DOCSLIB.ORG

Precise Values for Critical Fields in Quantum Electrodynamics *

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Precise Values for Critical Fields in Quantum Electrodynamics * Precise Values for Critical Fields in Quantum Electrodynamics * Gerhard Soff and Berndt Müller Institut für Theoretische Physik der Universität Frankfurt am Main. Germany and Johann Rafelski ** Department of Physics, University of Pennsylvania, Philadelphia, Pa 19174 (Z. Natur forsch. 29 a, 1267-1275 [1974] ; received May 30, 1974) A careful investigation of different corrections to binding energies of electrons in almost critical fields is performed. We investigate quantitatively the influence of the nuclear charge parameters, nuclear mass, degree of ionization on the value of the critical charge of the nucleus. Rather quali- tative arguments are given to establish the contribution of the quantumelectrodynamic corrections, which are found to be small. Some phenomenological modifications of QED are quantitatively in- vestigated and found to be of negligible influence on the value of the critical field. For heavy ion collisions with ZlJrZ2^> Zc r the critical separations between ions are given as results of precise solutions of the relativistic two coulomb center problem. Corrections due to electron-electron inter- action are considered. We find (with present theoretical accuracy) Zcr = 173 + 2, in the heavy ion collisions RCT(U-U) = 34.7 ±2 fm and RCT (U-C f) = 47.7 ± 2 fm. We shortly consider the pos- sibility of spontaneous muon production in muonic supercritical fields. 1. Introduction where r0 = 1.2fm. The atomic mass A has been approximated for superheavy elements by If the binding energy of a state is greater than 2 me c2, while this state is vacant spontaneous free A = 0.00733 Z2 + 1.3 Z + 63.6. (1.2) positron creation occurs and the vacuum becomes charged1-3. The external electromagnetic potential The Is i/o state was found for values of Z 169. for which the spontaneous positron creation may At this point the binding energy of the lsi/2 state occur is called critical. In order to investigate the reached 2 me c2. For the 2pi/o electron the critical positron creation in a heavy ion collision it is value was found to be Z = 182. If the critical value necessary to know the precise least nuclear charge of Z could be increased by more than 15 units due which produces the corresponding critical potential. to nuclear properties, the electron - electron inter- The solution of the Dirac equation in the case of action and quantumelectrodynamical effects the as- a pure Coulomb potential leads to the Sommerfeld sociated experiments1'2 may become impossible. finestructure formula which allows for lsj/o electrons The exact theoretical and experimental determina- only solutions for Z < a-1 = h c/e2 = 137.03602. tion of the critical charge seems also to be a valuable Beyond this value of Z no lsi/9 state is found in test for relativistic quantum mechanics and quan- the discrete spectrum. tum electrodynamics of strong fields. By a precise For extended nuclei the Dirac equation was first measurement of electronic transition energies in solved by Pomeranchuk and Smorodinsky 4 (Their superheavy atoms or quasiatoms formed during the work suffered from various numerical errors) and collision of heavy ions with Zx + Z.2~ Zcr, it may by Pieper and Greiner 5. The nuclear charge distri- be possible to decide if selfenergy- and vacuum- bution was given by a homogeneous charge sphere polarization corrections are described well within with a radius the frame of the generalized Hartree-Fock equation R (1.1) given by Reinhard 6 and also Rafelski and Müller 7 (r I*p(z)yK&p(z) o(J d3zyK 8 71 \x-z 3 K F(\x-z\, ;(p)) (1.3) fd z!y <Pp(x)<Pp(z)yK ®q{z) v x-z\ Reprint requests to Dr. G. Soff, Institut für Theoretische * Work supported by Deutsche Forschungsgemeinschaft, Physik der Universität Frankfurt/M., D-6000 Frankfurt/ by Bundesministerium für Bildung und Wissenschaft, Main 1, Robert-Mayer-Straße 8—10. and by Gesellschaft für Schwerionenforschung (GSI). ** Supported by US-Atomic Energy Comission. 1268 G. Soff et al. • Precise Values for Critical Fields in Quantum Electrodynamics with sin(<jr' |X — Z |) F(\x-z\, ep-eq, sg(p)) = f d?