The electron self-energy in QED at two loops revisited

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 $p^2/m^2$.

45 pages, v2: version to be published, v3: eq.(40) and (41) corrected

References in corpus (26)

• Lectures on differential equations for Feynman integrals
• Feynman Diagrams and Differential Equations
• The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms
• Elliptic polylogarithms and iterated integrals on elliptic curves I: general formalism
• Two-loop planar master integrals for Higgs$\to 3$ partons with full heavy-quark mass dependence
• Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral
• The Elliptic Double-Box Integral: Massless Amplitudes Beyond Polylogarithms
• Elliptic Feynman integrals and pure functions
• The kite integral to all orders in terms of elliptic polylogarithms
• The $\epsilon$-form of the differential equations for Feynman integrals in the elliptic case
• A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
• Five-loop quark mass and field anomalous dimensions for a general gauge group
• Five-loop renormalisation of QCD in covariant gauges
• Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series
• An Elliptic Generalization of Multiple Polylogarithms
• Analytic results for the planar double box integral relevant to top-pair production with a closed top loop
• The planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularisation parameter
• Solving differential equations for Feynman integrals by expansions near singular points
• Automated Solution of First Order Factorizable Systems of Differential Equations in One Variable
• Rationalizing Loop Integration
• BOKASUN: a fast and precise numerical program to calculate the Master Integrals of the two-loop sunrise diagrams
• Elliptic Functions and Maximal Unitarity
• Evaluating `elliptic' master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points
• Four-loop wave function renormalization in QCD and QED
• Differential equations for loop integrals in Baikov representation
• Rationalizing roots: an algorithmic approach

Cited by in corpus (20)

• The unequal mass sunrise integral expressed through iterated integrals on ${\overline{\mathcal{M}}}_{1,3}$
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• RationalizeRoots: Software Package for the Rationalization of Square Roots
• Numerical evaluation of iterated integrals related to elliptic Feynman integrals
• Bananas of equal mass: any loop, any order in the dimensional regularisation parameter
• The three-loop equal-mass banana integral in $\epsilon$-factorised form with meromorphic modular forms
• The elliptic double box and symbology beyond polylogarithms
• Symbology for elliptic multiple polylogarithms and the symbol prime
• Canonical Differential Equations Beyond Genus One
• Three-loop contributions to the $\rho$ parameter and iterated integrals of modular forms
• Modular transformations of elliptic Feynman integrals
• Bootstrapping elliptic Feynman integrals using Schubert analysis
• Cutting the traintracks: Cauchy, Schubert and Calabi-Yau
• On $\epsilon$-factorised bases and pure Feynman integrals
• Gravitational corrections to two-loop beta function in quantum electrodynamics
• Master integrals for $\mathcal{O}(\alpha {\alpha }_s)$ corrections to $H\to Z{Z}^{\ast }$
• Electroweak double-box integrals for Moller scattering
• Sunrise integral with two internal masses and pseudo-threshold kinematics in terms of elliptic polylogarithms
• A Feynman integral depending on two elliptic curves
• Two-loop master integrals for pseudo-scalar quarkonium and leptonium production and decay