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Elliptic polylogarithms and Feynman parameter integrals

Johannes Broedel, Claude Duhr, Falko Dulat, Brenda Penante, Lorenzo Tancredi

TL;DR

This work extends the direct-integration approach for Feynman integrals beyond multiple polylogarithms by employing elliptic polylogarithms (eMPLs) to represent two-loop integrals with elliptic curves. By carefully choosing Feynman-parameter orderings and constructing pure bases, the authors express two-loop non-planar triangles, an electroweak form factor, and the kite integral with three masses in terms of pure eMPLs with uniform weight. The results provide concrete evidence that a basis of pure elliptic master integrals exists for these cases and suggest a generalizable framework for higher-loop or higher-point elliptic integrals. This advances both practical computations in collider phenomenology and the theoretical understanding of transcendental structures in perturbative QFT.

Abstract

In this paper we study the calculation of multiloop Feynman integrals that cannot be expressed in terms of multiple polylogarithms. We show in detail how certain types of two- and three-point functions at two loops, which appear in the calculation of higher order corrections in QED, QCD and in the electroweak theory (EW), can naturally be expressed in terms of a recently introduced elliptic generalisation of multiple polylogarithms by direct integration over their Feynman parameter representation. Moreover, we show that in all examples that we considered a basis of pure Feynman integrals can be found.

Elliptic polylogarithms and Feynman parameter integrals

TL;DR

This work extends the direct-integration approach for Feynman integrals beyond multiple polylogarithms by employing elliptic polylogarithms (eMPLs) to represent two-loop integrals with elliptic curves. By carefully choosing Feynman-parameter orderings and constructing pure bases, the authors express two-loop non-planar triangles, an electroweak form factor, and the kite integral with three masses in terms of pure eMPLs with uniform weight. The results provide concrete evidence that a basis of pure elliptic master integrals exists for these cases and suggest a generalizable framework for higher-loop or higher-point elliptic integrals. This advances both practical computations in collider phenomenology and the theoretical understanding of transcendental structures in perturbative QFT.

Abstract

In this paper we study the calculation of multiloop Feynman integrals that cannot be expressed in terms of multiple polylogarithms. We show in detail how certain types of two- and three-point functions at two loops, which appear in the calculation of higher order corrections in QED, QCD and in the electroweak theory (EW), can naturally be expressed in terms of a recently introduced elliptic generalisation of multiple polylogarithms by direct integration over their Feynman parameter representation. Moreover, we show that in all examples that we considered a basis of pure Feynman integrals can be found.

Paper Structure

This paper contains 14 sections, 113 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Review of Elliptic Polylogarithms
  3. A non-planar triangle with a massive loop
  4. First master integral
  5. Second master integral
  6. Summary
  7. Summary
  8. Electroweak form factor
  9. Kite with three distinct masses
  10. Conclusions
  11. An example of analytic continuation
  12. Length 1 example
  13. Length 2 example
  14. Result for top-production first master in the Euclidean region

Figures (5)

  • Figure 1: Three possible configurations of the branch points in the complex plane such that $y^2 = P_4(x) \in \mathbb{R}$. $(i)$ All branch points are real and ordered. $(ii)$ Two branch points are real and two branch points are complex conjugate to each other. $(iii)$ All branch points are complex and pairwise complex conjugate to each other.
  • Figure 2: Triangle with massive loop.
  • Figure 3: Location of the ranch points of the final expression for the top-production triangle integral \ref{['eq:integral_final_v1']} in terms of eMPLs $\text{E}_4$ in the region $0<a<1/16$.
  • Figure 4: Triangle with massive loop.
  • Figure 5: Kite integral with three internal massive propagators with masses $m_1$, $m_2$ and $m_3$.