< si' (1.4) + sg(p) {q'-ia) + 1 if P>F, 1.5 units of the critical value for positron creation ;(P) = -1 if p<F, when including the electron-electron interaction in the Thomas-Fermi approximation. Numerically it is F means the Fermi energy. possible to find bound state wavefunctions up to This paper is organized as follows. In the next the region Z = Zer — 0.01. In Fig. 1 one can see the section we discuss the one center problem in the frame of the Dirac equation. We present several numerical results concerning the influence of dif- Z = 172 .U, E = -1021.966 keV ferent parameters on the critical potential. In Sect. 3 we solve the two center problem and approximate the electron-electron interaction and obtain nu- IWI2-102 merical results for critical radii. In Sect. 4 we then consider the quantum electrodynamical corrections for one and two center potentials. In the following section we discuss the possibility of muonic super- critical states and show, that they cannot be created in heavy ion collisions. We close our discussion of the possible effects which could change the value of the critical nuclear charge discussing several phe- nomenological modifications of classical electro- dynamics and QED in Section 6. We show that no essential change of the critical quantities is to be Fig. 1. The radial density | r |2 of the lsj/j electron and the nuclear charge distribution is shown for an assumed expected from such modifications if the values of nucleus with Z = 172.44. The binding energy differs only by the phenomenological parameters are restricted by 44 eV from the critical value —2 m c2 . present experiments. In the last section we sum- marize our results and give a short outlook on radial density of the lsj/9 electron of Z = 172.44 future calculations. (c == 9.9 fm, t = 2.2 fm, £=-1021.964 keV) in comparison to the nuclear charge density assumed. A very remarkable fact is the relatively localized 2. Zcr within the Framework of the Dirac appearance of the wave function. For the diving Equation velocity defined as = AE/dZ\z=Zor we obtain 30.1 keV/unit charge. In our calculations we use a realistic nuclear charge distribution of a Fermi type with the surface In Fig. 2 the maximum of the radial density 2 thickness parameter t fixed usually to 2.2 fm and ' xpr of the lsj/o electron as function of the nuclear the half density parameter c which was derived charge is shown. For Z ZCT it reaches a value from the equivalent radius of about 25 fm. In a high energy collision of heavy naked nuclei = r0 Äh (2.1) with Z = Z1 + Z2> 137 one could assume because of with r0 = 1.2fm. The nuclear mass A was chosen compression of nuclear matter that it is possible to according to Eq. (1.2) of Pieper and Greiner4. In observe positron creation for example in a Pb + Pb order to take into account the electronic interaction (Zx + Z2 = 164 <Zcr) collision or with even lighter we first employed a Thomas-Fermi-approximation8. projectiles. In Fig. 3 we show the binding energy We found numerical solutions for the lsi/o state for of the lsj/2 electron as a function of Z where we Z 172 (c = 9.234 fm), the binding energy for have assumed as charge distribution two concentric Z = 172 is -1018.139 keV. That means a shift by superposed homogeneously charged spheres with in- 1269 G. Soff et al. • Precise Values for Critical Fields in Quantum Electrodynamics max If r| [fm] cal equivalent radius of a fermi charge distribution with t = 2.2 fm as function of Z is shown. We note 1s-electron that a change in the nuclear radius of 10% corre- sponds to a shift in the diving point by one unit. As the equivalent radius is determined from the atomic mass, we can also plot the A dependence of the energy of the lsj/2 state — see Figure 4 b. In 160 Z Fig. 2. The maximum of | ip r ]2 of the lsx/2 electron as function of Z. Fig. 4. a) The critical equivalent radius of a fermi charge distribution with t = 2.2 fm as function of Z. b) and c) Energies of the lsj/2 state as function of the atomic mass number A and of the surface thickness t of the nucleus for Z = 171. Fig. 4 c one can further see the energy as function of the surface thickness parameter t, where c was fixed to 9.218 fm and Z=171. With the value of the diving velocity = 30.1 keV/charge unit it is easy to recover the corresponding effect on the shift in the value of Zcr. The determination of Zcr seems to be uncertain by one unit due to uncertainties in Fig. 3. The energy eigenvalue of the lsj/o electron as func- the nuclear charge distribution. As an example of a tion of Z in a high energy heavy ion collision. The nuclear charge distribution consists of superposed homogeneous numerical solution to an overdimensioned nucleus charge spheres of the collision partners with increasing we show the radial density \iyr 2 of the lsj/g elec- charge in the smaller nucleus. Diving occurs at Z=167. tron for Z = 184 (92U + 92U) in Fig. 5 with a as- creasing charge in the smaller particle. Diving then sumed nuclear radius corresponding to c = 20.3 fm.
