1 Introduction↩︎

In Quantum Electrodynamics (QED), the paradigm of constant and homogeneous electromagnetic fields provides the theoretical foundations for the widely used locally constant field approximation (LCFA); see, e.g. [1]–[6] and references therein. Such an approximation is applicable to any quantum field theory (QFT) model that involvesan interaction with the \(U\left( 1\right)\) gauge field ( “electric-like field”). Consequently, strong-field QED methods can be systematically extended to other models, which justifies the scientific significance of this approach within a broader framework of quantum theories. A strong constant electric field (\(\sim E_{c}=m^{2}c^{3}/e\hbar\), where \(-e\)is the electron charge and \(m\)is the mass of an electron) can create electron-positron pairs from a vacuum [7]. This phenomenon, known in the literature as the Schwinger effect, has been extensively investigated in many field-theoretic models, ranging fromhigh-energy physics, astrophysics, and thephysics of nanostructures; see, for instance, the recent review [8]. In this article, we propose an advancement of the LCFA method applicable to radiative processes accompanied bythe Schwinger effect.

Pair creation from the vacuum induced by an externalelectric field is a transient phenomenon. Nevertheless, there exists a window in the parameter range in whichthe back-reaction of the created pairs on the external field can be neglected (see Ref. [9] for details). Therefore, a physically meaningful formulation of the pair creation problem presupposes an external field of finite time duration. Such a version of the LCFA is proposed in Ref. [6]. Using this approach, we consider the \(3+1\) dimensional QED in the presence of the \(T\)-constant uniform electric field that exists during a macroscopic large time period \(T\), assumed to be much larger than the characteristic time scale \(\Delta t_{\mathrm{st}}=\left( eEc/\hbar \right) ^{-1/2}\), \(E>0\). This field switches-on at \(t_{1}=-T/2\) and -off at \(t_{2}=T/2\) abrubtly and remains constant with amplitude \(E\) within this interval \(\Delta t=t_{2}-t_{1}=T\). While switching-on and -off effects of the external field may influence vacuum instability in general [10], these can be neglected ifthe time interval \(T\) is sufficiently large, namely\[T/\Delta t_{\mathrm{st}}>\max \left\{ 1,E_{c}/E\right\} \,. \label{time-condition}\tag{1}\]

Following a nonperturbative formulation of strong-field QED developed in Refs. [11]–[15] and summarized in the book [16], we study the emission of a high-frequency photon (\(\omega \gg T^{-1}\)) accompanying vacuum instability under the action of a quasi-constant strong electric field \(E\) of finite duration \(T\). It is worth noting that the probabilities amplitudesfor the emission of a photon from a single-electron (positron) state and electron-positron creation by a photon in a constant electric field were previously considered by Nikishov in Ref. [17] (see also Ref. [18] where a photon emission from the vacuum accompanied by an electron-positron pair creation was considered). However,these amplitudes present a satisfactory description of photon emission only in the regime where the external field is not very strong and the effect of the pair creation is tini. In the case of a strong field, one has to use a perturbation theory with respect to the radiative interaction for average values, which is quite different from the technique that issuitable for calculating amplitudes. In this article we present an example of the perturbation theory with respect to the radiative interaction for average values that allows incorporating theLCFA.

The article is organized as follows: In Sect. 2 we present the effective perturbation theory of the photon emission in the presence of a strong electric field that is suitable for the LCFA. In Sect. 3 we explicitly derivethe total probability of one photon emission from the vacuum in a constant electric field of finite duration. We establish the domain of the applicability of the LCFA for the photon emission and show that there are characteristic angular and polarization properties of the emission in the range of high frequencies.

In what follows, we use the relativistic units \(\hslash =c=1\)in which the fine structure constant is \(\alpha =e^{2}/c\hslash =e^{2}\).

2 Effective perturbation theory of the photon emission↩︎

In the presence of the external fields that violate the vacuum stability,radiative processes corresponding to the emission of a single photon might occur either from an electron (a positron) or from the vacuum accompanied by the creation of an electron-positron pair. In the framework of the generalized Furry representation [16], the probability that a photon with momentum \(\mathbf{k}\) and polarization \(\vartheta\) is emitted from the vacuum accompanied by an electron-positron pair has the form:\[\begin{align} &&\mathcal{P}_{1}\left( \mathbf{k}\vartheta |0\right) =\sum_{n^{\prime },n}\left\vert w^{\left( 1\right) }\left( \overset{+}{n^{\prime }}\overset{-}{n};\mathbf{k}\vartheta |0\right) \right\vert ^{2}\left\vert c_{v}\right\vert ^{2}\,, \notag \\ &&w^{\left( 1\right) }\left( \overset{+}{n^{\prime }}\overset{-}{n};\mathbf{k}\vartheta |0\right) =\left\langle 0,\mathrm{out}|a_{n^{\prime }}\left( \mathrm{out}\right) b_{n}\left( \mathrm{out}\right) c_{\mathbf{k}\vartheta }S^{\left( 1\right) }|0,\mathrm{in}\right\rangle c_{v}^{-1}\,. \label{et0} \end{align}\tag{2}\] Here, the \(a\)’s, \(b\)’s, and \(c\)’s are annihilation operators of initial and final electrons, positrons, and photons, respectively. Their adjoints are creation operators. \(c_{v}=\langle 0,\mathrm{out}|0,\mathrm{in}\rangle\)is the vacuum to vacuum transition amplitude. The indices \(n=\left( \mathbf{p},\sigma \right)\) and \(n^{\prime }=\left( \mathbf{p}^{\prime },\sigma ^{\prime }\right)\) denote the complete set of particle’s and antiparticle’s quantum numbers, where \(\mathbf{p}\) is momentum and \(\sigma =\pm 1\) is the spin polarization. The external field under considerationis directed along the \(x\)-axis and is described by the vector potential with only one nonzero component:\[A_{x}^{\mathrm{ext}}(t)=-E\left\{ \begin{array}{ll} t_{1}, & t\in \mathrm{I}=(-\infty ,t_{1}),\;t_{1}=-T/2\, \\ t, & t\in \mathrm{Int}=[t_{1},t_{2}]\, \\ t_{2}, & t\in \mathrm{II}=(t_{2},\infty )\,,\;t_{2}=T/2\,\end{array}\right. .\]It is assumed that for \(t<t_{1}\) and for \(t>t_{2}\), the electric field is absent, therefore theinitial \(|0,\mathrm{in}\rangle\) and final \(|0,\mathrm{out}\rangle\) are vacuum states of free \(\mathrm{in}\)- and \(\mathrm{out}\)- charged particles, respectively. These vacua are different due to a difference of initial and final values of external electromagnetic field potentials and \(c_{v}=\left\langle 0,\mathrm{out}|0,\mathrm{in}\right\rangle\) is the vacuum to vacuum transition amplitude. During the time interval \(t_{2}-t_{1}=T\), the Dirac field interacts with the external field. Moreover, in Eq. (2 ), \(S^{\left( 1\right) }\) stands for the \(S\)-matrix truncated at first-order with respect to the radiative interaction,\[\begin{align} &&S=\mathcal{T}\exp \left[ -i\int \hat{\jmath}_{\mu }\left( x\right) \hat{A}^{\mu }\left( x\right) dx\right] \approx 1+S^{\left( 1\right) }\,, \notag \\ &&S^{\left( 1\right) }=-i\int \hat{\jmath}_{\mu }\left( x\right) \hat{A}^{\mu }\left( x\right) dx\,,\;\;dx=dtd\mathbf{r}\,,\;\;d\mathbf{r}=dx^{1}dx^{2}dx^{3}\,, \label{d2} \end{align}\tag{3}\] wherein \(\mathcal{T}\) denotes the time-ordering symbol, \(\hat{\jmath}^{\mu }\left( x\right) =-\left( e/2\right) \left[ \hat{\Psi}^{\dagger }\left( x\right) \gamma ^{0}\gamma ^{\mu },\hat{\Psi}\left( x\right) \right]\) is the current density field operator, and \(\gamma ^{\mu }\) are Dirac’s matrices. The Dirac field operators \(\hat{\Psi}\left( x\right)\), \(\hat{\Psi}^{\dagger }\left( x\right)\), and the electromagnetic field operator \(\hat{A}^{\mu }\left( x\right)\)are in the interaction representation. The Dirac field operators obey the Dirac equation with the potential \(\mathbf{A}^{\mathrm{ext}}(t)\).

