Schwinger pairs production in a soft-wall model

The Schwinger pairs production rate is calculated numerically in the soft-wall model with the help of a simpler method in determining the soft-wall’s position beyond which probe strings connecting the Schwinger pairs do not fall into. The critical behavior of the production rate and linear part in the middle region are both studied carefully. The latter manifests interesting new features. The results are compared with those in previous hard-wall models.

Figure 1

The parametrization for the cup-like minimal surface.

Figure 2

The radius of the minimal surface’s boundary is denoted as $x$. $z_c$ is the maximal value of $z$ on the string world sheet. The size relationship between $z_0,z_c$ and $z_t$ is also shown here.

Figure 3

The upper part is the production rate of Schwinger pairs in the soft-wall model. The lower part is that in the hard-wall model. Critical behavior is shown on the right side. The vertical axis is the production rate $P$. The horizontal axis is the normalized field strength $E/E_c$.

Figure 4

Production rate vs. abstract field strength relation. The left is for soft-wall model, while the right, hard- wall model. In both models, more smaller $z_0/z_t$ corresponds to more heavier Schwinger pairs and more larger upper-critical field strength.

Figure 5

The linear fitting of $P-E$ relations as $P\in [0.2,0.8]$. The vertical axis is the production rate $P$. The horizontal axis is dimensionless field strength $E/(\frac{T_F{L}^2}{z_t^2})$. The left-hand side is for the soft-wall model, in which all fitting lines intersect at $(E,P)=(2.5,-1.2)$. The right is for the hard-wall model, in which all fitting lines intersect at $(1,-1.2)$.