Chemical potential and the gap equation

In general, the kernel of QCD’s gap equation possesses a domain of analyticity upon which the equation’s solution at nonzero chemical potential is simply obtained from the in-vacuum result through analytic continuation. On this domain the single-quark number- and scalar-density distribution functions are $\mu$ independent. This is illustrated via two models for the gap equation’s kernel. The models are alike in concentrating support in the infrared. They differ in the form of the vertex, but qualitatively the results are largely insensitive to the Ansatz. In vacuum both models realize chiral symmetry in the Nambu-Goldstone mode, and in the chiral limit, with increasing chemical potential, they exhibit a first-order chiral symmetry restoring transition at $\mu \approx M(0)$, where $M(p^2)$ is the dressed-quark mass function.

Figure 1

Schwinger function in Eq. (32) computed from the gap equation defined via Eqs. (22, 23, 30). Upper panel—from bottom to top, $\mu =0$, 0.4, 0.5 GeV. Solid curves: our numerical results. Dashed curves: fit obtained via Eq. (38), with $M$, $\Gamma$ given in Table . Lower panel—${\mu }_a<\mu =0.53$, 0.54, 0.55 GeV.

Figure 2

Evolution with $\mu$ of the dimensionless quantities $A(p=0;\mu )$, $C(p=0;\mu )$, and $M(p=0;\mu )$ measured in GeV, all computed from the gap equation defined via Eqs. (22, 23, 30).

Figure 3

Chemical potential dependence of the vacuum quark condensate computed from the solutions of the gap equation defined via Eqs. (22, 23, 30).

Figure 4

Difference in pressure, Eq. (46), between the Nambu-Goldstone and Wigner solutions of the gap equation defined through Eqs. (22, 23, 30). It is noteworthy that $\mathcal{B}(0)=(0.16\mathrm{GeV}{)}^4$ since a typical phenomenological value for the bag constant is $\sim (0.15\mathrm{GeV}{)}^4$.

Figure 5

Upper panel—Single-particle number-density distribution function, $f_1(|\overrightarrow{p}|,\mu )$, introduced in Eq. (17), computed at $\mu =0.38$, 0.42 from the Wigner solution of the gap equation defined through Eqs. (22, 23, 30). This function is identically zero for $\mu <{\mu }_c^{\chi \mathrm{rb}}$. Lower panel—Single-particle scalar-density distribution function, $f_2(|\overrightarrow{p}|,\mu )$, introduced in Eq. (20), computed at $\mu =0.3$ from the Nambu-Goldstone solution of the aforementioned gap equation. This function is $\mu$ independent for $\mu <{\mu }_c^{\chi \mathrm{rb}}$ and identically zero otherwise.

Figure 6

Schwinger function in Eq. (32) computed from the gap equation defined via Eqs. (22, 24, 30). Solid curves—top to bottom, $\mu =0$, 0.2, 0.25 GeV. Dotted curve—$\mu =0.31\mathrm{GeV}$.

Figure 7

Chiral susceptibilities defined in Eq. (50) and computed from the gap equation defined via Eqs. (22, 24, 30). Upper panel—Nambu-Goldstone solution; lower panel—Wigner solution.

Figure 8

Chemical potential dependence of the chiral-limit vacuum quark condensate computed from the gap equation defined via Eqs. (22, 24, 30). Recall that ${\mu }_a^{\mathrm{bc}}=0.30\mathrm{GeV}$.

Figure 9

Upper panel—Single-fermion number-density distribution function, $f_1(|\overrightarrow{p}|,\mu )$, computed from the Wigner solution of the gap equation defined via Eqs. (22, 24, 30); lower panel—single-fermion scalar-density distribution function computed from this gap equation.