A correction to this paper has been published: https://doi.org/10.1007/JHEP08(2022)295

Switzerland

Gilberto Colangelo

Martin Hoferichter

Joachim Monnard

Spain

Jacobo Elvira

Erratum to: Radiative corrections to the forward-backward asymmetry in

Jacobo Ruiz de Elvira

Erratum to: Radiative corrections to the forward-backward asymmetry in e + e  −  → π + π  −

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 <math><mrow><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><mtext>hadrons</mtext></mrow></math> 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 <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup><mo>→</mo><mn>3</mn><mi>π</mi></math> 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 <math><msup><mi>e</mi><mo>+</mo></msup><msup><mi>e</mi><mo>−</mo></msup></math> data therefore cannot be explained by the IB contributions in the lattice-QCD calculations.

Bai-Long Hoid

Germany

Bastian Kubis

Dominic Schuh

Dominik Stamen

Peter Stoffer

Phenomenological Estimate of Isospin Breaking in Hadronic Vacuum Polarization

Isospin-breaking (IB) effects in the two-pion contribution to hadronic vacuum polarization (HVP) can be resonantly enhanced, if related to the interference of the ρ(770) and ω(782) resonances. This particular IB contribution to the pion vector form factor and thus the line shape in e + e  −  → π + π  −  can be described by the residue at the ω pole — the ρ-ω mixing parameter ϵ  ω . Here, we argue that while in general analyticity requires this parameter to be real, the radiative channels π 0 γ, ππγ, ηγ can induce a small phase, whose size we estimate as δ  ϵ  = 3.5(1.0)° by using a narrow-width approximation for the intermediate-state vector mesons. We then perform fits to the e + e  −  → π + π  −  data base and study the consequences for the two-pion HVP contribution to the anomalous magnetic moment of the muon, its IB part due to ρ-ω mixing, and the mass of the ω resonance. We find that the global fit does prefer a non-vanishing value of δ  ϵ  = 4.5(1.2)°, close to the narrow-resonance expectation, but with a large spread among the data sets, indicating systematic differences in the ρ-ω region.

Isospin-breaking effects in the two-pion contribution to hadronic vacuum polarization

We present a calculation of the -odd radiative corrections to in a dispersive formalism, concentrating on the leading pion-pole contribution in the virtual box diagrams. In particular, we show how the effect of a general pion vector form factor in the loop integral can be incorporated in a model-independent way and how the cancellation of infrared singularities proceeds in this case. The numerical results, dominated by the infrared enhanced contributions, indicate significant corrections beyond scalar QED, essentially confirming recent findings in generalized vector-meson-dominance models.

Radiative corrections to the forward-backward asymmetry in

We study the pion-mass dependence of the two-pion channel in the hadronic-vacuum-polarization (HVP) contribution to the anomalous magnetic moment of the muon <math><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mtext>HVP</mtext></mrow></msubsup></math>, by using an Omnès representation for the pion vector form factor with the phase shift derived from the inverse-amplitude method (IAM). Our results constrain the dominant isospin-1 part of the isospin-symmetric light-quark contribution, and should thus allow one to better control the chiral extrapolation of <math><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mtext>HVP</mtext></mrow></msubsup></math>, required for lattice-QCD calculations performed at larger-than-physical pion masses. In particular, the comparison of the one- and two-loop IAM allows us to estimate the associated systematic uncertainties and show that these are under good control.

Malwin Niehus

Chiral extrapolation of hadronic vacuum polarization

At low energies hadronic vacuum polarization (HVP) is strongly dominated by two-pion intermediate states, which are responsible for about 70% of the HVP contribution to the anomalous magnetic moment of the muon, <math><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mtext>HVP</mtext></mrow></msubsup></math>. Lattice-QCD evaluations of the latter indicate that it might be larger than calculated dispersively on the basis of <math><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo></math> hadrons data, at a level which would contest the long-standing discrepancy with the <math><msub><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow></msub></math> measurement. In this Letter we study to which extent this 2π contribution can be modified without, at the same time, producing a conflict elsewhere in low-energy hadron phenomenology. To this end we consider a dispersive representation of the <math><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><mn>2</mn><mi>π</mi></math> process and study the correlations which thereby emerge between <math><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mtext>HVP</mtext></mrow></msubsup></math>, the hadronic running of the fine-structure constant, the P-wave ππ phase shift, and the charge radius of the pion. Inelastic effects play an important role, despite being constrained by the Eidelman–Łukaszuk bound. We identify scenarios in which <math><msubsup><mrow><mi>a</mi></mrow><mrow><mi>μ</mi></mrow><mrow><mtext>HVP</mtext></mrow></msubsup></math> can be altered substantially, driven by changes in the phase shift and/or the inelastic contribution, and illustrate the ensuing changes in the <math><msup><mrow><mi>e</mi></mrow><mrow><mo>+</mo></mrow></msup><msup><mrow><mi>e</mi></mrow><mrow><mo>−</mo></mrow></msup><mo>→</mo><mn>2</mn><mi>π</mi></math> cross section. In the combined scenario, which minimizes the effect in the cross section, a uniform shift around 4% is required. At the same time both the analytic continuation into the space-like region and the pion charge radius are affected at a level that could be probed in future lattice-QCD calculations.

