The $ρ$ parameter at three loops and elliptic integrals

Abstract

We describe the analytic calculation of the master integrals required to compute the two-mass three-loop corrections to the ρ parameter. In particular, we present the calculation of the master integrals for which the corresponding differential equations do not factorize to first order. The homogeneous solutions to these differential equations are obtained in terms of hypergeometric functions at rational argument. These hypergeometric functions can further be mapped to complete elliptic integrals, and the inhomogeneous solutions are expressed in terms of a new class of integrals of combined iterative non-iterative nature.

The SAGEX review on scattering amplitudes Chapter 4: Multi-loop Feynman integrals

“…For complete physical processes more recently elliptic integrals were needed. This is the case in massive three-loop calculations for the QCD corrections of the ρ parameter [98,99] 2017 and in massless three-three loop calculations 2018 and later [235,236] 9 . There is a series of well-known examples of individual integrals of a certain structure in the literature as the sun-rise integral, cf e.g.…”

“…For second order linear differential equations van Hoeij proposed algorithms in [97] that can find hypergeometric series solutions ( p F q 's) in terms of certain rational function arguments. These advanced tools turned out to be instrumental to deal with the ρ-parameter in [98,99] and related quantities.…”

“…The same differential equations rule the non first-order factorizing cases in the calculation of the massive three-loop operator matrix element A Qg in the single mass case[234], found together with the calculation[99] 10. Very naturally, as now the technical aspects on complete elliptic integrals are very well known in particle physics, many applications find these contributions, cf e.g [243][244][245],.…”

The inverse Mellin transform via analytic continuation

“…The solutions in terms of complete elliptic integrals have been applied in the first analytic calculation of the three-loop ρ-parameter of the Standard Model [181], which had been calculated semi-analytically in [182] before. 14 The emergence of the 2 F 1 -solutions in the present context is related to contributions of the so-called two-loop massive sun-rise integral, related also to the kite-integral, on which a very extensive literature exists.…”

Large scale analytic calculations in quantum field theories

“…e.g. [151][152][153][154][155][156][157], the kite integral [158][159][160], the three-loop QCDcorrections to the ρ-parameter [161][162][163], and the three-loop QCD corrections to the massive operator matrix element A Qg [114]. After separating the first-order factorizing factors a Heun differential equation [164] remains in the case of the ρ-parameter.…”