Spatial structures of QCD pair condensates at various densities

The spatial structures of the chiral symmetry $\text{breaking}\left(\chi \mathrm{SB}\right)-\text{quark}-\mathrm{antiquark}\left(q\overline{q}\right)$ pair and two-color $\text{superconducting(2SC)}-\text{quark}-\mathrm{quark}\left(qq\right)$, $\text{antiquark}-\mathrm{antiquark}\left(\overline{q}\overline{q}\right)$, and $\text{hole}-\mathrm{hole}\left(hh\right)$ pairs are investigated. At low density, it is found that the $q\overline{q}$ pair is well localized with average bond length of the order $1\mathrm{fm}$. It is then suggested that the pions, which are excitations arising from flavor-space phase fluctuations, exhibit the spatial structure of the underlying $q\overline{q}$-paired ground state. At intermediate density where the quarks form a 2SC state, it is found that the $qq$ and $hh$ pairs are extended and oscillating in real space while the $\overline{q}\overline{q}$ pair remains well localized.

Figure 1

Probability density $P\left(\tau \right)$ vs. bond radius $\tau$ for $\nu =1$. The average bond radius is of the order $1\mathrm{fm}$ which is the same order of magnitude as the pion radius. Note that $P\left(\tau \right)$ vanishes when $\tau =0$. That is, the $q$ and $\overline{q}$ that form the pair do not coincide in real space.

Figure 2

Probability density $P\left(\tau \right)$ vs. bond radius $\tau$ for $\nu =0.5$. The average bond radius is of the order $1\mathrm{fm}$ which is the same order of magnitude as the pion radius. Note that $P\left(\tau \right)$ vanishes when $\tau =0$. That is, the $q$ and $\overline{q}$ that form the pair do not coincide in real space.

Figure 3

Probability density $P\left(\tau \right)$ vs. bond radius $\tau$. The average bond radii for $qq, \overline{q}\overline{q}$, and $hh$, are 11, 4, and $15\mathrm{fm},$ respectively. The interparticle distance at this density is $\approx 10\mathrm{fm}$. The $qq$ and $hh$ pairs form oscillating and extended wave functions while the $\overline{q}\overline{q}$ pair remains well localized. The oscillations can be attributed to the existence of sharp Fermi surfaces.

Figure 4

Probability density $P\left(\tau \right)$ vs. bond radius $\tau$. The average bond radius for quarks is $12\mathrm{fm}$. The interparticle distance at this density is $\approx 3\mathrm{fm}$. Note that the $qq$ and $hh$ pairs form oscillating and extended wave functions while the $\overline{q}\overline{q}$ pair remains well localized. The oscillations can be attributed to the existence of sharp Fermi surfaces.