Timelike and spacelike kernel functions for the hadronic vacuum polarization contribution to the muon anomalous magnetic moment

The complete set of relations, which mutually express the spacelike and timelike kernel functions for the hadronic vacuum polarization contribution to the muon anomalous magnetic moment $a^{\text{HVP}}_{\mu}$ in terms of each other, is obtained. By making use of the derived relations the explicit expression for the next-to-leading order spacelike kernel function, which enters the representation for $a^{\text{HVP}}_{\mu}$ involving the hadronic vacuum polarization function, is obtained. The corresponding next-to-leading order spacelike kernel function, which appears in the representation for $a^{\text{HVP}}_{\mu}$ involving the Adler function, is calculated numerically. The obtained results can be employed in the assessments of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment in the framework of the spacelike methods, such as lattice studies, MUonE project, and others.