Advertisement

Theoretical and Mathematical Physics

, Volume 198, Issue 3, pp 331–362 | Cite as

Essentially Nonperturbative Vacuum Polarization Effects in a Two-Dimensional Dirac–Coulomb System with Z > Zcr: Vacuum Charge Density

  • K. A. SveshnikovEmail author
  • Yu. S. Voronina
  • A. S. Davydov
  • P. A. Grashin
Article
  • 4 Downloads

Abstract

For a planar Dirac–Coulomb system with a supercritical axially symmetric Coulomb source with the charge Z > Zcr,1 and radius R0, we consider essentially nonperturbative vacuum-polarization effects. Based on a special combination of analytic methods, computer algebra, and numerical calculations used in our previous papers to study analogous effects in the one-dimensional “hydrogen atom,” we study the behavior of both the vacuum density ρVP(r⃗) and the total induced charge and also the vacuum-polarization energy EVP. We mainly focus on divergences of the theory and the corresponding renormalization, on the convergence of partial series for ρVP(r⃗) and ɛVP, on the integer-valuedness of the total induced charge, and on the behavior of the vacuum energy in the overcritical region. In particular, we show that the renormalization via the fermion loop with two external legs turns out to be a universal method, which removes the divergence of the theory in the purely perturbative and essentially nonperturbative modes for ρVP and ɛVP. The most important result is that for Z ≫ Zcr,1 in such a system, the vacuum energy becomes a rapidly decreasing function of the source charge Z, which reaches large negative values and whose behavior is estimated from below (in absolute value) as ~ −|ηeffZ3|/R0. We also study the dependence of polarization effects on the cutoff of the Coulomb asymptotic form of the external field. We show that screening the asymptotic value significantly changes the structure and properties of the first partial channels with mj = ±1/2,±3/2. We consider the nonperturbative calculation technique and the behavior of the induced density and the integral induced charge QVP in the overcritical region in detail.

