Addendum: Timelike and spacelike kernel functions for the hadronic vacuum polarization contribution to the muon anomalous magnetic moment (2022 J. Phys. G: Nucl. Part. Phys. 49 055001)

TL;DR

This addendum refines the mathematical relations between spacelike and timelike kernel functions for the hadronic vacuum polarization contribution to the muon g-2, providing new insights into their infrared behavior and specific contributions from leptonic and hadronic loops.

Contribution

It introduces an equivalent form of the relation linking spacelike and timelike kernel functions and analyzes the infrared limits and higher-order contributions in muon g-2 calculations.

Findings

Infrared limits of kernel functions are identical for spacelike and timelike cases.

New representations for next-to-leading order contributions are provided.

Kernel functions for electron and tau loops are explicitly presented.

Abstract

This addendum provides results complementary to those obtained in [J. Phys. G49, 055001 (2022)]. Specifically, an equivalent form of the relation, which binds together the "spacelike" kernel functions for the hadronic vacuum polarization contribution to the muon anomalous magnetic moment $a^{\text{HVP}}_{\mu}$, is obtained. It is shown that the infrared limiting value of the "spacelike" and "timelike" kernel functions, which enter the representations for $a^{\text{HVP}}_{\mu}$ involving the Adler function and the $R$-ratio, is identical to the corresponding QED contribution to the muon anomalous magnetic moment of the preceding order in the electromagnetic coupling. The next-to-leading order contributions $a^{\text{HVP}(3b)}_{\mu}$ (which includes the leptonic and hadronic insertions) and $a^{\text{HVP}(3c)}_{\mu}$ (which includes the double hadronic insertion), are studied. The three kernel functions appearing in the representations for $a^{\text{HVP}(3b)}_{\mu}$, which involve the hadronic vacuum polarization function, Adler function, and the $R$-ratio, are presented for the cases of the electron and $\tau$-lepton loop insertions.