Numerical scattering amplitudes with pySecDec

“…It is a 2-loop pentabox with an internal mass, massive legs, and the total of 7 scales. The integral is evaluated in 6 − 2ε space-time dimensions (where it is finite in ε) up to O(ε 4 ); a prefactor of Γ(2 + 2ε) is divided out to match the configuration of Section 6.4 of [14], where the same integral is calculated numerically via tropical integration.…”

“…Other public programs for the numerical evaluation of multi-loop integrals based on sector decomposition within dimensional regularisation [6,7] are sector_decomposition [8] and Fiesta [9,10,11,12,13]. The program Feyntrop [14] provides a numerical approach for evaluating quasi-finite Feynman integrals using tropical sampling [15]. Other analytic/semi-analytic approaches include DiffExp [16,17], AMFlow [18] and SeaSyde [19] which calculate Feynman integrals by solving differential equations using series expansions.…”

Integration-by-parts identities and differential equations for parametrised Feynman integrals

“…More generally, for the first time in several decades we are witnessing a rapid growth of our understanding of the mathematical properties of Feynman integrals, in particular with regards to analyticity and monodromy (see, for example, [35,[47][48][49][50], and the lectures in ref. [51]), with potential applications to questions of phenomenological interest, such as the study of infrared singularities [52] and the development of efficient methods of numerical integration [53,54].…”