ρ 0–γ mixing in the neutral channel pion form factor and its role in comparing e + e − with τ spectral functions

Fred Jegerlehner; Robert Szafron

doi:10.1140/epjc/s10052-011-1632-3

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Particles and Fields

Eur. Phys. J. C (2011) 71: 1632
https://doi.org/10.1140/epjc/s10052-011-1632-3

Regular Article - Theoretical Physics

ρ ⁰–γ mixing in the neutral channel pion form factor $F^{(e)}_{\pi}(s)$ and its role in comparing e ⁺ e ⁻ with τ spectral functions

Fred Jegerlehner¹^,2^* and Robert Szafron³

¹ Institut für Physik, Humboldt-Universität zu Berlin, Newtonstrasse 15, 12489, Berlin, Germany
² Deutsches Elektronen-Synchrotron (DESY), Platanenallee 6, 15738, Zeuthen, Germany
³ Institute of Physics, University of Silesia, ul. Uniwersytecka 4, 40007, Katowice, Poland

^* e-mail: fjeger@physik.hu-berlin.de

Received: 16 January 2011
Revised: 16 March 2011
Published online: 1 May 2011

Abstract

We study the effect of ρ ⁰–γ mixing in e ⁺ e ⁻→π ⁺ π ⁻ and its relevance for the comparison of the square modulus of the pion from-factor $|F_{\pi }^{(e)}(s)|^{2}$ , as measured in e ⁺ e ⁻ annihilation experiments, and $|F_{\pi }^{(\tau )}(s)|^{2}$ the corresponding quantity obtained after accounting for known isospin breaking effects by an isospin rotation from the τ-decay spectra. After correcting the τ data for the missing ρ–γ mixing contribution, besides the other known isospin symmetry violating corrections, the ππ I=1 part of the hadronic vacuum polarization contribution to the muon g−2 are fully compatible between τ based and e ⁺ e ⁻ based evaluations. τ data thus confirm result obtained with e ⁺ e ⁻ data. Our evaluation of the leading order vacuum polarization contribution, based on all e ⁺ e ⁻ data including more recent BaBar and KLOE data, yields $a_{\mu}^{\mathrm{had,LO}}[e]=690.75(4.72)\times 10^{-10}$ (e ⁺ e ⁻ based), while including τ data we find $a_{\mu}^{\mathrm{had,LO}}[e,\tau]=690.96(4.65)\times 10^{-10}$ (e ⁺ e ⁻+τ based). This backs the ∼3σ deviation between $a_{\mu}^{\mathrm{experiment}}$ and $a_{\mu}^{\mathrm{theory}}$ . For the τ di-pion branching fraction we find $B^{\mathrm{CVC}}_{\pi\pi^{0}}=25.20\pm0.17\pm0.28$ from e ⁺ e ⁻+CVC, while $B_{\pi\pi^{0}}=25.34\pm0.06\pm0.08$ is evaluated directly from the τ spectra.