Production of Dirac Particles in External Electromagnetic Fields

TL;DR

This paper investigates the creation of Dirac particles in external electromagnetic fields by solving the Dirac equation exactly and analyzing the resulting particle production, highlighting the dependence on field strengths.

Contribution

It provides exact solutions to the Dirac equation in electromagnetic fields and calculates particle creation rates, advancing understanding of quantum field effects in external fields.

Findings

Particle creation depends on electric and magnetic field strengths.

Exact solutions enable precise calculation of Bogoliubov coefficients.

Results clarify the relationship between field parameters and particle production.

Abstract

Pair creation of spin- 1/2 particles in Minkowski spacetime is investigated by obtaining exact solu- tions of the Dirac equation in the presence of electromagnetic fields and using them for determining the Bogoliubov coefficients. The resulting particle creation number density depends on the strength of the electric and magnetic fields.

Equations

$$\omega =\frac{(eE{)}^2}{4{\pi }^3}\sum _{n=1}^{+\infty }\frac{1}{n^2}exp(-\frac{n\pi m^2}{eE})$$

$$A_{\nu }=B_0\tau [1+\tanh (x/\tau )]{\delta }_{\nu }^2-E_0(\Gamma +\Lambda t){\delta }_{\nu }^3$$

$$j_{\nu }=\frac{1}{4\pi }[\frac{2|\overrightarrow{B}|}{\tau }\tanh (x/\tau )]{\delta }_{\nu }^2$$

$${\zeta }^{ϵ\theta }[\frac{\partial S}{\partial x^{ϵ}}-e{A}_{ϵ}][\frac{\partial S}{\partial x^{\theta }}-e{A}_{\theta }]+m^2=0$$

$$F^{\pi \varrho }{F}_{\pi \varrho }=2(B^2-E^2)=2B_0^2{\mathrm{sech}}^4(x/\tau )-E_0^2{\Lambda }^2$$

$$F^{\pi \varrho }{F}_{\pi \varrho }^{\ast }=4\overrightarrow{E}\cdot \overrightarrow{B}=4\Lambda E_0{B}_0{\mathrm{sech}}^2(x/\tau )$$

$$S(t,\overrightarrow{x})=P(x)+Q(t)+(y{k}_y+z{k}_z)$$

$${\dot{Q}}^2-{\acute{P}}^2-[k_z+e{E}_0(\Gamma +\Lambda t){]}^2$$

$$-[k_y-e{B}_0\tau (1+\tanh (x/\tau )){]}^2+m^2=0$$

$${\dot{Q}}^2-[e{E}_0(\Gamma +\Lambda t){]}^2-2k_{z}e{E}_0(\Gamma +\Lambda t)+m^2-k_z^2=v^2$$

$${\acute{P}}^2+\{e{B}_0\tau [1+\tanh (x/\tau )]{\}}^2-2k_{y}e{B}_0\tau [1+\tanh (x/\tau )]=v^2$$

$$S(\varrho ,\overrightarrow{x})$$

$$+\frac{e{E}_0}{\Lambda }{\int }_0^{\varrho }\sqrt{{\varrho }^2+\frac{2k_z}{e{E}_0}\varrho +(\frac{k_z^2+v^2-m^2}{e^2{E}_0^2})}d\varrho$$

$$=(\frac{\varrho e{E}_0+k_z}{2\Lambda })\sqrt{{\varrho }^2+\frac{2k_z}{e{E}_0}\varrho +(\frac{k_z^2+v^2-m^2}{e^2{E}_0^2})}$$

$$\displaystyle+(\frac{v^{2}-m^{2}}{2eE_{0}\Lambda})\ln\Bigg{\{}2\varrho+\frac{2k_{z}}{eE_{0}}$$

$$\displaystyle+2\sqrt{\varrho^{2}+\frac{2k_{z}}{eE_{0}}{\varrho}+(\frac{k_{z}^{2}+v^{2}-m^{2}}{e^{2}E_{0}^{2}})}\Bigg{\}}+S_{0}(0,\overrightarrow{x})$$

