Improved calculation of radiative corrections to $\boldsymbol{τ\toππν_τ}$ decays

Abstract

A reliable calculation of radiative corrections to $τ\toππν_τ$ decays is an important prerequisite for using hadronic $τ$ decays for a data-driven evaluation of the hadronic-vacuum-polarization contribution to the anomalous magnetic moment of the muon, $a_μ^\text{HVP, LO}[ππ,τ]$. In this Letter, we present an improved model-independent analysis of these radiative corrections, including, for the first time, effects beyond point-like pions in the evaluation of the loop diagrams. These structure-dependent corrections, implemented via a dispersive representation of the pion form factor, lead to significant changes compared to previous calculations due to enhancements near the $ρ(770)$ resonance. We also devise strategies for the matching to chiral perturbation theory and a stable implementation of the real corrections down to the two-pion threshold, which shows that some higher-order isospin-breaking corrections need to be kept due to a strong threshold enhancement. Finally, we perform dispersive fits to the currently available $τ\toππν_τ$ spectra and discuss the consequences for isospin-breaking corrections in the evaluation of $a_μ^\text{HVP, LO}[ππ,τ]$.

Figures (5)

• Figure 1: Virtual photonic corrections to $\tau\to\pi\pi\nu_\tau$ in ChPT. Solid lines denote leptons, dashed lines pions, and wiggly lines photons. Not shown are self-energy diagrams and contact terms.
• Figure 2: Box diagram in a dispersive approach. The gray blobs denote the pion form factor, in the neutral and charged channel, respectively. The short-dashed line indicates that the intermediate-state pion is taken on-shell.
• Figure 3: Sample diagrams for the real corrections from the WZW anomaly as well as the exchange of vector and axial-vector meson resonances (indicated by the double lines). Otherwise notation as in Figs. \ref{['fig:diagrams_ChPT']} and \ref{['fig:diagrams_disp']}.
• Figure 4: Global fit to the $\tau\to\pi\pi\nu_\tau$ spectrum, including the data sets from Belle Belle:2008xpe, ALEPH ALEPH:2005qgpDavier:2013sfa, CLEO CLEO:1999dln, and OPAL OPAL:1998rrm.
• Figure 5: Final result for $G_\text{EM}(s)$ and comparison to previous work, Flores-Baéz et al. (2006) Flores-Baez:2006yiq, Miranda, Roig (2020) Miranda:2020wdg, and using ChPT instead of dispersion relations for the box diagram (based on Refs. Cirigliano:2001erCirigliano:2002pv).