The Discrepancy Between tau and e+e- Spectral Functions Revisited and the Consequences for the Muon Magnetic Anomaly

TL;DR

The paper revisits the link between the tau and $e^+e^-$ spectral functions for the $\pi\pi$ final state, applying updated isospin-breaking corrections to reconcile the two datasets. By integrating Belle tau data with ALEPH, CLEO, and OPAL using HVPTools and refining radiative and hadronic corrections, the authors obtain a revised $a_\mu^{\rm had,LO}$ that reduces the discrepancy between tau- and $e^+e^-$-based evaluations. They report a tau-based $a_\mu$ that is about $1.9\sigma$ below the experimental value and demonstrate improved consistency in the CVC-predicted $\mathcal{B}_{\pi\pi^0}$ with the measured $\tau$ branching fraction. The results highlight the continued importance of low-energy $e^+e^-$ measurements and the need for precise LBLS calculations to solidify the Standard Model prediction of $a_\mu$.

Abstract

We revisit the procedure for comparing the pi pi spectral function measured in tau decays to that obtained in e+e- annihilation. We re-examine the isospin-breaking corrections using new experimental and theoretical input, and find improved agreement between the tau- --> pi- pi0 nu_tau branching fraction measurement and its prediction using the isospin-breaking-corrected e+e- --> pi+pi- spectral function, though not resolving all discrepancies. We recompute the lowest order hadronic contributions to the muon g-2 using e+e- and tau data with the new corrections, and find a reduced difference between the two evaluations. The new tau-based estimate of the muon magnetic anomaly is found to be 1.9 standard deviations lower than the direct measurement.

Figures (6)

• Figure 1: Relative comparison between the $\tau^-\to\pi^-\xspace\pi^0\xspace\nu_{\tau}$ invariant mass-squared measurements from ALEPH, CLEO, OPAL, Belle (data points) and the combined result (shaded band).
• Figure 2: Left: Isospin-breaking corrections from $G_{\rm EM}\xspace$, FSR, $\beta^3_0(s)/\beta^3_-(s)$ and $|F_0(s)/F_-(s)|^2$. Right: Isospin-breaking corrections in the ratio of $I=1$ components of the form factors $|F_0(s)/F_-(s)|^2$ due to the $\pi$ mass splitting $\delta m_\pi=m_{\pi^\pm}-m_{\pi^0}$, the $\rho$ mass splitting $\delta m_\rho=m_{\rho^\pm}-m_{\rho^0_{\rm bare}}$, and the difference $\delta\Gamma_\rho$ in the $\rho$ meson widths.
• Figure 3: Relative comparison between $e^+e^-$ and $\tau$ spectral functions, expressed in terms of the difference between neutral and charged pion form factors. Isospin-breaking (IB) corrections are applied to $\tau$ data with its uncertainties, although hardly visible, included in the error band.
• Figure 4: Fit of the pion form factor from $4m^2_\pi$ to $0.3\,{\rm GeV}^2$ using a third order expansion with the constraints $F(0)=1$ and using the measured pion charge radius-squared from space-like data amendolia86. The result of the fit to the $\tau$ data (left) and to $e^+e^-$ data (right) is integrated only up to $0.13\,{\rm GeV}^2$, beyond which we directly integrate over the data points.
• Figure 5: Compilation of recently published results for $a_\mu^{\rm SM}\xspace$ (in units of $10^{-11}$), subtracted by the central value of the experimental average bnlpdgg-2rev. The shaded band indicates the experimental error. The SM predictions are taken from: DEHZ 03 dehz03, HMNT 07 hmnt, J 07 jeger, and the present $\tau$- and $e^+e^-$-based predictions using $\tau$ and $e^+e^-$ spectral functions.