Non-planar elliptic vertex

TL;DR

This paper explores advanced analytical techniques for calculating higher-order elliptic master integrals, introducing power series solutions and exact expressions in special functions, enhancing precision in quantum field theory computations.

Contribution

It demonstrates the use of power series solutions and exact hypergeometric function representations for non-planar elliptic vertex integrals, extending existing methods.

Findings

Power series solutions for differential equations of elliptic integrals.

Exact results expressed in hypergeometric and Kampé de Fériet functions.

Enhanced analytical tools for higher-order epsilon calculations.

Abstract

We consider the problem of obtaining higher order in regularization parameter $\epsilon$ analytical results for master integrals with elliptics. The two commonly employed methods are provided by the use of differential equations and direct integration of parametric representations in terms of iterated integrals. Taking non-planar elliptic vertex as an example we show that in addition to two mentioned methods one can use analytical solution of differential equations in terms of power series. Moreover, in the last case it is possible to obtain the exact in $\epsilon$ results expressible either in terms of generalized hypergeometric or Kamp\'e de F\'eriet functions