Nonperturbative pair production in interpolating fields

We compare the effects of timelike, lightlike and spacelike one-dimensional inhomogeneities on the probability of nonperturbative pair production in strong fields. Using interpolating coordinates we give a unifying picture in which the effect of the inhomogeneity is encoded in branch cuts and poles circulated by complex worldline instantons. For spacelike inhomogeneities the length of the cut is related to the existence of critical points, while for lightlike inhomogeneities the cut contracts to a pole and the instantons become contractible to points, leading to simplifications particular to the lightlike case. We calculate the effective action in fields with up to three nonzero components, and investigate its behavior under changes in field dependence.

Figure 1

The complex instantons $q(\tau )$ in a time-dependent ${\mathrm{sech}}^2$ electric field, $a=c=\gamma =1$ and various imaginary ${\tau }_0$. As $\mathrm{Im}{\tau }_0\to 0$ the instantons become purely imaginary (purely real in Euclidean space) and sit on the branch cut, shown as the dashed black line. The instantons cannot cross the poles in the field at $\pm i\pi /2$, shown as dots.

Figure 2

The complex instantons $q(\tau )$ from Fig. 1, plotted at a height of $\mathrm{Im}\omega \dot{q}$ using (22), in order to to show how they circle the branch cut in the instanton velocity $\dot{q}$.

Figure 3

The instantons $t(\tau )$ and $z(\tau )$ for $c=1/2$, i.e. an electric field $E(t+z\sqrt{3})$, and $\gamma =a=1$. Even for ${\tau }_0=0$ the instantons have both real and imaginary parts.

Figure 4

Streamplots in the complex $q$–plane for the Sauter pulse instantons. Top to bottom: $c{\gamma }^2=3,0.1,0,-0.1,-0.5$. Arrows show the direction of the velocity $\dot{q}$.

Figure 5

Complex instantons for $c=-1/2$, i.e. an electric field depending on a spacelike coordinate $E(t\sqrt{3}+z)$. A fixed ${\tau }_0=i/300$ is chosen, and $\gamma$ varies up to the cutoff value of $\gamma =\sqrt{2}$ from (24), beyond which there are no periodic solutions.

Figure 6

Contour plot of the effective action $\mathrm{Im}\hspace{0.17em}\hspace{0.17em}\mathrm{\Gamma }/V^3$ as in (59), in units of the electron mass. In order to show structure, the frequency is fixed to the high value of $\omega =m/2$. For $c<0$, when the field inhomogeneity is spacelike, the electric field strength must obey $E_0/E_S>\sqrt{|c|}/2$ in order to be capable of producing pairs. We see that to achieve a given pair production probability it is preferable for the field to have temporal rather than spatial inhomogeneities.

Figure 7

Contour plot of the effective action $\mathrm{Im}\hspace{0.17em}\hspace{0.17em}\mathrm{\Gamma }/V^3$ as in (59), in units of the electron mass. Here $E=E_S/10$, and the critical curve is at $100c(\omega /m{)}^2=-1$.

Figure 8

Effect of a transverse field on the criticality condition, compare (64) with (60). Parameters as in Fig. 6. For fields with spacelike inhomogeneities $c<0$, turning on transverse electric and magnetic fields reduces the parameter region in which pair production is possible.

Figure 9

Complex instantons $\omega t$, $\omega x_1$ and $\omega x_2$ for $c=\gamma =1$, with $\omega \overline{t}=1$ and initial condition $\omega t(0)=-1.75$.

Figure 10

Contour plot of the effective action (72) with $E_0=E_S$. There is no pair production for $c<0$.