The \(\mathrm{in}\)- and \(\mathrm{out}\)- sets of creation and annihilation operatorsof electrons and positronsare defined by the two representations of the quantum Dirac field \(\hat{\Psi}\left( x\right)\) as\[\begin{align} \hat{\Psi}\left( x\right) &=&\sum_{n}\left[ a_{n}\left( \mathrm{in}\right) \; _{+}\psi _{n}\left( x\right) +b_{n}^{\dagger }\left( \mathrm{in}\right) \; _{-}\psi _{n}\left( x\right) \right] \,, \notag \\ &=&\sum_{n}\left[ a_{n}\left( \mathrm{out}\right) \;^{+}\psi _{n}\left( x\right) +b_{n}^{\dagger }\left( \mathrm{out}\right) \;^{-}\psi _{n}\left( x\right) \right] \,, \label{d4} \end{align}\tag{4}\] where \(\;_{\zeta }\psi _{n}\left( x\right)\) and \(\;^{\zeta }\psi _{n}\left( x\right)\) are orthonormal \(\mathrm{in}\)- and \(\mathrm{out}\)-solutions of the Dirac equation with the potential \(\mathbf{A}^{\mathrm{ext}}(t)\). These solutions have a well-defined sign of frequency \(\zeta\) (\(\zeta =+\) for electrons and \(\zeta =-\) for positrons) either before the field switches-on or after it switches-off, respectively. They arerelated by a linear transformation of the form\[\begin{align} \;^{\zeta }\psi _{n}\left( x\right) &=&g_{n}\left( _{+}|^{\zeta }\right) \,_{+}\psi _{n}\left( x\right) +g_{n}\left( _{-}|^{\zeta }\right) \,_{-}\psi _{n}\left( x\right) \,, \notag \\ \;_{\zeta }\psi _{n}\left( x\right) &=&g_{n}\left( ^{+}|_{\zeta }\right) \,^{+}\psi _{n}\left( x\right) +g_{n}\left( ^{-}|_{\zeta }\right) \,^{-}\psi _{n}\left( x\right) \,, \label{f7} \end{align}\tag{5}\] where the decomposition coefficients are complex, \(g_{n}\left( _{\zeta ^{\prime }}|^{\zeta }\right) =g_{n}\left( ^{\zeta }|_{\zeta ^{\prime }}\right) ^{\ast }\). Because these coefficients obey certain unitarity relations, all coefficients can be expressed in terms of two of them, e.g. of \(g\left( _{+}\left\vert ^{+}\right. \right)\) and \(g\left( _{-}\left\vert ^{+}\right. \right)\). However, even the latter coefficients are not completely independent as they satisfy the condition\[\left\vert g_{n}\left( _{-}\left\vert ^{+}\right. \right) \right\vert ^{2}+\left\vert g_{n}\left( _{+}\left\vert ^{+}\right. \right) \right\vert ^{2}=1\,. \label{f9}\tag{6}\] Then a linear canonical transformation (Bogolubov transformation) between \(\mathrm{in}\)- and \(\mathrm{out}\)- operators which follows from Eq. (4 ) is defined by these coefficients\[\begin{align} a_{n}\left( \mathrm{out}\right) & =g_{n}\left( ^{+}|_{+}\right) a_{n}(\mathrm{in})+g_{n}\left( ^{+}|_{-}\right) b_{n}^{\dagger }(\mathrm{in}), \notag \\ b_{n}^{\dagger }\left( \mathrm{out}\right) & =g_{n}\left( ^{-}|_{+}\right) a_{n}(\mathrm{in})+g_{n}\left( ^{-}|_{-}\right) b_{n}^{\dagger }(\mathrm{in}). \label{f10} \end{align}\tag{7}\]

Using relations (7 ), one finds that the differential mean number of the pairs created is \[N_{n}^{\mathrm{cr}}=\left\vert g_{n}\left( {}_{-}|{}^{+}\right) \right\vert ^{2}. \label{f14}\tag{8}\]

The decomposition of the operator \(\mathbf{\hat{A}}\left( x\right)\)in terms of creation and annihilation operators of free photons, \(c_{\mathbf{k}\vartheta }^{\dagger }\)and \(c_{\mathbf{k}\vartheta }\), reads:\[\mathbf{\hat{A}}(x)=\sum_{\mathbf{k,}\vartheta }\sqrt{\frac{2\pi }{V\omega }}\boldsymbol{\epsilon }_{\mathbf{k}\vartheta }\left[ c_{\mathbf{k}\vartheta }\,e^{i(\mathbf{kr}-\omega t)}+c_{\mathbf{k}\vartheta }^{\dagger }\,e^{-i(\mathbf{kr}-\omega t)}\right] \,, \label{d3}\tag{9}\] where \(\vartheta =1,2\)denotes a polarization index, \(\epsilon _{\mathbf{k}\vartheta }\)are mutual orthogonal unit polarization vectors transversal to a wave vector \(\mathbf{k}\), \(\omega =\left\vert \mathbf{k}\right\vert\), and \(V\) is the volume of the box regularization.

