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The electron self-energy in QED at two loops revisited

Ina Hönemann, Kirsten Tempest, Stefan Weinzierl

TL;DR

This work revisits the two-loop electron self-energy in QED, addressing elliptic integrals that arise beyond polylogarithms. The authors express all relevant master integrals as iterated integrals of modular forms and evaluate them numerically via convergent $q$-series around four cusps, enabling precise results for all real $p^2/m^2$. They demonstrate that truncating the $q$-series at ${\mathcal{O}}(q^{30})$ yields a relative precision better than $10^{-20}$ for the finite part, offering analytic and computational advantages over purely numerical approaches. The methodology—combining $\,\varepsilon$-form differential equations, modular-forms iterated integrals, and multi-cusp $q$-expansions—promises broad applicability to elliptic Feynman integrals in perturbative quantum field theory.

Abstract

We reconsider the two-loop electron self-energy in quantum electrodynamics. We present a modern calculation, where all relevant two-loop integrals are expressed in terms of iterated integrals of modular forms. As boundary points of the iterated integrals we consider the four cases $p^2=0$, $p^2=m^2$, $p^2=9m^2$ and $p^2=\infty$. The iterated integrals have $q$-expansions, which can be used for the numerical evaluation. We show that a truncation of the $q$-series to order ${\mathcal O}(q^{30})$ gives numerically for the finite part of the self-energy a relative precision better than $10^{-20}$ for all real values $p^2/m^2$.

The electron self-energy in QED at two loops revisited

TL;DR

This work revisits the two-loop electron self-energy in QED, addressing elliptic integrals that arise beyond polylogarithms. The authors express all relevant master integrals as iterated integrals of modular forms and evaluate them numerically via convergent -series around four cusps, enabling precise results for all real . They demonstrate that truncating the -series at yields a relative precision better than for the finite part, offering analytic and computational advantages over purely numerical approaches. The methodology—combining -form differential equations, modular-forms iterated integrals, and multi-cusp -expansions—promises broad applicability to elliptic Feynman integrals in perturbative quantum field theory.

Abstract

We reconsider the two-loop electron self-energy in quantum electrodynamics. We present a modern calculation, where all relevant two-loop integrals are expressed in terms of iterated integrals of modular forms. As boundary points of the iterated integrals we consider the four cases , , and . The iterated integrals have -expansions, which can be used for the numerical evaluation. We show that a truncation of the -series to order gives numerically for the finite part of the self-energy a relative precision better than for all real values .

Paper Structure

This paper contains 24 sections, 143 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Lagrange density and renormalisation
  3. Definitions and notation for the master integrals
  4. Iterated integrals
  5. The two-loop self-energy
  6. The bare two-loop self-energy
  7. The counterterms from renormalisation
  8. Evaluation of the master integrals
  9. Transcendental constants
  10. Integrals expressible in terms of harmonic polylogarithms
  11. The elliptic master integrals
  12. The cusp $p^2=0$
  13. The cusp $p^2=m^2$
  14. The cusp $p^2=9m^2$
  15. The cusp $p^2=\infty$
  16. ...and 9 more sections

Figures (11)

  • Figure 1: The Feynman graphs contributing to the two-loop electron self-energy.
  • Figure 2: The kite graph. This graph is equivalent to the second graph in fig. \ref{['fig_diagrams']}.
  • Figure 3: The Feynman graphs corresponding to the counterterms. In the first graph we take the ${\mathcal{O}}(\alpha^2)$-term.
  • Figure 4: The path in $q_{1,0}$-space as $x$ ranges over ${\mathbb R}$. We always have $|q_{1,0}| \le 1$ and $|q_{1,0}| = 1$ only at $x \in \{1,9,\infty\}$.
  • Figure 5: The path in $q_{6,1}$-space as $x$ ranges over ${\mathbb R}$. We always have $|q_{6,1}| \le 1$ and $|q_{6,1}| = 1$ only at $x \in \{0,9,\infty\}$.
  • ...and 6 more figures