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We present a detailed analysis of $e^+e^-topi^+pi^-$ data up to $sqrt{s}=1,text{GeV}$ in the framework of dispersion relations. Starting from a family of $pipi$ $P$-wave phase shifts, as derived from a previous Roy-equation analysis of $pipi$ scattering, we write down an extended Omn`es 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 $omega$ 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 $a_mu^{pipi}|_{leq 1,text{GeV}}=495.0(1.5)(2.1)times 10^{-10}$ and $a_mu^{pipi}|_{leq 0.63,text{GeV}}=132.8(0.4)(1.0)times 10^{-10}$. 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 $pipi$ $P$-wave, valuable input for future global analyses of low-energy $pipi$ scattering, as well as a determination of the pion charge radius, $langle r_pi^2 rangle = 0.429(1)(4),text{fm}^2$.
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We address the contribution of the $3pi$ channel to hadronic vacuum polarization (HVP) using a dispersive representation of the $e^+e^-to 3pi$ amplitude. This channel gives the second-largest individual contribution to the total HVP integral in the anomalous magnetic moment of the muon $(g-2)_mu$, both to its absolute value and uncertainty. It is largely dominated by the narrow resonances $omega$ and $phi$, but not to the extent that the off-peak regions were negligible, so that at the level of accuracy relevant for $(g-2)_mu$ an analysis of the available data as model independent as possible becomes critical. Here, we provide such an analysis based on a global fit function using analyticity and unitarity of the underlying $gamma^*to3pi$ amplitude and its normalization from a chiral low-energy theorem, which, in particular, allows us to check the internal consistency of the various $e^+e^-to 3pi$ data sets. Overall, we obtain $a_mu^{3pi}|_{leq 1.8,text{GeV}}=46.2(6)(6)times 10^{-10}$ as our best estimate for the total $3pi$ contribution consistent with all (low-energy) constraints from QCD. In combination with a recent dispersive analysis imposing the same constraints on the $2pi$ channel below $1,text{GeV}$, this covers nearly $80%$ of the total HVP contribution, leading to $a_mu^text{HVP}=692.3(3.3)times 10^{-10}$ when the remainder is taken from the literature, and thus reaffirming the $(g-2)_mu$ anomaly at the level of at least $3.4sigma$. As side products, we find for the vacuum-polarization-subtracted masses $M_omega=782.63(3)(1),text{MeV}$ and $M_phi=1019.20(2)(1),text{MeV}$, confirming the tension to the $omega$ mass as extracted from the $2pi$ channel.
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, $a_mu^text{HVP}$. Lattice-QCD evaluations of the latter indicate that it might be larger than calculated dispersively on the basis of $e^+e^-totext{hadrons}$ data, at a level which would contest the long-standing discrepancy with the $a_mu$ measurement. In this Letter we study to which extent this $2pi$ 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 $e^+e^- to 2pi$ process and study the correlations which thereby emerge between $a_mu^text{HVP}$, the hadronic running of the fine-structure constant, the $P$-wave $pipi$ phase shift, and the charge radius of the pion. Inelastic effects play an important role, despite being constrained by the Eidelman-Lukaszuk bound. We identify scenarios in which $a_mu^text{HVP}$ can be altered substantially, driven by changes in the phase shift and/or the inelastic contribution, and illustrate the ensuing changes in the $e^+e^-to 2pi$ 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.
Hadronic vacuum polarization (HVP) is not only a critical part of the Standard Model (SM) prediction for the anomalous magnetic moment of the muon $(g-2)_mu$, but also a crucial ingredient for global fits to electroweak (EW) precision observables due to its contribution to the running of the fine-structure constant encoded in $Deltaalpha^{(5)}_text{had}$. We find that with modern EW precision data, including the measurement of the Higgs mass, the global fit alone provides a competitive, independent determination of $Delta alpha^{(5)}_text{had}big|_text{EW}=270.2(3.0)times 10^{-4}$. This value actually lies below the range derived from $e^+e^-totext{hadrons}$ cross-section data, and thus goes into the opposite direction as would be required if a change in HVP were to bring the SM prediction for $(g-2)_mu$ into agreement with the Brookhaven measurement. Depending on the energy where the bulk of the changes in the cross section occurs, reconciling experiment and SM prediction for $(g-2)_mu$ by adjusting HVP would thus not necessarily weaken the case for physics beyond the SM (BSM), but to some extent shift it from $(g-2)_mu$ to the EW fit. We briefly explore some options of BSM scenarios that could conceivably explain the ensuing tension.
The $pi^0$ pole constitutes the lowest-lying singularity of the hadronic light-by-light (HLbL) tensor, and thus provides the leading contribution in a dispersive approach to HLbL scattering in the anomalous magnetic moment of the muon $(g-2)_mu$. It is unambiguously defined in terms of the doubly-virtual pion transition form factor, which in principle can be accessed in its entirety by experiment. We demonstrate that, in the absence of a direct measurement, the full space-like doubly-virtual form factor can be reconstructed very accurately based on existing data for $e^+e^-to 3pi$, $e^+e^-to e^+e^-pi^0$, and the $pi^0togammagamma$ decay width. We derive a representation that incorporates all the low-lying singularities of the form factor, matches correctly onto the asymptotic behavior expected from perturbative QCD, and is suitable for the evaluation of the $(g-2)_mu$ loop integral. The resulting value, $a_mu^{pi^0text{-pole}}=62.6^{+3.0}_{-2.5}times 10^{-11}$, for the first time, represents a complete data-driven determination of the pion-pole contribution with fully controlled uncertainty estimates. In particular, we show that already improved singly-virtual measurements alone would allow one to further reduce the uncertainty in $a_mu^{pi^0text{-pole}}$.
In this third paper of a series dedicated to a dispersive treatment of the hadronic light-by-light (HLbL) tensor, we derive a partial-wave formulation for two-pion intermediate states in the HLbL contribution to the anomalous magnetic moment of the muon $(g-2)_mu$, including a detailed discussion of the unitarity relation for arbitrary partial waves. We show that obtaining a final expression free from unphysical helicity partial waves is a subtle issue, which we thoroughly clarify. As a by-product, we obtain a set of sum rules that could be used to constrain future calculations of $gamma^*gamma^*topipi$. We validate the formalism extensively using the pion-box contribution, defined by two-pion intermediate states with a pion-pole left-hand cut, and demonstrate how the full known result is reproduced when resumming the partial waves. Using dispersive fits to high-statistics data for the pion vector form factor, we provide an evaluation of the full pion box, $a_mu^{pitext{-box}}=-15.9(2)times 10^{-11}$. As an application of the partial-wave formalism, we present a first calculation of $pipi$-rescattering effects in HLbL scattering, with $gamma^*gamma^*topipi$ helicity partial waves constructed dispersively using $pipi$ phase shifts derived from the inverse-amplitude method. In this way, the isospin-$0$ part of our calculation can be interpreted as the contribution of the $f_0(500)$ to HLbL scattering in $(g-2)_mu$. We argue that the contribution due to charged-pion rescattering implements corrections related to the corresponding pion polarizability and show that these are moderate. Our final result for the sum of pion-box contribution and its $S$-wave rescattering corrections reads $a_mu^{pitext{-box}} + a_{mu,J=0}^{pipi,pitext{-pole LHC}}=-24(1)times 10^{-11}$.
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