Load more

Recommended publications
  • Self-Energy Screening Approximations in Multi-Electron Atoms
    Radiation Physics and Chemistry 85 (2013) 118–123 Contents lists available at SciVerse ScienceDirect Radiation Physics and Chemistry journal homepage: www.elsevier.com/locate/radphyschem Self-energy screening approximations in multi-electron atoms J.A. Lowe a, C.T. Chantler a,n, I.P. Grant b a School of Physics, University of Melbourne, Australia b Mathematical Institute, Oxford University, Oxford, UK HIGHLIGHTS c We develop a self-energy screening approximation suitable for multi-electron atoms. c This approximation is tested in a number of few- and many-electron systems. c We obtain superior agreement with experiment compared with existing approximations. c An implementation of this approximation is provided for use with GRASP2K. article info abstract Article history: Atomic structure calculations have reached levels of accuracy which require evaluation of many- Received 31 October 2012 electron QED contributions. Since exact analytic solutions do not exist, a number of heuristics have Accepted 3 January 2013 been used to approximate the screening of additional electrons. Herein we present an implementation Available online 11 January 2013 for the widely used GRASP atomic-structure code based on Welton’s concept of the electron self- Keywords: energy. We show that this implementation provides far superior agreement compared with a range of QED other theoretical predictions, and that the discrepancy between the present implementation and that Self-energy previously used is of comparable magnitude to other sources of error in high-accuracy atomic GRASP calculations. This improvement is essential for ongoing studies of complex atomic systems. Screening & 2013 Elsevier Ltd. All rights reserved. Atomic structure 1. Quantum electrodynamics and self-energy electron with the quantised electromagnetic field.
    [Show full text]
  • Bastian Sikora
    Dissertation submitted to the Combined Faculties of the Natural Sciences and Mathematics of the Ruperto-Carola-University of Heidelberg, Germany for the degree of Doctor of Natural Sciences put forward by Bastian Sikora born in Munich, Germany Oral examination: April 18th, 2018 a Quantum field theory of the g-factor of bound systems Referees: Honorarprof. Dr. Christoph H. Keitel Prof. Dr. Maurits Haverkort a Abstract In this thesis, the theory of the g-factor of bound electrons and muons is presented. For light muonic ions, we include one-loop self-energy as well as one- and two-loop vacuum polarization corrections with the interaction with the strong nuclear potential taken into account to all orders. Furthermore, we include effects due to nuclear structure and mass. We show that our theory for the bound-muon g-factor, combined with possible future bound-muon experiments, can be used to improve the accuracy of the muon mass by one order of magnitude. Alternatively, our approach constitutes an independent access to the controversial anomalous magnetic moment of the free muon. Furthermore, two-loop self-energy corrections to the bound-electron g-factor are investigated theoretically to all orders in the nuclear coupling strength parameter Zα. Formulas are derived in the framework of the two-time Green's function method, and the separation of divergences is performed by dimensional regularization. Our numerical evaluation by treating the nuclear Coulomb interaction in the intermediate-state propagators to zero and first order show that such two-loop terms are mandatory to take into account in stringent tests of quantum electrodynamics with the bound-electron g-factor, and in projected near-future determinations of fundamental constants.
