Center vortex model of confinement

The dual superconductor model relies on the condensation of monopoles, codimension 3 objects to generate a confining force between quarks and/or gluons. It was recognized by a variety of people, including Nielsen and Olesen, 't Hooft, Cornwall, and Mack that condensed and percolated magnetic vortex lines can also disorder Wilson loops sufficiently to lead to an area law. Since gauge bosons are also invariant under the transfromation of the center of the group,

$$U_kA_\mu(x)U_k^\dagger +\frac{i}{g}U\partial_\mu U^\dagger=A_\mu(x),$$

if $U\in Z_N$, the pure gauge theories are really $SU(N)/Z_n$ theories. Of course, in continuum this does not mean too much, as $Z_N$ factors cannot change from point to point, due to continuity. Note, however that continuum models of these vortices have been constructed using singular gauge transformations by Faber, Greensite, and Olejnik. Still, in what follows we will use lattice gauge theory arguments.

In a $SU(N)/Z_N$ gauge theory vortices labeled by the center of the group may exist. This points to the importance of the center of the group. If the center of the group is important than one would expect that transforming the gauge theory to a center gauge, where the gauge field is brought as close as possible to the element of the center, which could be written for $SU(2)$ as

$$U_\mu=e^{ik\pi\sigma_3}=(-1)^k,\ \ \ k=0,1,$$

and for $SU(3)$ as

$$U_\mu=e^{ik\pi\lambda_8/\sqrt{3}}=e^{i2k\pi/3},\ \ \ k=0,1,2.$$

In center gauge gauge fields are completely fixed, except for the center of the gauge group. This amounts to fixing all three independent parameters of the $SU(2)$ gauge group. There are some points however, where the gauge field is diagonal. At these points only one parameter of the gauge group can be fixed (at these points the $Z_N$ symmetry of the gauge fixed theory becomes $U(1)$). The points at which the gauge field is diagonal form a codimension 2 set (2 nondiagonal components vanish). Thus the corresponding lines form strings (vortices) in 3$D$ space or surfaces in spacetime. At these points the center gauge becomes singular. One would expect that the effects of this singularity to taper off with the distance from the core. The domain that differs markedly from the vacuum configuration is called a thick vortex. Since the important objects are the $Z_N$ vortices after center gauge fixing one can project to the $Z_N$ group, just like after Abelian gauge fixing one projects to the Abelian subgroup. Then one obtains a lattice consisting of $U_\mu\in Z_N$ objects on the lattice. The theory still retains a discrete $Z_N$ gauge symmetry.

A $Z_N$ gauge theory is nothing else but a theory of vortices. A nontrivial value of an individual link variable, $U_\mu(x)=e^{i\pi\sigma_3}=-1$ has no physical significance as a gauge transformation at either ends of the link by $V=e^{i\pi\sigma_3}=-1$ can change it to $U_\mu(x)=1$. Plaquettes are, however gauge invariant. Thus the fact whether the plaquette

$$U^P_{\mu\nu}(x)=U_\mu(x)U_\nu(x+\mu)U^\dagger_\mu(x+\nu)U_\nu^\dagger(x)$$

is $1$ or -1 is a gauge invariant statement.

Consider now an elementary lattice cube in $D=3$. Its boundary consists of 6 plaquettes. The product of these 6 plaquettes is always 1, This is so because in the product every one of the 12 links in the edges of the cube appears twice. This means that the number of odd plaquettes in every cube is even, most of the time 0 and sometimes 2 and even more rarely 4 or 6. Then connecting the middle of the odd plaquettes one obtains a continuous line that must run through the lattice, either closing to itself or running out to infinity (when in a cube there are 4 frustrated plaquettes then two strings meet). In four dimensional space these strings are two dimensional surfaces. On a very cold lattice (weak gauge coupling) the plaquettes are almost uniformly equal to 1 as the action is just $-\sum U^P$. Then vortices may form only very short loops. At increasing coupling (`temperature') the average size of vortex loops increases and at a critical value the vortices percolate (a macroscopic fraction of them will rund through the whole lattice).

