https://www.zora.uzh.ch/id/eprint/191493/1/1-s2.0-S0370157320302556-main.pdf

The anomalous magnetic moment of the muon in the Standard Model

We review the present status of the Standard Model calculation of the anomalous magnetic moment of the muon. This is performed in a perturbative expansion in the fine-structure constant $\alpha$ and is broken down into pure QED, electroweak, and hadronic contributions. The pure QED contribution is by far the largest and has been evaluated up to and including $\mathcal{O}(\alpha^5)$ with negligible numerical uncertainty. The electroweak contribution is suppressed by $(m_\mu/M_W)^2$ and only shows up at the level of the seventh significant digit. It has been evaluated up to two loops and is known to better than one percent. Hadronic contributions are the most difficult to calculate and are responsible for almost all of the theoretical uncertainty. The leading hadronic contribution appears at $\mathcal{O}(\alpha^2)$ and is due to hadronic vacuum polarization, whereas at $\mathcal{O}(\alpha^3)$ the hadronic light-by-light scattering contribution appears. Given the low characteristic scale of this observable, these contributions have to be calculated with nonperturbative methods, in particular, dispersion relations and the lattice approach to QCD. The largest part of this review is dedicated to a detailed account of recent efforts to improve the calculation of these two contributions with either a data-driven, dispersive approach, or a first-principle, lattice-QCD approach. The final result reads $a_\mu^\text{SM}=116\,591\,810(43)\times 10^{-11}$ and is smaller than the Brookhaven measurement by 3.7$\sigma$. The experimental uncertainty will soon be reduced by up to a factor four by the new experiment currently running at Fermilab, and also by the future J-PARC experiment. This and the prospects to further reduce the theoretical uncertainty in the near future-which are also discussed here-make this quantity one of the most promising places to look for evidence of new physics.

We address the contribution of the $3\pi$ channel to hadronic vacuum polarization (HVP) using a dispersive representation of the $e^+e^-\to 3\pi$ 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 $\gamma^*\to3\pi$ amplitude and its normalization from a chiral low-energy theorem, which, in particular, allows us to check the internal consistency of the various $e^+e^-\to 3\pi$ data sets. Overall, we obtain $a_\mu^{3\pi}|_{\leq 1.8\,\text{GeV}}=46.2(6)(6)\times 10^{-10}$ as our best estimate for the total $3\pi$ contribution consistent with all (low-energy) constraints from QCD. In combination with a recent dispersive analysis imposing the same constraints on the $2\pi$ 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 $3.4\sigma$. 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 $2\pi$ channel.

/pdf/three-pion-contribution-to-hadronic-vacuum-polarization-3iaox9z950.pdf

Three-pion contribution to hadronic vacuum polarization

We address the contribution of the 3π channel to hadronic vacuum polarization (HVP) using a dispersive representation of the e+e− → 3π amplitude. This channel gives the second-largest individual contribution to the total HVP integral in the anomalous magnetic moment of the muon (g − 2)μ, both to its absolute value and uncertainty. It is largely dominated by the narrow resonances ω and ϕ, but not to the extent that the off-peak regions were negligible, so that at the level of accuracy relevant for (g − 2)μ 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 γ∗ → 3π 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− → 3π data sets. Overall, we obtain $$ {a}_{\mu}^{3\pi } $$
 |≤1.8 GeV = 46.2(6)(6) × 10−10 as our best estimate for the total 3π contribution consistent with all (low-energy) constraints from QCD. In combination with a recent dispersive analysis imposing the same constraints on the 2π channel below 1 GeV, this covers nearly 80% of the total HVP contribution, leading to $$ {a}_{\mu}^{\mathrm{HVP}} $$
 = 692.3(3.3) × 10−10 when the remainder is taken from the literature, and thus reaffirming the (g−2)μ anomaly at the level of at least 3.4σ. As side products, we find for the vacuum-polarization-subtracted masses Mω = 782.63(3)(1) MeV and Mϕ = 1019.20(2)(1) MeV, confirming the tension to the ω mass as extracted from the 2π channel.

/pdf/two-pion-contribution-to-hadronic-vacuum-polarization-505bs1gcpn.pdf

Two-pion contribution to hadronic vacuum polarization

We employ a mathematical framework based on rational approximants in order to calculate the pseudoscalar-pole piece of the hadronic light-by-light contribution to the anomalous magnetic moment of the muon, ${a}_{\ensuremath{\mu}}^{\mathrm{HLbL};P}$. The method is systematic and data based, profiting from over 13 different collaborations, and able to ascribe, for the first time, a systematic uncertainty which provides for the model independence. As a result, we obtain ${a}_{\ensuremath{\mu}}^{\mathrm{HLbL};P}=94.3(5.3)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}$, whose uncertainty is well below the one foreseen at future experiments measuring the $({g}_{\ensuremath{\mu}}\ensuremath{-}2)$.

Pseudoscalar-pole contribution to the (g μ -2 ) : A rational approach

We present the mini-proceedings of the workshops Hadronic contributions to the muon anomalous magnetic moment: strategies for improvements of the accuracy of the theoretical prediction and $(g-2)_{\mu}$: Quo vadis?, both held in Mainz from April 1$^{\rm rst}$ to 5$^{\rm th}$ and from April 7$^{\rm th}$ to 10$^{\rm th}$, 2014, respectively.

Hadronic contributions to the muon anomalous magnetic moment Workshop. $(g-2)_{\mu}$: Quo vadis? Workshop. Mini proceedings

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Hadronic contributions to the muon anomalous magnetic moment Workshop. $(g-2)_{\mu}$: Quo vadis? Workshop. Mini proceedings