Keywords

planar Dirac–Coulomb system vacuum polarization essentially nonperturbative effects for Z > Zcr effects of Coulomb asymptotic screening 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    J. Reinhardt and W. Greiner, “Quantum electrodynamics of strong fields,” Rep. Progr. Phys., 40, 219–295 (1977).ADSCrossRefGoogle Scholar
  2. 2.
    W. Greiner, B. Müller, and J. Rafelski, Quantum Electrodynamics of Strong Fields, Springer, Berlin (1985).CrossRefGoogle Scholar
  3. 3.
    G. Plunien, B. Müller, and W. Greiner, “The Casimir effect,” Phys. Rep., 134, 87–193 (1986).ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    R. Ruffini, G. Vereshchagin, and S.-S. Xue, “Electron–positron pairs in physics and astrophysics: From heavy nuclei to black holes,” Phys. Rep., 487, 1–140 (2010); arXiv:0910.0974v3 [astro-ph.HE] (2009).ADSCrossRefGoogle Scholar
  5. 5.
    W. Greiner and J. Reinhardt, Quantum Electrodynamics, Springer, Berlin (2012).zbMATHGoogle Scholar
  6. 6.
    V. M. Kuleshov, V. D. Mur, N. B. Narozhnyi, A. M. Fedotov, Yu. E. Lozovik, and V. S. Popov, “Coulomb problem for a Z > Zcr,” Phys. Usp., 58, 785–791 (2015).ADSCrossRefGoogle Scholar
  7. 7.
    J. Rafelski, J. Kirsch, B. Müller, J. Reinhardt, and W. Greiner, “Probing QED vacuum with heavy ions,” arXiv:1604.08690v1 [nucl-th] (2016).Google Scholar
  8. 8.
    S. I. Godunov, B. Machet, and M. I. Vysotsky, “Resonances in positron scattering on a supercritical nucleus and spontaneous production of e+e − pairs,” Eur. Phys. J. C, 77, 782 (2017); arXiv:1707.07497v2 [hep-ph] (2017).ADSCrossRefGoogle Scholar
  9. 9.
    M. I. Katsnelson, “Nonlinear screening of charge impurities in graphene,” Phys. Rev. B, 74, 201401 (2006); arXiv:cond-mat/0609026v3 [cond-mat.mes-hall] (2006).ADSCrossRefGoogle Scholar
  10. 10.
    A. V. Shytov, M. I. Katsnelson, and S. Levitov, “Vacuum polarization and screening of supercritical impurities in graphene,” Phys. Rev. Lett., 99, 236801 (2007); arXiv:0705.4663v2 [cond-mat.mes-hall] (2007).ADSCrossRefGoogle Scholar
  11. 11.
    K. Nomura and A. H. MacDonald, “Quantum transport of massless Dirac fermions,” Phys. Rev. Lett., 98, 076602 (2007).ADSCrossRefGoogle Scholar
  12. 12.
    V. N. Kotov, V. M. Pereira, and B. Uchoa, “Polarization charge distribution in gapped graphene: Perturbation theory and exact diagonalization analysis,” Phys. Rev. B, 78, 075433 (2008).ADSCrossRefGoogle Scholar
  13. 13.
    V. M. Pereira, V. N. Kotov, and A. H. Castro Neto, “Supercritical Coulomb impurities in gapped graphene,” Phys. Rev. B, 78, 085101 (2008); arXiv:0803.4195v2 [cond-mat.mes-hall] (2008).ADSCrossRefGoogle Scholar
  14. 14.
    I. F. Herbut, “Topological insulator in the core of the superconducting vortex in graphene,” Phys. Rev. Lett., 104, 066404 (2010).ADSCrossRefGoogle Scholar
  15. 15.
    Y. Wang, D. Wong, A. V. Shytov, V. W. Brar, S. Choi, Q. Wu, H.-Z. Tsai, W. Regan, A. Zettl, R. K. Kawakami, S. G. Louie, L. S. Levitov, and M. F. Crommie, “Observing atomic collapse resonances in artificial nuclei on graphene,” Science, 340, 734–737 (2013); arXiv:1510.02890v1 [cond-mat.mes-hall] (2015).ADSCrossRefGoogle Scholar
  16. 16.
    Y. Nishida, “Vacuum polarization of graphene with a supercritical Coulomb impurity: Low-energy universality and discrete scale invariance,” Phys. Rev. B, 90, 165411 (2014); arXiv:1405.6299v2 [cond-mat.mes-hall] (2014).ADSCrossRefGoogle Scholar
  17. 17.
    R. Barbieri, “Hydrogen atom in superstrong magnetic fields: Relativistic treatment,” Nucl. Phys. A, 161, 1–11 (1991).ADSCrossRefGoogle Scholar
  18. 18.
    V. P. Krainov, “A hydrogen-like atom in a superstrong magnetic field,” Sov. Phys. JETP, 37, 406 (1973).ADSGoogle Scholar
  19. 19.
    A. E. Shabad and V. V. Usov, “Positronium collapse and the maximum magnetic field in pure QED,” Phys. Rev. Lett., 96, 180401 (2006); arXiv:hep-th/0605020v1 (2006).ADSCrossRefGoogle Scholar
  20. 20.
    A. E. Shabad and V. V. Usov, “Bethe–Salpeter approach for relativistic positronium in a strong magnetic field,” Phys. Rev. D, 73, 125021 (2006); arXiv:hep-th/0603070v2 (2006).ADSCrossRefGoogle Scholar
  21. 21.
    A. E. Shabad and V. V. Usov, “Electric field of a pointlike charge in a strong magnetic field and ground state of a hydrogenlike atom,” Phys. Rev. D, 77, 025001 (2008); arXiv:0707.3475v3 [astro-ph] (2007).ADSCrossRefGoogle Scholar
  22. 22.
    V. N. Oraevskii, A. I. Rez, and V. B. Semikoz, “Spontaneous production of positrons by a Coulomb center in a homogeneous magnetic field,” Sov. JETP, 45, 428–435 (1977).ADSGoogle Scholar
  23. 23.