$${\psi }_{(t\to \mp \infty )}=e^{iS(t)}\to e^{\pm i(\frac{e{E}_0\Lambda }{2})t^2\pm i(\frac{v^2-m^2}{2e{E}_0\Lambda })\ln (2\Lambda |t|)}$$

$$[i{\gamma }^{\nu }{\partial }_{\nu }+e{A}_{\nu }{\gamma }^{\nu }-m]\psi =0$$

$$[(\overrightarrow{P}-e\overrightarrow{A}{)}^2+m^2-e\overrightarrow{\sigma }\cdot (\overrightarrow{B}+i\overrightarrow{E})]\varphi =(p_0-e{A}_0{)}^2\varphi$$

$$\psi=\left(\begin{array}[]{c}{[{\overrightarrow{\sigma}\cdot(\overrightarrow{P}-e\overrightarrow{A})+(p_{0}-eA_{0})+m}]}\phi\\ {[{\overrightarrow{\sigma}\cdot(\overrightarrow{P}-e\overrightarrow{A})+(p_{0}-eA_{0})-m}]}\phi\\ \end{array}\right)$$

$$\varphi =e^{i(y{k}_y+z{k}_z)}\left(\begin{array}{c}{\chi }_1(x)T_1(t)\\ {\chi }_2(x)T_2(t)\end{array}\right)$$

$$\{-\frac{d^2}{d{x}^2}-e{B}_0(e{B}_0{\tau }^2+s){\mathrm{sech}}^2(x/\tau )$$

$$+2e{B}_0(e{B}_0-\tau k_y)\tanh (x/\tau )+2e{B}_0\tau (e{B}_0\tau -k_y)$$

$$+m^2+k_y^2+k_z^2+\frac{d^2}{d{t}^2}+(e{E}_0\Lambda t{)}^2+2e{E}_0\Lambda (e{E}_0\Gamma +k_z)t$$

$$+e{E}_0\Gamma (e{E}_0\Gamma +2k_z)-ise{E}_0\Lambda \}{\chi }_s(x)T_s(t)=0$$

$$[\hat{F}(x)+\hat{Q}(t)]{\chi }_s(x)T_s(t)=0$$

$$\hat{F}(x)=-\frac{d^2}{d{x}^2}-e{B}_0(e{B}_0{\tau }^2+s){\mathrm{sech}}^2(x/\tau )$$

$$+2e{B}_0(e{B}_0-\tau k_y)\tanh (x/\tau )$$

$$+2e{B}_0\tau (e{B}_0\tau -k_y)+m^2+k_y^2+k_z^2$$

$$\hat{Q}(t)=\frac{d^2}{d{t}^2}+(e{E}_0\Lambda t{)}^2+2e{E}_0\Lambda (e{E}_0\Gamma +k_z)t$$

$$\displaystyle eE_{0}\Gamma(eE_{0}\Gamma+2k_{z})-iseE_{0}\Lambda\$$

$$[\hat{F}(x)-{\varpi }^2]{\chi }_s(x)=0$$

$$[\hat{Q}(t)+{\varpi }^2]T_s(t)=0$$

$$[\frac{d^2}{d{r}^2}+\Sigma {\mathrm{sech}}^{2}r-{\rm Y}\tanh r-\epsilon ]{\chi }_s(x)=0$$

$$\Sigma =e{B}_0(e{B}_0{\tau }^2+s),{\rm Y}=2e{B}_0{\tau }^2(e{B}_0-\tau k_y)$$

$$\epsilon =2e{B}_0{\tau }^3(e{B}_0\tau -k_y)+{\tau }^2(m^2+k_y^2+k_z^2-{\varpi }^2)$$

Full text

PACS numbers

13.40.-f, 23.20 Ra.

Electromagnetic interactions, pair production.

