Effects of the density-dependent weak form factors on the neutrino reaction via neutral current for the nucleon in nuclear medium and $^{12}$C

The nucleon form factors in free space are usually thought to be modified when a nucleon is bound in a nucleus or immersed in a nuclear medium. We investigate effects of the density-dependent axial and weak-vector form factors on the electro-neutrino ($ u_e$) and anti-electro-neutrino $({bar u_e})$ reactions via neutral current (NC) for a nucleon in nuclear medium or $^{12}$C. For the density-dependent form factors, we exploit the quark-meson-coupling (QMC) model, and apply them to the $ u_e$ and ${bar u_e}$ induced reactions by NC. About 12% decrease of the total cross section by $ u_e$ reaction on the nucleon is obtained at normal density, $rho = rho_0 sim 0.15 {fm}^{-3} $, as well as about 18% reduction of total ${ u}_e$ cross section on $^{12}$C, by the modification of the weak form factors of the bound nucleon. However, similarly to the charged current reaction, effects of the nucleon property change in the ${bar u}_e$ reaction reduce significantly the cross sections about 30% for the nucleon in matter and $^{12}$C cases. Such a large asymmetry in the ${bar u}_e$ cross sections is addressed to originate from the different helicities of ${bar u}_e$ and ${ u}_e$.