Numerical calculation of the full two-loop electroweak corrections to muon ($g-2$)

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 counterterms are calculated to get the renormalized quantity. As for the numerical calculation, we adopt the 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 $a_{\mu }^{\mathrm{EW}:\mathrm{OS}}[1+2\text{−}\mathrm{loop}]=151.2(\pm 1.0)\times {10}^{-11}$, 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 $a_{\mu }^{\mathrm{EW}}[1+2\text{−}\mathrm{loop}]=152.9(\pm 1.0)\times {10}^{-11}$. It is in agreement with the previous works given in PDG [Phys. Rev. D 98, 030001 (2018)] within errors.

Figure 1

Types of topology.

Figure 2

Classification of diagrams.

Figure 3

Sample diagrams with $W$.

Figure 4

All the two-loop diagrams containing IR-divergence.

Figure 5

Diagrams containing IR-divergence through $\delta Z$. The straight lines and wavy lines represent fermion and boson, respectively. The circle represent $\delta Z_{\mu }^{1/2}$ or $\delta Z_W^{1/2}$. In the case where the corner includes $\nu$-particle, both of $\delta Z_{\mu }^{1/2}$, $\delta Z_W^{1/2}$ are taken into account. The gray circles and white circles correspond to self CT and vertex CT in Table 3, respectively.

Figure 6

Diagram containing both UV- and IR-divergences. $C{T}_Z$, $C{T}_1\sim C{T}_5$ represent the CT-terms.