††preprint: APS/123-QED

I Introduction

After the pioneering works of Sauter 1 , Heisenberg and Euler 2 on the particle creation by the strong electromagnetic fields, Schwinger formulated the following pair creation probability per unit volume and time by obtaining the one-loop effective action in a constant and homogeneous classical electric field (in natural units, $\hbar=c=1$) 3 :

[TABLE]

where $m$ and $e$ are the mass and charge of the electron, $E$ is the electric field, respectively. Since then, this process is called Schwinger mechanism and has become an important problem in the quantum field theory (QFT). Such kind of a classical electric field is assumed to be order of $E\sim 10^{16}V/cm$ 4 which is very difficult to generate by the current technology. Strong fields arising from the collisions between the relativistic high energy particles and heavy-ions are called color electric fields and have ability to create particles from the vacuum. These type of collisions are generated at the modern colliders, i.e at CERN. Schwinger mechanism is attributed to the hadronic particle creation and on the base of Color Glass Condensate (CGS), this phase is called as Glasma.

The Schwinger mechanism have been studied in the presence of various stationary and non-stationary external fields 5 -9 . The studies about the Schwinger mechanism in gauge fields having both electric and magnetic field components have revealed that electric field has a dominant influence in creating the particles. Therefore, the pair creation mechanism is totally attributed to the pure electric field 10 . This quantum effect of the classical electromagnetic fields is carried out to the curved spacetime as well 11 -13 .

There is a considerable point in some of the studies in the literature that they support the magnetic field has a reduction effect in the particle creation process. One of the aims of this study is to investigate this phenomena for a particular choice of the electromagnetic gauge field that has both electric and magnetic field components.

The calculation of fermionic particle creation rate requires to define the positive and negative frequency energy states, namely the ”in” and ”out” mode vacuum solutions. For the motion of the relativistic charged particles moving in an external field, analysis of mode functions as positive and negative frequency solutions is not easy since the Lagrangian of the corresponding system completely depends on space-time coordinates. Namely, particle concept becomes indefinite owing to interaction with the external fields. For this reason we require a condition to define ”particle” concept. In the present study we will apply a quasiclassical method. We obtain exact solutions of the Hamilton-Jacobi (HJ) equation and discuss their asymptotic behavior in the infinite past and future . Then, asymptotic behavior of the solutions of the Dirac equation in the neighborhood of the time singularities will be identified. With the help of this analysis and comparison of asymptotic solutions of both HJ and Dirac equations in the infinite past and future, the particle picture will be identified.

We define positive and negative frequency mode functions in such a way that the positive frequency mode function approaches $e^{iS(t)}$ and the negative frequency one $e^{-iS(t)}$ in asymptotic regions 4 , where $S(t)$ is the solution of the HJ equation for the presence of a $4$-vector electromagnetic potential given as

[TABLE]

where $\tau$, $\Gamma$ and $\Lambda$ are constants. This new suggested form of the vector potential generates parallel stationary electric 5 and Sauter type magnetic fields 14 that are persuaded in the Glasma flux tube model of high energy heavy ion collisions.

Magnetic current emerging is found to be

[TABLE]

The outline of the paper is as follow: In Section $2$ we solve the relativistic HJ equation and obtain the asymptotic behavior of the solutions. In Section $3$ we solve the Dirac equation for the considered electromagnetic fields and obtain the asymptotic limits of the solutions to define the vacuum ”in” and ”out” modes by referring the asymptotic solutions of the HJ equation. We use the Bogoliubov transformation technique to relate the solutions at the boundaries and calculate the particle creation number density for fermions in Section $4$. Finally, in Section $5$ we discuss the results we obtained. Throughout the paper the natural units, \hbar̄=c=1 are used.

II Solutions of the Hamilton-Jacobi Equation

The relativistic HJ equation for the action S is given by 11 :

[TABLE]

where $\zeta^{\epsilon\theta}=(1,-1,-1,-1)$ is the Minkowski metric, $m$ is the mass of the particle and $A_{\nu}$ is the $4$-vector electromagnetic potential.

