Abstract
Numerical calculation of two-loop electroweak corrections to the muon anomalous magnetic moment ($g$-2) is done based on, on shell renormalization scheme (OS) and free quark model (FQM). The GRACE-FORM system is used to generate Feynman diagrams and corresponding amplitudes. Total 1780 two-loop diagrams and 70 one-loop diagrams composed of counter terms are calculated to get the renormalized quantity. As for the numerical calculation, we adopt trapezoidal rule with Double Exponential method (DE). Linear extrapolation method (LE) is introduced to regularize UV- and IR-divergences and to get finite values. The reliability of our result is guaranteed by several conditions. The sum of one and two loop electroweak corrections in this renormalization scheme becomes $a_\mu^{EW:OS}[1{\rm+}2{\rm -loop}]= 151.2 (\pm 1.0)\times 10^{-11}$, where the error is due to the numerical integration and the uncertainty of input mass parameters and of the hadronic corrections to electroweak loops. By taking the hadronic corrections into account, we get $a_\mu^{EW}[1{\rm+}2 {\rm -loop}]= 152.9 (\pm 1.0)\times 10^{-11}$. It is in agreement with the previous works given in PDG within errors.
Highlights
In order to get a sign of beyond the standard model physics from high precision experimental data, we need higher order radiative corrections within the Standard Model (SM)
(3) GRACE system generates the amplitude of each diagram in accordance with Feynman rules for ELWK theory with nonlinear gauge (NLG) [35]
As for counterterms, GRACE has a library of renormalization constants at one-loop level based on OSrenormalization scheme
Summary
In order to get a sign of beyond the standard model physics from high precision experimental data, we need higher order radiative corrections within the Standard Model (SM) For this purpose our group has been developing the automatic calculation system GRACE [1] since the late 1980s. The number of diagrams is huge and the numerical integration requires the big CPU-power to achieve the required accuracy, we must wait until various environments are improved During these days, several groups did the approximate calculations [13,14,15,16,17,18] and the approximate value of the twoloop ELWK correction is widely accepted [19,20]. As new experiments at FNALE989 [27] will announce their first result in 2019 and J-PARC-E034 [28] is planning the new experiment, we can expect to have new data soon
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