Effect of an imaginary part of the Schwinger-Dyson equation at finite temperature and density

We examined the effect of an imaginary part of the ladder approximation Schwinger-Dyson equation. We show that the imaginary part enhances the effect of the first-order transition and affects a tricritical point. In particular, a chemical potential at the tricritical point is displaced about 200 MeV. Thus, one should not ignore the imaginary part. However, because the imaginary part is small away from the tricritical point, one should be able to ignore the imaginary part. In addition, we also examined the contribution of the wave-function renormalization constant.

Figure 1

The chemical potential dependence of $\mathrm{Re}{B}_0\left(0\right)$ in cases (A1) and (B1). To facilitate visualization, we depict the results of $V(B\ne 0)-V(B=0)\le 0$.

Figure 2

The chemical potential dependence of $V(B\ne 0)-V(B=0)$ in cases (A1) and (B1).

Figure 3

The chemical potential dependence of $B_0\left(0\right)$ in case (A1). $T=0.135$ and 0.14 GeV.

Figure 4

The chemical potential dependence of $V(B\ne 0)-V(B=0)$ in case (A1). $T=0.14$ GeV.

Figure 5

The temperature dependence of $B_0\left(0\right)$ at $\mu =0$ in case (A1).

Figure 6

The temperature dependence of $V(B\ne 0)-V(B=0)$ at $\mu =0$ in case (A1).

Figure 7

The chemical potential dependence of $\mathrm{Re}{B}_0\left(0\right)$ in cases (A2) and (B2).

Figure 8

The chemical potential dependence of $V(B\ne 0)-V(B=0)$ in cases (A2) and (B2).

Figure 9

Tricritical points in cases (A1), (A2), (B1), and (B2). The crosses indicate each tricritical point. The other symbols are the phase transition points of first order.

Figure 10

The chemical potential dependence of $C_0\left(0\right)$ and $A_0\left(0\right).T=0.14$ GeV.

Figure 11

The difference of $\mathrm{Re}{B}_0\left(0\right)$ and $\mathrm{Im}{B}_0\left(0\right)$. Critical points are ($100,268$), ($125,133$), and ($140,48$) MeV.

Figure 12

The angular dependence of $\mathrm{Re}{\langle \overline{\psi }\psi \rangle }_{\phi }$ and $\mathrm{Im}{\langle \overline{\psi }\psi \rangle }_{\phi }.T=150$ MeV. $\mu =120$ MeV.

Figure 13

The temperature dependence of the chiral condensate ${\langle \overline{\psi }\psi \rangle }_{\pi }$, the dressed Polyakov loop ${\Sigma }_1$, and $\partial {\Sigma }_1/\partial T.{\langle \overline{\psi }\psi \rangle }_{\pi }=0$ indicates the chiral symmetry restoration. ${\langle \overline{\psi }\psi \rangle }_{\pi }=0$ and the peak of $\partial {\Sigma }_1/\partial T$ is identical within 1 MeV.

Figure 14

The chemical potential dependence of the chiral condensate, the dressed Polyakov loop ${\Sigma }_{\pm 1}$, and $\partial {\Sigma }_{\pm 1}/\partial \mu$ at $T=150$ MeV. The critical chemical potential obtained by the effective potential for the chiral phase transition is 127 MeV.