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The $ρ$ parameter at three loops and elliptic integrals

J. Blümlein, A. De Freitas, M. van Hoeij, E. Imamoglu, P. Marquard, C. Schneider

TL;DR

The work delivers an analytic treatment of the six non-factorizable master integrals required for the two-mass three-loop corrections to the $ ho$ parameter. It combines differential equations with hypergeometric and elliptic-function theory to obtain homogeneous solutions, and introduces iterated non-iterative integrals for the inhomogeneous part. By recasting elliptic structures in terms of $ ext{eta}$-functions and modular forms via Legendre-Jacobi transformations, the authors achieve uniform, analytic representations and boundary expansions that align with known limits. The results enable a fully analytic description of the two-mass three-loop contributions to $ ho$, offering structural insights into elliptic appearances in multi-loop Feynman integrals and providing a path toward modular-function based representations.

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 $ρ$ parameter at three loops and elliptic integrals

TL;DR

The work delivers an analytic treatment of the six non-factorizable master integrals required for the two-mass three-loop corrections to the parameter. It combines differential equations with hypergeometric and elliptic-function theory to obtain homogeneous solutions, and introduces iterated non-iterative integrals for the inhomogeneous part. By recasting elliptic structures in terms of -functions and modular forms via Legendre-Jacobi transformations, the authors achieve uniform, analytic representations and boundary expansions that align with known limits. The results enable a fully analytic description of the two-mass three-loop contributions to , offering structural insights into elliptic appearances in multi-loop Feynman integrals and providing a path toward modular-function based representations.

Abstract

We describe the analytic calculation of the master integrals required to compute the two-mass three-loop corrections to the 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.

Paper Structure

This paper contains 4 sections, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The differential equations
  3. Solutions
  4. $\eta$-ratios and $\mathbf{q}$-series representations

Figures (2)

  • Figure 1: The six master integrals required for the calculation of the two-mass three-loop contributions to the $\rho$ parameter which obey differential equations that are not factorizable to first order. Dashed lines represent massless propagators, while solid lines represent massive propagators (thin lines for the smaller mass and thick lines for the larger one). A dot on a line represents a squared propagator.
  • Figure 2: The two-mass contributions to $\delta^{(2)}$ as a function of $x$.