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)

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.