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Analytic results for two-loop planar master integrals for Bhabha scattering

Claude Duhr, Vladimir A. Smirnov, Lorenzo Tancredi

TL;DR

This work completes the analytic evaluation of the planar master integrals for two-loop QED Bhabha scattering with full electron-mass dependence by employing differential equations in a canonical basis. It demonstrates that all integrals in the second planar family admit representations in terms of multiple polylogarithms except for one, which requires elliptic generalisations, and it provides a compact elliptic MPL expression for a master integral in the first planar family as well. The authors develop and deploy a canonical basis with a differential-system structure $df = oldsymbol{} dA f$, where four square roots enter the alphabet, enabling systematic analytic (MPL and eMPL) representations and robust numerical evaluation via DiffExp. A key outcome is the explicit, compact eMPL expression for the difficult integral $f_{14}$ and a parallel compact eMPL form for the first planar family integral $B(s,t,m^2)$, illustrating the power of elliptic generalisations in two-loop massive Bhabha scattering. The results pave the way toward complete two-loop QED predictions for a standard candle process and showcase a framework that can be extended to non-planar topologies.

Abstract

We analytically evaluate the master integrals for the second type of planar contributions to the massive two-loop Bhabha scattering in QED using differential equa- tions with canonical bases. We obtain results in terms of multiple polylogarithms for all the master integrals but one, for which we derive a compact result in terms of elliptic mul- tiple polylogarithms. As a byproduct, we also provide a compact analytic result in terms of elliptic multiple polylogarithms for an integral belonging to the first family of planar Bhabha integrals, whose computation in terms of polylogarithms was addressed previously in the literature.

Analytic results for two-loop planar master integrals for Bhabha scattering

TL;DR

This work completes the analytic evaluation of the planar master integrals for two-loop QED Bhabha scattering with full electron-mass dependence by employing differential equations in a canonical basis. It demonstrates that all integrals in the second planar family admit representations in terms of multiple polylogarithms except for one, which requires elliptic generalisations, and it provides a compact elliptic MPL expression for a master integral in the first planar family as well. The authors develop and deploy a canonical basis with a differential-system structure , where four square roots enter the alphabet, enabling systematic analytic (MPL and eMPL) representations and robust numerical evaluation via DiffExp. A key outcome is the explicit, compact eMPL expression for the difficult integral and a parallel compact eMPL form for the first planar family integral , illustrating the power of elliptic generalisations in two-loop massive Bhabha scattering. The results pave the way toward complete two-loop QED predictions for a standard candle process and showcase a framework that can be extended to non-planar topologies.

Abstract

We analytically evaluate the master integrals for the second type of planar contributions to the massive two-loop Bhabha scattering in QED using differential equa- tions with canonical bases. We obtain results in terms of multiple polylogarithms for all the master integrals but one, for which we derive a compact result in terms of elliptic mul- tiple polylogarithms. As a byproduct, we also provide a compact analytic result in terms of elliptic multiple polylogarithms for an integral belonging to the first family of planar Bhabha integrals, whose computation in terms of polylogarithms was addressed previously in the literature.

Paper Structure

This paper contains 12 sections, 60 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Canonical differential equations
  3. From $d\log$-forms to multiple polylogarithms
  4. Rational alphabets without square roots
  5. Alphabets with a single elliptic square root
  6. Integration of the differential equations in terms of MPLs
  7. Compact analytic results in terms of eMPLs
  8. An analytic result for $f_{14}$ in terms of eMPLs
  9. A compact analytic result for the first planar family
  10. Conclusions
  11. Canonical basis
  12. Elliptic multiple polylogarithms

Figures (3)

  • Figure 1: Planar graphs of the first and the second type for two-loop Bhabha scattering. Solid (dashed) lines indicate massive (massless) propagators. All the external momenta are incoming.
  • Figure 2: The graph associated with the master integral which we evaluate in terms of elliptic polylogarithms.
  • Figure 3: The graph associated with the integral $B(s,t,m^2)$ in eq. \ref{['eq:B_def']}.