    B. M. Karnakov and V. S. Popov, “A hydrogen atom in a superstrong magnetic field and the Zeldovich effect,” JETP, 97, 890–914 (2003).ADSCrossRefGoogle Scholar
  24. 24.
    M. I. Vysotskii and S. I. Godunov, “Critical charge in a superstrong magnetic field,” Phys. Usp., 57, 194–198 (2014).ADSCrossRefGoogle Scholar
  25. 25.
    A. Davydov, K. Sveshnikov, and Yu. Voronina, “Vacuum energy of one-dimensional supercritical Dirac–Coulomb system,” Internat. J. Modern Phys. A, 32, 1750054 (2017); arXiv:1709.04239v1 [hep-th] (2017).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Yu. S. Voronina, A. S. Davydov, and K. A. Sveshnikov, “Vacuum effects for a one-dimensional ‘hydrogen atom’ with Z > Zcr,” Theor. Math. Phys., 193, 1647–1674 (2017).CrossRefzbMATHGoogle Scholar
  27. 27.
    Yu. Voronina, A. Davydov, and K. Sveshnikov, “Nonperturbative effects of vacuum polarization for a quasi-onedimensional Dirac–Coulomb system with Z > Zcr,” Phys. Part. Nucl. Lett., 14, 698–712 (2017).CrossRefGoogle Scholar
  28. 28.
    Yu. S. Voronina, A. S. Davydov, K. A. Sveshnikov, and P. A. Grashin, “Essential nonperturbative vacuumpolarization effects in a two-dimensional Dirac–Coulomb system for Z > Zcr: Vacuum-polarization energy,” Theor. Math. Phys. (2019 in press).Google Scholar
  29. 29.
    P. Gärtner, U. Heinz, B. Müller, and W. Greiner, “Limiting charge for electrostatic point sources,” Z. Phys. A, 300, 143–155 (1981).ADSCrossRefGoogle Scholar
  30. 30.
    I. Aleksandrov, G. Plunien, and V. Shabaev, “Nuclear recoil and vacuum-polarization effects on the binding energies of supercritical H-like ions,” Eur. Phys. J. D, 70, 18 (2016); arXiv:1511.04346v1 [physics.atom-ph] (2015).ADSCrossRefGoogle Scholar
  31. 31.
    B. L. Voronov, D. M. Gitman, and I. V. Tyutin, “The Dirac Hamiltonian with a superstrong Coulomb field,” Theor. Math. Phys., 150, 34–72 (2007).MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    D. M. Gitman, I. V. Tyutin, and B. L. Voronov, Self-Adjoint Extensions in Quantum Mechanics (Progr. Math. Phys., Vol. 62), Springer, New York (2012).CrossRefzbMATHGoogle Scholar
  33. 33.
    D. Gitman, A. Levin, I. Tyutin, and B. L. Voronov, “Electronic structure of super heavy atoms revisited,” Phys. Scr., 87, 038104 (2013).ADSCrossRefzbMATHGoogle Scholar
  34. 34.
    V. R. Khalilov and I. V. Mamsurov, “Planar density of vacuum charge induced by a supercritical Coulomb potential,” Phys. Lett. B, 769, 152–158 (2017); arXiv:1604.01271v1 [hep-th] (2016).ADSCrossRefzbMATHGoogle Scholar
  35. 35.
    A. Davydov, K. Sveshnikov, and Yu. Voronina, “Nonperturbative vacuum polarization effects in two-dimensional supercritical Dirac–Coulomb system: I. Vacuum charge density,” Internat. J. Modern Phys. A, 33, 1850004 (2018); arXiv:1712.02704v1 [hep-th] (2017).ADSCrossRefzbMATHGoogle Scholar
  36. 36.
    P. J. Mohr, G. Plunien, and G. Soff, “QED corrections in heavy atoms,” Phys. Rep., 293, 227–369 (1998).ADSCrossRefGoogle Scholar
  37. 37.
    E. H. Wichmann and N. M. Kroll, “Vacuum polarization in a strong Coulomb field,” Phys. Rev., 101, 843–859 (1956).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Y. Hosotani, “Spontaneously broken Lorentz invariance in three-dimensional gauge theories,” Phys. Lett. B, 319, 332–338 (1993); arXiv:hep-th/9308045v1 (1993).ADSCrossRefGoogle Scholar
  39. 39.
    V. R. Khalilov and I. V. Mamsurov, “Vacuum polarization of planar charged fermions with Coulomb and Aharonov–Bohm potentials,” Modern Phys. Lett. A, 31, 1650032 (2016); arXiv:1509.02775v2 [cond-mat.meshall] (2015).ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vol. 1, McGraw-Hill, New York (1953).zbMATHGoogle Scholar
  41. 41.
    M. Gyulassy, “Higher order vacuum polarization for finite radius nuclei,” Nucl. Phys. A, 244, 497–525 (1975).ADSCrossRefGoogle Scholar
  42. 42.
    U. Fano, “Effects of configuration interaction on intensities and phase shifts,” Phys. Rev., 124, 1866–1878 (1962).ADSCrossRefzbMATHGoogle Scholar
  43. 43.
    Yu. Voronina, K. Sveshnikov, P. Grashin, and A. Davydov, “Essentially non-perturbative and peculiar polarization effects in planar QED with strong coupling,” Phys. E, 106, 298–311 (2019); arXiv:1805.10688v2 [cond-mat.mes-hall] (2018).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  • K. A. Sveshnikov
    • 1
    • 2
    Email author
  • Yu. S. Voronina
    • 1
    • 2
  • A. S. Davydov
    • 1
    • 2
  • P. A. Grashin
    • 1
    • 2
  1. 1.Faculty of PhysicsLomonosov Moscow State UniversityMoscowRussia
  2. 2.Bogolyubov Institute of Theoretical Problems of the MicroworldLomonosov Moscow State UniversityMoscowRussia

Personalised recommendations