Particle production at a finite potential step:
Transition from Euler-Heisenberg to Klein paradox

We consider the potential step

where subscript ‘1’ denotes a single step, and step height

The electric field has one $z$-direction component

where ${\mathcal{E}}_{0}={\mathcal{E}}(0)$ is the field at the center of the step. The limit $L\to 0$ is as indicated a $\delta$-function. The SS requires the presence of a dipole charge density

which behaves like a $\delta^{\prime}$-function. This behavior agrees with that of a self-consistent potential step obtained with back-reaction effects in charged vacuum solutions [34, 35].

The Sauter step potential has an exact solution to the Dirac equation [1], allowing ‘Klein paradox’ pair production under the condition

We obtain the vacuum decay rate by pair production evaluating the vacuum persistence probability $P$

which is less than unity only when when $\mathrm{Im}[\Gamma]>0$. We recall that the vacuum decay rate can be related to the rate of pair production, see [36, 37].

The physically relevant quantity in the study of a sharp localized potential step is the action density ${\mathcal{L}}$ per unit of time $T$ and surface $a_{\perp}$, instead of the usual action per time and volume: ${\mathcal{L}}={\Gamma}/{a_{\perp}T}$. The vacuum decay rate probability $w$ per unit of $a_{\perp}T$ follows from

We use here the results for the imaginary parts of effective action $\mathrm{Im}[{\mathcal{L}}]$ presented by Kim [25]. The two cases of interest are:

• •

  Spin $1/2$: The imaginary part of effective action ${\mathcal{L}}^{1/2}_{\mathrm{SS}}(\Delta V,m,L)$, from Eq. (22) of [25]

  $\displaystyle\mathrm{Im}[{\mathcal{L}}^{1/2}_{\mathrm{SS}}]=-\int_{D}\frac{d\omega d^{2}k_{\perp}}{(2\pi)^{3}}\ln\Big{[}\frac{\sinh[\pi\Omega_{+}/2]\sinh[\pi\Omega_{-}/2]}{\sinh[\pi\Delta_{+}/2]\sinh[\pi\Delta_{-}/2]}\Big{]},$

  (8)

  and

• •

  Spin-$0$: The imaginary part of effective action ${\mathcal{L}}^{0}_{\mathrm{SS}}(\Delta V,m,L)$, from Eq. (23) of [25]

  $\displaystyle 2\mathrm{Im}[{\mathcal{L}}^{0}_{\mathrm{SS}}]=\!\!\int_{D}\frac{d\omega d^{2}k_{\perp}}{(2\pi)^{3}}\ln\Big{[}\frac{\cosh[\pi\Omega_{+}/2]\cosh[\pi\Omega_{-}/2]}{\cosh[\pi\Delta_{+}/2]\cosh[\pi\Delta_{-}/2]}\Big{]}\;,$

  (9)

  where:

and spin-$1/2$ spin projections $\sigma=\pm 1/2$, and for spin $0$ $\sigma=0$. As shown in Fig. 1, the domain of integration $D$ in Eq. (8) describes the region in which particle states of energy $\omega$, with transverse momentum $k_{\perp}$, are capable of tunneling through the ($2m$) mass gap

The Klein paradox limit is obtained by taking the limit $L\to 0$ in Eq. (8), see Fig. 1. In the opposite limit we obtain the known form of EHS effective action following Nikishov [22], see also [25] and [26], evaluating Eq. (8) in the $L\to\infty$ limit: As $L$ increases, the potential step height increases, while the electric field ${\mathcal{E}}_{0}$ stays quasi-constant in an ever larger space domain $V_{\mathrm{EHS}}\propto L\times a_{\perp}$; thus we must also find $\mathrm{Im}[{\mathcal{L}}^{1/2}_{\mathrm{SS}}]\propto L$.

The EHS-LCF approximation arises from the EHS analytical result upon integration over $z$: For spin $1/2$

and for spin $0$

Only where the electric field ${\mathcal{E}}(z)$ of the SS given by Eq. (3) is nonvanishing does a contribution, dominated by the domain of maximum field, arise. The integral diverges in the $L\to 0$ KP limit due the quadratic in the field pre-factor.

