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

Samuel Abreu, Matteo Becchetti, Claude Duhr, Robin Marzucca

TL;DR

This paper provides fully analytic three‑loop results for the SM $\rho$ parameter with two massive quark flavors in terms of elliptic polylogarithms and iterated Eisenstein integrals. By exploiting a differential equation structure identical to the sunrise graph, all elliptic master integrals are shown to belong to the same function class, enabling analytic continuation across all kinematic regions and fast, convergent numerical evaluations. The authors construct explicit representations via elliptic‑function periods and iterated Eisenstein integrals for Topology A and extend them to Topology B through a mass‑ratio transformation, delivering a compact expression for the elliptic part of $\delta^{(2)}(t)$ and a precise numerical value at the physical point. The work reinforces that elliptic Feynman integrals at high loops can be handled with established modular‑form machinery, with implications for other precision observables in the electroweak sector.

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.

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

TL;DR

This paper provides fully analytic three‑loop results for the SM parameter with two massive quark flavors in terms of elliptic polylogarithms and iterated Eisenstein integrals. By exploiting a differential equation structure identical to the sunrise graph, all elliptic master integrals are shown to belong to the same function class, enabling analytic continuation across all kinematic regions and fast, convergent numerical evaluations. The authors construct explicit representations via elliptic‑function periods and iterated Eisenstein integrals for Topology A and extend them to Topology B through a mass‑ratio transformation, delivering a compact expression for the elliptic part of and a precise numerical value at the physical point. The work reinforces that elliptic Feynman integrals at high loops can be handled with established modular‑form machinery, with implications for other precision observables in the electroweak sector.

Abstract

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

Paper Structure

This paper contains 17 sections, 129 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Notations and computational setting
  3. eMPLs on the torus and on the elliptic curve
  4. Elliptic curves and elliptic functions
  5. Elliptic multiple polylogarithms
  6. Evaluating $f^{(2)}_8$ in terms of eMPLs
  7. eMPLs and iterated Eisenstein integrals
  8. Modular forms: a brief introduction
  9. Iterated Eisenstein integrals
  10. The master integrals for Topology $A$ in the region $0<t<1$
  11. Analytic continuation and numerical evaluation
  12. Analytic continuation
  13. Numerical evaluation of iterated integrals of modular forms
  14. Three-loop contributions to the $\rho$ parameter
  15. Conclusion
  16. ...and 2 more sections

Figures (2)

  • Figure 1: Master integrals for the family of integrals defined in eq. \ref{['Topo1']}. Dots denote squared propagators, blue (thick) lines denote propagators with mass $m_1^2$, green (thin) lines denote propagators with mass $m_2^2$, and dashed lines denote massless propagators.
  • Figure 2: Three-loop corrections to the $\rho$ parameter, for $N_c=3$, $C_F=4/3$ and $n_l=4$ massless quarks, as a function of $t=m_2^2/m_1^2$ which we vary between 0 and 1.