Peaks of sound velocity in two color dense QCD: Quark saturation effects and semishort range correlations

We discuss stiffening of dense matter in two color QCD (${\mathrm{QC}}_2\mathrm{D}$) where hadrons are mesons and diquark baryons. We study two models which describe a transition of matter from the Bose-Einstein-condensation regime at low density to the Bardeen-Cooper-Schrieffer regime at high density. The first model is based on coherent states of diquarks, and the second is the Nambu-Jona-Lasinio model with diquark pairing terms. We particularly focus on how quark states are occupied as baryon density increases. We find that, due to the occupied quark levels, the ideal gas picture of diquarks breaks down at density significantly less than the density where baryon cores overlap. The saturation of quark states at low momenta stiffens equations of state. We also study the effects of interactions which depend on the quark occupation probability. We argue that equations of state become very stiff when the bulk part of the quark Fermi sea has the effective repulsion but the Fermi surface enjoys the attractive correlations. This disparity for different momentum domains is possible due to the strong channel dependence in gluon exchanges with momentum transfer of 0.2–1 GeV. These concepts can be transferred from ${\mathrm{QC}}_2\mathrm{D}$ to QCD in any numbers of colors.

Figure 1

$|C{|}^2$ vs $n_B$, each normalized by $n_0=0.16{\mathrm{fm}}^{-3}$. The value of $|C{|}^2$ corresponds to $n_B$ in the ideal Bose gas limit in which the composite nature is neglected or $\mathrm{\Lambda }$ is taken to be infinity.

Figure 2

The occupation probability $f_q(k)$ as a function of momentum $k$. The cases $n_B/n_0=0.5$, 1.0, 2.0, and 4.0 cases are shown.

Figure 3

The $f_q(k)/f_q^{\text{ideal}}(k)$ ratio for various $n_B$. The ratio measures how the quark Fermi sea differs from naive extrapolation of the ideal gas estimate.

Figure 4

The (dimensionless) diquark amplitude $d_{\mathit{k}}$ as a function of momentum $k$. The cases $n_B/n_0=0.5$, 1.0, 2.0, and 4.0 cases are shown.

Figure 5

$ϵ$ vs $P$ for various $C_A$. We set $\mathrm{\Lambda }=0.2\mathrm{GeV}$, $M_q=0.3\mathrm{GeV}$, and $C_S=0$. The arrows indicate the location of ${ϵ}^{\mathrm{id}\mathrm{sat}}=M_B{n}_B^{\mathrm{id}\mathrm{sat}}$. For $\mathrm{\Lambda }=0.2\mathrm{GeV}$, the ideal gas picture should have the quark saturation at $n_B^{\mathrm{id}\mathrm{sat}}\simeq 0.56n_0$.

Figure 6

The square of sound velocity $c_s^2$ as a function of $n_B$. We set $\mathrm{\Lambda }=0.2\mathrm{GeV}$, $M_q=0.3\mathrm{GeV}$, and $C_S=0$. The conformal value $c_s^2=1/3$ is also shown as a guide.

Figure 7

$ϵ$ vs $P$ for various $C_A$. We set $\mathrm{\Lambda }=0.2\mathrm{GeV}$, $M_q=0.3\mathrm{GeV}$, and $C_S=0$. For $\mathrm{\Lambda }=0.2\mathrm{GeV}$, the ideal gas picture should have the quark saturation at $ϵ\simeq 0.018\mathrm{GeV}{\mathrm{fm}}^{-3}$ ($n_B^{\mathrm{id}\mathrm{sat}}=0.56n_0$).

Figure 8

The square of sound velocity $c_s^2$ as a function of $n_B$. We set $\mathrm{\Lambda }=0.2\mathrm{GeV}$, $M_q=0.3\mathrm{GeV}$, and $C_S=C_A$. The conformal value $c_s^2=1/3$ is also shown as a guide.

Figure 9

The Dirac mass ($M$), diquark gap ($\mathrm{\Delta }$), and normalized baryon number density $n_B/n_0$ with $n_0=0.16{\mathrm{fm}}^{-3}$. The thin lines are the results for $H=0$ with which $\mathrm{\Delta }=0$.

Figure 10

The $ϵ$ vs $P$ for $H/G=1.0$ and 0.0. The results for $ϵ=3P$ and $P$ are also shown as guide lines.

Figure 11

The $n_B/n_0$ vs $c_s^2$ for $H/G=1.0$ and 0.0.

Figure 12

The occupation probability of particle states, $f_{\mathrm{p}}^{\mathrm{MF}}(k)$, for various $n_B/n_0$.

Figure 13

The occupation probability of antiparticle states, $f_{\mathrm{a}}^{\mathrm{MF}}(k)$, for various $n_B/n_0$.

Figure 14

The relations among $f_{\mathit{q}}(k)$ and EOS. When $f_{\mathit{q}}$ evolves as in an ideal baryon gas picture, then the quark saturation radically stiffens EOS and makes a peak in the sound velocity $c_s^2=\mathrm{d}P/\mathrm{d}ϵ$. If the baryon gas picture breaks down at lower density, stiffening mildly occurs from low density and the peak structure is modest or absent.