Dynamics of monopoles and flux tubes in two-flavor dynamical QCD

We investigate the confining properties of the QCD vacuum with $N_f=2$ flavors of dynamical quarks, and compare the results with the properties of the quenched theory. We use nonperturbatively $\mathcal{O}(a)$ improved Wilson fermions to keep cutoff effects small. We focus on color magnetic monopoles. Among the quantities we study are the monopole density and the monopole screening length, the static potential and the profile of the color electric flux tube. We furthermore derive the low-energy effective monopole action. Marked differences between the quenched and dynamical vacuum are found.

Figure 1

The histogram of closed monopole loops of length $L$ in full QCD on the ${24}^{3}48$ lattice at $\beta =5.29$, $\kappa =0.1355$. The bin size is 200.

Figure 2

The same as in Fig. 1, but on the ${16}^{3}32$ lattice at $\beta =5.29$, $\kappa =0.135$. The bin size is 60.

Figure 3

The monopole density in full and quenched QCD. The quenched results refer to $\beta =6.0$ and are listed at $m_{\pi }/m_{\rho }=1$.

Figure 4

Comparison of the Abelian and nonAbelian potential in full QCD on the ${16}^{3}32$ lattice at $\beta =5.29$, $\kappa =0.135$. The solid lines are fits of the form (15).

Figure 5

Same as in Fig. 4, but for the quenched theory at $\beta =6.0$.

Figure 6

Decomposition of the Abelian potential into monopole and photon parts on the ${16}^{3}32$ lattice at $\beta =5.29$, $\kappa =0.135$ in full QCD. The solid lines are fits of the form (15).

Figure 7

The same as in Fig. 6, but for the quenched theory at $\beta =6.0$.

Figure 8

Data and fit of the magnetic flux on the ${16}^{3}32$ lattice in full and quenched QCD.

Figure 9

The monopole density on the ${16}^{3}32$ lattice at $\beta =5.29$, $\kappa =0.135$, and its dependence on the number of Gribov copies.

Figure 10

The Abelian static potential on the ${16}^{3}32$ lattice at $\beta =5.29$, $\kappa =0.135$, and its dependence on the number of Gribov copies. Also shown is the result of the iterative gauge fixing.

Figure 11

The Abelian string tension on the ${16}^{3}32$ lattice at $\beta =5.29$, $\kappa =0.135$, and its dependence on the number of Gribov copies, together with the result of the iterative gauge fixing ($\circ$).

Figure 12

The action density ${\rho }_A^{\mathcal{W}}(s)r_0^4$ of the Abelian flux tube as a function of $x=s_1$, $y=s_2$ at $z=s_3=0$ on the ${16}^{3}32$ lattice in full (top) and quenched QCD (bottom) at $\beta =5.20$, $\kappa =0.1355$ and $\beta =6.0$, respectively.

Figure 13

The action density ${\rho }_A^{\mathcal{W}}(s)r_0^4$ of Fig. 12 plotted across the flux tube at $x=0$.

Figure 14

Distribution of the color electric field ${\overrightarrow{E}}^{\mathcal{W}}$ in full (top) and quenched QCD (bottom) in the ($x$,$y$) plane for the same lattices as in Fig. 12. The magnitude of $E^{\mathcal{W}}$ is indicated by the length of the arrows.

Figure 15

The color electric field $E_1^{\mathcal{W}}$ of Fig. 14 plotted across the flux tube at $x=0$.

Figure 16

The monopole density ${\rho }_M^{\mathcal{W}}(s)r_0^3$ as a function of $x$, $y$ at $z=0$ in full (top) and quenched QCD (bottom) for the same lattices as in Fig. 12.

Figure 17

The solenoidal monopole current $k^{\mathcal{W}}{r}_0^3$ in the ($y$,$z$) plane (i.e. perpendicular to the flux tube) at $x=0$ in full (left) and quenched QCD (right) for the same lattices as in Fig. 12.

Figure 18

Test of dual Ampère’s law in full (top) and quenched QCD (bottom) for the same lattices as in Fig. 12.

Figure 19

Monopole (top) and photon part (bottom) of the action density ${\rho }_A^{\mathcal{W}}{r}_0^4$ on the (quenched) ${16}^{3}32$ lattice at $\beta =6.0$.

Figure 20

Distribution of the monopole (top) and photon part of the color electric field ${\overrightarrow{E}}^{\mathcal{W}}$ (bottom) on the ${16}^{3}32$ lattice at $\beta =6.0$. For better visibility the monopole part is enhanced by a factor of 2 relative to the photon part.

Figure 21

Monopole and photon part of the color electric field $E_1^{\mathcal{W}}$ in the ($y$,$z$) plane at $x=0$ (left) and parallel to the flux tube (right).

Figure 22

The monopole part of the color electric field ${\overrightarrow{E}}^{\mathcal{W}}$ on the ${24}^{3}48$ lattice. Only the region $-18\le x\le 17$, $-10\le y\le 9$ is shown.

Figure 23

The monopole self-coupling $G_1$, as a function of the physical length scale $b$. The symbols are: $n=1$ ($•$),$n=2$ ($\blacksquare$), $n=3$ ($⧫$), $n=4$ ($▴$) in full QCD, and $n=1$ ($\circ$), $n=2$ ($\square$), $n=3$ ($◊$), $n=4$ ($△$) in quenched QCD.

Figure 24

The dependence of $G_1$ on the ratio $m_{\pi }/m_{\rho }$ for $b=0.09\mathrm{f}\mathrm{m}$. The entry at $m_{\pi }/m_{\rho }=1$ corresponds to the quenched theory.

Figure 25

The coupling constants $G_2$ and $G_3$ as a function of $b$. The symbols are as in Fig. 23.

Figure 26

The coupling constants $G_{28}$ and $G_{29}$ of the four-point and six-point interactions, respectively, as a function of $b$. The symbols are as in Fig. 23.

Figure 27

The two-point coupling $G$ of the monopole currents $k(s,\mu )$ and $k(s^{\prime },\mu )$ as a function of distance $d=\sqrt{\sum _{\mu }(s_{\mu }-s_{\mu }^{\prime }{)}^2}$ in full ($•$) and quenched QCD ($\blacksquare$), compared to the Coulomb propagator ($◊$).

Figure 28

The self-coupling $G_{\mathrm{s}\mathrm{e}\mathrm{l}\mathrm{f}}$ and the Coulomb coupling $g_{\mathrm{m}}$ as a function of $b$. The symbols are as in Fig. 23.