Wilson loops from multicentre and rotating branes, mass gaps and phase structure in gauge theories

Abstract

Within the AdS/CFT correspondence we use multicentre D3-brane metrics to investigate Wilson loops and compute the associated heavy quark-antiquark potentials for the strongly coupled SU(N) super-Yang-Mills gauge theory, when the gauge symmetry is broken by the expectation values of the scalar fields. For the case of a uniform distribution of D3-branes over a disc, we find that there exists a maximum separation beyond which there is no force between the quark and the antiquark, i.e. the screening is complete. We associate this phenomenon with the possible existence of a mass gap in the strongly coupled gauge theory. In the finite-temperature case, when the corresponding supergravity solution is a rotating D3-brane solution, there is a class of potentials interpolating between a Coulombic and a confining behaviour. However, above a certain critical value of the mass parameter, the potentials exhibit a behaviour characteristic of statistical systems undergoing phase transitions. The physical path preserves the concavity property of the potential and minimizes the energy. Using the same rotating-brane solutions, we also compute spatial Wilson loops, associated with the quark-antiquark potential in models of three-dimensional gauge theories at zero temperature, with similar results.

Figures (7)

• Figure 1: Curve (a) corresponds to the quark--antiquark potential as computed using (\ref{['comstru']}) and (\ref{['comst1']}). Curve (b) corresponds to the same potential as computed using (\ref{['lle']}) and (\ref{['eenq']}). Lengths and energies are measured in units of ${\pi R^2\over r_0}$ and $r_0$, respectively. Both curves demonstrate that there is a maximum value $L_{\rm max}$, given by (\ref{['gew']}) and (\ref{['maxx']}), where the energy becomes zero and the screening of charges is complete.
• Figure 2: Curves (a), (b) and (c) correspond to the distance between a quark and an antiquark as a function of $U_0$, as computed using (\ref{['jsa']}) for three different values of $\mu=0$, $3$ and $10$, respectively. Lengths and energies are measured in units of ${R^2\over r_0}$ and $r_0$, respectively. All three curves approach $L=0$ as $U_0\to U_H$ according to (\ref{['kpw11']}). For large $U_0$ they unify according to (\ref{['sjhd']}) irrespectively of the value of $\mu$. Curve (a) can be obtained using (\ref{['lle1']}). Curve (c) is approximately what we would obtain using (\ref{['hds']}).
• Figure 3: Curves (a), (b) and (c) correspond to the quark--antiquark potential as a function of $U_0$ as computed using (\ref{['dqp']}) for three different values of $\mu=0$, $3$ and $10$, respectively. Energies are measured in units of $r_0$. All three curves approach $U=0$ as $U_0\to U_H$ according to (\ref{['kp11w1']}). For large $U_0$ the curves become parallel as they follow (\ref{['qqD3']}) with the energy shifted in its curve by ${U_H\over \pi}$. Curve (a) can be obtained using (\ref{['eenq1']}). Curve (c) is approximately what we would obtain using (\ref{['hds1']}).
• Figure 4: Curves (a), (b) and (c) correspond to the quark--antiquark potential as computed using (\ref{['jsa']}) and (\ref{['dqp']}) for the three different values of $\mu=0$, $3$ and $10$, respectively. Lengths and energies are measured in units of ${R^2\over r_0}$ and $r_0$, respectively. For small separations $L\to 0$, all three curves approach the Coulombic law. Curve (a) can be obtained using (\ref{['lle1']}) and (\ref{['eenq1']}). Curve (c) is approximately what we would obtain using (\ref{['hds']}) and (\ref{['hds1']}).
• Figure 5: Curves (a), (b) correspond to the quark--antiquark potential as computed using (\ref{['jk1d']}) and (\ref{['jewh']}) for the two different values of $\mu=2$ and $0$, respectively, corresponding to $\lambda<\lambda_{\rm cr}\simeq 2.85$. Lengths and energies are measured in units of ${R^2\over r_0}$ and $r_0$, respectively. For small separation $L\to 0$, both curves approach the Coulombic law. For large separation we obtain a linear behaviour. Curve (b) can also be obtained using (\ref{['tora1']}) and (\ref{['hwp1']}).