The $\eta$ decay into 3$\pi$ in asymmetric nuclear medium

We explore how the $\eta$-$\pi^0$ mixing angle and the $\eta$ meson decay into $\pi^{+}\pi^{-}\pi^0$ and 3$\pi^{0}$ are modified in the nuclear medium on the basis of the in-medium chiral effective theory with the isospin asymmetry $\alpha$ varied, where $\alpha\equiv \delta\rho/\rho$ with $\delta \rho=\rho_n-\rho_p$ and $\rho=\rho_n+\rho_p$. We find that the larger the isospin asymmetry $\delta \rho$ and smaller the total density $\rho$, the more enhanced the mixing angle. We show that the decay width in the nuclear medium has an additional density dependence which cannot be renormalized into that of the mixing angle: The additional term originates from the vertex proportional to a low energy constant $c_1$, which only comes into play in the nuclear medium but not in the free space. It turns out that the resultant density effect on the decay widths overwhelms that coming from the isospin asymmetry, and the higher the $\rho$, the more enhanced the decay widths; the width for the $\pi^{+}\pi^{-}\pi^0$ decay is enhanced with a factor two to three at the normal density $\rho_0$ with a minor increase due to $\delta \rho$, while that for the 3$\pi^0$ decay shows only a small increase of around 10 percent even at $\rho_0$. We mention the possible relevance of the partial restoration of chiral symmetry to the unexpected density effect on the decay widths in the nuclear medium.