The electromagnetic potential satisfy the Lorentz gauge and the Lorentz invariants are determined from the electromagnetic field tensor as follows

[TABLE]

and

[TABLE]

Because of the space-time dependence of the considered electromagnetic field, the solution of the HJ equation can be separated as follow:

[TABLE]

where $k_{y}$ and $k_{z}$ can be viewed as the conserved momenta that exist given the symmetries chosen for the electromagnetic gauge (2). By using (7) in Eq. (4) we obtain

[TABLE]

where dot and acute denote derivatives with respect to $t$ and $x$, respectively.

We obtain two first order differential equations as follows:

[TABLE]

and

[TABLE]

where $v^{2}$ is the constant of separation.

Time-dependent external fields cause unstable vacuum and this results in the pair creation by vacuum. For this reason the dynamics involving spatial coordinates effect the solutions only by a constant and we obtain the solution of the HJ equation for electromagnetic gauge (2) as follow

[TABLE]

where $\varrho=(\Gamma+\Lambda t)$.

The dependence of the solution on time is derived by $\psi\rightarrow e^{iS(t)}$ and we arrive the following expressions for the asymptotic behavior of the relativistic wave function:

[TABLE]

where the upper and lower signs represent the negative and positive-frequency states, respectively.

III Solutions of the Dirac Equation

The Dirac equation in external electromagnetic fields is given by 15

[TABLE]

where $\gamma^{\nu}$ are Dirac matrices, $A_{\nu}$ is the $4$-vector electromagnetic potential, $m$ is the mass of electron, $e$ is the charge of the electron and $\psi$ is the four-component spinor.

The Dirac equation yields four coupled differential equations for the spinor and usually it is difficult to obtain the exact analytical solutions, in particular for mathematically complicated external fields. This difficulty of the problem has been accomplished by Feynmann and Gell-Mann by considering a two-component form of the Dirac equation in the presence of electromagnetic fields as follow 16

[TABLE]

where $\overrightarrow{\sigma}$ are usual Pauli matrices and \phi=(\begin{array}[]{c}\phi_{1}\\ \phi_{2}\end{array}) are the solutions of the two-component equation. The four-component spinor can be derived from $\phi$ as follow

[TABLE]

Thence, for the purpose of obtaining the analytic solutions we follow up the two-component formalism and consider the electromagnetic gauge (2). Because the given gauge field depends on $x$ coordinate and $t$, both $k_{y}$ and $k_{z}$ are constants of the motion and solutions can be written in the form

[TABLE]

Therefore, with the usage of Eqs. (2) and (16), Eq.(14) becomes

[TABLE]

where the spin index $s$ has the $\pm 1$ eigenvalues corresponding to the spinors $\phi_{1}$ and $\phi_{2}$, respectively. This equation can be written in a simpler form as

[TABLE]

with the following definitions

[TABLE]

[TABLE]

Eq.(18) has a separable form, so we get the following two equations

[TABLE]

[TABLE]

where $\varpi^{2}$ is the constant of separation.

By defining $x=\tau r$, Eq. (21) becomes

[TABLE]

where the definitions

[TABLE]

were made.

Following Rosen and Morse 17 , we set $\chi_{s}(r)=e^{ra}\cosh^{-b}rf_{s}(r)$ and obtain the following equation

[TABLE]

In order $\chi/f$ to be finite in the range $-\infty\leq r\leq+\infty$, $(a^{2}+b^{2}-\varepsilon)=0$ and $(2ab+\Upsilon)=0$ conditions are necessary 17 . From these conditions we derive the following expressions for $a$ and $b$:

[TABLE]

and

[TABLE]

Therefore, keeping these expressions and by introducing $\eta=\frac{1}{2}(1+\tanh r)$ we arrive

[TABLE]

which is the differential equation satisfied by the hypergeometric functions. The hypergeometric function remaining finite at $\eta=0$ will provide this equation and solution will be given as 18

[TABLE]

So, we obtain

[TABLE]

For this solution to be convergent at infinity the following condition must be satisfied 17

[TABLE]

Then

[TABLE]

and

[TABLE]

The constant of separation $\varpi$ can be easily derived from $(a^{2}+b^{2}-\varepsilon)=0$.