    [Show full text]
  • Renormalization of QED in an External Field
    EPJ manuscript No. (will be inserted by the editor) Renormalization of QED in an external field Christian Brouder Laboratoire de Min´eralogie-Cristallographie, CNRS UMR7590, UPMC/UDD/IPGP, Case 115, 4 place Jussieu 75252 Paris cedex 05, France [email protected] November 7, 2018 Abstract. The Schwinger equations of QED are rewritten in three different ways as integral equations involving functional derivatives, which are called weak field, strong field, and SCF quantum electrodynam- ics. The perturbative solutions of these equations are given in terms of appropriate Feynman diagrams. The Green function that is used as an electron propagator in each case is discussed in detail. The general renormalization rules for each of the three equations are provided both in a non perturbative way (Dyson relations) and for Feynman diagrams. PACS. 12.20.-m Quantum electrodynamics – 11.10.Gh Renormalization 1 Introduction The standard approach to QED comes clearly from particle physics, where the S matrix is most useful, and where in and out states of the scattering experiments This paper is a step towards the calculation of photon are well defined. In solid state physics, measurements are and electron spectroscopies of matter based on quantum usually based on a different principle. The spectroscopist electrodynamics. shines on the sample a beam of electrons or photons com- Starting from quantum electrodynamics, which is a ing from a classical source. By classical we mean that the most accurate and successful theory, garanties that the ba- source is not influenced by the system being measured. sis of the calculation is sound. Although such an approach After its interaction with the sample, another beam of may look too true to be beautiful, it seems adequate be- electrons or photons is measured.
    [Show full text]
  • III-2: Vacuum Polarization
    2012 Matthew Schwartz III-2: Vacuum polarization 1 Introduction In the previous lecture, we calculated the Casimir effect. We found that the energy of a system involving two plates was infinite; however the observable, namely the force on the plates, was finite. At an intermediate step in calculating the force we needed to model the inability of the plates to restrict ultra-high frequency radiation. We found that the force was independent of the model and only determined by radiation with wavelengths of the plate separation, exactly as physical intuition would suggest. More precisely, we proved the force was independent of how we modeled the interactions of the fields with the plates as long as the very short wavelength modes were effectively removed and the longest wavelength modes were not affected. Some of our models were inspired by physical arguments, as in a step-function cutoff representing an atomic spacing; others, such as the ζ-function regulator, were not. That the calculated force is indepen- dent of the model is very satisfying: macroscopic physics (the force) is independent of micro- scopic physics (the atoms). Indeed, for the Casimir calculation, it doesn’t matter if the plates are made of atoms, aether, phlogiston or little green aliens. The program of systematically making testable predictions about long-distance physics in spite of formally infinite short distance fluctuations is known as renormalization. Because physics at short- and long-distance decouples, we can deform the theory at short distance any way we like to get finite answers – we are unconstrained by physically justifiable models.
    [Show full text]
  • Bremsstrahlung and X-Ray Spectra for Kaonic and Pionic Hydrogen and Nitrogen
    UDÑ 539.192 T. A. FLORKO, O. YU. KHETSELIUS, YU. V. DUBROVSKAYA, D. E. SUKHAREV Odessa National Polytechnical University, Odessa Odessa State Environmental University, Odessa I. I. Mechnikov Odessa National University, Odessa BREMSSTRAHLUNG AND X-RAY SPECTRA FOR KAONIC AND PIONIC HYDROGEN AND NITROGEN The level energies, energy shifts and transition rates are estimated for pionic and kaonic atoms of hydrogen and nitrogen on the basis of the relativistic perturbation theory with an account of nuclear and radiative effects. New data about spectra of the exotic atomic systems can be considered as a new tool for sensing the nuclear structure and creation of new X-ray sources too. 1. INTRODUCTION orities in the experimental research activities. In the last papers (look, for example, [5,6] and [10] too) this At present time, the light hadronic (pionic, kaonic problem is physically reasonably solved. In the theory etc.) atomic systems are intensively studied and can of the kaonic and pionic atoms there is an important be considered as a candidate to create the new low- task, connected with a direct calculation of the radia- energy X-ray standards [1-12]. In the last few years tive transition energies within consistent relativistic transition energies in pionic [1] and kaonic atoms [2] quantum mechanical and QED methods (c.f.[13- have been measured with an unprecedented precision. 15]). The multi-configuration Dirac-Fock (MCDF) Besides, an important aim is to evaluate the pion mass approximation is the most reliable approach for multi- using high accuracy X-ray spectroscopy [1-10]. Simi- electron systems with a large nuclear charge; in this lar endeavour are in progress with kaonic atoms.
