A Schematic Model for $\rho$-$a_1$ Mixing at Finite Density and In-Medium Effective Lagrangian

Abstract

Based on schematic two-level models extended to $a_1$-meson degrees of freedom, we investigate possible mechanisms of chiral restoration in the vector/axialvector channels in cold nuclear matter. In the first part of this article we employ the massive Yang-Mills framework to construct an effective chiral Lagrangian based on low-energy mesonic modes at finite density. The latter are identified through nuclear collective excitations of `meson'-sobar type such as $\pi \leftrightarrow [\Delta (1232)N^{-1}]\equiv \hat{\pi }$, $\rho \leftrightarrow [N^{\ast }(1520)N^{-1}]\equiv \hat{\rho }$, etc.. In a mean-field type treatment the in-medium gauge coupling $\hat{g}$, the (axial-) vector meson masses and ${\hat{f}}_{\pi }$ are found to decrease with density indicating the approach towards chiral restoration in the language of in-medium effective fields. In the second part of our analysis we evaluate the (first) in-medium Weinberg sum rule which relates vector and axialvector correlators to the pion decay constant. Using in-medium $\rho$/$a_1$ spectral functions (computed in the two-level model) also leads to a substantial reduction of the pion decay constant with increasing density.

Comments: 12 pages, 6 figures