Austria

United States

Constraints on the two-pion contribution to hadronic vacuum polarization

While the low-energy part of the hadronic light-by-light (HLbL) tensor can be constrained from data using dispersion relations, for a full evaluation of its contribution to the anomalous magnetic moment of the muon (g − 2) μ  also mixed- and high-energy regions need to be estimated. Both can be addressed within the operator product expansion (OPE), either for configurations where all photon virtualities become large or one of them remains finite. Imposing such short-distance constraints (SDCs) on the HLbL tensor is thus a major aspect of a model-independent approach towards HLbL scattering. Here, we focus on longitudinal SDCs, which concern the amplitudes containing the pseudoscalar-pole contributions from π 0, η, η′. Since these conditions cannot be fulfilled by a finite number of pseudoscalar poles, we consider a tower of excited pseudoscalars, constraining their masses and transition form factors from Regge theory, the OPE, and phenomenology. Implementing a matching of the resulting expressions for the HLbL tensor onto the perturbative QCD quark loop, we are able to further constrain our calculation and significantly reduce its model dependence. We find that especially for the π 0 the corresponding increase of the HLbL contribution is much smaller than previous prescriptions in the literature would imply. Overall, we estimate that longitudinal SDCs increase the HLbL contribution by   <math> <mi>Δ</mi> <msubsup> <mi>a</mi> <mi>μ</mi> <mtext>LSDC</mtext> </msubsup> <mo>=</mo> <mn>13</mn> <mfenced> <mn>6</mn> </mfenced> </math>  $$ \varDelta {a}_{\mu}^{\mathrm{LSDC}}=13(6) $$  × 10-11. This number does not include the contribution from the charm quark, for which we find   <math> <msubsup> <mi>a</mi> <mi>μ</mi> <mrow> <mi>c</mi> <mo>−</mo> <mtext>quark</mtext> </mrow> </msubsup> </math>  $$ {a}_{\mu}^{c- quark} $$  = 3(1) × 10 −11.

Franziska Hagelstein

Laetitia Laub

Longitudinal short-distance constraints for the hadronic light-by-light contribution to (g − 2) μ  with large-N  c  Regge models

A key ingredient in the evaluation of hadronic light-by-light (HLBL) scattering in the anomalous magnetic moment of the muon <math><mo>(</mo><mi>g</mi><mo>−</mo><mn>2</mn><msub><mo>)</mo><mi>μ</mi></msub></math> concerns short-distance constraints that follow from QCD by means of the operator product expansion. Here we concentrate on the most important such constraint, in the longitudinal amplitudes, and show that it can be implemented efficiently in terms of a Regge sum over excited-pseudoscalar states, constrained by phenomenological input on masses, two-photon couplings, as well as short-distance constraints on HLBL scattering and the pseudoscalar transition form factors. Our estimate of the effect of the longitudinal short-distance constraints on the HLBL contribution is <math><mi>Δ</mi><msubsup><mi>a</mi><mi>μ</mi><mrow><mi>LSDC</mi></mrow></msubsup><mo>=</mo><mn>13</mn><mo>(</mo><mn>6</mn><mo>)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>−</mo><mn>11</mn></mrow></msup></math>. This is significantly smaller than previous estimates, which mostly relied on an ad-hoc modification of the pseudoscalar poles and led to up to a 40% increase with respect to the nominal pseudoscalar-pole contributions, when evaluated with modern input for the relevant transition form factors. We also comment on the status of the transversal short-distance constraints and, by matching to perturbative QCD, argue that the corresponding correction will be significantly smaller than its longitudinal counterpart.

Short-distance constraints on hadronic light-by-light scattering in the anomalous magnetic moment of the muon

We present a detailed analysis of e + e − → π + π − data up to   <math> <msqrt> <mi>s</mi> </msqrt> <mo>=</mo> <mn>1</mn> </math>  $$ \sqrt{s}=1 $$  GeV in the framework of dispersion relations. Starting from a family of ππ P-wave phase shifts, as derived from a previous Roy-equation analysis of ππ scattering, we write down an extended Omnès representation of the pion vector form factor in terms of a few free parameters and study to which extent the modern high-statistics data sets can be described by the resulting fit function that follows from general principles of QCD. We find that statistically acceptable fits do become possible as soon as potential uncertainties in the energy calibration are taken into account, providing a strong cross check on the internal consistency of the data sets, but preferring a mass of the ω meson significantly lower than the current PDG average. In addition to a complete treatment of statistical and systematic errors propagated from the data, we perform a comprehensive analysis of the systematic errors in the dispersive representation and derive the consequences for the two-pion contribution to hadronic vacuum polarization. In a global fit to both time- and space-like data sets we find a   μ   ππ  |≤ 1 GeV = 495.0(1.5)(2.1) × 10− 10 and a   μ   ππ  |≤ 0.63 GeV = 132.8(0.4)(1.0) × 10− 10. While the constraints are thus most stringent for low energies, we obtain uncertainty estimates throughout the whole energy range that should prove valuable in corroborating the corresponding contribution to the anomalous magnetic moment of the muon. As side products, we obtain improved constraints on the ππ P-wave, valuable input for future global analyses of low-energy ππ scattering, as well as a determination of the pion charge radius, 〈r   π  2 〉 = 0.429(1)(4) fm2.