^{1}

^{2}

^{1}

^{1}

^{1}

^{2}

^{3}.

Using the AdS/CFT correspondence, we investigate the Schwinger effect in a confining D3-brane background with chemical potential. The potential between a test particle pair on the D3-brane in an external electric field is obtained. The critical field

Schwinger effect is known as the pair production in an external electric field in QED [

AdS/CFT, namely, the duality between type IIB superstring theory formulated on

As we know, a finite temperature of the gauge theory is on the gravity side of the duality represented by a black hole horizon in the bulk located at

The organization of this paper is as follows. In the next section, the background with chemical potential is briefly introduced. In Section

In the framework of the AdS/CFT duality,

The chemical potential

Note that when

We should point out that the chemical potential implemented in this way is not the quark (or baryon) chemical potential of QCD but a chemical potential that refers to the

We now perform the potential analysis for confining D3-brane background with chemical potential using the metric equation (

By using the static gauge,

Then the Lagrangian density becomes

Now that

The boundary condition at

By integrating (

Plugging (

To ensure that the potential analysis is right and consistent with the DBI result, we should pause here to gain the critical field by DBI action at hand.

The DBI action is

In terms of the metric equation (

After considering

Here the electric field is only turned on

Plugging (

To avoid the action equation (

So the range of electric field is

Finally, the critical field

Note that

To proceed, we compute the total potential. For convenience, we introduce a dimensionless parameter

Then, from (

We have checked that the total potential

Before discussing the results, let us recall the potential analysis in confining D3-brane background. There exist two critical values of the electric field,

We now discuss the results. The total potential can be plotted versus the separate distance of the test particle pair on the D3-brane numerically. Here Figure

The total potential versus distance

In order to see the effect of chemical potential clearly, we plot the potential versus

The total potential versus distance

In this section, we study the pair production rate. Here we consider an AdS soliton background [

The holographic pair production rate is [

To evaluate the pair production rate, one needs to compute the action or the expectation value of a circular Wilson loop on the probe brane. In [

Then the string action can be expressed as

The boundary conditions of (

To proceed further, we need to turn to numerical methods. To compare with the case in [

To see the effect of the chemical potential on Schwinger effect, we plot

Schwinger effect in confining gauge theories may be an important ingredient in looking for new aspects of QCD in the presence of strong external fields. In this paper, we have studied the effect of chemical potential on the holographic Schwinger effect by considering a confining D3-brane background with charge. The potential of a test particle pair was obtained by calculating the Nambu-Goto action of string attaching the rectangular Wilson loop at the probe D3-brane. The production rate for various cases was evaluated numerically. The effect of the chemical potential on the production rate was qualitatively shown. From the results, we can indeed see that there exist two critical values of the electric field. In addition, both the potential analysis and the numerical results of the production rate suggest that the presence of chemical potential tend to suppress the Schwinger effect. Interestingly, the instanton effects on the Schwinger effect have been studied in [

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