Toichiro Kinoshita’s research while affiliated with Cornell University and other places

The anomalous magnetic moment of the electron ae measured in a Penning trap occupies a unique position among high precision measurements of physical constants in the sense that it can be compared directly with the theoretical calculation based on the renormalized quantum electrodynamics (QED) to high orders of perturbation expansion in the fine structure constant α, with an effective parameter α/π. Both numerical and analytic evaluations of ae up to (α/π)4 are firmly established. The coefficient of (α/π)5 has been obtained recently by an extensive numerical integration. The contributions of hadronic and weak interactions have also been estimated. The sum of all these terms leads to ae(theory) = 1159652181.606(11)(12)(229)×10−12, where the first two uncertainties are from the tenth-order QED term and the hadronic term, respectively. The third and largest uncertainty comes from the current best value of the fine-structure constant derived from the cesium recoil measurement: α−1(Cs)=137.035999046(27). The discrepancy between ae(theory) and ae((experiment)) is 2.4σ. Assuming that the standard model is valid so that ae(theory) = ae(experiment) holds, we obtain α−1(ae)=137.0359991496(13)(14)(330), which is nearly as accurate as α−1(Cs). The uncertainties are from the tenth-order QED term, hadronic term, and the best measurement of ae, in this order.

In order to improve the theoretical prediction of the electron anomalous magnetic moment $a_e$ we have carried out a new numerical evaluation of the 389 integrals of Set V, which represent 6,354 Feynman vertex diagrams without lepton loops. During this work, we found that one of the integrals, called X024, was given a wrong value in the previous calculation due to an incorrect assignment of integration variables. The correction of this error causes a shift of $-1.25$ to the Set~V contribution, and hence to the tenth-order universal (i.e., mass-independent) term $A_1^{(10)}$. The previous evaluation of all other 388 integrals is free from errors and consistent with the new evaluation. Combining the new and the old (excluding X024) calculations statistically, we obtain $7.606~(192) (\alpha/\pi)^5$ as the best estimate of the Set V contribution. Including the contribution of the diagrams with fermion loops, the improved tenth-order universal term becomes $A_1^{(10)}=6.678~(192)$. Adding hadronic and electroweak contributions leads to the theoretical prediction $a_e (\text{theory}) =1~159~652~182.032~(720)\times 10^{-12}$. From this and the best measurement of $a_e$, we obtain the inverse fine-structure constant $\alpha^{-1}(a_e) = 137.035~999~1491~(331)$. The theoretical prediction of the muon anomalous magnetic moment is also affected by the update of QED contribution and the new value of $\alpha$, but the shift is much smaller than the theoretical uncertainty.

In order to improve the theoretical prediction of the electron anomalous magnetic moment $a_e$ we have carried out a new numerical evaluation of the 389 integrals of Set V, which represent 6,354 Feynman vertex diagrams without lepton loops. During this work, we found that one of the integrals, called X024, was given a wrong value in the previous calculation due to an incorrect assignment of integration variables. The correction of this error causes a shift of $-1.25$ to the Set~V contribution, and hence to the tenth-order universal (i.e., mass-independent) term $A_1^{(10)}$. The previous evaluation of all other 388 integrals is free from errors and consistent with the new evaluation. Combining the new and the old (excluding X024) calculations statistically, we obtain $7.587~(193) (\alpha/\pi)^5$ as the best estimate of the Set V contribution. Including the contribution of the diagrams with fermion loops, the improved tenth-order universal term becomes $A_1^{(10)}=6.678~(192)$. Adding hadronic and electroweak contributions leads to the theoretical prediction $a_e (\text{theory}) =1~159~652~182.032~(720)\times 10^{-12}$. From this and the best measurement of $a_e$, we obtain the inverse fine-structure constant $\alpha^{-1}(a_e) = 137.035~999~1491~(331)$. The theoretical prediction of the muon anomalous magnetic moment is also affected by the update of QED contribution and the new value of $\alpha$, but the shift is much smaller than the theoretical uncertainty.

