NumExp: Numerical epsilon expansion of hypergeometric functions

Abstract

It is demonstrated that the well-regularized hypergeometric functions can be evaluated directly and numerically. The package NumExp is presented for expanding hypergeometric functions and/or other transcendental functions in a small regularization parameter. The hypergeometric function is expressed as a Laurent series in the regularization parameter and the coefficients are evaluated numerically by using the multi-precision finite difference method. This elaborate expansion method works for a wide variety of hypergeometric functions, which are needed in the context of dimensional regularization for loop integrals. The divergent and finite parts can be extracted from the final result easily and simultaneously. In addition, there is almost no restriction on the parameters of hypergeometric functions.

Resummed photon spectrum from dark matter annihilation for intermediate and narrow energy resolution

“…The factorization formula for the intermediate resolution case given in (40) and (41) puts constraints on the anomalous dimensions, since the physical photon energy spectrum has to be independent of the virtuality and rapidity factorization scales µ and ν. This independence on the scales manifests itself in the two consistency equations…”

“…It is instructive to separate the integrated photon energy spectrum σv (E γ res ) into the contributions due to the different Sommerfeld factors in (40). Thus, we write…”

“…and with the help of relations between polylogarithms of different arguments in intermediate steps. We used the package NumExp [40] for the numerical expansion of the Appell F 1 function in (199) to check (202). In total we find…”

HYPERgeometric functions DIfferential REduction (HYPERDIRE): MATHEMATICA based packages for differential reduction of generalized hypergeometric functions: FD and Kalmykov1, Moch2 2014Computer Physics Communications350180View full textAdd to dashboardCite

“…When for some values of i and k both coefficients, defined by Eqs. (34) and (35) are zero, a further simplification can be performed, so that the rank of system is reduced to six or to an even smaller number.…”

“…Purely numerical approaches [34] can be applied to arbitrary values of the parameters. However this technique typically does not produce stable numerical result around the region of singularities of hypergeometric function.…”