Supersymmetry and $R$-symmetry breaking in metastable vacua at finite temperature and density

We study a metastable supersymmetry-breaking vacuum in a generalized O’Raifeartaigh model at finite temperature and chemical potentials. Fields in the generalized O’Raifeartaigh model possess different $R$-charges to realize $R$-symmetry breaking. Accordingly, at finite density and temperature, the chemical potentials have to be introduced in a nonuniform way. Based on the formulation elaborated in our previous work, we study the one-loop thermal effective potential including chemical potentials in the generalized O’Raifeartaigh model. We perform a numerical analysis and find that the $R$-symmetry-breaking vacua, which exist at zero temperature and zero chemical potential, are destabilized for some parameter regions. In addition, we find that there are parameter regions where new $R$-symmetry-breaking vacua are realized even at high temperature by the finite density effects.

Figure 1

The plot shows the parameter region in the $(r,y)$ plane where $R$-symmetry is broken (shown in black). The left region of the plot (shown in light grey) shows metastable vacua at $X=0$, and $R$-symmetry is preserved at the vacuum. In the right-upper region (shown in dark grey), the potential monotonically decreases for (3.8).

Figure 2

Plots of allowed regions (shown in black) where the $R$-symmetry is broken at metastable vacuum for (a) $T=0.2$, (b) $T=0.5$, and (c) $T=0.8$ with $\mathrm{\Lambda }=1$. The left region of each plot (shown in light grey) shows the metastable vacuum at $X=0$, and $R$-symmetry is preserved at the vacuum. In the right-upper region (shown in dark grey) the potential monotonically decreases for (3.8).

Figure 3

Plots of the potentials for $(r,y)=(1.2,0.4)$ and $T=0.2$ (left), $(r,y)=(1.3,0.2)$ and $T=0.5$ (middle), and $(r,y)=(2.6,0.4)$ and $T=0.8$ (right).

Figure 4

Plots of the potentials for $(r,y)=(5,0.2)$ and $T=0.2$ (left), $(r,y)=(4,0.24)$ and $T=0.5$ (middle), and $(r,y)=(6,0.24)$ and $T=0.8$ (right).

Figure 5

Plots of the potentials for $T=0.2$ (left), $T=0.5$ (middle), and $T=0.8$ (right) with $(r,y)=(6,0.01)$.