To construct a perturbation theory for the probability amplitude\[W=\left\langle \mathrm{out}\left\vert \mathcal{S}\right\vert \mathrm{in}\right\rangle \,,\]for transition from an initial to a final state, one needs to reduce the \(S\)-matrix to a generalized normal form with respect to the vacua \(\left\langle 0,\mathrm{out}\right\vert\) and \(|0,\mathrm{in}\rangle\). This is exactly how expression (2 ) was obtained. It presents a satisfactory description of photon emission in the regime of weak external field, in which the differential number of pairs created from the vacuum is naturally small, \(N_{n}^{\mathrm{cr}}=\left\langle 0,\mathrm{in}\left\vert a_{n}^{\dagger }\left( \mathrm{out}\right) a_{n}\left( \mathrm{out}\right) \right\vert 0,\mathrm{in}\right\rangle \ll 1\). Probabilities of such a kind were considered by Nikishov for the case of a constant electric field [18]. However, in the case of a strong field, emission processes can be followed by the creation ofelectron-positron pairs from the vacuum. The resulting total probability of one photon emission from the vacuum in this case admits a complicated representation. Nevertheless, this difficulty can be circumvented by employing a perturbative approach formulated for average values. Specifically, the generating functional of mean values has the form\[\left\langle \mathcal{F}\left( t\right) \right\rangle =\left\langle \mathrm{in}\left\vert \mathcal{S}^{-1}\mathcal{TF}\left( t\right) \mathcal{S}\right\vert \mathrm{in}\right\rangle \,,\]where the symbol \(\mathcal{T}\) acts on both sides: it orders field operators to the right of it and antiorders them to the left. For physically admissible external fields, when the density of the number of pairs created from a vacuum is finite and the final and initial vacuum are not orthogonal, \(\langle 0,\mathrm{out}|0,\mathrm{in}\rangle \neq 0\) andthere is an unitary operator \(\mathcal{V}\) that relates the in and out- representations of the Fock space via \(|\mathrm{in}\rangle =V|\mathrm{out}\rangle\).

The total probability of one photon emission from the vacuum followed by any number of created electron-positron pairs, \(\mathcal{P}\left( \mathbf{k}\vartheta |0,\mathrm{in}\right)\), can be represented as a trace of the operators \(c_{\mathbf{k}\vartheta }\mathcal{S}\left\vert 0,\mathrm{in}\right\rangle \left\langle 0,\mathrm{in}\right\vert \mathcal{S}^{-1}c_{\mathbf{k}\vartheta }^{\dagger }\) with respect to the out-basis, \[\mathcal{P}\left( \left. \mathbf{k}\vartheta \right\vert 0,\mathrm{in}\right) =\mathrm{tr\,}\left[ c_{\mathbf{k}\vartheta }\mathcal{S}\left\vert 0,\mathrm{in}\right\rangle \left\langle 0,\mathrm{in}\right\vert \mathcal{S}^{-1}c_{\mathbf{k}\vartheta }^{\dagger }\right] \;. \label{f42b}\tag{10}\] By using the unitary transformation \(\mathcal{V}\), we can pass from the basis of the final states to the basis of the initial states and represent the trace (10 ) as an average value of the photon number operator,\[\mathcal{P}\left( \left. \mathbf{k}\vartheta \right\vert 0,\mathrm{in}\right) =\left\langle 0,\mathrm{in}\right\vert \mathcal{S}^{-1}c_{\mathbf{k}\vartheta }^{\dagger }c_{\mathbf{k}\vartheta }\mathcal{S}\left\vert 0,\mathrm{in}\right\rangle \;, \label{ph95number}\tag{11}\] where the \(S\)-matrix is truncated at first-order, \(\mathcal{S}\approx 1+S^{\left( 1\right) }\). Generally speaking, one can find similar expressions for emission probabilities when some charged particles are already present in the initial state. For example, the total probabilities of one photon emission from a single-electron state \(\mathcal{P}\left( \mathbf{k}\vartheta |\overset{+}{n}\right)\) can be presented as\[\mathcal{P}\left( \mathbf{k}\vartheta |\overset{+}{n}\right) =\left\langle 0,\mathrm{in}\left\vert a_{n}\left( \mathrm{in}\right) \mathcal{S}^{-1}c_{\mathbf{k}\vartheta }^{\dagger }c_{\mathbf{k}\vartheta }\mathcal{S}a_{n}^{\dagger }\left( \mathrm{in}\right) \right\vert 0,\mathrm{in}\right\rangle . \label{ph95number-b}\tag{12}\]

In course of constructing a perturbation theory with respect to the radiative interaction for average values unlike the case of the probability amplitudesone needs to reorganize the \(S\)-matrix in a normal form \(:\ldots :\) with respect to the in-vacuum. In the first-order approximation, it is sufficient to represent only the operator \(\mathbf{\hat{\jmath}}\left( x\right)\) in such a form,\[\mathbf{\hat{\jmath}}\left( x\right) =\;:\mathbf{\hat{\jmath}}\left( x\right) :+\;\langle \mathbf{j}\left( x\right) \rangle _{\mathrm{in}}\;,\;\; \langle \mathbf{j}\left( x\right) \rangle _{\mathrm{in}}\;=\langle 0,\mathrm{in}\left\vert \mathbf{\hat{\jmath}}\left( x\right) \right\vert 0,\mathrm{in}\rangle \;. \label{in-currentA}\tag{13}\] The vacuum mean current \(\langle \mathbf{j}\left( x\right) \rangle _{\mathrm{in}}\) is a sum of a vacuum polarization current and of a current of created particles. It is a nontrivial object in a slowly varying electric field and depends on the definition of the initial vacuum, \(|0,\mathrm{in}\rangle\) and on the evolution of the electric field from the initial time\(t_{1}\)of switching onto the time instant \(t\). After switching off the electric field at time\(t_{2}\), the term \(\langle \mathbf{j}\left( x\right) \rangle _{\mathrm{in}}\) represents the current density of the created pairs of charged particles. This current is a source in the Maxwell equations for a mean electromagnetic field. Such a mean field is a slowly varying crossed field emitted perpendicular to the axis of the external electric field. In the frequency range of the photon emission \(\omega \gg T^{-1},\) which is interesting to us, the contribution due to the current \(\langle \mathbf{j}\left( x\right) \rangle _{\mathrm{in}}\) can be neglected. In the Fock space, the identity operator can be represented as a sum of projection operators onto states with a certain number of initial particles and antiparticles. Inserting this operator between the operators \(c_{\mathbf{k}\vartheta }^{\dagger }\) and \(c_{\mathbf{k}\vartheta }\), we can represent the total probability (11 ) as follows:\[\begin{align} &&\mathcal{P}\left( \mathbf{k}\vartheta |0\right) =\sum_{n^{\prime },n}\left\vert w_{\mathrm{in}}^{\left( 1\right) }\left( \overset{-}{n}\overset{+}{n^{\prime }};\mathbf{k}\vartheta |0\right) \right\vert ^{2}\,, \notag \\ &&w_{\mathrm{in}}^{\left( 1\right) }\left( \overset{-}{n}\overset{+}{n^{\prime }};\mathbf{k}\vartheta |0\right) =i\sqrt{\frac{2\pi }{V\omega }}\int \mathbf{j}_{\mathrm{in}}\left( \overset{-}{n}\overset{+}{n^{\prime }}|0\right) \boldsymbol{\epsilon }_{\mathbf{k}\vartheta }e^{i\left( \omega t-\mathbf{kr}\right) }dx\,, \notag \\ &&\mathbf{j}_{\mathrm{in}}\left( \overset{-}{n}\overset{+}{n^{\prime }}|0\right) =\left\langle 0,\mathrm{in}\left\vert b_{n}\left( \mathrm{in}\right) a_{n^{\prime }}\left( \mathrm{in}\right) :\mathbf{\hat{\jmath}}\left( x\right) :\right\vert 0,\mathrm{in}\right\rangle \,. \label{ga24614} \end{align}\tag{14}\] Similarly, we obtain the total probability of one photon emission from an electron.

