Many-body theory of ρ-ω mixing

We calculate the tensor describing ρ-ω mixing making use of an extended Nambu–Jona-Lasinio (NJL) model that we have developed in recent years. We use the definition of the ρ and ω fields that arises upon a momentum-space bosonization of the extended NJL model. A quantity of interest is the on-shell (${\mathit{q}}^2$=${\mathit{m}}_{\mathrm{\omega }}^2$) matrix element that describes the coupling of the ρ and ω fields, 〈ρ‖${\mathit{H}}_{\mathit{SB}}$‖ω〉. That quantity has been determined to be 〈ρ‖${\mathit{H}}_{\mathit{SB}}$‖ω〉=-(4520±600) ${\mathrm{MeV}}^2$ in a study of the two-pion decay of the ω meson. If corrected for an electromagnetic process, the strong interaction contribution is 〈ρ‖${\mathit{H}}_{\mathit{SB}}^{\mathrm{st}}$‖ω〉=-(5130±600) ${\mathrm{MeV}}^2$. Our calculation of 〈ρ‖${\mathit{H}}_{\mathit{SB}}^{\mathrm{st}}$‖ω〉 is sensitive to the difference of the current quark masses, ${\mathit{m}}_{\mathit{d}}^0$-${\mathit{m}}_{\mathit{u}}^0$. Our results may be put into agreement with the data, if we use ${\mathit{m}}_{\mathit{d}}^0$-${\mathit{m}}_{\mathit{u}}^0$=3.1±0.3 MeV. (That value is reduced to ${\mathit{m}}_{\mathit{d}}^0$-${\mathit{m}}_{\mathit{u}}^0$=2.7±0.3 MeV, if we use a subtraction scheme that causes the polarization tensor to be equal to zero at ${\mathit{q}}^2$=0.) The momentum-space bosonization procedure naturally leads to momentum-dependent coupling constants, ${\mathit{g}}_{\mathrm{\omega }\mathit{qq}}$(${\mathit{q}}^2$) and ${\mathit{g}}_{\mathrm{\rho }\mathit{qq}}$(${\mathit{q}}^2$). The value of these constants increases by about a factor of √2, when one goes from ${\mathit{q}}^2$=${\mathit{m}}_{\mathrm{\omega }}^2$ (or ${\mathit{q}}^2$=${\mathit{m}}_{\mathrm{\rho }}^2$) to ${\mathit{q}}^2$=0. The values at ${\mathit{q}}^2$=0 reproduce the strength of the relevant components of the nucleon-nucleon interaction at small momentum transfer, as was demonstrated in an earlier work. © 1996 The American Physical Society.