By introducing a variable $\xi=\sqrt{\frac{2}{eE_{0}\Lambda}}(eE_{0}\Lambda t+eE_{0}\Gamma+k_{z})$ we obtain the following equation from Eq.(22)

[TABLE]

Solutions of this differential equation are parabolic cylinder functions 18

[TABLE]

where $\tilde{a}=(\frac{iesE_{0}\Lambda+k_{z}^{2}-\varpi^{2}}{2eE_{0}\Lambda})$.

Therefore, exact solutions are obtained and all components of the Dirac spinor can be found with the insertion of Eqs. (30) and (35) into Eq.(16).

IV Particle Creation via Bogoliubov Transformation Method

Due to difficulty of the direct observation of the pair creation in a constant field 10 , because the typical $|eE|$ is smaller than $m^{2}$, the particle creation will be induced by the time-dependent components of the wave-function (28), namely the parabolic cylinder functions.

Two solutions of the Eq. (34) are given as:

[TABLE]

and

[TABLE]

These are not the only solutions and any of the remaining two-sets can be constructed via Bogoliubov coefficients as follow:

[TABLE]

and

[TABLE]

The Bogoliubov transformation method is a technique that associates a canonical commutation relation algebra or a canonical anti-commutation relation algebra into another representation, caused by an isomorphism 19 .

In the Minkowskian QFT, eigenfunctions of the field equation, $\psi$, can be written with the help of the mode solutions as 19 -20 .

[TABLE]

where we have the relations $(\varphi_{i},\varphi_{j})=\delta_{ij}$, $(\varphi_{i}^{\ast},\varphi_{j}^{\ast})=\delta_{ij}$, $(\varphi_{i},\varphi_{j}^{\ast})=0$ and $(\Theta_{i},\Theta_{j})=\delta_{ij}$, $(\Theta_{i}^{\ast},\Theta_{j}^{\ast})=\delta_{ij}$, $(\Theta_{i},\Theta_{j}^{\ast})=0$ for $\varphi$ and $\Theta$ are mode solutions. The $\varphi$ and $\Theta$ can be expanded in terms of each other.

The creation and annihilation operators $a_{n}^{\dagger},b_{k}^{\dagger}$ and $a_{n},b_{k}$ are in correlation by the following expressions

[TABLE]

[TABLE]

$\alpha_{kn}$ and $\beta_{kn}$ are Bogoliubov coefficients determined by $\alpha_{ij}=(\Theta_{i},\varphi_{j})$, $\beta_{ij}=-(\Theta_{i},\varphi_{j}^{\ast})$. They are related as

[TABLE]

[TABLE]

Let $|0_{a}\rangle$ and $|0_{b}\rangle$, are two states of vacuum in the Fock space and are related to each particle notion in (30). They are represented for all $n$ and $k$ as

[TABLE]

[TABLE]

If $|0_{b}\rangle$ is introduced as the usual vacuum, then $|0_{a}\rangle$ is regarded as a many-particle state. Therefore, the number of $\Theta_{n}$-mode particles in the state of $|0_{a}\rangle$ is

[TABLE]

If the $\varphi_{n}(x)$ are defined as positive frequency modes and the $\Theta_{n}(x)$ modes are linear unification of them, then $\beta_{jk}=0$. Then, $b_{k}|0_{b}\rangle=0$ and $a_{k}|0_{a}\rangle=0$. Hence, $\varphi_{j}$ and $\Theta_{k}$ modes have a common vacuum state. If $\beta_{jk}\neq 0$, then $\Theta_{k}$ contain a combination of positive-$\varphi_{k}$ and negative-$\varphi_{k}^{\ast}$ frequency modes.