Because the probability (14 ) describes the process of one photon emission from the vacuum,we expect it is proportional to the total number density of pairs produced, \(n^{\mathrm{cr}}\). Assuming that the electron density in the initial state is small, we see that the emission from the vacuum provides the main contribution to the emission process. Therefore, the contribution of processes with initial particles is not considered here.

3 LCFA↩︎

3.1 Photon emission in a constant electric field of finite duration↩︎

In which follows, it is convenient to separate the components of the momentum directed along the field and orthogonal to it as \(\mathbf{p}=\left( p_{x},\mathbf{p}_{\bot }\right)\). The electric field acting during the time \(T\) creates a considerable number of pairs from vacuum only in a finite range in the momentum space,\[D:\Delta t_{st}^{2}p_{\bot }^{2}<T/\Delta t_{st}-\tau ,\;\Delta t_{st}\left\vert p_{x}\right\vert <\frac{1}{2}T/\Delta t_{st}-\tau , \label{e3}\tag{15}\] where \(\tau\) is an arbitrary number satisfying the condition (see Ref. [19] for details)\[T/\Delta t_{st}\gg \tau \gg \,\max \left\{ 1,E_{c}/E\right\} . \label{Tcond}\tag{16}\] In this range, the differential mean number of the pairs created is identical with that of the constant electric field, \[N_{n}^{\mathrm{cr}}=e^{-\pi \lambda }\,, \label{N95cr}\tag{17}\] and solutions of the Dirac equation can be represented as:\[\begin{align} &&_{\pm }\psi _{n}\left( x\right) =\exp \left( i\mathbf{pr}\right) \;_{\pm }\psi _{n}\left( t\right) , \notag \\ &&_{\pm }\psi _{n}\left( t\right) =\sqrt{eE}\left[ \left( \pm 1+i\right) \;_{\pm }\varphi _{n,\mp 1}(t)+B\left( \mathbf{p}\right) \;_{\pm }\varphi _{n,\pm 1}(t)\right] v_{\pm 1,\sigma }, \notag \\ &&B\left( \mathbf{p}\right) =\left( eE\right) ^{-1/2}\left( \alpha ^{2}p_{y}+\alpha ^{3}p_{z}+\gamma ^{0}m\right) ,\;\boldsymbol{\alpha }=\gamma ^{0}\boldsymbol{\gamma }\mathbf{,} \notag \\ &&_{+}\varphi _{n,\varkappa }(t)=CD_{\nu -(1+\varkappa )/2}\left[ -(1-i)\xi \right] ,\;\;_{-}\varphi _{n,\varkappa }(t)=CD_{-\nu -(1-\varkappa )/2}\left[ -(1+i)\xi \right] \,, \notag \\ &&\xi =\frac{eEt-p_{x}}{\sqrt{eE}},\;\nu =i\lambda /2,\;\lambda =\frac{p_{\bot }^{2}+m^{2}}{eE},\;C=\frac{\exp (-\pi \lambda /8)}{\sqrt{2eEV}}\,, \label{24610} \end{align}\tag{18}\] where \(D\)’s are the linearly independent Weber parabolic cylinder functions (WPCFs) [20], and \(v_{\varkappa ,\sigma }\) is a set of constant orthonormalized spinors, \[\begin{align} \gamma ^{0}\gamma ^{1}v_{\varkappa ,\sigma } &=&\varkappa v_{\varkappa ,\sigma },\;\varkappa =\pm 1; \notag \\ \;i\gamma ^{2}\gamma ^{3}v_{\varkappa ,\sigma } &=&\sigma v_{\varkappa ,\sigma },\;\sigma =\pm 1,\;v_{\varkappa ,\sigma }^{\dagger }v_{\varkappa ^{\prime },\sigma ^{\prime }}=\delta _{\varkappa \varkappa ^{\prime }}\delta _{\sigma \sigma ^{\prime }}\;. \label{s1} \end{align}\tag{19}\] We call the inequality (15 ) as the range of the stabilization for a creation process.

Expressing the volume element \(dk\) in spherical coordinates, \(dk=\omega ^{2}d\omega d\Omega\), where \(\omega\)is the frequency of the radiated photon in a region enclosed by the solid angle \(d\Omega\), one can write the probability of one photon emission with a given polarization \(\vartheta\) per unit frequency and solid angle, which is accompanied by pair production from the vacuum, as\[\begin{align} &&\frac{d\mathcal{P}\left( \mathbf{k}\vartheta |0\right) }{d\omega d\Omega }=\alpha \frac{\omega \Delta t_{st}^{2}}{\left( 2\pi \right) ^{2}}\sum_{\mathbf{p}}\sum_{\sigma ,\sigma ^{\prime }=\pm 1}\left. \left\vert M_{n^{\prime }n}^{0}\right\vert ^{2}\right\vert _{\mathbf{p}^{\prime }=\mathbf{p}-\mathbf{k}}\;, \notag \\ &&M_{n^{\prime }n}^{0}=-\frac{V}{\Delta t_{st}}\int_{t_{1}}^{t_{2}}\;_{+}\bar{\psi}_{n^{\prime }}(t)\mathbf{\gamma }\boldsymbol{\epsilon }_{\mathbf{k}\vartheta }\;_{-}\psi _{n}(t)e^{i\omega t}dt\;, \label{ga4} \end{align}\tag{20}\] where representation (4 ) is used and integral over the space volume \(V\) is fulfilled.

In the stabilization range and neglecting switching-on and -off effects, one can see that \(M_{n^{\prime }n}^{0}\) is a linear combination of the following integrals\[Y_{j^{\prime }j}^{0}\left( t_{2},t_{1}\right) =\int_{u_{1}}^{u_{2}}D_{-\nu ^{\prime }-j^{\prime }}\left[ -(1+i)u_{-}\right] D_{-\nu -j}\left[ -(1+i)u_{+}\right] e^{iu_{0}u}du, \label{ga6}\tag{21}\] where \[\begin{align} &&u=\Delta t_{st}\left[ eEt-\frac{1}{2}\left( p_{x}+p_{x}^{\prime }\right) \right] ,\;u_{1,2}=\left. u\right\vert _{t=t_{1,2}}\;, \notag \\ &&u_{x}=\Delta t_{st}\left( p_{x}^{\prime }-p_{x}\right) ,\;u_{\pm }=u\pm u_{x}/2,\quad u_{0}=\Delta t_{st}\omega \;, \notag \\ &&\nu ^{\prime }=i\lambda ^{\prime }/2,\;\lambda ^{\prime }=\left. \lambda \right\vert _{\mathbf{p}^{\prime }=\mathbf{p}-\mathbf{k}}\;. \label{e10} \end{align}\tag{22}\]

In the case of the \(T\)-constant electric field, there is the natural range of the very low frequency of emission, \(\omega \lesssim \omega ^{\mathrm{IR}}=2\pi T^{-1}\). In this range, generally speaking, the radiation must be treated in the mean field approximation. We assume that the low energy emission in this range does not exhaust the external field. Note that the frequencies of soft photons, whose nature is associated with the impossibility of separating a charged particle from its radiation field, are lower than \(\omega ^{\mathrm{IR}}\). For our purposes, it is enough to restrict the applicability of the perturbation theory with respect of the photon emission by the condition \(\omega >\omega ^{\mathrm{IR}}\), which is convenient to represent as:\[u_{0}>u_{0}^{\mathrm{IR}},\;u_{0}^{\mathrm{IR}}=2\pi \Delta t_{st}T^{-1}. \label{IR1}\tag{23}\] This condition provides the domain of the applicability of the \(T\)-constant electric field model in the framework of the perturbation theory.

