Geodesic
Essentially nonperturbative vacuum polarization effects in a~two-dimensional Dirac--Coulomb system for $Z>Z_{\text{cr}}$: Vacuum polarization effects
Teoretičeskaâ i matematičeskaâ fizika, Tome 199 (2019) no. 1, pp. 69-103

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For a planar Dirac–Coulomb system with a supercritical extended axially symmetric Coulomb source with a charge $Z>Z_{\mathrm{cr},1}$ and radius $R_0$, we consider essentially nonperturbative vacuum polarization effects in the overcritical region. Using results obtained in our preceding paper for the induced charge density $\rho_{\scriptscriptstyle\mathrm{VP}}(\vec{r}\,)$, we thoroughly consider the calculation of the vacuum energy $\mathcal{E}_{\scriptscriptstyle\mathrm{VP}}$ based on the renormalization, the convergence of the partial expansion for $\rho_{\scriptscriptstyle\mathrm{VP}}(\vec{r}\,)$, and the behavior of the integral induced charge $Q_{\scriptscriptstyle\mathrm{VP}}$ in the overcritical region. In particular, we show that the renormalization using the fermionic loop with two external lines turns out to be a universal technique, which eliminates the divergence of the theory in the purely perturbative and essentially nonperturbative modes for $\rho_{\scriptscriptstyle\mathrm{VP}}(\vec{r}\,)$ and $\mathcal{E}_{\scriptscriptstyle\mathrm{VP}}$. The most significant result is that for $Z\gg Z_{\mathrm{cr},1}$ in such a system, the vacuum energy becomes a rapidly decreasing function of the source charge $Z$ reaching large negative values; its behavior is estimated from below (in absolute value) as $\sim-|\eta_{\mathrm{eff}}Z^3|/R_0$. We also study the dependence of the effect of the decrease in $\mathcal{E}_{\scriptscriptstyle\mathrm{VP}}$ on the cutoff of the Coulomb asymptotics of the external field at different scales $R_1>R_0$ and $R_1\gg R_0$.
Mots-clés : planar Dirac–Coulomb system
Keywords: vacuum polarization, essentially nonperturbative effects for $Z>Z_{\mathrm{cr}}$, effects of screening the Coulomb asymptotics. 
@article{TMF_2019_199_1_a4,
     author = {K. A. Sveshnikov and Yu. S. Voronina and A. S. Davydov and P. A. Grashin},
     title = {Essentially nonperturbative vacuum polarization effects in a~two-dimensional {Dirac--Coulomb} system for $Z>Z_{\text{cr}}$: {Vacuum} polarization effects},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {69--103},
     publisher = {mathdoc},
     volume = {199},
     number = {1},
     year = {2019},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2019_199_1_a4/}
}
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K. A. Sveshnikov; Yu. S. Voronina; A. S. Davydov; P. A. Grashin. Essentially nonperturbative vacuum polarization effects in a~two-dimensional Dirac--Coulomb system for $Z>Z_{\text{cr}}$: Vacuum polarization effects. Teoretičeskaâ i matematičeskaâ fizika, Tome 199 (2019) no. 1, pp. 69-103. http://geodesic.mathdoc.fr/item/TMF_2019_199_1_a4/