^{1}

^{1}

^{2}

^{1}

^{1}

^{2}

^{3}.

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

AdS/CFT [

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 [

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

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 [

Let us begin with a brief review of the background in [

The geometries are flat when

By using a radical redefinition

In the presence of hyperscaling violations, the energy scale is

For the generalized scaling solutions of (

Also, to consider the thermodynamic stability, one needs

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

The generalizations of (

The Hawking temperature is

In the holographic description, the heavy quark potential is given by the expectation value of the static Wilson loop

On the other hand, the expectation value of Wilson loop in (

We now analyze the heavy quark potential using the metric of (

Using the parametrization

Plugging (

We now identify the Lagrangian as

Note that

This constant can be found at special point

By integrating (

On the other hand, plugging (

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 (

Note that the potential

Before evaluating the heavy quark potential of (

In Figure

Plots of

To study how the potential changes with the temperature

Plots of

Moreover, to see the short distance behavior of the potential, we take the limit

One can see that the potential is dependent on

In this section, we study the effect of a constant electric field on the heavy quark potential following the method proposed in [

The constant

Parallel to the case of the previous section, we have

We call again the separation length and the heavy quark potential as

To see the effect of the constant electric field

Plots of

In this paper, we have investigated the heavy quark potential in the backgrounds with hyperscaling violation at finite temperature. These theories are strongly coupled with anisotropic scaling symmetry in the time and a spatial direction. Although the theories are not directly applicable to QCD, the features of them are similar to QCD. Therefore one can expect that the results obtained from these theories provide qualitative insights into analogous questions in QCD. In addition, an understanding of how the heavy quark potential changes by these theories may be useful for theoretical predictions.

In Section

Finally, it is interesting to mention that the drag force [

The authors declare that they have no competing interests.

This research is partly supported by the Ministry of Science and Technology of China (MSTC) under the 973 Project no. 2015CB856904