It is easy to show that a finite denstity of percolated vortex lines in the $Z_N$ gauge theory leads to confinement. Consider a Wilson loop in $Z_2$ gauge theory. Lay an arbitrary surface over the Wilson loop. Then the product of plaquettes, $U^P$ over the surface is equal to the value of the Wilson loop, because contributions from inner lines of the surface all cancer each other. The product of plaquettes is equal to

$$W_k(a)=(-1)^k,$$

where $k$ is the number of vortices intersecting the surface of are $a$. Clearly this is a gauge invariant quantity. If we neglect interactions of the vortices then the distribution of vortices is random. Suppose that the average density of vortices (number of vortices intersecting unit area) is $\sigma/2$. and the area of the loop is $a$, Let us embed the area $A$ into a much larger area, $A$, for which relative fluctuations can already be neglected. Suppose the expected number of vortices over the area $A$ is $n$, then the probability that exactly $k$ vortices intersects area $a$ follows a binomial distribution,

$$P_k(a)=\frac{a^k(A-a)^{n-k}}{A^n}\left(\begin{array}{c} n\\ k\end{array}\right).$$

The average over Wilson loops is

$$\langle W(a)\rangle =\sum_k W_k(a)P_k(a)=\left(\frac{A-2a}{A}\right)^n\simeq e^{-2a n/A}=e^{-\sigma a},$$

because $n/A=\sigma/2$, the density of vortices. Clearly, random vortices with a finite density imply an area law and the density of vortices is just half of the string tension.

Notice that we assumed in the above derivation that the distribution of vortices is completely random. There are two reasons why this approximation may not be correct. First of all, vortices interact. This is not terribly important as, at least in Higgs theories, one can estimate the interaction energy. This is an exponentially decreasing function of distance, controlled by the smaller of the gauge boson or Higgs mass. The second reason is more serious. We commented earlier that vortices, if not percolated form closed loops. Suppose that the average length of the loops is $2L$. Then every vortex intersecting a surface will intersect it again within a distance $L$. This means that there is no problem with Wilson loops much smaller in linear size than $L$. On the other hand if $a>>L^2$ then chances are, unless a vortex is close to the loop itself, that the vortex intersects the loop twice. A vortex intersecting the loop twice gives a contribution 1 to the Wilson loop. Thus, only vortices intersecting the loop in a strip of width $L$ around the perimeter will contribute to a nontrivial value of the Wilson loop. In effect, when calculating the average of the Wilson loop $a$ should be substituted by $sL$, where $s$ is the perimeter of the Wilson loop. Then for every loop of area $a\ge L^2$ one would obtain a perimeter law while for smaller loops $a\le L^2$ an area law. It is clear that the area law is to be the correct asymptotic behavior of Wilson loops percolated vortices are a neccessity.

The question could be asked whether it is possible to have percolated vortices at all. The energy of vortices is roughly proportional to their length. Percolated vortices on infinite lattices have infinite energy, so one would think that the probability of having such vortices is zero. This argument is, however, false due to entropy considerations. Note that vortices are undulating. Just consider a lattice vortex of $D$ dimension. Then in every step the vortex has $2^D$ choices to turn into a new direction (or if one does not allow them to turn back, this number is $2^D-1$. After $L$ steps this amounts to an entropy of $S=L\log(2^D-1)$. If the energy of the vortex is $L\rho$ then the total free energy is $F=L[\log(2^D-1)-\rho]$ (we set the `temperature' equal to 1). Clearly, there is a phase transition at $\log(2^D-1)=\rho_c$. If $\rho>\rho_c$ than the vortices do not percolate. If, however, $\rho<\rho_0$ then it is advantageous to create a new percolated vortex. It is a numerical question whether the energy of non-Abelian $Z_N$ vortices is such that they percolate. This is determined by the coupling constant of the effective $Z_N$ gauge theory obtained after center gauge fixing and center projection.