Dynamically assisted pair production in subcritical potential step and particle--anti-particle interpretations

TL;DR

The paper examines vacuum pair production in a spatially inhomogeneous external field by contrasting two particle–antiparticle frameworks (A) and (B) for a subcritical step potential, and then adds a weak oscillating field to study dynamically assisted production. Using Furry-picture perturbation theory, it shows that pair creation can occur at second order when the total energy $V_0+oldsymbol{ extomega}$ exceeds $2m$, even if $V_0<2m$, with richer momentum structures emerging as $oldsymbol{ extomega}$ crosses thresholds. The authors derive explicit expressions for differential particle numbers in both pictures and demonstrate that, while the two frameworks agree in some regimes, they yield quantitatively different results at second order, especially for $oldsymbol{ extomega}>2m$. These findings indicate a sensitivity of pair-production observables to the chosen particle–antiparticle interpretation and motivate numerical and experimental tests, as well as extensions to gauge-invariant formulations and higher-order perturbative effects.

Abstract

Particle--anti-particle interpretation under spatially inhomogeneous external fields within the framework of quantum field theory is a nontrivial problem. In this paper, we focus on the two interpretations established in [Phys. Rev. D 93, 045002 (2016)] and [Prog. Theor. Exp. Phys. 2022, 073B02 (2022)], both of which give consistent results of vacuum instability and pair production. To shed light on their differences, a pair production under a potential step assisted by a weak and oscillating electric field is discussed. It is shown that the potential step and the oscillating field, each insufficient for vacuum decay, can produce pairs when combined. In addition, the two pictures give rise to quantitative differences in the number of created pairs at the second-order perturbation of the oscillating field. It might provide a clue to investigate the correct particle--anti-particle interpretation by comparing the result with numerical simulations or experiments.

Figures (5)

• Figure 1: The scattering wave functions belonging to the energy regions (i)--(iv) under the subcritical potential step (with height $V_0 < 2m$). Blue arrows represent directions of incident, reflected, and transmitted waves in the scattering wave functions. Gray-shaded regions are a mass gap, where oscillating solutions do not exist. The left half corresponds to the left-incident case, while the right half corresponds to the right-incident case.
• Figure 2: The "in" mode functions (B) in the energy regions (i)--(iv) under the same potential step. The left half of the figures shows $\varphi_s^{(E)}$, while the right half shows $\chi_s^{(E)}$. The directions of plane waves included in the mode functions are expressed in blue or red arrows. The blue arrows in the regions (ii) and (iii) are the same as those in Fig. \ref{['fig::psi_phi']}.
• Figure 3: (Left panel) The momentum distribution of the differential particle number density for $V_0 = 1.5m$ and $\omega = 1.5m$. The other parameters are $\mathcal{E}_z = 0.01m^2/e$ and $l = 2m^{-1}, 4m^{-1}, 6m^{-1}, 8m^{-1}$ and $\infty m^{-1}$. The results for each $l$ are drawn in the blue, orange, green, and red solid lines and purple dashed lines, respectively. (Right panel) The schematic picture of particle production from the Dirac sea when $V_0 < 2m$ and $\omega < 2m$. The created particle moving to the left is drawn in the orange ball with the orange arrow, while the hole in the Dirac sea corresponds to the dashed orange circle. The particle in the site of the hole jumps up and tunnels to the positive-frequency area in $z < 0$ along the blue and red dashed arrows.
• Figure 4: (Left panel) The momentum distribution of the particle number $V_0 = 1.5m$ and $\omega = 2.5m$. The other parameters are the same as Fig. \ref{['fig::n(k,s)_subcritical']}. (Right panel) The schematic picture of particle production from the Dirac sea when $V_0 < 2m$ and $2m < \omega < V_0 + 2m$.
• Figure 5: Comparison of the momentum distributions of the number density of particles within the particle--anti-particle picture (A) and (B), each of which is expressed in the blue solid line and orange dashed line, respectively. The parameters are $V_0 = 1.5m$, $\omega = 2.5m$, $l = 6m^{-1}$, and $\mathcal{E}_z = 0.01m^2/e$.