Vacuum polarization contribution to muon g−2 as an inverse problem

Hsiang-Nan Li,Hiroyuki Umeeda

doi:10.1103/physrevd.102.094003

Abstract

We analyze the electromagnetic current correlator at an arbitrary photon invariant mass $q^2$ by exploiting its associated dispersion relation. The dispersion relation is turned into an inverse problem, via which the involved vacuum polarization function $\Pi(q^2)$ at low $q^2$ is solved with the perturbative input of $\Pi(q^2)$ at large $q^2$. It is found that the result for $\Pi(q^2)$, including its first derivative $\Pi^\prime(q^2=0)$, agrees with those from lattice QCD, and its imaginary part accommodates the $e^+e^-$ annihilation data. The corresponding hadronic vacuum polarization contribution $a^{\rm HVP}_\mu= (641^{+65}_{-63})\times 10^{-10}$ to the muon anomalous magnetic moment $g-2$, where the uncertainty arises from the variation of the perturbative input, also agrees with those obtained in other phenomenological and theoretical approaches. We point out that our formalism is equivalent to imposing the analyticity constraint to the phenomenological approach solely relying on experimental data, and can improve the precision of the $a^{\rm HVP}_\mu$ determination in the Standard Model.

Highlights

How to resolve the discrepancy between the theoretical prediction for the muon anomalous magnetic moment aμ 1⁄4 ðgμ − 2Þ=2 in the Standard Model and its experimental data has been a long standing mission
The major uncertainty in the former arises from the vacuum polarization function Πðq2Þ defined by an electromagnetic current correlator at a photon invariant mass q2, to which various phenomenological and theoretical approaches have been attempted
In this paper we have extended a new formalism for extracting nonperturbative observables to the study of the hadronic vacuum polarization (HVP) contribution aHμ VP to the muon anomalous magnetic moment g − 2

Summary

INTRODUCTION

How to resolve the discrepancy between the theoretical prediction for the muon anomalous magnetic moment aμ 1⁄4 ðgμ − 2Þ=2 in the Standard Model and its experimental data has been a long standing mission. The measured cross section for eþe− annihilation into hadrons has been employed to determine the hadronic vacuum polarization (HVP) contribution in a dispersive approach, giving aHμ VP 1⁄4 ð693.9Æ4.0Þ×10−10 [1] [see aHμ VP 1⁄4 ð692.78 Æ 2.42Þ × 10−10 in [2] ] This value, consistent with earlier similar observations [2,3,4,5], corresponds to a 3.3σ deviation between the Standard Model prediction for aμ and the data [6], aeμxp − aSμM 1⁄4 ð26.1 Æ 7.9Þ × 10−10. We evaluate the HVP contribution to the muon anomalous magnetic moment numerically in Sec. III, and compare our prediction aHμ VP 1⁄4 ð641þ−6635Þ × 10−10 from the ρ, ω, and φ resonances, where the uncertainty comes from the variation of the perturbative input, with those from other phenomenological and LQCD approaches.

THE FORMALISM

HVP contribution

The hybrid approach

Analyticity constraint

CONCLUSION