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Polarization of an Electron–Positron Vacuum by a Strong Magnetic Field

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Polarization of an Electron–Positron Vacuum by a Strong Magnetic Field Journal of Experimental and Theoretical Physics, Vol. 98, No. 3, 2004, pp. 395–402. Translated from Zhurnal Éksperimental’noÏ i TeoreticheskoÏ Fiziki, Vol. 125, No. 3, 2004, pp. 453–461. Original Russian Text Copyright © 2004 by Rodionov. NUCLEI, PARTICLES, AND THEIR INTERACTION Polarization of an Electron–Positron Vacuum by a Strong Magnetic Field with an Allowance Made for the Anomalous Magnetic Moments of Particles V. N. Rodionov Moscow State Geological Prospecting University, Moscow, 118873 Russia e-mail: [email protected] Received June 10, 2003 Abstract—Given the anomalous magnetic moments of electrons and positrons in the one-loop approximation, we calculate the exact Lagrangian of an intense constant magnetic field that replaces the Heisenberg–Euler Lagrangian in traditional quantum electrodynamics (QED). We have established that the derived generalization of the Lagrangian is real for arbitrary magnetic fields. In a weak field, the calculated Lagrangian matches the standard Heisenberg–Euler formula. In extremely strong fields, the field dependence of the Lagrangian com- pletely disappears and the Lagrangian tends to a constant determined by the anomalous magnetic moments of the particles. © 2004 MAIK “Nauka/Interperiodica”. 1. INTRODUCTION for large momentum transfers and the processes in intense electromagnetic fields [2–17]. In fact, the over- The quantum corrections to the Maxwellian lapping of seemingly distinctly different areas of phys- Lagrangian of a constant electromagnetic field were ics is not accidental and is suggested by simple dimen- first calculated by Heisenberg and Euler [1] in 1936. sion considerations. Radiative corrections corresponding to polarization of Allowance for the electromagnetic field intensity an electron–positron vacuum by external electromag- based on the exact integrability of the equations of netic fields with diagrams containing different numbers motion is known to play an important role in studying of electron loops are still the focus of attention [2–4]. the quantum effects of the interaction between charged Estimates suggest that the quantum (radiative) correc- particles and the electromagnetic field. In particular, the tions could reach the Maxwellian energy density of the standard Schwinger correction to the Bohr magneton electromagnetic field only in exponentially strong elec- 1 π α e tromagnetic fields (Fc ~ exp(3 / )Hc) [5]. The calcu- µ = ------- , lations by Heisenberg and Euler are known to contain 0 2m no approximations in the intensity of external electro- which is called the anomalous magnetic moment of a magnetic fields, and their results have been repeatedly particle, confirmed by calculations performed in terms of differ- α ent approaches. On this basis, several authors have ∆µ µ = 0------ , identified the field intensity Fc with the validity bound- 2π ary of universally accepted QED. However, it is clear that, although such quantities were greatly overesti- manifests itself only in the nonrelativistic limit for mated because the corresponding scale lengths are weak quasi-static fields [7]. Indeed, when the influence many orders of magnitude smaller than not only the of an intense external field is accurately taken into scale on which weak interactions manifest themselves, account, the anomalous magnetic moment of a particle but also the Planck length, determining the validity calculated in QED as a one-loop radiative correction range of traditional QED is currently of fundamental decreases with increasing field intensity and increasing importance. While on the subject of the physics of energy of the moving particles from the Schwinger extremely small distances, we cannot but say that there value to zero. In particular, for magnetic fields H ~ Hc, is a strong analogy2 between the phenomena that arise the anomalous magnetic moment of an electron is described by the asymptotic formula [7, 10] 1 ប 2 | | Here, we use a system of units with = c = 1, Hc = m / e = α Hc 2H 13 ∆µ()H = µ------ ------ ln------- . (1) 4.41 × 10 G is the characteristic scale of the electromagnetic 02π H H field intensity in QED, e and m are the electron charge and mass, c and α = e2 = 1/137 is the fine-structure constant. It follows from (1) that ∆µ(H) becomes zero only at one 2 This analogy was first pointed out by Migdal [6] and Ritus [2]. point while decreasing with increasing field. A similar 1063-7761/04/9803-0395 $26.00 © 2004 MAIK “Nauka/Interperiodica” 396 RODIONOV expression for the anomalous magnetic moment of an terms. Thus, an electron in modified QED has an anom- electron in an intense constant crossed field E ⊥ H alous magnetic moment from the outset: ӷ (E = H) at Hp⊥ mHc, where p⊥ is the electron momentum component perpendicular to E × H, can be 2 ∆µ µ µ µ m represented as [11] ==– 0 0 1 +1------- – . (4) M2 αΓ() –2/3 ∆µ() µ 1/3 3 p⊥ H E = 0----------------------------------- . (2) An important aspect of the problem under consider- 