The probability of the photon emission given by Eq. (20 ), can be integrated over \(\mathbf{k}\) only between such limits that leave the integral probability much smaller than unity. Let us demonstrate that for the integration over \(\omega\) there is a natural cutoff from above. Let us consider the high frequency case,\[u_{0}\gtrsim \tau _{\gamma }, \label{lim3}\tag{24}\] where \(\tau _{\gamma }\) is an arbitrary given number, \(\tau _{\gamma }\gg 1\). In the case under consideration, it is reasonable to assume that, for example, \(\tau _{\gamma }\gtrsim 3\). The probability (20 ) is the linear combination of integrals (21 ). There are intervals where main contributions to the integrals are formed. We can find this intervals using the saddle-point method.

Let us consider integral \(Y_{j^{\prime }j}^{0}\left( t_{2},t_{1}\right)\). Under condition (24 ), the mentioned saddle-point is situated in the range where absolute values of arguments of both WPCF’s involved in the integral are big, \[\left\vert u_{\pm }\right\vert \gg \max \left\{ 1,\lambda \right\} . \label{lim3b}\tag{25}\] In this case, if \(u_{\pm }<0\), one uses the following asymptotic expansion [(8.4.(1)) from Ref. [20]]:

\[D_{p}\left( z\right) =e^{-z^{2}/4}z^{p}\left[ 1+O\left( \left\vert z\right\vert ^{-2}\right) \right] \;\mathrm{if}\;\left\vert \arg z\right\vert <\frac{3\pi }{4}. \label{asy95exp}\tag{26}\] If \(u_{\pm }>0\), one uses a relation between WPCF’s (see (8.2.(7)) in Ref. [20]) and then applies Eq. (26 ). Thus one finds that the saddle-point is \(u=u_{0}/2\). Since \(u_{0}\) is positive, the saddle-point can be situated only in the range \(u_{\pm }>0\).

In this range, the longitudinal kinetic momenta of a created electron is negative, \(P_{x}\left( t\right) =p_{x}-eEt<0\), and the longitudinal kinetic momenta of a created positron is positive, \(P_{x}^{\prime \left( p\right) }\left( t\right) =-\left( p_{x}^{\prime }-eEt\right) >0\), and the modulus of these momenta are large. This means that such electrons and positrons can be treated as final particles, they acquire final longitudinal kinetic momenta at the moment of time \(t_{2}\). The kinetic energies of these particles are determined mainly by their longitudinal kinetic momenta \(\left\vert P_{x}\left( t\right) \right\vert\) and \(\left\vert P_{x}^{\prime }\left( t\right) \right\vert\). We see that the saddle-point equation represents a conservation law of the kinetic energy, \[\left\vert P_{x}\left( t\right) \right\vert +\left\vert P_{x}^{\prime }\left( t\right) \right\vert =\omega , \label{en95cons}\tag{27}\] where \(p_{x}^{\prime }=p_{x}-k_{x}\). In the neighborhood of the saddle-point, \(t=\) \(t_{c}\), the corresponding kernels have Gaussian forms with maxima at the time instant \(t_{c}\),\[t_{c}=\frac{1}{2}\left( \Delta t_{st}^{2}\omega +\frac{p_{x}+p_{x}^{\prime }}{eE}\right) , \label{tc}\tag{28}\] and with the standard deviation \[\Delta t_{sd}=\Delta t_{st}/\sqrt{2}. \label{sd}\tag{29}\] The time \(t_{c}\) corresponds to the position of the center of the formation interval \(\Delta t\) for given \(\omega\), \(p_{x}\), and \(p_{x}^{\prime }\) in the range \(u_{\pm }>0\). The width of the formation interval \(\Delta t\) must be large enough to accommodate the points \(u_{+}\) and \(u_{-}\). In addition, the formation interval must overlap the interval \(\Delta t_{sd}\), \(\Delta t_{sd}<\Delta t\). It is natural to assume that \(\Delta t\sim \Delta t_{st}\). Thisimplies the following condition:\[\left\vert u_{x}\right\vert <1\;. \label{lim4}\tag{30}\] Under condition (24 ) \(\left\vert u_{x}\right\vert \ll u_{0}\), it follows from Eq. (28 ) that\[t_{c}\approx \frac{1}{2}\Delta t_{st}^{2}\omega +\frac{p_{x}}{eE}\;. \label{lim6}\tag{31}\] This means that for the photon emission of a given frequency\(\omega\), the dependence of the effect on\(p_{x}\)comes down to just shifting of the center of the formation interval.On the other hand, for a given momentum \(p_{x}\), photon with higher frequency is formed later.

Thus, in the case of high frequencies, the width of the formation interval does not depend on the frequency \(\omega\) and on the momenta of the particles and is determined entirely by the electric field \(E\). The variationoftheexternal electricfieldactingontheparticle withintheformationlengthcanbeneglected,which allows us to use the LCFA.The obtained results can be easily extended to the study of the emission in any slowly varying field configuration assuming that the electric field \(E\left( t\right) >0\) is uniform and time-dependent. In this case, the kinetic energies in the conservation law (27 ) and expressions derived from it have to be given by general forms,\[\left\vert P_{x}\left( t\right) \right\vert =\left\vert p_{x}+eA_{x}\left( t\right) \right\vert ,\;\left\vert P_{x}^{\prime }\left( t\right) \right\vert =\left\vert p_{x}^{\prime }+eA_{x}\left( t\right) \right\vert , \label{gen95kin}\tag{32}\] where \(A_{x}\left( t\right)\) is a potential step of a slowly varying field. If the electric field decreases quickly enough beyond the formation interval, the upper limitation to the intensity of the constant electric field (see Ref. [9] for details) can be significantly weakened.