    [Show full text]
  • For Reference ·Stanley J~ Brodsky ..An,D Peter J
    /- ' ' - ' ·: '. ·'-• ·. ' To be .p~b1i$h~d: in 1 '#iavyion Atomic-· -LBL-6087 , c.\ Physics,!'lva'f:lA; Sellip,, ed., (/\ SLAC-PUB-1889 :Spri11ger-Ve:rlag;' (19.77) .·· ·.-.: >;: -. ;·-:. QUANTUM'ELECTRODYNAMICS iN STRON6 AND SUPERCRITICAL FIELDS "- . -: ' -, ~ for Reference ·Stanley J~ Brodsky ..an,d Peter J. Mohr Not to be taken from this room Febxuary 19 77 ·, · ... Prepared for the u: .S.· :E:;ne:rgy Resear.ch and. Developrnent Administration' )..lnd~;r:~c ont:ract w;. 7 40 5-ENG ... 48 • • • • - ~-- • • ~ • • • < • ,,- • - • ' • - • - •• )•' DISCLAIMER This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of Califomia. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Govemment or any agency thereof or the Regents of the University of California. 0 LBL-6087 SLAC-PUB-1889 QUANTUM ELECTRODYNAMICS IN STRONG AND SUPERCRITICAL FIELDS Stanley J. Brodsky Stanford Linear Accelerator Center ... ... Stanford University Stanford, California 94305 Peter J.
    [Show full text]
  • On the Properties of the Uehling Potential
    On the properties of the Uehling potential Alexei M. Frolov∗ Department of Chemistry University of Western Ontario, London, Ontario N6H 5B7, Canada (Dated: October 8, 2013) Abstract A number of properties of the Uehling potential are investigated. In particular, we determine the Fourier spatial resolution of the Uehling potential. The lowest-order correction on vacuum polarisation is re-written in terms of the electron density distribution function. We also discuss the consecutive approximations of the perturbation theory developed for the short-range Uehling potential in the Coulomb few-body systems (atoms). The cusp problem is formulated for few-body systems in which particles interact with each other by the mixed (Coulomb + Uehling) potential. arXiv:1210.6737v8 [physics.atom-ph] 24 Mar 2013 ∗E–mail address: [email protected] 1 I. INTRODUCTION As is well known (see, e.g., [1], [2]) in the lowest order approximation the effect of vacuum polarisation between two interacting electric charges is described by the Uehling potential U(r) [3]. In [4] we have derived the closed analytical formula for the Uehling potential. For atomic systems this formula is written in the following three-term form (in atomic units h¯ =1, me =1, e = 1) + 2 2αQ 1 ∞ 1 1 √ξ 1 U(2br)= exp( 2α− ξr) 1+ − dξ 3π · r hZ1 − 2ξ2 ξ2 i 2αQ b2r2 br b2r2 5 = 1+ K0(2br) Ki1(2br) + Ki2(2br) , (1) 3πr h 3 − 6 − 3 6 i 1 e2 1 where the notation Q stands for the electric charge of the nucleus, b = α− and α = hc¯ ≈ 137 h is the dimensionless fine-structure constant.
