[Paper Review] Isospin-breaking effects in the three-pion contribution to hadronic vacuum polarization

Martin Hoferichter, Bai-Long Hoid+2 more|arXiv (Cornell University)|Jul 5, 2023

Particle physics theoretical and experimental studies4 citations

TL;DR

This paper develops a dispersive formalism to compute leading isospin-breaking effects in the $e^{+}e^{-} \to 3\pi$ process, including final-state radiation and $\rho$--$\omega$ mixing, generalizing the $\eta_{2\pi}$ correction factor to $\eta_{3\pi}$. The global fit to BaBar data yields $a_{\mu}^{3\pi}|_{\leq 1.8\,\text{GeV}} = 45.91(53)\times 10^{-10}$, with $0.51(1)\times 10^{-10}$ from FSR and $-2.68(70)\times 10^{-10}$ from $\rho$--$\omega$ mixing, showing significant cancellation with the $2\pi$ channel contribution.

ABSTRACT

Isospin-breaking (IB) effects are required for an evaluation of hadronic vacuum polarization at subpercent precision. While the dominant contributions arise from the $e^+e^- oπ^+π^-$ channel, also IB in the subleading channels can become relevant for a detailed understanding, e.g., of the comparison to lattice QCD. Here, we provide such an analysis for $e^+e^- o 3π$ by extending our dispersive description of the process, including estimates of final-state radiation (FSR) and $ρ$-$ω$ mixing. In particular, we develop a formalism to capture the leading infrared-enhanced effects in terms of a correction factor $η_{3π}$ that generalizes the analog treatment of virtual and final-state photons in the $2π$ case. The global fit to the $e^+e^- o 3π$ data base, subject to constraints from analyticity, unitarity, and the chiral anomaly, gives $a_μ^{3π}|_{\leq 1.8\, ext{GeV}}=45.91(53) imes 10^{-10}$ for the total $3π$ contribution to the anomalous magnetic moment of the muon, of which $a_μ^ ext{FSR}[3π]=0.51(1) imes 10^{-10}$ and $a_μ^{ρ ext{-}ω}[3π]=-2.68(70) imes 10^{-10}$ can be ascribed to IB. We argue that the resulting cancellation with $ρ$-$ω$ mixing in $e^+e^- o 2π$ can be understood from a narrow-resonance picture, and provide updated values for the vacuum-polarization-subtracted vector-meson parameters $M_ω=782.70(3)\, ext{MeV}$, $M_ϕ=1019.21(2)\, ext{MeV}$, $Γ_ω=8.71(3)\, ext{MeV}$, and $Γ_ϕ=4.27(1)\, ext{MeV}$.

Motivation & Objective

• To develop a systematic framework for isospin-breaking effects in the $e^{+}e^{-} \to 3\pi$ channel, essential for subpercent precision in $a_\mu$.
• To quantify the impact of final-state radiation (FSR) and $\rho$--$\omega$ mixing on the hadronic vacuum polarization contribution to the muon $g-2$.
• To improve the precision of the $3\pi$ contribution to $a_\mu$ by incorporating the latest BaBar data and analytic constraints.
• To test the consistency of $\rho$--$\omega$ mixing parameters across $2\pi$ and $3\pi$ channels via a unified dispersive description.

Proposed method

• A dispersive representation of the $e^{+}e^{-} \to 3\pi$ amplitude is constructed, enforcing analyticity, unitarity, and the chiral anomaly.
• The formalism generalizes the $\eta_{2\pi}$ FSR correction factor to $\eta_{3\pi}(q^2)$, capturing infrared-enhanced radiative corrections in the $3\pi$ channel.
• A coupled-channel model for $e^{+}e^{-}$, $\pi^+\pi^-$, and $3\pi$ is used to describe $\rho$--$\omega$ mixing while preserving unitarity and analyticity.
• The $\rho$--$\omega$ mixing parameter $\epsilon_\omega$ is constrained by fitting to BaBar data, with sensitivity tested via variation of the mixing phase.
• Global fits to the $3\pi$ data base are performed, including uncertainties from the $\rho$ line shape and $\epsilon_\omega$ variation.
• Resonance parameters for $\omega$ and $\phi$ are extracted from the global fit, consistent with dispersive constraints.

Experimental results

Research questions

• RQ1 How do final-state radiation effects in $e^{+}e^{-} \to 3\pi$ contribute to the anomalous magnetic moment of the muon?
• RQ2 To what extent does $\rho$--$\omega$ mixing in the $3\pi$ channel affect the hadronic vacuum polarization and its comparison with the $2\pi$ channel?
• RQ3 Can a unified dispersive formalism consistently describe both FSR and $\rho$--$\omega$ mixing in $3\pi$ final states while preserving analyticity and unitarity?
• RQ4 What is the quantitative impact of isospin-breaking effects in $3\pi$ on the total $a_\mu^{3\pi}$, and how do they compare to the $2\pi$ channel?
• RQ5 Is the $\rho$--$\omega$ mixing parameter $\epsilon_\omega$ in the $3\pi$ channel consistent with that in the $2\pi$ channel?

Key findings

• The total $3\pi$ contribution to the muon anomalous magnetic moment is $a_{\mu}^{3\pi}|_{\leq 1.8\,\text{GeV}} = 45.91(53)\times 10^{-10}$, with uncertainties reduced by nearly a factor of two compared to previous estimates.
• The final-state radiation contribution is $a_{\mu}^{\text{FSR}}[3\pi] = 0.51(1)\times 10^{-10}$, consistent with expectations from the relative size of the $2\pi$ and $3\pi$ channels.
• The $\rho$--$\omega$ mixing contribution is $a_{\mu}^{\rho\text{--}\omega}[3\pi] = -2.68(70)\times 10^{-10}$, large and negative, indicating significant cancellation with the $2\pi$ channel.
• The $\rho$--$\omega$ mixing parameter $\epsilon_\omega$ extracted from the $3\pi$ data is slightly smaller than expected from the $2\pi$ channel but consistent within uncertainties, supporting a universal description.
• The global fit yields updated vector-meson parameters: $M_\omega = 782.70(3)\,\text{MeV}$, $\Gamma_\omega = 8.71(3)\,\text{MeV}$, $M_\phi = 1019.21(2)\,\text{MeV}$, $\Gamma_\phi = 4.27(1)\,\text{MeV}$.
• The inclusion of $\rho$--$\omega$ mixing significantly improves the fit quality to BaBar data, confirming its visibility despite the broad $\rho$ width.