It follows from Eq. (31 ) that for any given \(p_{x}\) the high frequency emission, \(\omega /\omega _{sc}\gtrsim \tau _{\gamma }\), \(\omega _{sc}=\Delta t_{st}^{-1}\), starts when the longitudinal kinetic momentum \(P_{x}\left( t\right)\) reaching its threshold value at \(t_{c}\sim t_{0}\) according to condition (24 ),\[\frac{2\left\vert P_{x}\left( t_{0}\right) \right\vert }{\omega _{sc}}\approx \tau _{\gamma }\;. \label{lim7}\tag{33}\] The minimal frequency for the region of high frequencies is: \[\omega _{\min }\approx \tau _{\gamma }\omega _{sc}\;. \label{e34}\tag{34}\]

The smallest possible value of the moment \(t_{0}\) , at which Eq. (34 ) holds true, is achieved at the smallest possible momentum value \(p_{x}\) from the finite range (15 ). Taking it into account, we find\[t_{0}-t_{1}\sim \left( \tau _{\gamma }/2+\tau \right) \Delta t_{st}. \label{e36}\tag{35}\]

The frequency \(\omega\) grows linearly, \(\frac{d\omega }{dt_{c}}=2eE\), from the minimum value \(\omega _{\min }\) as long as the electric field is acting and reaches the maximum possible for a given \(p_{x}\) frequency \(\omega _{2}\) at the time instant \(t_{c}\sim t_{2}\), when the electric field switches off. Photon with such a frequency is emitted during the formation interval preceding the moment \(t_{2}\) of switching off the electric field. It follows from Eq. (31 ) that\[\omega _{2}\approx 2\left\vert P_{x}\left( t_{2}\right) \right\vert . \label{e32}\tag{36}\] Absolute maximum among all possible frequencies \(\omega _{2}\) with different momenta \(p_{x}\) satisfying Eq. (15 ) is:\[\omega _{\mathrm{\max }}\approx 2\Delta t_{st}^{-2}\left[ t_{2}+\max \left( -p_{x}/eE\right) \right] \approx 2\left( T/\Delta t_{st}\right) \omega _{sc}\;. \label{e33}\tag{37}\] A frequency range between \(\omega _{\mathrm{\max }}\) and \(\omega _{\min }\) does exists if \[\frac{\omega _{\mathrm{\max }}}{\omega _{\min }}\approx \frac{2T}{\Delta t_{st}\tau _{\gamma }}>1, \label{e33a}\tag{38}\] which means that the field duration time \(T\) satisfying Eq. (1 ) and the upper limitation to the intensity is sufficiently large.

We see that in the range of high-frequencies, the domain of the applicability of the LCFA for a photon emission accompanying pair creation from a vacuum is \(\omega _{\min }<\omega <\omega _{\mathrm{\max }}\). The obtained results can be extended to a general slowly varying field configuration assuming that the kinetic energies in the conservation law (27 ) are given by general forms (32 ). In this case,\[\omega _{\min }/\sqrt{eE\left( t_{0}\right) }=2\left\vert P_{x}\left( t_{0}\right) \right\vert /\sqrt{eE\left( t_{0}\right) }\approx \tau _{\gamma },\;\omega _{2}\approx 2\left\vert p_{x}+eA_{x}\left( t_{2}\right) \right\vert >\omega _{\min }. \label{lim11}\tag{39}\] We find that the integral of the probability of one photon emission, given by Eq. (20 ), over frequency \(\omega\) is bounded above by the value of \(\omega _{\mathrm{\max }}\).

3.2 Large duration limit↩︎

For the momenta \(p_{x}\) and \(p_{x}^{\prime }\) satisfying condition (15 ) and for finite \(u_{0}\ll \min \left( \left\vert u_{1}\right\vert ,\left\vert u_{2}\right\vert \right)\), one can use limit \(T\rightarrow \infty\) in integrals (21 ). We denote the corresponding limits as:\[Y_{j^{\prime }j}^{0}\left( \rho ,\varphi \right) =\left. Y_{j^{\prime }j}^{0}\left( t_{2},t_{1}\right) \right\vert _{T\rightarrow \infty }\;. \label{e16}\tag{40}\] Here, assuming \(u_{0}^{2}-u_{x}^{2}\neq 0,\) the hyperbolic coordinates \(\rho\) and \(\varphi\),\[\begin{align} &&u_{0}=\rho \cosh \varphi ,\;u_{x}=\rho \sinh \varphi \;, \notag \\ &&\rho =\sqrt{u_{0}^{2}-u_{x}^{2}},\;\tanh \varphi =\frac{u_{x}}{u_{0}}\;, \label{e17} \end{align}\tag{41}\] are introduced. We have \(p_{x}^{\prime }=p_{x}-k_{x}\). Therefore, in any frequency range the ratio \(\left\vert u_{x}\right\vert /u_{0}\)is bounded above, \[\frac{\left\vert u_{x}\right\vert }{u_{0}}=\frac{\left\vert k_{x}\right\vert }{\omega }\leq 1. \label{lim5}\tag{42}\] The \(\varphi\) dependence of integrals (40 ) can be factorized as (see Appendix B in Ref. [21] for details) \[Y_{j^{\prime }j}^{0}\left( \rho ,\varphi \right) =e^{-i\beta \varphi }Y_{j^{\prime }j}^{0}\left( \rho ,0\right) ,\;i\beta =\left( \nu -\nu ^{\prime }+j-j^{\prime }\right) /2, \label{a6b}\tag{43}\]

WPCF’s in the integral \(Y_{j^{\prime }j}^{0}\left( \rho ,0\right)\) are solutions of the same differential equation. Taking it into account and performing integrations by parts one finds that the function \(Y_{j^{\prime }j}^{0}\left( \rho ,0\right)\) satisfies the differential equation for the confluent hypergeometric functions. An explicit form of \(Y_{j^{\prime }j}^{0}\left( \rho ,0\right)\) can be fixed by boundary conditions at \(\rho \rightarrow 0\) corresponding to the original integral; see Appendix B in Ref. [21] for details. The following relations take place:\[\begin{align} &&\mathcal{J}_{j^{\prime }j}^{0}\left( \rho \right) =Y_{j^{\prime }j}^{0}\left( \rho ,0\right) =e^{i\pi \left( \nu +\nu ^{\prime }+j+j^{\prime }\right) /2}I_{j^{\prime }j}(\rho )\,, \notag \\ &&I_{j^{\prime },j}(\rho )=\sqrt{\pi }e^{-i\pi /4}e^{i\rho ^{2}/4}Z^{i\beta }\Psi \left( \nu +j,1+2i\beta ;Z\right) ,\;\;Z=e^{-i\pi /2}\rho ^{2}/2\;, \label{e21a} \end{align}\tag{44}\] where \(\Psi\) is the confluent hypergeometric function (CHF) (we use notation of Ref. [22]).

