Quantum kinetic approach to the Schwinger production of scalar particles in an expanding universe

Abstract

We study the Schwinger pair creation of scalar charged particles by a homogeneous electric field in an expanding universe in the quantum kinetic approach. We introduce an adiabatic vacuum for the scalar field based on the Wentzel–Kramers–Brillouin solution to the mode equation in conformal time and apply the formalism of Bogolyubov coefficients to derive a system of quantum Vlasov equations for three real kinetic functions. Compared to the analogous system of equations previously reported in the literature, the new one has two advantages. First, its solutions exhibit a faster decrease at large momenta which makes it more suitable for numerical computations. Second, it predicts no particle creation in the case of conformally coupled massless scalar field in the vanishing electric field, i.e., it respects the conformal symmetry of the system. We identify the ultraviolet divergences in the electric current and energy–momentum tensor of produced particles and introduce the corresponding counterterms in order to cancel them.

Appendix A Expansion in the inverse powers of momentum

In this Appendix, we derive asymptotical expressions for the kinetic functions $\mathcal {F}(\eta ,\varvec{p})$, $\mathcal {G}(\eta ,\varvec{p})$, and $\mathcal {H}(\eta ,\varvec{p})$ in the limit of large momenta. For this, we need the corresponding expansions of the quantities $\omega (\eta ,\varvec{p})$ and $Q(\eta ,\varvec{p})$ which are the coefficients in the system of quantum Vlasov equations (24):

$$\begin{aligned}{} & {} \omega (\eta ,\varvec{p})=p+\omega ^{(-1)}+ \mathcal {O}(p^{-3}), \qquad \omega ^{(-1)}=\frac{m^{2}a^{3}+(6\xi -1)a''}{2ap}, \end{aligned}$$

(A1)

$$\begin{aligned}{} & {} Q(\eta ,\varvec{p})=Q^{(-1)}+Q^{(-2)}+ \mathcal {O}(p^{-3}), \end{aligned}$$

(A2)

$$\begin{aligned}{} & {} Q^{(-1)}=\frac{e\,a^{2}(\varvec{p}\!\cdot \!\varvec{E})}{p^{2}}, \quad Q^{(-2)}=\frac{1}{p^{2}}\left[ m^{2}aa'+\frac{6\xi -1}{2}\left( \frac{a'''}{a}-\frac{a'a''}{a^{2}} \right) \right] .\nonumber \\ \end{aligned}$$

(A3)

Let us represent the total derivative operator with respect to conformal time as the sum of two operators $\hat{L}^{(0)}$ and $\hat{L}^{(-1)}$:

$$\begin{aligned} \frac{d}{d\eta }=\hat{L}^{(0)}+\hat{L}^{(-1)},\quad \text {where} \quad \hat{L}^{(0)}=\frac{\partial }{\partial \eta },\quad \hat{L}^{(-1)}=e\,a^{2}\varvec{E}\frac{\partial }{\partial \varvec{p}}. \end{aligned}$$

(A4)

Obviously, the operator $\hat{L}^{(0)}$ does not change the asymptotical behavior of a given term at large momenta while $\hat{L}^{(-1)}$ reduces the power of momentum by one.

Further, we represent the kinetic functions as power series in inverse momentum: for $\mathcal {F}(\eta ,\varvec{p})$, $\mathcal {G}(\eta ,\varvec{p})$, and $\mathcal {H}(\eta ,\varvec{p})$ the series starts from the term $\propto p^{-4}$, $\propto p^{-3}$, and $\propto p^{-2}$, respectively. Then, substituting these expansions together with Eqs. (A1)–(A4) into the system of quantum Vlasov equations in Eq. (24), we obtain the following set of equations:

$$\begin{aligned} \hat{L}^{(0)}\mathcal {F}^{(-4)}&=Q^{(-1)}\mathcal {G}^{(-3)}\, , \end{aligned}$$

(A5)

$$\begin{aligned} 0&=\frac{1}{2}Q^{(-1)}+2p\,\mathcal {H}^{(-2)}\, , \end{aligned}$$

