Scalar and pseudoscalar meson pole terms in the hadronic light-by-light contributions to $a_{\mu }^{had}$

Abstract

Third QED order hadronic light-by-light (LBL) contributions $a_{\mu }^{LBL}(M)$ to the anomalous magnetic moment of the muon $a_{\mu }^{had}$ from the pole terms of scalar $\sigma$, $a_0(980)$ and pseudoscalar ${\pi }^0$, $\eta$, ${\eta }^{\prime }$ mesons (M) in the framework of the linearized extended Nambu-Jona-Lasinio model are evaluated. The off-shell structure of the photon-photon-meson vertices is taken into account by means of constituent quark triangle loops. The mass of the quark is taken to be $m_u=m_d=m_q=(280\pm 20)$ MeV. The unknown strong coupling constants of ${\pi }^0,\eta ,{\eta }^{\prime }$ and $a_0$ mesons with quarks are evaluated in a comparison of the corresponding theoretical two-photon widths calculated in the framework of our approach with experimental ones. The $\sigma$-meson coupling constant is taken to be equal to ${\pi }_0$-meson coupling constant as it follows from the linearized Nambu-Jona-Lasinio model Lagrangian. Then one obtains $a_{\mu }^{LBL}({\pi }_0)$=$(81.83\pm 16.50)\times 10^{-11}$, $a_{\mu }^{LBL}(\eta )$=$(5.62\pm 1.25)\times 10^{-11}$, $a_{\mu }^{LBL}({\eta }^{\prime })$=$(8.00\pm 1.74)\times 10^{-11}$, $a_{\mu }^{LBL}(\sigma )$=$(11.67\pm 2.38)\times 10^{-11}$ and $a_{\mu }^{LBL}(a_0)$=$(0.62\pm 0.24)\times 10^{-11}$. The total contribution of meson poles in LBL is $a_{\mu }^{LBL}(M)$=$(107.74\pm 16.81)\times 10^{-11}$.

Comments: 16 LaTeX pages, 2 figures, revised paper