Four-loop corrections with two closed fermion loops to fermion self energies and the lepton anomalous magnetic moment

TL;DR

This work advances the analytic evaluation of four-loop on-shell integrals by targeting corrections with two or three closed massless fermion loops to the muon anomalous magnetic moment $a_\mu$ and to QCD on-shell renormalization constants $Z_m^{\rm OS}$ and $Z_2^{\rm OS}$. It develops a comprehensive on-shell four-loop framework, identifying five integral families and reducing them to 13 master integrals, with the non-trivial masters computed via the Dimensional Recurrence and Analyticity method and cross-validated by numerical tools such as FIESTA and Mellin-Barnes techniques. The paper provides explicit analytic expressions for the $n_l^2$ and $n_l^3$ contributions to $Z_m^{\rm OS}$ and $Z_2^{\rm OS}$ in QCD, and presents the corresponding $n_l^2$ and $n_l^3$ results for $a_\mu$ in QED, including detailed numerical values for the dominant terms $a_\mu^{(43)}$, $a_\mu^{(42)a}$, and $a_\mu^{(42)b}$. These results, together with the described suite of computational tools and cross-checks, establish a path toward complete four-loop on-shell calculations and set the stage for extending the analysis to $n_l^1$ and non-fermionic contributions; Appendix A and B provide the analytic master integrals and the relation between schemes respectively.

Abstract

We compute the eighth-order fermionic corrections involving two and three closed massless fermion loops to the anomalous magnetic moment of the muon. The required four-loop on-shell integrals are classified and explicit analytical results for the master integrals are presented. As further applications we compute the corresponding four-loop QCD corrections to the mass and wave function renormalization constants for a massive quark in the on-shell scheme.

Figures (4)

• Figure 1: Sample Feyman diagrams for the photon-muon vertex contributing to $a_\mu$. Wavy and straight lines represent photons and fermions, respectively. In this paper we consider the contribution where at least two of the closed loops correspond to massless fermions. The last diagram in the second line is a representative of the so-called "light-by-light" contribution.
• Figure 2: Sample Feynman diagrams for the QCD corrections to the fermion propagator contributing to $Z_m^{\rm OS}$ and $Z_2^{\rm OS}$. Curly and straight lines represent gluons and fermions, respectively. In this paper we consider the contribution where at least two of the closed loops correspond to massless fermions.
• Figure 3: Master integrals for the $n_l^2$ and $n_l^3$ contribution which are easily obtained by applying one- and two-loop formulae, see e.g., Ref. Smirnov:2013. Solid lines carry the mass $M$ and dashed lines are massless. For $L_1$ to $L_6$ we have $q^2=M^2$ where $q$ is the external momentum; $L_7$ is a vacuum integral.
• Figure 4: Non-trivial master integrals contributing to the $n_l^2$ contribution. The same notation as in Fig. \ref{['fig::MI1']} has been used.