Revisit of the electromagnetic correction to $τ\toππν_τ$ and its implication for muon $g-2$ based on $τ$ data

TL;DR

The paper revisits the leading-order hadronic vacuum polarization contribution to the muon anomalous magnetic moment $a_{\mu}$ from the $\pi\pi$ channel by exploiting experimental $\tau\to\pi\pi\nu_{\tau}$ data and a refined long-distance electromagnetic correction $G_{\rm EM}$. It computes $G_{\rm EM}$ from the full $\tau^-\to\pi^-\pi^0\nu_\tau\gamma$ amplitude within resonance chiral theory, including both even- and odd-intrinsic-parity resonance operators, and determines the resonance coupling $d_4$ by analyzing $\omega\to\pi^0\pi^0\gamma$ with both vector and scalar resonances; two solutions emerge, Sol-A with $d_4=-0.42$ and Sol-B with $d_4=1.01$. The negative solution aligns with previous $O(p^4)$ results, while the positive solution aligns with some $O(p^6)$ findings, and the authors adopt Sol-A as the baseline, treating the difference as a systematic. Using the complete isospin-breaking corrections in $R_{\rm IB}(t)$ and the tau data from Belle, ALEPH, CLEO, and OPAL, they obtain $a_{\mu}^{\rm HVP,LO}|_{\pi\pi,\tau\text{data}} = 516.0(5.4)\times 10^{-10}$ and, when combined with other hadronic channels, $a_{\mu}^{\rm HVP,LO}|_{\tau\text{data}} = 702.1(5.4)\times 10^{-10}$; after incorporating the WP25 updates, the implied SM deviation is $\Delta a_{\mu} = (14.9 \pm 5.6)\times 10^{-10}$, corresponding to about $2.7\sigma$. The analysis demonstrates the sensitivity of the EM corrections to the $d_4$ parameter and provides an independent cross-check of the $a_{\mu}$ evaluation using $\tau$ data, highlighting a remaining tension with the current world average.

Abstract

In this work we focus on the evaluation of the leading-order hadronic vacuum polarization contribution from the $ππ$ channel to the muon anomalous magnetic moment $a_μ$ by using the experimental $τ\toππν_τ$ data. The isospin breaking corrections play the decisive role in this approach of computing $a_μ$. One of such important isospin breaking sources is the long-distance electromagnetic correction factor $G_{\rm EM}$ of the $τ\toππν_τ$ process from the real photon radiation. The latter effect can be calculated from the $τ\toππν_τγ$ amplitude, which is revised in this work within the resonance chiral theory by simultaneously including the even-intrinsic-parity and odd-intrinsic-parity resonance operators. We update the determination of the only unknown resonance coupling through the $ω\toπ^0π^0γ$ decay by including contributions from both the vector and scalar resonances. By taking other remaining contributions from the muon $g-2$ White Paper 2025, we further revise the complete value of $a_μ$, which turns out to deviate from the newest world average result after Fermilab's measurement at the level of 2.7 $σ$.

Figures (3)

• Figure 1: Illustration of relevant Feynman diagrams for the scalar resonance contribution to the $\omega\to\pi^0\pi^0\gamma$ decay. The thick solid black line and the wiggle line denote the $\omega$ resonance and the photon, respectively, while the dashed lines stand for the pions.
• Figure 2: Illustration of different results for EM corrected function $G_{\text{EM}}(t)$: Sol-A (black dashed line, with negative value of $d_4$), Sol-B (orange dashed line, with positive value of $d_4$), $G_{\text{EM}}^{\text{SI}}(t)$ (red long-dashed line, SI term only), the leading Low approximation for $G_{\text{EM}}^{(0)}(t)$ (blue dotted line) and the result with the full amplitude from Ref. Cirigliano:2002pv (blue solid line, labeled as CEN).
• Figure 3: Full IB correction functions $R_{\text{IB}}(t)$ with complete EM corrections $G_{\text{EM}}\left(t\right)$.