Vector-Meson Interaction Hamiltonian

Abstract

The vector mesons are described in terms of the antisymmetric second-rank tensor field $T_{\mu \nu }$. The vector-meson-baryon interaction Lagrangian, ${\mathbf{L}}_{\mathrm{Int}}=\frac{1}{2}{g}_1\overline{\mathbf{B}}{\sigma }_{\mu \nu }{\mathbf{T}}_{\mu \nu }\mathbf{B}+i{g}_2\overline{\mathbf{B}}{\gamma }_{\nu }{\partial }_{\mu }{\mathbf{T}}_{\mu \nu }\mathbf{B}=\frac{1}{2}{\mathbf{J}}_{\mu \nu }{\mathbf{T}}_{\mu \nu }+{\mathbf{J}}_{\nu }{\partial }_{\mu }{\mathbf{T}}_{\mu \nu }$, gives the following interaction Hamiltonian, in the interaction representation: $H_{\mathrm{Int}}=-L_{\mathrm{Int}}+\frac{1}{4}\left[\right(\frac{1}{m_T^{}{}_{}{}^2})J_{\mu \nu }{J}_{\mu \nu }+2\mathrm{}{J}_{\mu }{J}_{\mu }]+\frac{1}{2}\left[\right(\frac{1}{m_{\mathrm{T}}^{}{}_{}{}^2})J_{\mu \rho }{J}_{\mu \sigma }+J_{\rho }{J}_{\sigma }]n_{\rho }{n}_{\sigma }$. The normal dependent terms which appear in the $S$ matrix vanish to any order in perturbation theory, and the $S$-matrix elements can be calculated by using the effective interaction Hamiltonian ${\left(H_{\mathrm{Int}}\right)}_{\mathrm{eff}}=-L_{\mathrm{Int}}+\frac{1}{4}\left[\right(\frac{1}{m_T^{}{}_{}{}^2})J_{\mu \nu }{J}_{\mu \nu }+2\mathrm{}{J}_{\mu }{J}_{\mu }]$.