Therefore, we can define the positive- and negative-frequency solutions in order to find the Bogoliubov coefficients. Asymptotic expansion of the parabolic cylinder functions is gives by 21

[TABLE]

Taking into account this relation for Eqs. (36),(37) in the limit $t\rightarrow+\infty$ (namely, $\xi\rightarrow+\infty$) and comparing the asymptotic expansion of them with Eq. (12), we see that the positive and negative-frequency mode solutions will be as follows respectively,

[TABLE]

and

[TABLE]

We conclude that the solutions behave as $T_{\pm}\approx e^{\pm iS(t)}$.

For $t\rightarrow-\infty$($\xi\rightarrow-\infty$), the solutions are in the form

[TABLE]

and

[TABLE]

so that their asymptotic behavior should be $T_{\pm}\approx e^{\pm iS(t)}$. It is clear that the solutions are different in the asymptotic regions and this is the nature of the particle creation. Therefore, the solutions for $t\rightarrow+\infty$ belong to vacuum ”out” mode whereas are vacuum ”in” mode for $t\rightarrow-\infty$.

The positive and negative frequency vacuum ”out” and ”in” modes can be related to each other with the Bogoliubov coefficients. By using Eq. (39), we can write

[TABLE]

Expanding the left side of this expression according to the below formula 21

[TABLE]

and using the result $[D_{\nu}(z)]^{\ast}=D_{-\nu-1}(-iz)$ that can be derived easily by taking the advantage of the relation between the parabolic cylinder function and Whittaker function given as 21

[TABLE]

we obtain the Bogoliubov coefficients $\alpha$ and $\beta$ as follows:

[TABLE]

and

[TABLE]

where $\breve{a}=(\frac{k_{z}^{2}-\varpi^{2}}{2eE_{0}\Lambda})$ and $|\alpha|^{2}+|\beta|^{2}=1$ condition is satisfied.

Then, we find the below expression for the Bogoliubov coefficients

[TABLE]

By considering the following formula for Gamma functions 17

[TABLE]

the number density of the created particles can be computed as follow

[TABLE]

where the parameter $\breve{a}$ in terms of the physical constants of four-vector potential (2) has been given as below.

[TABLE]

V Conclusion

In this study, we used the two-component formalism for the Dirac equation that is proposed by Feynmann and Gell-Mann. This approach to the problem removes the complexity of obtaining the exact solutions. One of the advantages of working with this form of the Dirac equation is that these solutions are valid for the Klein-Gordon particles in the case of $s=0$. Thus the results can be used both for scalar and fermionic particles.

Mechanism of particle production by strong electric fields is significant in order to figure out the early stages of the heavy-ion collisions, for example their effect on the thermalization of quarks and gluons. For the analysis of our problem we take account a strong constant electric field and a space-dependent hyperbolic magnetic field. Exact solutions of the Dirac equation were identified in terms of the parabolic cylinder and hypergeometric functions.

Existence of the strong electric fields cause to unstable vacuum that is asymptotically static at future. The ”in” and ”out” vacuum states were determined with the help of the asymptotic solutions of relativistic HJ equation. They were related by the Bogoliubov coefficients that are used to calculate the particle creation number density in Eq.(60). This expression depends on the parameters of electric and magnetic fields and is not in Fermi-Dirac thermal form. As it is seen by analyzing the formula and also from the Figure $1$, selected form of the magnetic field has a reduction effect on the creation of fermionic particles. This situation is compatible with previous obtained results. Also it can be seen from Figure $1$, particle creation rate increases due to electric field strength, $(E_{0}\Lambda)$.

Acknowledgements.

This study is supported by the Research Fund of Mersin University in TURKEY with project number: 2016-1-AP4-1425.