(A6)

$$\begin{aligned} 0&=\frac{1}{2}Q^{(-2)}+2p\,\mathcal {H}^{(-3)},\end{aligned}$$

(A7)

$$\begin{aligned} \hat{L}^{(0)}\mathcal {H}^{(-2)}&=-2p\,\mathcal {G}^{(-3)}\, ,\end{aligned}$$

(A8)

$$\begin{aligned} \hat{L}^{(0)}\mathcal {H}^{(-3)}+\hat{L}^{(-1)}\mathcal {H}^{(-2)}&=-2p\,\mathcal {G}^{(-4)}\, . \end{aligned}$$

(A9)

From Eqs. (A6) and (A7) we immediately find expressions for the terms $\mathcal {H}^{(-2)}$ and $\mathcal {H}^{(-3)}$:

$$\begin{aligned} \mathcal {H}^{(-2)}&=-\frac{1}{4p}Q^{(-1)}=-\frac{e\,a^{2}(\hat{\varvec{p}}\!\cdot \!\varvec{E})}{4p^{2}}\, ,\end{aligned}$$

(A10)

$$\begin{aligned} \mathcal {H}^{(-3)}&=-\frac{1}{4p}Q^{(-2)}=-\frac{1}{4p^{3}}\left[ m^{2}aa'+\frac{6\xi -1}{2}\left( \frac{a'''}{a}-\frac{a'a''}{a^{2}} \right) \right] \, , \end{aligned}$$

(A11)

where $\hat{\varvec{p}}=\varvec{p}/p$ is the unit vector in the direction of $\varvec{p}$.

Then, the terms $\mathcal {G}^{(-3)}$ and $\mathcal {G}^{(-4)}$ can be found from Eqs. (A8) and (A9), respectively:

$$\begin{aligned} \mathcal {G}^{(-3)}&=-\frac{1}{2p}\hat{L}^{(0)}\mathcal {H}^{(-2)}=\frac{ea^2}{8p^{3}}\,\hat{\varvec{p}}\cdot \Big (\varvec{E}'+2\frac{a'}{a}\varvec{E} \Big ) \, ,\end{aligned}$$

(A12)

$$\begin{aligned} \mathcal {G}^{(-4)}&=-\frac{1}{2p}\left[ \hat{L}^{(0)}\mathcal {H}^{(-3)}+\hat{L}^{(-1)}\mathcal {H}^{(-2)} \right] =\frac{e^{2}a^{4}\big [\varvec{E}^{2}-3(\hat{\varvec{p}}\cdot \varvec{E})^2\big ]}{8p^{4}} \nonumber \\&\quad + \frac{1}{8p^{4}}\!\bigg [m^{2}(a^{\prime 2}+aa'') + (6\xi -1)\left( \frac{a^{\textrm{IV}}}{2a}-\frac{a^{\prime \prime 2}}{2a^{2}}-\frac{a'a'''}{a^{2}} + \frac{a^{\prime 2}a''}{a^{3}} \right) \bigg ]\, . \end{aligned}$$

(A13)

Finally, the term $\mathcal {F}^{(-4)}$ can be found by solving differential equation (A5). It is easy to see that the following function is a solution to this equation:

$$\begin{aligned} \mathcal {F}^{(-4)}=\frac{e^{2}a^{4}(\hat{\varvec{p}}\!\cdot \!\varvec{E})^{2}}{16p^{4}}. \end{aligned}$$

(A14)

Thus, we derived a few first terms in the Laurent series for the kinetic functions $\mathcal {F}(\eta ,\varvec{p})$, $\mathcal {G}(\eta ,\varvec{p})$, and $\mathcal {H}(\eta ,\varvec{p})$ at large momenta $p\rightarrow \infty$. However, it is not convenient to use them in the computations in Sect. 4 because they lead to spurious infrared divergences in the integrals for physical observables. In order to overcome this problem, it is more convenient to perform expansion in inverse powers of $\epsilon _{\varvec{p}}$ instead of p. Therefore, in Eqs. (A10)–(A14) we replace $p\rightarrow \epsilon _{\varvec{p}}$ in denominators and get Eqs. (37)–(39) in the main text.