skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Vacuum polarization in Coulomb field revisited

Abstract

Simplified derivation of Wichmann–Kroll term is presented. The derivation uses two formulas for hypergeometric functions, but otherwise is elementary. It is found that Laplace transform of the vacuum charge density diverges at zero momentum transfer. This divergence has nothing to do with known ultraviolet divergence. The latter is related to the large momentum behavior of the pertinent integral, while the former to the small momentum behavior. When these divergences are removed, the energy shift caused by vacuum polarization for an ordinary hydrogen obtained here is in an exact agreement with the result obtained by Wichmann and Kroll. Also, for muonic hydrogen the result obtained here reasonably agrees with that given in literature.

Authors:
;
Publication Date:
OSTI Identifier:
22617492
Resource Type:
Journal Article
Resource Relation:
Journal Name: Annals of Physics; Journal Volume: 379; Other Information: Copyright (c) 2017 Elsevier Science B.V., Amsterdam, The Netherlands, All rights reserved.; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; BOUND STATE; CHARGE DENSITY; COULOMB FIELD; HYPERGEOMETRIC FUNCTIONS; LAPLACE TRANSFORMATION; MOMENTUM TRANSFER; QUANTUM ELECTRODYNAMICS; RENORMALIZATION; ULTRAVIOLET DIVERGENCES; VACUUM POLARIZATION

Citation Formats

Zamastil, J., E-mail: zamastil@karlov.mff.cuni.cz, and Šimsa, D.. Vacuum polarization in Coulomb field revisited. United States: N. p., 2017. Web. doi:10.1016/J.AOP.2017.02.008.
Zamastil, J., E-mail: zamastil@karlov.mff.cuni.cz, & Šimsa, D.. Vacuum polarization in Coulomb field revisited. United States. doi:10.1016/J.AOP.2017.02.008.
Zamastil, J., E-mail: zamastil@karlov.mff.cuni.cz, and Šimsa, D.. Sat . "Vacuum polarization in Coulomb field revisited". United States. doi:10.1016/J.AOP.2017.02.008.
@article{osti_22617492,
title = {Vacuum polarization in Coulomb field revisited},
author = {Zamastil, J., E-mail: zamastil@karlov.mff.cuni.cz and Šimsa, D.},
abstractNote = {Simplified derivation of Wichmann–Kroll term is presented. The derivation uses two formulas for hypergeometric functions, but otherwise is elementary. It is found that Laplace transform of the vacuum charge density diverges at zero momentum transfer. This divergence has nothing to do with known ultraviolet divergence. The latter is related to the large momentum behavior of the pertinent integral, while the former to the small momentum behavior. When these divergences are removed, the energy shift caused by vacuum polarization for an ordinary hydrogen obtained here is in an exact agreement with the result obtained by Wichmann and Kroll. Also, for muonic hydrogen the result obtained here reasonably agrees with that given in literature.},
doi = {10.1016/J.AOP.2017.02.008},
journal = {Annals of Physics},
number = ,
volume = 379,
place = {United States},
year = {Sat Apr 15 00:00:00 EDT 2017},
month = {Sat Apr 15 00:00:00 EDT 2017}
}
  • We continue our study of the behavior of the vacuum polarization near the nucleus, the region of importance for heavy muonic atoms. We show that the vacuum polarization potential induced by a point nuclear charge can be expanded in certain integral and nonintegral powers of the radius r, and expansion involving r$sup -1$, r, r , and higher terms. The coefficient of the r -1 term was computed in the previous paper, analytically, to all orders of Zα. In this paper, we compute the coefficients of the r and r$sup 2 terms analytically, to all orders in Zα. Ourmore » results agree in third order with earlier calculations, which had been done only to this order. Parts of our calculation are considerably simplified by an expansion in powers of the electron mass using formal operator and determinantal techniques.« less
  • This is the first of a series of three papers concerned with the short- distance behavior of the vacuum polarization potential. It is this region that gives the principal energy level shift in heavy muonic atoms. In this paper, we compute the induced vacuum polarization point charge that appears in the limit of a point nuclear charge source. Our result, computed analytically to all orders of Z$alpha$, agrees with an earlier calculation of Wichmann and Kroll. We have been able to simplify the calculation considerably by taking advantage of a zero- electron-mass limit and by using formal operator and determinantalmore » techniques.« less
  • We compute the change in the vacuum polarization near a high-Z nucleus arising from the finite extent of the nuclear charge density. A massless- electron technique is exploited to permit an analytical calculation of the effect to all orders in Z$alpha$. Applications to muonic atoms are of particular interest as tests of quantum electrodynamics. The calculation of muonic energy level shifts is carried out to lowest order in the ratio of the nuclear radius to the radius of the muonic orbit. This approximation is appropriate to many muonic states with large orbits. Our analytical results are in excellent agreement withmore » numerical studies by Gyulassy and give as a special case the order $alpha$ (Z$alpha$)$sup 3$ calculation of Arafune. The effect on muonic x rays such as those produced in the 5g$sub 9$/$sub 2$ $Yields$ 4f$sub 7$/$sub 2$ transition in $sup 208$Pb is about 5 eV, increasing slightly the discrepancy between theory and experiment in such systems.« less