The Mellin-Barnes Approach to Hadronic Vacuum Polarization and $g_{\mu}-2$

TL;DR

This paper introduces a Mellin-Barnes approximant method to accurately compute the hadronic vacuum polarization contribution to muon g-2 using derivatives at zero momentum, matching traditional experimental results.

Contribution

It presents a novel Mellin-Barnes approximant technique for evaluating the hadronic vacuum polarization contribution to muon g-2 from limited derivatives data.

Findings

The MBa method converges rapidly to experimental values.

Accurate g-2 estimates can be obtained from minimal derivatives data.

The approach is validated with QED and experimental spectral functions.

Abstract

It is shown that with a precise determination of a few derivatives of the hadronic vacuum polarization (HVP) self-energy function $\Pi(Q^2)$ at $Q^2=0$, from lattice QCD (LQCD) or from a dedicated low-energy experiment, one can obtain an evaluation of the lowest order HVP contribution to the anomalous magnetic moment of the muon $a_{\mu}^{\rm HVP}$ with an accuracy comparable to the one reached using the $e^+ e^-$ annihilation cross-section into hadrons. The technique of Mellin-Barnes approximants (MBa) that we propose is illustrated in detail with the example of the two loop vacuum polarization function in QED. We then apply it to the first few moments of the hadronic spectral function obtained from experiment and show that the resulting MBa evaluations of $a_{\mu}^{\rm HVP}$ converge very quickly to the full experimental determination.