93 mHc ation is that the current state of the art in the develop- ∆µ ≠ ment of laser physics [14] allows one to carry out a Note that (E) 0 in the entire range of parameters. number of optical experiments to directly measure the Numerous calculations of the Lagrangian for an contributions from the nonlinear vacuum effects pre- electromagnetic field (see, e.g., [1–3, 5–7, 11]) have dicted by various generalizations of Maxwellian elec- been performed by assuming that the magnetic moment trodynamics [15]. Therefore, it should be emphasized of electrons is exactly equal to the Bohr magneton,3 i.e., that experimental verification of the nonlinear vacuum at ∆µ = 0. However, the following question is of consid- effects with a high accuracy in the presence of rela- erable importance in elucidating the internal closeness tively weak electromagnetic fields can also provide of QED: What effects will allowance for the anomalous valuable information about the validity of QED predic- magnetic moments of electrons and positrons produce tions at small distances [16, 17]. Note in passing that when calculating the polarization of an electron– precision measurements of various quantities (e.g., the positron vacuum by intense electromagnetic fields? anomalous magnetic moments of an electron and a muon) at nonrelativistic energies, together with studies Thus, it is of interest to take the radiative corrections of the particle interaction at high energies, are of cur- to the Maxwellian Lagrangian of a constant field calcu- rent interest in the same sense. lated by the traditional method and compare them with the results that can be obtained from similar calcula- tions by taking into account the nonzero anomalous 2. CORRECTION TO THE LAGRANGIAN magnetic moments of particles. The fact that the OF AN ELECTROMAGNETIC FIELD Lagrangian replacing the Heisenberg–Euler Lagran- WITH ALLOWANCE gian with nonzero anomalous magnetic moments can FOR ANOMALOUS MAGNETIC MOMENTS be calculated by retaining the method of exact solutions OF PARTICLES of the Dirac equation in arbitrarily intense electromag- netic fields also deserves serious attention. In the Let us consider the correction to the Lagrangian of approach under development, the suggested theoretical an electromagnetic field attributable to the polarization generalization initially contains no constraints on the of an electron–positron vacuum in the presence of an electromagnetic field intensity. arbitrarily strong constant magnetic field by taking into account the nonzero anomalous magnetic moments of It should be noted that nonzero anomalous magnetic the particles. To solve this problem, it is convenient, as moments also appear in some of the modified quantum in the standard approach [5], to represent the electron– field theories (QFTs) that also describe the electromag- positron vacuum as a system of electrons that fill “neg- netic interactions. In particular, this is true for a gener- ative” energy levels. For a constant uniform magnetic alization of the traditional QFT known as the theory field, the Dirac–Pauli equation containing the interac- with “fundamental mass” (see, [12, 13] and references tion of a charged lepton with the field (including the therein). The starting point of this theory is the condi- anomalous magnetic moment of the particle) has an tion that the mass spectrum of elementary particles is exact solution [18]. In this case, the energy eigenvalues limited. This condition can be represented as explicitly depend on the spin orientation with respect to ≤ the axis of symmetry specified by the magnetic field mM, (3) direction. Thus, the energy spectrum of an electron that where the new universal parameter M is called the fun- moves in an arbitrarily intense constant uniform mag- damental mass. Relation (3) is used as an additional netic field is fundamental physical principle that underlies the new 2 (),,ξ p QFT. A significant deviation from traditional calcula- En pH = m ------ tions is the fact that the charged leptons in QED with m2 fundamental mass have magnetic moments that are not (5) µµ– H 2 1/2 equal to the Bohr magneton. This is because, apart from + H ()12++n ξ + 1 + ξ--------------0 ------ , ------ 2µ H the traditional “minimal” term, the new Lagrangian of Hc 0 c the electromagnetic interaction includes “nonminimal” where p is the electron momentum component along 3 The authors of [4] took into account the anomalous magnetic the external field H; n = 0, 1, 2, … is the quantum num- moments of particles when analyzing the equilibrium processes ber of the Landau levels; and ξ = ±1 characterizes the in a degenerate electron–nucleon gas in a strong magnetic field. electron spin component along the magnetic field. JOURNAL OF EXPERIMENTAL AND THEORETICAL PHYSICS Vol. 98 No. 3 2004 POLARIZATION OF AN ELECTRON–POSITRON VACUUM 397 Noting that the radiative correction to the classical ral to call the quantity density of the Lagrangian is equal, to within the sign, to 2 2 the total energy density of the electron–positron vac- M M Hc* ==------- -------Hc (10) uum in the presence of an external field [5], e m2 a fundamental field.
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