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

TL;DR

This paper derives relations between spacelike and timelike kernel functions for the hadronic vacuum polarization contribution to the muon g-2, providing explicit and numerical results useful for lattice and experimental assessments.

Contribution

It introduces explicit relations between spacelike and timelike kernel functions and computes the next-to-leading order spacelike kernel functions for muon g-2 evaluations.

Findings

Derived relations between spacelike and timelike kernels.

Explicit expression for the next-to-leading order spacelike kernel.

Numerical calculation of the spacelike kernel involving the Adler function.

Abstract

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.