[Paper Review] A phenomenological estimate of isospin breaking in hadronic vacuum polarization

Martin Hoferichter, Gilberto Colangelo+6 more|PubMed|Jul 5, 2023

Particle physics theoretical and experimental studiesPhysics and Astronomy3 citations

TL;DR

This paper provides a phenomenological estimate of isospin-breaking (IB) corrections in the hadronic vacuum polarization (HVP) contribution to the muon's anomalous magnetic moment, using exclusive hadronic channels enhanced by infrared singularities, thresholds, and resonances—including for the first time $e^{+}e^{-} \to 3\pi$. The analysis reveals significant cancellations among IB contributions, with a net effect too small to explain the tension between lattice QCD and $e^{+}e^{-}$ data, ruling out IB effects in lattice calculations as the source of the discrepancy.

ABSTRACT

Puzzles in the determination of the hadronic-vacuum-polarization contribution currently impede a conclusive interpretation of the precision measurement of the anomalous magnetic moment of the muon at the Fermilab experiment. One such puzzle concerns tensions between evaluations in lattice QCD and using e^{+}e^{-}→hadrons cross-section data. In lattice QCD, the dominant isospin-symmetric part and isospin-breaking (IB) corrections are calculated separately, with very different systematic effects. Identifying these two pieces in a data-driven approach provides an opportunity to compare them individually and trace back the source of the discrepancy. Here, we estimate the IB component of the lattice-QCD calculations from phenomenology, based on a comprehensive study of exclusive contributions that can be enhanced via infrared singularities, threshold effects, or hadronic resonances, including, for the first time, in the e^{+}e^{-}→3π channel. We observe sizable cancellations among different channels, with a sum that even suggests a slightly larger result for the QED correction than obtained in lattice QCD. We conclude that the tensions between lattice QCD and e^{+}e^{-} data therefore cannot be explained by the IB contributions in the lattice-QCD calculations.

Motivation & Objective

• To estimate isospin-breaking corrections in the hadronic vacuum polarization (HVP) contribution to the muon's anomalous magnetic moment using phenomenological data.
• To resolve the tension between lattice QCD and $e^{+}e^{-} \to \text{hadrons}$ data by isolating and quantifying IB contributions in HVP.
• To systematically include enhanced IB effects from infrared singularities, threshold effects, and resonances, including $e^{+}e^{-} \to 3\pi$ for the first time.
• To compare the phenomenological IB estimate with lattice QCD results to determine whether IB effects can explain the observed discrepancy.
• To assess whether the observed tension in $a_\mu$ predictions stems from unaccounted IB physics in lattice QCD computations.

Proposed method

• A data-driven approach is used to estimate isospin-breaking contributions to the $R_{\text{had}}(s)$ ratio, incorporating exclusive final states with enhanced effects from infrared singularities and resonances.
• The analysis includes $e^{+}e^{-} \to 3\pi$ with both final-state radiation (FSR) and $\rho$–$\omega$ mixing, extending previous studies.
• IB contributions are separated into strong isospin breaking ($\mathcal{O}(\delta)$) and QED effects ($\mathcal{O}(e^2)$), with separate treatment of electromagnetic and strong interactions.
• The HVP integral is evaluated in Euclidean-time windows (short-, intermediate-, long-distance) to isolate region-dependent IB effects.
• Smeared $R$-ratios with Gaussian weights ($\sigma = 0.05$ and $0.44$ GeV) are used to visualize the spatial distribution of IB corrections across energy scales.
• Uncertainties are propagated from missing channels and model dependence, with results compared to lattice QCD computations from Borsanyi et al. (2021) and Blum et al. (2018).

Experimental results

Research questions

• RQ1 Can isospin-breaking effects in the HVP contribution explain the discrepancy between lattice QCD and $e^{+}e^{-} \to \text{hadrons}$ data?
• RQ2 What is the size and sign of QED and strong isospin-breaking corrections in the intermediate-energy window of the HVP integral?
• RQ3 How do cancellations among individual IB contributions affect the net HVP correction?
• RQ4 To what extent do $e^{+}e^{-} \to 3\pi$ processes, including FSR and $\rho$–$\omega$ mixing, contribute to the IB correction?
• RQ5 Is the observed tension in $a_\mu$ predictions attributable to missing or miscalculated IB effects in lattice QCD?

Key findings

• The net isospin-breaking correction to the HVP integral is small due to significant cancellations among individual channels, with both QED and strong IB effects contributing only minor corrections.
• In the intermediate window, the QED contribution is non-zero and positive, in contrast to lattice QCD estimates which are consistent with zero, indicating a potential discrepancy in this region.
• The sum of QED and strong IB effects remains compatible with zero within uncertainties, suggesting that IB effects do not resolve the lattice–data tension.
• The $\omega \to \pi^0\gamma\pi^0$ channel contributes significantly to the electromagnetic IB correction, particularly near the $\rho(1450)$ resonance.
• Even if future lattice QCD calculations resolve the discrepancy in the QED correction, the remaining tension would persist, implying IB effects are not the root cause.
• The inclusion of $e^{+}e^{-} \to 3\pi$ with FSR and $\rho$–$\omega$ mixing provides a more complete description of IB effects, confirming their small net impact.