Window quantities for the hadronic vacuum polarization contributions to the muon anomalous magnetic moment in spacelike and timelike domains

TL;DR

This paper explores the relationships between different window quantities for the hadronic vacuum polarization contributions to the muon g-2 in spacelike and timelike domains, providing formulas to accurately compare and evaluate these contributions using various data inputs.

Contribution

It derives explicit expressions for window edge effects, enabling consistent evaluation of muon g-2 contributions from diverse data sources and domains.

Findings

Mutual equivalence of window quantities requires accounting for edge effects.

Derived formulas allow hybrid evaluation using multiple functions and data types.

Facilitates comparison of different experimental and theoretical approaches.

Abstract

The relations between the window quantities for the hadronic vacuum polarization contributions to the muon anomalous magnetic moment $a^{\text{HVP}}_{\mu}$ in spacelike and timelike domains are studied. Two types of window functions (abrupt and smooth) as well as two kinds of kinematic intervals (symmetric and asymmetric with respect to the spacelike/timelike flip) are addressed. It is shown that the window quantities for $a^{\text{HVP}}_{\mu}$ represented in terms of the hadronic vacuum polarization function $\bar\Pi(Q^2)$, the Adler function $D(Q^2)$, and the $R$-ratio of electron-positron annihilation into hadrons are mutually equivalent only if the additional contributions due to the window edge effects are properly taken into account and the explicit expressions for such contributions are derived. The obtained results enable one to evaluate $a^{\text{HVP}}_{\mu}$ by making simultaneous use of the inputs for functions $\bar\Pi(Q^2)$, $D(Q^2)$, and $R(s)$ at various energies and an example of such hybrid assessment is provided. The obtained results also enable one to accurately compare the window quantities for $a^{\text{HVP}}_{\mu}$ based, e.g., on MUonE or lattice data with the ones based on $R$-ratio data, even if the window function covers different kinematic ranges in spacelike and timelike domains.