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Register a new userReview of Recent Calculations of the Hadronic Vacuum Polarisation Contribution
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Zhiqing Zhang
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Recent calculations of the hadronic vacuum polarisation contribution are reviewed. The focus is put on the leading-order contribution to the muon magnetic anomaly involving $e^+e^-$ annihilation cross section data as input to a dispersion relation approach. Alternative calculation including tau data is also discussed. The $tau$ data are corrected for various isospin-breaking sources which are explicitly shown source by source.
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Precise data on e^+e^- to hadrons have recently become available and are used to compute the lowest-order hadronic vacuum polarisation contribution to the muon magnetic anomaly through dispersion relations. This is the case for the dominant pi+ pi- channel, but the most significant progress comes from the near completion of the BABAR program of measuring exclusive processes below 2 GeV with the initial-state radiation method which allows an efficient coverage of a large range of energies.. In this paper we briefly review the data treatment, the achieved improvements, and the result obtained for the full Standard Model prediction of the muon magnetic anomaly. The value obtained, a_mu (had~LO)=(692.6 +- 3.3)x 10^{-10} is 20% more precise than our last estimate in 2010. It deviates from the direct experimental determination by (27.4 +- 7.6)x 10^{-10} (3.6 sigma). Perpectives for further improvement are discussed.
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.
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, $langle r_pi^2 rangle = 0.429(1)(4),text{fm}^2$.
We compute the vacuum polarisation on the lattice in quenched QCD using non-perturbatively improved Wilson fermions. Above Q^2 of about 2 GeV^2 the results are very close to the predictions of perturbative QCD. Below this scale we see signs of non-perturbative effects which we can describe by the use of dispersion relations. We use our results to estimate the light quark contribution to the muons anomalous magnetic moment. We find the result 446(23) x 10^{-10}, where the error only includes statistical uncertainties. Finally we make some comments on the applicability of the Operator Product Expansion to our data.
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.
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