Abstract
We compute fully analytic results for the three-loop diagrams involving two different massive quark flavours contributing to the ρ parameter in the Standard Model. We find that the results involve exactly the same class of functions that appears in the well-known sunrise and banana graphs, namely elliptic polylogarithms and iterated integrals of modular forms. Using recent developments in the understanding of these functions, we analytically continue all the iterated integrals of modular forms to all regions of the parameter space, and in each region we obtain manifestly real and fast-converging series expansions for these functions.
Highlights
Are well understood, including their analytic continuation to arbitrary kinematic regions and their efficient numerical evaluation for arbitrary complex arguments [4,5,6,7]
Since elliptic functions seem to be a feature of many two-loop diagrams with massive propagators, it is natural to expect that these functions prominently show up when performing calculations in the electroweak sector of the Standard Model (SM), where the gauge bosons and the fermions acquire a mass through the Higgs mechanism
In this paper we have presented for the first time fully analytic results in terms of elliptic multiple polylogarithms (eMPLs) and iterated Eisenstein integrals for the three-loop corrections to the ρ parameter in the SM with two massive quark flavours
Summary
We review the background to the three-loop QCD corrections to the electroweak ρ parameter with two massive quark flavours. It is convenient to transform this system into a second-order equation satisfied by f8(,2h), Dt2f8(,2h) = 0 , This differential operator is the same one as in the case of the sunrise integral with a massive external leg and three massive propagators of equal mass [13]. Before we discuss the computation of f8(2), f9(2) and f10, let us mention that the system of differential equations satisfied by f8(2) and f9(2) was analysed in refs. Our strategy for the computation of f8(2), f9(2) and f10 relies on obtaining an analytic solution for f8(2) from its Feynman parametric representation Before we discuss this in detail, we review some of the mathematical background needed to perform all the integrals analytically. We keep the review to a strict minimum, and we refer to the literature for a more detailed discussion (see, e.g., refs. [58,59,60] and references therein)
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