Correlators of circular Wilson loops from holography

We study the correlators of two circular Wilson loops of different radii at strong coupling. In our setup one Wilson loop is located inside the other. We use holography to calculate the connected two-point function. Both an anti-de Sitter background and a confining background are considered. As the computation for the confining case cannot be carried out analytically, we solve the problem numerically. In the anti-de Sitter case, our results agree with similar holographic calculations. In the case of a confining background, we find an asymptotic area law, in agreement with the result of the lattice strong coupling expansion. We also elaborate on the subtle issue of the interplay between connected and disconnected string world sheets.

Figure 1

Two circular Wilson loops of different radii $a>b$ located one inside the other.

Figure 2

The holographic setup. The connected correlator of two Wilson loops is given by the world sheet of an open string that terminates on the Wilson loops themselves.

Figure 3

The $(r,z)$ plane. The string is hung from ($r=b$, $z=z_{\Lambda }$) and ($r=a$, $z=z_{\Lambda }$). The tip of the string is at ($r=r_{\mathrm{IR}}$, $z=z_{\mathrm{IR}}$).

Figure 4

The $(r,z)$ plane. The confining string “rests” in close proximity of $z=z_0$.

Figure 5

Various solutions of the classical equations in the AdS background in the $(x,z)$ place. We defined the Minkowski variable $x=r\cos \varphi$ and show here a section of the world sheet of the strings.

Figure 6

Three-dimensional rendering of two of the world sheets obtained numerically in the AdS background. On the left, a leading contribution to the connected correlation function. On the right, a configuration for which the choices of $r_{\mathrm{IR}}$ and $z_{\mathrm{IR}}$ make it resemble the disconnected diagram, in which the two hemispheres are connected by a narrow throat near the origin.

Figure 7

The (subtracted) Nambu-Goto action $S$ of a set of configurations with $r_{\mathrm{IR}}=0.55$, ${\alpha }^{\prime }=1$, for $0.01<z_{\mathrm{IR}}<2$ and $z_{\Lambda }={10}^{-5}$ computed numerically as a function of the ratio of the radii $a/b$ (dots). By comparison, we show also the approximation in Eq. (14) (dashed line) and the result of the disconnected circular Wilson loops in Eq. (22) (continuum line).

Figure 8

The result of the single circular Wilson loop. In the top panel, we show the shape of the configurations in the $(x,z)$ place. In the bottom panel, we show the subtracted action $S$ obtained numerically, as a function of the radius $b$ of the loop (squares), together with the approximation valid in the case of AdS background (continuum line) and an approximation of the asymptotic behavior at large-$b$, in which the string tension is $\sigma =\frac{1}{2\pi }$ (dashed line).

Figure 9

The subtracted action $S$ in of the strings probing the confining background, as a function of $a^2-b^2$ (dots). The slope is the linear behavior with $\sigma =\frac{1}{2\pi z_0^2}$. The plot has been obtained with the numerical choices $z_0=1.5$, $r_{\mathrm{IR}}=75$ and $z_{\Lambda }={10}^{-6}$.

Figure 10

The result of the two circular Wilson loops. Top panel shows the shape of the string in the $(x,z)$ plane. The middle and bottom panels are illustrating the regularized action $S$ as a function of the smaller radius $b$ (circles). Notice that all the configurations have $a=6$ and $z_0=1$. We also show the result of the disconnected configuration with the same value for $a$ and $b$ (squares).