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Advances in High Energy Physics

Volume 2017 (2017), Article ID 8276534, 7 pages

https://doi.org/10.1155/2017/8276534

## Heavy Quark Potential with Hyperscaling Violation

^{1}School of Mathematics and Physics, China University of Geosciences, Wuhan 430074, China^{2}Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE), Central China Normal University, Wuhan 430079, China

Correspondence should be addressed to Zi-qiang Zhang

Received 19 December 2016; Accepted 23 January 2017; Published 6 March 2017

Academic Editor: Song He

Copyright © 2017 Zi-qiang Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

We investigate the behavior of the heavy quark potential in the backgrounds with hyperscaling violation. The metrics are covariant under a generalized Lifshitz scaling symmetry with the dynamical Lifshitz parameter and hyperscaling violation exponent . We calculate the potential for a certain range of and and discuss how it changes in the presence of the two parameters. Moreover, we add a constant electric field to the backgrounds and study its effects on the potential. It is shown that the heavy quark potential depends on the nonrelativistic parameters. Also, the presence of the constant electric field tends to increase the potential.

#### 1. Introduction

AdS/CFT [1–3], which relates a -dimensional quantum field theory with its dual gravitational theory, living in dimensions, has yielded many important insights into the dynamics of strongly coupled gauge theories. For reviews, see [4–15] and references therein.

Due to the broad application of this characteristic, many authors have considered the generalizations of the metrics dual to field theories. One of such generalizations is to use metric with hyperscaling violation. Usually, the metric is considered to be an extension of the Lifshitz metric and has a generic Lorentz violating form [16–20]. As we know, Lorentz symmetry represents a foundation of both general relativity and the standard model, so one may expect new physics from Lorentz invariance violation. For that reason, the metrics with hyperscaling violation have been used to describe the string theory [21–25] and holographic superconductors [26–29] as well as QCD [30–32].

The heavy quark potential of QCD is an important quantity that can probe the confinement mechanism in the hadronic phase and the meson melting in the plasma phase. In addition, it has been measured in great detail in lattice simulations. The heavy quark potential for SYM theory was first obtained by Maldacena in his seminal work [33]. Interestingly, it is shown that for the space the energy shows a purely Coulombian behavior which agrees with a conformal gauge theory. This proposal has attracted lots of interest. After [33], there are many attempts to address the heavy quark potential from the holography. For example, the potential at finite temperature has been studied in [34, 35]. The subleading order correction to this quantity is discussed in [36, 37]. The potential has also been investigated in some AdS/QCD models [38, 39]. Other important results can be found, for example, in [40–44].

Although the theories with hyperscaling violation are intrinsically nonrelativistic, we can use them as toy models for quarks from the holography point of view. In addition, one can expect that the results obtained from these theories provide qualitative insights into analogous questions in QCD. In this paper, we will investigate the heavy quark potential in the Lifshitz backgrounds with hyperscaling violation. We want to know what will happen to the potential if we have the quark-antiquark pair in such backgrounds? More specifically, we would like to see how the potential changes in the presence of the nonrelativistic parameters. In addition, we will add a constant electric field to the backgrounds and study how it affects the potential. These are the main motivations of the present work.

We organize the paper as follows. In the next section, the backgrounds of the hyperscaling violation theories in [25] are briefly reviewed. In Section 3, we study the heavy quark potential in these backgrounds in terms of the and parameters. In Section 4, we investigate a constant electric field effect on the heavy quark potential. The last part is devoted to conclusion and discussion.

#### 2. Hyperscaling Violation Theories

Let us begin with a brief review of the background in [25]. It has been argued that the Lorentz invariance is broken in this background metric. Although charge densities induce a trivial (gapped) behavior at low energy/temperature, there still exist special cases where there are nontrivial IR fixed points (quantum critical points) where the theory is scale invariant. Usually, the metric is expressed as [21]where with being the IR scale. The above metric is covariant under a generalized Lifshitz scaling symmetry; that is, where is called the dynamical Lifshitz parameter or the dynamical critical exponent which characterizes the behavior of system near the phase transition. stands for the hyperscaling violation exponent which is responsible for the nonstand scaling of physical quantities and controls the transformation of the metric. The scalar curvature of these geometries is

The geometries are flat when and . The geometry is Ricci flat when and . The geometry is in Rindler coordinates when and . Usually, the above special solutions violate the Gubser bound conditions [25]. In addition, the pure Lifshitz case is related to .

By using a radical redefinition and rescaling and , we have the following metric: with .

In the presence of hyperscaling violations, the energy scale is

For the generalized scaling solutions of (5), the Gubser bound conditions are as follows:

Also, to consider the thermodynamic stability, one needs

More discussions about other generalized Lifshitz geometries can be found in [25].

The generalizations of (5) to include finite temperature can be written aswhere , , and .

The Hawking temperature is

#### 3. Heavy Quark Potential

In the holographic description, the heavy quark potential is given by the expectation value of the static Wilson loop where is a closed loop in a 4-dimensional space time and the trace is over the fundamental representation of the SU(*N*) group. is the gauge potential and enforces the path ordering along the loop . The heavy quark potential can be extracted from the expectation value of this rectangular Wilson loop in the limit :

On the other hand, the expectation value of Wilson loop in (12) is given by where is the regularized action. Therefore, the heavy quark potential can be expressed as

We now analyze the heavy quark potential using the metric of (9). The string action can reduce to the Nambu-Goto action:where is the determinant of the induced metric on the string world sheet embedded in the target space; that is,where and are the target space coordinates and the metric, and with parameterize the world sheet.

Using the parametrization , , and , we extremize the open string worldsheet attached to a static quark at and an antiquark at . Then the induced metric of the fundamental string is given bywith , .

Plugging (17) into (15), the Euclidean version of Nambu-Goto action in (9) becomes

We now identify the Lagrangian as

Note that does not depend on explicitly, so the corresponding Hamiltonian will be a constant of motion; that is,

This constant can be found at special point , where , as and then a differential equation is derived: with

By integrating (22) the separation length of quark-antiquark pair becomes

On the other hand, plugging (22) into the Nambu-Goto action of (18), one finds the action of the heavy quark pair:

This action is divergent, but the divergences can be avoided by subtracting the inertial mass of two free quarks, given by

Subtracting this self-energy, the regularized action is obtained:

Applying (14), we end up with the heavy quark potential with hyperscaling violation:

Note that the potential in the Lifshitz space-time [45, 46] can be derived from (28) if we neglect the effect of the hyperscaling violation exponent by plugging in (28). Also, in the limit (), (28) can reduce to the finite temperature case in [34, 35].

Before evaluating the heavy quark potential of (28), we should pause here to determine the allowed region for and at hand. The space boundary is considered at , consequently . To avoid (28) being ill-defined, it is required that . Moreover, one should consider the Gubser conditions of (7) and the thermodynamic stability condition of (8). With these restrictions, one finds and then one can choose the values of and in such a range.

In Figure 1, we plot the potential versus distance with different , . In Figure 1(a), the dynamical exponent is and from top to bottom the hyperscaling violation exponent is , respectively. In Figure 1(b), and from top to bottom , respectively. From the figures, we can see clearly that by increasing the potential decreases. One finds also that increasing leads to increasing the potential. In other words, increasing and had different effects on the potential. Then one can change the potential by changing the values of these parameters. Therefore, the heavy quark potential depends on the nonrelativistic parameters.