We find that \(M_{n^{\prime }n}^{0}\) has the form:\[\begin{align} M_{n^{\prime }n}^{0} &\approx &-\frac{\mu }{2}\left[ 2i\tilde{\chi}_{\sigma ^{\prime }\sigma }^{\vartheta \left( 0,0\right) }Y_{00}^{0}\left( \rho ,\varphi \right) +\left( 1-i\right) \tilde{\chi}_{\sigma ^{\prime }\sigma }^{\vartheta \left( 0,1\right) }Y_{01}^{0}\left( \rho ,\varphi \right) \right. \notag \\ &+&\left. \left( -1+i\right) \tilde{\chi}_{\sigma ^{\prime }\sigma }^{\vartheta \left( 1,0\right) }Y_{10}^{0}\left( \rho ,\varphi \right) +\tilde{\chi}_{\sigma ^{\prime }\sigma }^{\vartheta \left( 1,1\right) }Y_{11}^{0}\left( \rho ,\varphi \right) \right] , \label{am24a} \end{align}\tag{45}\] where \[\begin{align} &&\tilde{\chi}_{\sigma ^{\prime }\sigma }^{\vartheta \left( 0,0\right) }=v_{+1,\sigma ^{\prime }}^{\dagger }\mathbf{\alpha }\boldsymbol{\epsilon }_{\mathbf{k}\vartheta }v_{-1,\sigma },\;\tilde{\chi}_{\sigma ^{\prime }\sigma }^{\vartheta \left( 1,1\right) }=v_{+1,\sigma ^{\prime }}^{\dagger }B\left( \mathbf{p}^{\prime }\right) \mathbf{\alpha }\boldsymbol{\epsilon }_{\mathbf{k}\vartheta }B\left( \mathbf{p}\right) v_{-1,\sigma }, \notag \\ &&\tilde{\chi}_{\sigma ^{\prime }\sigma }^{\vartheta \left( 0,1\right) }=v_{+1,\sigma ^{\prime }}^{\dagger }\mathbf{\alpha }\boldsymbol{\epsilon }_{\mathbf{k}\vartheta }B\left( \mathbf{p}\right) v_{-1,\sigma },\;\tilde{\chi}_{\sigma ^{\prime }\sigma }^{\vartheta \left( 1,0\right) }=v_{+1,\sigma ^{\prime }}^{\dagger }B\left( \mathbf{p}^{\prime }\right) \mathbf{\alpha }\boldsymbol{\epsilon }_{\mathbf{k}\vartheta }v_{-1,\sigma }, \notag \\ &&\mu =e^{-\pi \left( \lambda +\lambda ^{\prime }\right) /8}\exp \left( i\omega \frac{p_{x}+p_{x}^{\prime }}{2eE}\right) . \label{am25a} \end{align}\tag{46}\]

To describe the angular distribution, we define the orthonormal triple\[\begin{align} \mathbf{k/}k &=&(\cos \phi ,\,\sin \theta \sin \phi ,\,\cos \theta \sin \phi )\,, \notag \\ \boldsymbol{\epsilon }_{\mathbf{k}1} &=&\mathbf{e}_{x}\times \mathbf{k/}\left\vert \mathbf{e}_{x}\times \mathbf{k}\right\vert ,\quad \boldsymbol{\epsilon }_{\mathbf{k}2}=\mathbf{k\times }\boldsymbol{\epsilon }_{\mathbf{k}1}/\left\vert \mathbf{k\times }\boldsymbol{\epsilon }_{\mathbf{k}1}\right\vert , \label{ga7} \end{align}\tag{47}\] such that\[\begin{align} \boldsymbol{\epsilon }_{\mathbf{k}1} &=&(0,\,-\cos \theta ,\sin \theta \,)\,, \notag \\ \boldsymbol{\epsilon }_{\mathbf{k}2} &=&(\sin \phi ,\,-\sin \theta \cos \phi ,-\,\cos \theta \cos \phi )\,, \label{ga8} \end{align}\tag{48}\] where \(0\leq \phi \leq \pi\), \(-\pi \leq \theta \leq +\pi\).

The matrix elements\(M_{n^{\prime }n}^{0}\) givenby Eq. (45 ) involve complicated combinations of \(\gamma\)-matricesmaking theangular and polarization distributions of the emitted photon difficult to analyse. However, in the domain of the applicability of the LCFA to the high frequency range (24 ), \(\omega _{\min }<\omega <\omega _{\mathrm{\max }}\), theanalysis becomes simpler.

To this end, we start with the fact that under condition (24 ) \(\left\vert u_{x}\right\vert \ll u_{0}\), it follows that \(\tanh \varphi \ll 1\) then \(\varphi \approx \cos \phi =k_{x}/\omega\), \(\left\vert \cos \phi \right\vert \ll 1\). We see that a contribution depending on parameter \(\varphi\) in the integral (43 ) is small and \(u_{0}\approx \rho\). The radiation is directed mainly near the plane orthogonal to the axis \(x\).

In the case of the high frequency, \(\rho \gtrsim \tau _{\gamma }\gg 1\), the both parameters \(\;c\) and \(Z\) of the CHF \(\Psi \left( a,c;Z\right)\) appearing in Eq. (44 ) are large and \(a\) and the ratio \(\eta =Z/c\) are fixed and positive, \(\eta >0\). Under these conditions, using appropriate asymptotic approximation (see sec. 13.8(ii) in Ref. [23]), we find that the integral \(Y_{j^{\prime }j}^{0}\left( \rho ,0\right)\) can be approximated as\[\begin{align} &&Y_{j^{\prime }j}^{0}\left( \rho ,0\right) \approx Ge^{i\pi \left( j+j^{\prime }\right) /2}Z^{-j^{\prime }}\left[ D_{-\left( \nu +j\right) }\left( 0\right) +O\left( \rho ^{-1}\right) \right] , \notag \\ &&G=\sqrt{\pi }e^{i\Theta }e^{-3\pi \left( \lambda +\lambda ^{\prime }\right) /8},\;\Theta =\left[ \rho ^{2}-\left( \lambda +\lambda ^{\prime }\right) \ln \rho ^{2}/2-\pi \right] /4, \notag \\ &&D_{-j-\nu }\left( 0\right) =2^{-\left( \nu +j\right) /2}\sqrt{\pi }\Gamma \left( 1/2+\left( \nu +j\right) /2\right) ^{-1}. \label{ga9461} \end{align}\tag{49}\] The leading contributions to the amplitude \(M_{n^{\prime }n}^{0}\) arise from the terms proportional to \(Y_{00}^{0}\) and \(Y_{01}^{0}\),\[Y_{0j}^{0}\left( \rho ,\varphi \right) =e^{-i\varphi j/2}e^{\left( \lambda -\lambda ^{\prime }\right) \varphi /4}Y_{0j}^{0}\left( \rho ,0\right) . \label{ga9462}\tag{50}\]

