Constraints on the two-pion contribution to hadronic vacuum polarization

Abstract

At low energies hadronic vacuum polarization (HVP) is strongly dominated by two-pion intermediate states, which are responsible for about $70\%$ of the HVP contribution to the anomalous magnetic moment of the muon, $a_{\mu }^{\text{HVP}}$. Lattice-QCD evaluations of the latter indicate that it might be larger than calculated dispersively on the basis of $e^{+}{e}^{-}\to \text{hadrons}$ data, at a level which would contest the long-standing discrepancy with the $a_{\mu }$ measurement. In this Letter we study to which extent this $2\pi$ contribution can be modified without, at the same time, producing a conflict elsewhere in low-energy hadron phenomenology. To this end we consider a dispersive representation of the $e^{+}{e}^{-}\to 2\pi$ process and study the correlations which thereby emerge between $a_{\mu }^{\text{HVP}}$, the hadronic running of the fine-structure constant, the $P$-wave $\pi \pi$ phase shift, and the charge radius of the pion. Inelastic effects play an important role, despite being constrained by the Eidelman-Lukaszuk bound. We identify scenarios in which $a_{\mu }^{\text{HVP}}$ can be altered substantially, driven by changes in the phase shift and/or the inelastic contribution, and illustrate the ensuing changes in the $e^{+}{e}^{-}\to 2\pi$ cross section. In the combined scenario, which minimizes the effect in the cross section, a uniform shift around $4\%$ is required. At the same time both the analytic continuation into the space-like region and the pion charge radius are affected at a level that could be probed in future lattice-QCD calculations.

9 pages, 9 figures; version published in PLB

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