    [Show full text]
  • Positronium Hyperfine Splitting Corrections Using Non-Relativistic QED
    Positronium Hyperfine Splitting Corrections Using Non-Relativistic QED Seyyed Moharnmad Zebarjad Centre for High Energy Physics Department of Physics, McGil1 University Montréal, Québec Novernber 1997 A Thesis submitted to the Faculty of Graduate Studies and Research in partial fulWment of the requirements of the degree of Doctor of Philosophy @ Seyyed Moh~unmadZebarjad, 1997 National Library Bibliothèque nationale of Canada du Canada Acquisitions and Acquisitions et Bibliographie Services services bibliographiques 395 Weiiington Street 395. nie Wellington OttawaON K1A ON4 Ottawa ON K 1A ON4 Canaâa Canada Your fïb Votre reiemnw Our fî& Notre reIPrmce The author has granted a non- L'auteur a accordé une Licence non exclusive licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sel1 reproduire, prêter, distribuer ou copies of ths thesis in microform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/film, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantiai extracts fiom it Ni la thèse ni des extraits substantiels may be prïnted or otherwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son pemksion. autorisation. Contents Abstract vii Résumé viii Acknowledgement s x Statement of Original Contributions xi 1 Motivation and Outline of this Thesis 1 2 Introduction to NRQED 7 2.1 NRQED Lagrangian. 7 2.2 Matching at Leading Order.
    [Show full text]
  • A Review Arxiv:2103.06034V1 [Physics.Atom-Ph] 10 Mar 2021
    Advances in Bremsstrahlung: A Review D. H. Jakubassa-Amundsen Mathematics Institute, University of Munich, Theresienstrasse 39, 80333 Munich, Germany Abstract Recent developments in bremsstrahlung from electrons colliding with atoms and nuclei at energies between 0.1 MeV and 500 MeV are reviewed. Considered are cross sections differential in the photon degrees of freedom, including coincidence geometries of photon and scattered electron. Also spin asymmetries and polarization transfer for polarized electron beams are investigated. An interpretation of the measurements in terms of the current bremsstrahlung theories is furnished. Contents 1. Introduction p. 2 2. Theory for electron bremsstrahlung p. 2 2.1 Plane-wave Born approximation 2 2.2 Sommerfeld-Maue approximation 9 2.3 Higher-order analytical theories 12 2.4 Relativistic partial-wave theory 17 2.5 The Dirac-Sommerfeld-Maue (DSM) model 25 2.6 Screening effects 32 2.7 Nuclear and QED effects 36 3. Polarization p. 37 3.1 Definition of the electron-photon polarization correlations 38 3.2 Triply differential cross section in coplanar geometry 41 3.3 Outlook into noncoplanar geometry 44 3.4 Sum rules for the polarization correlations 45 3.5 Correspondence to the spin asymmetries in elastic scattering 47 4. Positron bremsstrahlung p. 49 4.1 Positron theory 50 arXiv:2103.06034v1 [physics.atom-ph] 10 Mar 2021 4.2 Results for positron versus electron impact 52 5. Experiment in comparison with theory p. 54 5.1 Cross sections 54 5.2 Spin asymmetries 61 6. Summary p. 70 References 77 1 1. Introduction The interaction of charged particles by means of electromagnetic potentials and their coupling to weak photon fields are basically well-understood processes.
    [Show full text]
  • Arxiv:Nucl-Th/9706081V1 27 Jun 1997 Tvr O Nrisadtetertclcluain Which Calculations Theoretical Lo the the Screening
    Small Effects in Astrophysical Fusion Reactions A.B. Balantekina, C. A. Bertulanib and M.S. Husseinc a University of Wisconsin, Department of Physics Madison, WI 53706, USA. E-mail: [email protected] b Instituto de F´ısica, Universidade Federal do Rio de Janeiro 21945-970 Rio de Janeiro, RJ, Brazil. E-mail: [email protected] c Universidade de S˜ao Paulo, Instituto de F´ısica 05389-970 S˜ao Paulo, SP, Brazil. E-mail: [email protected] Abstract We study the combined effects of vacuum polarization, relativity, Bremsstrahlung, and atomic polarization in nuclear reactions of astrophys- ical interest. It is shown that these effects do not solve the longstanding differences between the experimental data of astrophysical nuclear reactions at very low energies and the theoretical calculations which aim to include electron screening. arXiv:nucl-th/9706081v1 27 Jun 1997 Typeset using REVTEX 1 I. INTRODUCTION Understanding the dynamics of fusion reactions at very low energies is essential to un- derstand the nature of stellar nucleosynthesis. These reactions are measured at laboratory energies and are then extrapolated to thermal energies. This extrapolation is usually done by introducing the astrophysical S-factor: 1 σ (E)= S(E) exp [ 2πη(E)] , (1.1) E − 2 where the Sommerfeld parameter, η(E), is given by η(E)= Z1Z2e /hv.¯ Here Z1, Z2, and v are the electric charges and the relative velocity of the target and projectile combination, respectively. The term exp( 2πη) is introduced to separate the exponential fall-off of the cross-section − due to the Coulomb interaction from the contributions of the nuclear force.
