Remarks on the Hadronic Effect in Muon $g-2$: Low Energy Behavior of $V^0 $-$\pi^+$ Scattering

Abstract

The behavior of the $V^0$-$\pi^+$ scattering amplitude (where $V^0 = \rho^0 , \omega$ , or $\phi$) in the low pion momentum limit is studied motivated by its relevance to the theory of muon $g - 2$. Current algebra analysis shows that its S-wave component must vanish in the chiral limit under general physical assumptions. We confirm this result with various low energy models of vector meson. Our result justifies the calculation of the charged pion loop part of the hadronic light-by-light scattering contribution to muon $ g-2 $ within the framework of hidden local symmetry model of low energy hadron dynamics.