Inhomogeneous chiral condensate in the quark-meson model

The two-flavor quark-meson model is used as a low-energy effective model for QCD to study inhomogeneous chiral condensates at finite baryon chemical potential ${\mu }_B$. The parameters of the model are determined by matching the meson and quark masses, and the pion decay constant to their physical values using the on-shell and modified minimal subtraction schemes. Using a chiral-density wave ansatz for the inhomogeneity, we calculate the effective potential in the mean-field approximation and the result is completely analytic. The size of the inhomogeneous phase depends sensitively on the pion mass and whether one includes the vacuum fluctuations or not. Finally, we briefly discuss the mean-field phase diagram.

Figure 1

Left: Dispersion relation $E_{-}$ for $\frac{q}{2}<\mathrm{\Delta }$. The horizontal orange line is for $\mu >\mathrm{\Delta }-\frac{q}{2}$. Right: Dispersion relation $E_{-}$ for $\frac{q}{2}<\mathrm{\Delta }$. The horizontal green line is for the case $\mu >\frac{q}{2}-\mathrm{\Delta }$ and the horizontal orange line is for the case $\mu <\frac{q}{2}-\mathrm{\Delta }$. See main text for discussion of the regions of integration in the different cases.

Figure 2

The phase diagram in the $\mu \text{−}T$ plane for $m_q=300\mathrm{MeV}$ and $m_{\sigma }=600\mathrm{MeV}$ in the chiral limit without quantum fluctuations. A dashed line indicates a second-order transition, while a solid line indicates a first-order transition. The region between the red lines is the inhomogeneous phase.

Figure 3

The phase diagram in the $\mu \text{−}T$ plane for $m_q=300\mathrm{MeV}$ and $m_{\sigma }=600\mathrm{MeV}$ in the chiral limit including vacuum fluctuations. A dashed line indicates a second-order transition, while a solid line indicates a first-order transition. The region between the red lines is the inhomogeneous phase.

Figure 4

Gap $\mathrm{\Delta }$ (solid blue line) and wave vector $q$ (dashed red line) as functions of the quark chemical potential $\mu$ in the chiral limit, at $T=0$, and for $m_{\sigma }=2m_q=600\mathrm{MeV}$. Left panel is without vacuum fluctuations and right panel with vacuum fluctuations.

Figure 5

Gap $\mathrm{\Delta }$ (solid blue line) and wave vector $q$ (dashed red line) as functions of the quark chemical potential $\mu$ at the physical point for $T=0$ and $m_{\sigma }=2m_q=600\mathrm{MeV}$. Left panel is without vacuum fluctuations and right panel with vacuum fluctuations.