Abstract
I present a new data driven update of the hadronic vacuum polarization effects for the muon and the electron $g-2$. For the leading order contributions I find $a_\mu^{\mathrm{had}(1)}=(686.99\pm 4.21)[687.19\pm 3.48]\times 10^{-10}$ based on $e^+e^-$data [incl. $\tau$ data], $a_\mu^{\mathrm{had}(2)}= (-9.934\pm 0.091) \times 10^{-10}$ (NLO) and $a_\mu^{\mathrm{had}(3)}= (1.226\pm 0.012) \times 10^{-10}$ (NNLO) for the muon, and $a_e^{\mathrm{had}(1)}=(184.64\pm 1.21)\times 10^{-14}$ (LO), $a_e^{\mathrm{had}(2)}=(-22.10\pm 0.14)\times 10^{-14}$ (NLO) and $a_e^{\mathrm{had}(3)}=(2.79\pm 0.02)\times 10^{-14}$ (NNLO) for the electron. A problem with vacuum polarization undressing of cross-sections (time-like region) is addressed. I also add a comment on properly including axial mesons in the hadronic light-by-light scattering contribution. My estimate here reads $a_\mu[a_1,f_1',f_1] \sim ({ 7.51 \pm 2.71}) \times 10^{-11}\,.$ With these updates $a_\mu^{\rm exp}-a_\mu^{\rm the}=(32.73\pm 8.15)\times 10^{-10}$ a 4.0 $\sigma$ deviation, while $a_e^{\rm exp}-a_e^{\rm the}=(-1.10\pm 0.82)\times 10^{-12}$ shows no significant deviation.
Highlights
A well known general problem in electroweak precision physics are the higher order contributions from hadrons at low energy scales
Considering the lepton anomalous magnetic moments one distinguishes three types of non-perturbative corrections: (a) Hadronic Vacuum Polarization (HVP) of order O(α2), O(α3), O(α4); (b) Hadronic Light-by-Light (HLbL) scattering at O(α3); (c) hadronic effects at O(αGF m2μ) in 2-loop electroweak (EW) corrections, in all cases quark-loops appear as hadronic “blobs”
Evaluation of non-perturbative effects is possible by using experimental data in conjunction with Dispersion Relations (DR), by low energy effective modeling via a Resonance Lagrangian Approach (RLA) ( Vector Meson Dominance (VMD) implemented in accord with chiral structure of QCD) [1,2,3], like the Hidden Local Symmetry (HLS) or the Extended Nambu Jona-Lasinio (ENJL) models, or by lattice QCD
Summary
The next-to-leading order (NLO) HVP is represented by diagrams in figure 8. With kernels from [53], the results of an updated evaluation are presented in table 2. Diagrams are shown in figure 8 and corresponding contributions evaluated with kernels from [54] are listed in table 3. The present results ahμad[LO VP] = (6889 ± 35) × 10−11 amount to +59.09 ±0.30 ppm, which poses the major challenge. 1.0) × 10−11 and ahμad[NNLO VP] = (12.4 ± 0.1) × 10−11 relevant will be known well enough These number compare with the well established weak aEμW = (154 ± 1) × 10−11 and the problematic HLbL estimated to contribute ahμad,LbL = [(105÷106)±(26÷39)]×10−11, which is representing a +0.90 ±0.28 ppm effect. Generation experiments require a factor 4 reduction of the uncertainty optimistically feasible should be a factor 2 we hope
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