Using the Dirac’s representation for the \(\gamma\)-matrices, one finds: \[\begin{align} \tilde{\chi}_{\sigma ^{\prime }\sigma }^{\vartheta \left( 0,0\right) } &=&\left\{ \begin{array}{ll} 0\,, & \mathrm{if}\;\;\sigma ^{\prime }=\sigma \, \\ \left( -i\epsilon _{\mathbf{k}\vartheta }^{2}+\sigma \epsilon _{\mathbf{k}\vartheta }^{3}\right) , & \mathrm{if}\;\;\sigma ^{\prime }\neq \sigma \,\end{array}\right. , \notag \\ \tilde{\chi}_{\sigma ^{\prime }\sigma }^{\vartheta \left( 0,1\right) } &=&\frac{\epsilon _{\mathbf{k}\vartheta }^{1}}{\sqrt{eE}}\left\{ \begin{array}{ll} -m, & \mathrm{if}\;\;\sigma ^{\prime }=\sigma \, \\ \left( -ip_{y}+\sigma p_{z}\right) & \mathrm{if}\;\;\sigma ^{\prime }\neq \sigma \,\end{array}\right. . \label{ga9462b} \end{align}\tag{51}\] Then terms mixingthese matrix elements contribute only if \(\sigma ^{\prime }\neq \sigma\) and \(\vartheta =2\). In the leading-order term approximation with respect to \(\rho\), we find:\[\sum_{\sigma ,\sigma ^{\prime }=\pm 1}\left. \left\vert M_{n^{\prime }n}^{0}\right\vert ^{2}\right\vert _{\mathbf{p}^{\prime }=\mathbf{p}-\mathbf{k}}\approx 2\pi e^{-\pi \left( \lambda +\lambda ^{\prime }\right) }\times \left\{ \begin{array}{c} \cosh \frac{\pi \lambda }{4},\;\mathrm{if}\;\vartheta =1 \\ \sinh \frac{\pi \lambda }{4}\left( 1+\mathcal{M}\cos \phi \right) ,\;\mathrm{if}\;\vartheta =2\end{array}\right. , \label{ga9463}\tag{52}\] where\[\begin{align} &&\mathcal{M}=1+2\pi \mathrm{Re}\Omega \left( \sinh \frac{\pi \lambda }{2}\right) ^{1/2}\left( \sinh \frac{\pi \lambda }{4}\right) ^{-1}, \\ &&\Omega =\frac{p_{\theta }}{\sqrt{eE}\sqrt{\lambda }}e^{i\Theta _{M}},\;\Theta _{M}=-\frac{\pi }{4}-\arg \left[ \Gamma \left( \frac{1}{2}-\frac{i\lambda }{4}\right) \Gamma \left( 1+\frac{i\lambda }{4}\right) \right] , \end{align}\]and remain the only terms linear with respect to \(\cos \phi\) . Here the projection of the vector \(\mathbf{p}_{\bot }\) onto the direction of the vector \(\mathbf{k}_{\bot }\) is denoted as \(p_{\theta }\). The projection of this vector onto the perpendicular direction is denoted as \(p_{\bar{\theta}}\), so that \(\mathbf{p}_{\bot }^{2}=p_{\theta }^{2}+p_{\bar{\theta}}^{2}\).

We see that the probability of the high-frequency radiation given by Eqs. (20 ) and (52 ) has two linear polarizations. The polarization \(\vartheta =2\) is in the plane \(kX\) spanned by the direction of propagation and the external electric field. The polarization \(\vartheta =1\) is in the plane orthogonal to the plane \(kX\). In the case of polarization \(\vartheta =2\), the probability has a small contribition proportional to \(\cos \phi =k_{x}/\omega\). The quantity given by Eq. (52 ) depends on the frequency of the photon due to the parameter \(\lambda ^{\prime }\), \[\lambda ^{\prime }=\lambda +\rho ^{2}-2\rho p_{\theta }/\left( eE\right) ^{-1/2}. \label{ga10}\tag{53}\]

Now we can estimate the probability of the high-frequency emission, given by Eq. (20 ). In the case of the \(T\)-constant field, the momentum range \(D\) is finite, given by Eq. (15 ). The probability (20 ) can be presented by an integral over the range \(D\). Taking into account that the contributions of large transverse momentum is exponentially suppressed, it is sufficient to choose the limits of integration over its components as \(-\varepsilon _{0}<p_{\theta }/\left( eE\right) ^{-1/2}<\varepsilon _{0}\) and \(-\varepsilon _{0}<p_{\bar{\theta}}/\left( eE\right) ^{-1/2}<\varepsilon _{0}\), where \(1\ll \varepsilon _{0}\ll \rho\). In this case, the approximation (52 ) is valid. Neglecting exponentially suppressed terms and a small contribition depending on \(\cos \phi\), we finally obtain that\[\begin{align} &&\frac{d\mathcal{P}\left( \mathbf{k}\vartheta |0\right) }{d\omega d\Omega }\approx \frac{\alpha n^{\mathrm{cr}}V}{16\pi ^{2}\omega _{sc}}\mathcal{R}\left( \mathbf{k,}\vartheta \right) , \notag \\ &&\mathcal{R}\left( \mathbf{k,}\vartheta \right) =e^{-\pi \left[ \lambda _{0}+\rho ^{2}\right] }\left( \delta _{\vartheta 1}\cosh \frac{\pi \lambda _{0}}{4}+\delta _{\vartheta 2}\sinh \frac{\pi \lambda _{0}}{4}\right) , \label{ga13} \end{align}\tag{54}\] where \(n^{\mathrm{cr}}\) is the density of the pairs created from the vacuum,\[n^{\mathrm{cr}}=r^{\mathrm{cr}}T,\;r^{\mathrm{cr}}=\frac{\left( eE\right) ^{2}}{4\pi ^{3}}e^{-\pi \lambda _{0}},\;\lambda _{0}=\frac{m^{2}}{eE}. \label{ga14}\tag{55}\]

We see that the dependence of the probability of the high-frequency emission on frequency is determined mainly by the value of \(e^{-\pi \rho ^{2}}\). In the case of not very strong electric field, we have \(\cosh \frac{\pi \lambda _{0}}{4}\approx \sinh \frac{\pi \lambda _{0}}{4}\approx \exp \frac{\pi \lambda _{0}}{4}\) then the probability of the emission for both polarization are the same. In the case of a strong field, \(\pi \lambda _{0}\lesssim 1\), the emission is polarized in the plane orthogonal to the plane \(kX\). The emission accompanying the pair creation from vacuum is distinguished by the fact of the cylindrical symmetry and its characteristic polarization properties.

The obtained exact result for the large duration limit, given by Eqs. (43 ) - (48 ), can be also used as a basis for LCFA in the range of anot very high frequencies, \(1/T\ll \omega <\omega _{\min }\). We will present the LCFA representation for the probability of the high-frequency emission that hold true for any slowly varying field of a constant direction somewhere else. Note that presented results can be useful to consider such kind of the emission for 3D Dirac semimetals where the energy gap plays the role of a mass term. When this gap is small enough, the critical field is available in laboratory conditions.

4 Conclusion↩︎

Following a nonperturbative formulation of strong-field QED we present the effective perturbation theory of the photon emission in the presence of a strong electric field that is suitable to the locally constant field approximation (LCFA). We construct closed formulas for the total probabilities. We explicitly derivethe total probability of one photon emission from the vacuum in a constant electric field of finite duration \(T\). We establish the domain of the applicability of the LCFA for the photon emission and show that in the case of high frequencies, the width of the formation interval does not depend on the frequency and on the momenta of the particles and is determined entirely by the electric field \(E\). The variationoftheexternal electricfieldactingontheparticle withintheformationlengthcanbeneglected,which allows us to use the LCFA.The frequency of the emission grows linearly and bounded above by the value of the work of the electric field \(2eET\). The radiation is directed mainly near the plane orthogonal to the axis of the electric field. The emission accompanying the pair creation from vacuum is distinguished by the fact of the cylindrical symmetry and its characteristic polarization properties. In the case of a strong field, the emission is polarized in the direction orthogonal to the wave vector and the electric field. The obtained results can be easily extended to the study of the emission in any slowly varying field configuration assuming that the electric field \(E\left( t\right) >0\) is uniform and time-dependent.

4.0.0.0.1 Acknowledgement

D. M. G. thanks FAPESP (Grant No. 21/10128-0) and CNPq for permanent support. The work of T. C. A. is funded by XJTLU Research Development Funding, award no. RDF-21-02-056.

4.0.0.0.2 Conflict of Interest

The authors declare that there is no conflict of interest, either existing or potential.

References↩︎