In Fig. 3 we present the ratios of EHS-LCF to exact SS imaginary actions for spin-$1/2$ and spin-0. We see that in the parameter domain first accessible to experiment, near threshold $e\Delta V\sim 2m$, the EHS-LCF approximation is not an acceptable approach to guide the design of future experiments. The domain in which the EHS-LCF approximation is valid is obtained by moving diagonally up the plots; both $L$ and $e\Delta V$ values increasing. The upper right hand corners of the plots (domains within contour = $1.2$) correspond to less than $20\%$ error. We find to our surprise that there is a finite optimal $L$, dependent on step height $e\Delta V$, at which the EHS-LCF is closest to the exact SS imaginary action.

We compute the SS transverse momentum spectra using the mean number of pairs produced [22, 25]

Summing over momentum and spin states, the pair production rate per unit cross sectional area and time

where the domain $D$ of tunneling states is rewritten as

equivalent to Eq. (11) with exchanged order of integration, and with $k_{\perp}$ written in terms of $m_{\perp}$. With the help of Eq. (26) we can write for Eq. (25)

with the transverse spectrum

The exact transverse spectrum Eq. (28) has a maximum value of $m_{\perp}$ beyond which no particle is produced. Such a finite upper $m_{\perp}$ limit does not arise in the EHS-LCF approximation [36, 37]

In the EHS limit of SS with $L\to\infty$ at a fixed field strength ${\mathcal{E}}_{0}=\Delta V/L$, see end of section II.2 for details, Eq. (28) and Eq. (29) are asymptotically equal.

In Fig. 4 we plot the transverse momentum spectra as functions of $m_{\perp}^{2}$ (straight lines correspond to Gaussian spectra). The small momentum ($m_{\perp}\sim m$) agreement between the exact SS and the EHS-LCF spectra is a consequence of normalization. In comparing the two spectra we are thus evaluating their distributions, comparing the relative suppression of large momentum states. Due to the need to satisfy the upper limit $m_{\perp\mathrm{max}}^{2}=4m^{2}$, Eq. (26) we find that the exact SS spectra with $L/\lambdabar_{C}=0.1,0.01$ drop much faster than the Gaussian EHS-LCF distributions. At larger $L$, see $L/\lambdabar_{C}=100$ example, the SS spectrum becomes Gaussian, yet the distribution remains more narrow than (below) the EHS result.

We now turn to comparison of spin-0 with spin-$1/2$ pair production. We define ratio $R$ as

the ratio of exact SS imaginary actions Eq. (9) and Eq. (8). It was noted in work by Gies and Torgrimsson [39] that the ratio may be nontrivial. We indeed find, surprisingly to us, that the spin-$0$ production dominates the spin-$1/2$ rate in the KP limit in the proximity of the pair threshold ($e\Delta V=2.01m$), despite spin multiplicity, Fig. 5. In the large $L$ EHS-LCF limit (dotted line), the spin-$0$ contribution becomes half the spin-$1/2$ rate.

We have established a relation between the Euler-Heisenberg-Schwinger and the spontaneous particle production from a potential well, creating in explicit computation an understanding of when the application of the EHS-locally-constant-field rate of pair production is allowable. Moreover, the following insights developed in this work extend prior effort on spontaneous particle production:

1.   The physical interpretation of the sharp-edge KP limit of spin-$1/2$ effective action requires consideration of the vacuum decay rate per unit-surface area:

2.   We quantify the domain of validity of the EHS-LCF approximation:

The domain of optimal step width $L$ and height $e\Delta V$, where the EHS-LCF imaginary action is closest to the exact SS rate, surprisingly, spans a finite range of values in $L$. As an example, in Fig. 3 of section III.2 we find that for the domain $2m<e\Delta V<5m$, EHS-LCF is closer to the exact SS result when $L=2\lambdabar_{C}$ than when $L=10\lambdabar_{C}$. Thus any application of EHS-LCF to static localized electric fields must take into account the extraordinary cases where stronger, sharper applied fields are more accurately represented than weaker, smoother fields spanning larger spatial domains.

3.   We demonstrate additional (compared to EHS) suppression of the produced particle spectra at high momentum:

4.   We restore self-adjointness to the spin-0 square well case by making one side smooth:

Our current results may find application:

1.   Peripheral ultra-relativistic heavy ion collisions [4, 5]: The electrical fields of colliding ions are a function of collision energy i.e. the Lorentz-factor $\gamma$. Already for moderately relativistic $\gamma\simeq 20$ the shape of the field at the instant of ions passing each other resembles the SS case considered here. The form of the field undergoes a shift as a function of time from the LCF limit for well separated ions, to the KP limit at the instant ions pass each other. In this condition we expect the dominant particle production contribution suitable for probing the KP vacuum decay limit to be ‘heavy’ particles such as muons.

