Partial restoration of chiral symmetry in the color flux tube

Using the quark eigenmodes computed on the lattice with the overlap-Dirac operator, we investigate the spatial distribution of the chiral condensate around static color sources corresponding to quark-antiquark and three-quark systems. A flux structure of chromo fields appears in the presence of such color charges. The magnitude of the chiral condensate is reduced inside the color flux, which implies partial restoration of chiral symmetry inside hadrons. Taking a static baryon source in a periodic box as a toy model of nuclear matter, we estimate the magnitude of the chiral symmetry restoration as a function of baryon matter density.

Figure 1

Eigenvalue (in MeV) of the low-lying eigenmodes on the $24^3\times 48$ lattice. By including 240 eigenmodes, we can cover the range of $\lambda \lesssim 300\mathrm{MeV}$.

Figure 2

Snapshots of (a) the local chiral condensate, (b) the action density, and (c) the topological charge distributions observed with the sum of 20 lowest-lying eigenmodes. These pictures show the same $T\text{−}X$ slice of a $24^3\times 48$ lattice on a representative gauge configuration. The chiral condensate local fluctuations are correlated with the local topological charge measurements.

Figure 3

Schematic picture of the flux-tube measurement. The static quark and antiquark are located at $(R/2,0)$ and $(-R/2,0)$ on the $XY$ plane.

Figure 4

Spatial profile of the local chiral condensate $⟨\overline{q}{q}^{(N)}(\overrightarrow{x}){⟩}_W$ around a Wilson loop $W(R,T)$ with $R=8$ and $T=4$. The positions of color sources are at $(X,Y)=(4,0)$ and $(-4,0)$, which are shown in the plot by white circles.

Figure 5

Ratio of the chiral condensate $r(x)$ for a separation $R=8$. The color sources are separated along the $X$ axis and set at $(X,Y)=(4,0)$ and $(-4,0)$. The plots show the cross section (a) along the $X$ axis at $Y=0$ and (b) along $Y$ axis at $X=0$.

Figure 6

The spatial profile of both the local chiral condensate $⟨\overline{q}{q}^{(\text{subt})}(\overrightarrow{x}){⟩}_W$ and UV Dirac mode truncated action density $⟨{\rho }^{(N)}(\overrightarrow{x}){⟩}_W$ around the color sources with a separation $R=8$ using 160 low-lying eigenmodes. For a comparison of their shape, both quantities are normalized at the origin. (a) cross section at $Y=0$. (b) cross section at $X=0$.

Figure 7

Same as Fig. 5 but with different numbers of eigenmodes included: $N=120$, 160, 200, and 240.(a) cross section at $Y=0$. (b) cross section at $X=0$.

Figure 8

Chiral condensate ratio $r(x)$ along the $X$ axis. The results with increasing separation $R$: $R=4$, 8, and 10. Color sources are located at $(R/2,0)$ and $(-R/2,0)$.

Figure 9

Ratio at the center of the flux $r(0)$ as a function of the separation $R$.

Figure 10

Cross section of the ratio along the $Y$ axis with different separations $R$ and cut points $X$.

Figure 11

A schematic picture of the construction for three-quark system with an isosceles right triangle configuration. Three Wilson lines $U_k$ correspond to the static color sources.

Figure 12

Condensate ratio $r_{3\mathrm{Q}}(\overrightarrow{x})$ with the color sources at $Q_1=(6,0)$, $Q_2=(0,6)$, and $Q_3=(0,0)$ on the $XY$ plane.

Figure 13

Chiral condensate ratio $r_{3\mathrm{Q}}(\overrightarrow{x})$ along the line of $X=Y$ with the color sources at $(R,0),(0,R)$, and (0, 0) on the $XY$ plane with $R=3$ and 6.

Figure 14

Reduction of the chiral condensate at finite density measured by $⟨\overline{q}q{⟩}_{\rho }/⟨\overline{q}q{⟩}_0$. The estimates for a given configuration of color sources, i.e., at (0, 0), $(R,0)$, and $(0,R)$ on the $XY$ plane, with $R=3$ and 6 are shown.