Multidimensional worldline instantons

We extend the worldline instanton technique to compute the vacuum pair production rate for spatially inhomogeneous electric background fields, with the spatial inhomogeneity being genuinely two or three dimensional, both for the magnitude and direction of the electric field. Other techniques, such as WKB, have not been applied to such higher dimensional problems. Our method exploits the instanton dominance of the worldline path integral expression for the effective action.

Figure 1

Numerical worldline instanton loop for the case of $f(k{x}_1,k{x}_2)=\frac{\tanh (k{x}_1+k{x}_2)}{1+(k{x}_1{)}^2+10(k{x}_2{)}^2}$. The first plot is the 3D parametrized plot of {$x_1(\tau )$, $x_2(\tau )$, $x_4(\tau )$}; the second plot is $x_1(\tau )$ vs $x_2(\tau )$; the third plot is $x_1(\tau )$ vs $x_4(\tau )$; and the last plot is $x_2(\tau )$ vs $x_4(\tau )$. The parameters used to generate these plots are $E=1$, $k=\sqrt{2}$, $a=0.593$, and ${\dot{x}}_1(0)=0.56240809104352223$.

Figure 2

Numerical worldline instanton loop for the case of $f(k{x}_1,k{x}_2,k{x}_3)=\frac{k{x}_1+k{x}_2+k{x}_3}{[1+(k{x}_1{)}^2+2(k{x}_2{)}^2+10(k{x}_3{)}^2{]}^2}$. The first plot is the 3D parametrized plot of {$x_1(\tau )$, $x_2(\tau )$, $x_3(\tau )$}; the second plot is {$x_1(\tau )$, $x_2(\tau )$, $x_4(\tau )$}; the third plot is {$x_2(\tau )$, $x_3(\tau )$, $x_4(\tau )$}; and the last plot is {$x_3(\tau )$, $x_1(\tau )$, $x_4(\tau )$}. The parameters used to generate these plots are $E=1$, $k=1$, ${\dot{x}}_1(0)=0.6$, ${\dot{x}}_2(0)=0.453000684$, and ${\dot{x}}_3(0)=0.20173312728$.

Figure 3

The electric field $\overrightarrow{E}(x_1,x_2)$ for the case of $f(k{x}_1,k{x}_2)=(k{x}_1+k{x}_2){\mathrm{e}}^{-(k{x}_1{)}^2-(k{x}_2{)}^2}$. The first plot is the vector plot of $\overrightarrow{E}$ as a function of ($k{x}_1$, $k{x}_2$); the second plot is the 3D plot of the magnitude of $\overrightarrow{E}$ as a function of ($k{x}_1$, $k{x}_2$). To generate these plots, we use the parameter $E=1$.

Figure 4

$\mathrm{Im}{\Gamma }^{\mathrm{WLI}}/\mathrm{Im}{\Gamma }^{\mathrm{LCF}}$ for the cases of $f(k{x}_1,k{x}_2)=\frac{k{x}_1+k{x}_2}{1+(k{x}_1{)}^2+(k{x}_2{)}^2}$, and $f(k{x}_1,k{x}_2)=(k{x}_1+k{x}_2){\mathrm{e}}^{-(k{x}_1{)}^2-(k{x}_2{)}^2}$. The parameters used to generate this plot are $E=1$, $e=1$, and $m=1$.

Figure 5

Vector plot of the electric field $\overrightarrow{E}(x_1,x_2,x_3)$ for the case of $f(k{x}_1,k{x}_2,k{x}_3)=\frac{k{x}_1+k{x}_2+k{x}_3}{[1+(k{x}_1{)}^2+(k{x}_2{)}^2+(k{x}_3{)}^2{]}^{3/2}}$.

Figure 6

$\mathrm{Im}{\Gamma }^{\mathrm{WLI}}/\mathrm{Im}{\Gamma }^{\mathrm{LCF}}$ for the cases of $f(k{x}_1,k{x}_2,k{x}_3)=\frac{k{x}_1+k{x}_2+k{x}_3}{[1+(k{x}_1{)}^2+(k{x}_2{)}^2+(k{x}_3{)}^2{]}^{3/2}}$ and $f(k{x}_1,k{x}_2,k{x}_3)=(k{x}_1+k{x}_2+k{x}_3){\mathrm{e}}^{-(k{x}_1{)}^2-(k{x}_2{)}^2-(k{x}_3{)}^2}$. The parameters used to generate this plot are $E=1$, $e=1$, and $m=1$.