2.   Non-relativistic heavy ion collisions [3, 2]: Spontaneous and induced production of positrons was explored several decades ago in the near adiabatic limit of slowly moving nuclei. We believe that the theory of these processes could be considerably advanced by developing further the context of particle emission through a surface proposed here. This step is essential to achieve a better understanding of positron production rates and their scaling behavior as a function of impact parameter and nuclear charge.

3.   The ultra-short laser pulse particle production [6, 7]: As the high-intensity light frontier advances, pulsed laser field configurations will reach the barely critical (for electrons) potential step regime. Our results demonstrate that actual pair production rates may be orders of magnitude smaller compared to locally constant field estimates, see Fig. 3: For a recent review of such work see [14, 15]. However, the incorporation of the dynamics of time dependent fields and light pulse motion may be key to more definitive understanding of this experimental case.

4.   Astrophysics [8, 9]: The electron-positron annihilation line at 511 keV is the most prominent spectral feature in the $\gamma$-ray spectrum of the Milky Way, where the origin of positrons remains an open question. Extreme field strengths are predicted for cataclysmic merger and collapse events in astrophysics. Implementation of the here advanced particle production per unit-surface SS model should allow a better estimation of the positron formation yields in such events and hence the expected positron emission, and annihilation radiation intensity.

In this work we have considered the case of fixed in space potential configurations. Thus we did not explore the assistance to particle production provided by the eventual time dependence of field configurations [44] – in early literature this effect was called ‘induced’ (by time dependence), see Refs. [3, 20]. Initial steps to accommodate the effect of time dependence for SS action were already made by Narozhnyi and Nikishov [23]. During the writing of this paper the authors became aware of the recently submitted work on sharp time-dependent steps [45], extending the exactly solvable class of field profiles such as the Sauter step to asymmetric (single step) configurations.

Accommodation of time dependence is without doubt a required further development of the results we present to allow application in actual experiments. For the spin-$0$ case, with a sharp-walled well potential, this is a critical step to obtain actual pair production ensuing from the break-up of the condensate formed in this situation. The remaining question is whether the charge neutral pair condensate releases pairs (sudden potential wall removal), or if the pairs self-annihilate (slow wall removal). In addition, when pair production is abundant it is also essential to discuss the effect of back reaction on the produced spectra of particles [34, 46].

Another potential future development relates to the analytical properties of the Sauter effective action which differ from the EHS result. The EHS action for constant fields possesses an essential singularity in the imaginary part for a vanishing electrical field. However, this is not the case for fields existing for finite time periods: Bialynicki-Birula, Rudnicki and Wienczek [47] have demonstrated that even an adiabatic switch-on suffices to assure the absence of an essential singularity for the pair production rate in time-dependent fields. A similar study for spatially varying SS fields would greatly improve understanding of this phenomenon. How the removal of the essential singularity affects the real part of the effective action is also an open question.

The EHS action (both its real and imaginary parts) was shown to possess an effective temperature representation obeying Bose statistics [29]. The imaginary part of the SS effective action has been connected to spin statistics [26, 27]. Whether this representation holds in general for fields of finite spacetime extent is of considerable interest.

There is long-standing interest [48, 49, 50, 51, 52, 53] in understanding how the effective action depends on the anomalous magnetic moment $g-2$: Generalizing to arbitrary spin $g$-factors is possible in the EHS action by implementing the Klein-Gordon-Pauli equation (second order Dirac equation with $g\neq 2$ [40]). We applied the same extension in the KP limit of SS studying the bound state structure in the well; spontaneous pair production requires $|g|>1$. This result predicts interesting $g-2$ behavior for SS effective action, a topic of potential future interest.

We are also interested in a better understanding of the non-perturbative vacuum response of spin-$1/2$ ($e^{+}-e^{-}$) pair fluctuations in strong rapidly moving EM-field configurations: This is in our opinion of essential importance for fully understanding the interaction of intense light pulses with the EM-vacuum. We expect a wealth of non-trivial behavior: We recall that the available real part of SS effective action [25] relies upon the imaginary part being nonzero, i.e. the potential step being capable of producing pairs: $e\Delta V>2m$. We further know that for small $|g|<1$, and in the spin-$0$ ($g=0$) case, for sharp-walled wells, the imaginary part of the action can be null for arbitrary step heights (no particle production), including the supercritical $e\Delta V>2m$. Understanding of the real part of the effective action where its imaginary part vanishes could rely on the analytical structure of results here considered.