Radiative corrections to $τ\toππν_τ$

TL;DR

The paper develops a dispersive, model-independent framework to compute radiative and isospin-breaking corrections to τ → ππν_τ, extending ChPT results with a pion-vector form factor that incorporates ρ, ρ′, and ρ″ resonances. By matching to ChPT and carefully treating real-emission contributions and endpoint/threshold singularities, they obtain a UV-finite, IR-consistent correction G_EM(s) and quantify the τ-specific IB corrections to a_mu^HVP,LO[ππ,τ], finding a notable ρ-region effect and a reduction in theoretical uncertainties, though scheme-matching with S_EW^{ππ} remains a dominant source of uncertainty. The work relies on fits to the τ spectral function using a dispersive f_+(s) with Omnès unitarization and conformal polynomials, enabling a coherent evaluation across the full kinematic range and highlighting tensions between threshold and resonance regions that motivate further data and lattice inputs. Overall, the analysis provides a more reliable long-range radiative correction for tau-based determinations of the two-pion HVP contribution to the muon g-2 and outlines the path toward a fully robust, data-driven HVP evaluation from hadronic τ decays.

Abstract

Hadronic $τ$ decays present an opportunity to determine the isovector part of the hadronic-vacuum-polarization contribution to the anomalous magnetic moment of the muon in a way complementary to $e^+e^-\to\text{hadrons}$ cross sections. However, the required isospin rotation is only exact in the isospin limit, and corrections need to be under control to draw robust conclusions, most notably for $τ\toππν_τ$ decays to determine the two-pion contribution, $a_μ^\text{HVP, LO}[ππ,τ]$. In this work, we present a novel analysis of the required radiative corrections using dispersion relations, thereby extending in a model-independent way the previous analysis in chiral perturbation theory (ChPT) beyond the threshold region. In particular, we include the dominant structure-dependent virtual corrections from pion-pole diagrams, leading to sizable changes in the vicinity of the $ρ(770)$ resonance. Moreover, we work out the matching to ChPT and devise a strategy for a stable numerical evaluation of real-emission contributions near the two-pion threshold, which proves important to capture isospin-breaking corrections enhanced by the threshold singularity. For the numerical analysis, we use a dispersive representation of the pion form factor including the $ρ'$, $ρ''$ resonances, perform fits to the available data sets for the $τ\toππν_τ$ spectral function, and calculate the corresponding radiative correction factor $G_\text{EM}(s)$ in a self-consistent manner. Based on these results, we evaluate the $τ$-specific isospin-breaking corrections to $a_μ^\text{HVP, LO}[ππ,τ]$.

Figures (19)

• Figure 1: Leading diagrams for the radiative corrections to $\tau^- (l_1) \to \pi^-(q_1) \pi^0(q_2) \nu_\tau(l_2)$, excluding wave-function renormalization.
• Figure 2: Counterterm diagram for the radiative corrections to $\tau^- \to \pi^- \pi^0 \nu_\tau$.
• Figure 3: Loop and counterterm diagrams of the external-leg contributions.
• Figure 4: Triangle/box diagram with $\pi\pi\gamma$ and $\tau\nu_\tau\pi\pi$ vertices dressed with the pion VFF.
• Figure 5: Initial- and final-state-radiations diagrams for $\tau^- \to \pi^- \pi^0 \nu_\tau$.