    [Show full text]
  • Effects of the Fermionic Vacuum Polarization in QED
    Eur. Phys. J. C (2018) 78:12 https://doi.org/10.1140/epjc/s10052-017-5498-x Regular Article - Theoretical Physics Effects of the fermionic vacuum polarization in QED M. F. X. P. Medeiros1,a, F. E. Barone2,b, F. A. Barone1,c 1 IFQ-Universidade Federal de Itajubá, Av. BPS 1303, Caixa Postal 50, Itajubá, MG CEP 37500-903, Brazil 2 Niterói, RJ, Brazil Received: 3 December 2017 / Accepted: 22 December 2017 / Published online: 9 January 2018 © The Author(s) 2018. This article is an open access publication Abstract Some effects of vacuum polarization in QED due radiative corrections to the Aharonov–Bohm scattering [10], to the presence of field sources are investigated. We focus the interaction between two solenoids [11], vacuum currents on effects with no counter-part in Maxwell electrodynamics. produced around a Dirac string [12,13], Bremsstrahlung and The the Uehling interaction energy between two stationary pair production in the Aharonov–Bohm potential [14] and so point-like charges is calculated exactly in terms of Meijer-G on. functions. Effects induced on a hydrogen atom by the vacuum The vacuum polarization can be modified by the presence polarization in the vicinity of a Dirac string are considered. of an external field. The most common situations studied in We also calculate the interaction between two parallel Dirac this context are the effects produced by the presence of an strings and corrections to the energy levels of a quantum par- external magnetic field in the Coulomb interaction and in the ticle constrained to move on a ring circumventing a solenoid.
    [Show full text]
  • Arxiv:1509.02594V2 [Hep-Th] 30 Dec 2015 Hoyfo Lsia on Fve,A Swl Nw.Nevertheless Discu Supers Been Known
    Remarks on nonlinear electrodynamics II 1, Patricio Gaete ∗ 1Departmento de F´ısica and Centro Cient´ıfico-Tecnol´ogico de Valpara´ıso, Universidad T´ecnica Federico Santa Mar´ıa, Valpara´ıso, Chile (Dated: December 31, 2015) We consider both massive Euler-Heisenberg-like and Euler-Heisenberg-like Electrodynamics in the approximation of the strong-field limit. Our analysis shows that massive Euler-Heisenberg- type Electrodynamics displays the vacuum birefringence phenomenon. Afterwards, we calculate the lowest-order modifications to the interaction energy for both classes of Electrodynamics, within the framework of the gauge-invariant path-dependent variables formalism. Interestingly enough, for massive Euler-Heisenbeg-like electrodynamics (Whichmann-Kroll) we obtain a new long-range (1/r3- type) correction, apart from a long-range (1/r5- type) correction to the Coulomb potential. However, Euler-Heisenberg-like Electrodynamics in the approximation of the strong-field limit (to the leading logarithmic order), display a long-range (1/r5- type) correction to the Coulomb potential. Again, for their non-commutative versions, the interaction energy is ultraviolet finite. PACS numbers: 14.70.-e, 12.60.Cn, 13.40.Gp I. INTRODUCTION The phenomenon of vacuum polarization in Quantum Electrodynamics (QED), arising from the polarization of virtual electron-positron pairs and leading to nonlinear interactions between electromagnetic fields, remain as exciting as in the early days of QED [1–5]. An example that illustrates this is the scattering of photons by photons, which despite remarkable progresses has not yet been confirmed [6–10]. Along the same line, we also recall that alternative scenarios such as Born-Infeld theory [11], millicharged particles [12] or axion-like particles [13–15] may have more significant contributions to photon-photon scattering physics.
    [Show full text]