DOI:https://doi.org/10.1103/PhysRevD.96.019901

This paper presents a detailed account of evaluation of the electron anomalous magnetic moment a_e which arises from the gauge-invariant set, called Set V, consisting of 6354 tenth-order Feynman diagrams without closed lepton loops. The latest value of the sum of Set V diagrams evaluated by the Monte-Carlo integration routine VEGAS is 8.726(336)(\alpha/\pi)^5, which replaces the very preliminary value reported in 2012. Combining it with other 6318 tenth-order diagrams published previously we obtain 7.795(336)(\alpha/\pi)^5 as the complete mass-independent tenth-order term. Together with the improved value of the eighth-order term this leads to a_e(theory)=1 159 652 181.643 (25)(23)(16)(763) \times 10^{-12}, where first three uncertainties are from the eighth-order term, tenth-order term, and hadronic and elecroweak terms. The fourth and largest uncertainty is from \alpha^{-1}=137.035 999 049(90), the fine-structure constant derived from the rubidium recoil measurement. a_e(theory) and a_e(experiment) agree within about one \sigma. Assuming the validity of the standard model, we obtain the fine-structure constant \alpha^{-1}(a_e)=137.035 999 1570(29)(27)(18)(331), where uncertainties are from the eighth-order term, tenth-order term, hadronic and electroweak terms, and the measurement of a_e. This is the most precise value of \alpha available at present and provides a stringent constraint on possible theories beyond the standard model.

This paper presents the current status of the theory of electron anomalous magnetic moment ae ≡(g-2)/2, including a complete evaluation of 12,672 Feynman diagrams in the tenth-order perturbation theory. To solve this problem, we developed a code-generator which converts Feynman diagrams automatically into fully renormalized Feynman-parametric integrals. They are evaluated numerically by an integration routine VEGAS. The preliminary result obtained thus far is 9.16 (58) (α/π)⁵, where (58) denotes the uncertainty in the last two digits. This leads to ae(theory) = 1.159 652 181 78 (77) ×10⁻³, which is in agreement with the latest measurement ae (exp:2008) = 1.159 652 180 73 (28) ×10⁻³. It shows that the Feynman–Dyson method of perturbative QED works up to the precision of 10⁻¹².

In this review, we summarize the results of our numerical work carried out over nearly ten years on the complete determination of the 10th-order contribution to the anomalous magnetic moments of leptons in the perturbation theory of quantum electrodynamics. Our approach is based on a reorganized renormalization method in which no divergent quantities appear explicitly in any part of the calculation, which is crucial for the feasibility of numerical integration. The enormous number of 10th-order diagrams and the complexity of the renormalization procedure are such that we could not have handled this problem without the development of an automated code-generating algorithm. The systematic approach to these problems is described in some detail.

We report the result of our calculation of the complete tenth-order QED terms of the muon g-2. Our result is a_\mu^{(10)} = 753.29 (1.04) in units of (\alpha/\pi)^5, which is about 4.5 s.d. larger than the leading-logarithmic estimate 663 (20). We also improved the precision of the eighth-order QED term of a_\mu, obtaining a_\mu^{(8)} = 130.8794(63) in units of (\alpha/\pi)^4. The new QED contribution is a_\mu(QED) = 116 584 718 951 (80) \times 10^{-14}, which does not resolve the existing discrepancy between the standard-model prediction and measurement of a_\mu.

This paper presents the complete QED contribution to the electron g-2 up to the tenth order. With the help of the automatic code generator, we have evaluated all 12672 diagrams of the tenth-order diagrams and obtained 9.16 (58)(\alpha/\pi)^5. We have also improved the eighth-order contribution obtaining -1.9097(20)(\alpha/\pi)^4, which includes the mass-dependent contributions. These results lead to a_e(theory)=1 159 652 181.78 (77) \times 10^{-12}. The improved value of the fine-structure constant \alpha^{-1} = 137.035 999 174 (35) [0.25 ppb] is also derived from the theory and measurement of a_e.

This paper reports the tenth-order QED contribution to the lepton g-2 from the gauge-invariant set, called Set III(c), which consists of 390 Feynman vertex diagrams containing an internal fourth-order light-by-light-scattering subdiagram. The mass-independent contribution of Set III(c) to the electron g-2 (a_e) is 4.9210(103) in units of (alpha/pi)^5. The mass-dependent contributions to a_e from diagrams containing a muon loop is 0.00370(37) (alpha/pi)^5. The tau-lepton loop contribution is negligible at present. Altogether the contribution of Set III(c) to a_e is 4.9247 (104) (alpha/pi)^5. We have also evaluated the contribution of the closed electron loop to the muon g-2 (a_mu). The result is 7.435(134) (alpha/pi)^5. The contribution of the tau-lepton loop to a_mu is 0.1999(28)(alpha/pi)^5. The total contribution of variousleptonic loops (electron, muon, and tau-lepton) of Set III(c) to a_mu is 12.556 (135) (alpha/pi)^5.