Three-loop contributions to the $\rho$ parameter and iterated integrals of modular forms

Abstract

We compute fully analytic results for the three-loop diagrams involving two different massive quark flavours contributing to the $\rho$ 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.

37 pages, 2 figures

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• Collider Physics at the Precision Frontier
• Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives
• Analytic Structure of all Loop Banana Amplitudes
• The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods
• The SAGEX Review on Scattering Amplitudes, Chapter 3: Mathematical structures in Feynman integrals
• Meromorphic modular forms and the three-loop equal-mass banana integral
• Two-loop leading colour QCD helicity amplitudes for top quark pair production in the gluon fusion channel
• Linear power corrections to $e^{+}{e}^{-}$ shape variables in the three-jet region
• Canonical Differential Equations Beyond Genus One
• The inverse Mellin transform via analytic continuation
• Classification of Feynman integral geometries for black-hole scattering at 5PM order
• The analytic leading color contribution to the Higgs-gluon form factor in QCD at NNLO
• Electroweak double-box integrals for Moller scattering
• Two-loop amplitude for mixed QCD-EW corrections to $gg\to Hg$
• The spectrum of Feynman-integral geometries at two loops
• Two-loop master integrals for pseudo-scalar quarkonium and leptonium production and decay
• Integral representation for three-loop banana graph
• A Feynman integral depending on two elliptic curves