Electron Mass Operator in a Strong Magnetic Field and Dynamical Chiral Symmetry Breaking

Abstract

The electron mass operator in a strong magnetic field is calculated. The contribution of higher Landau levels of virtual electrons, along with the ground Landau level, is shown to be essential in the leading log approximation. The effect of the electron dynamical mass generation by a magnetic field is investigated. In a model with N charged fermions, it is shown that some critical number Ncr exists for any value of the electromagnetic coupling constant α, such that the fermion dynamical mass is generated with a doublet splitting for N < Ncr, and the dynamical mass does not arise at all for N > Ncr, thus leaving the chiral symmetry unbroken.PACS numbers: 11.30. Rd, 11.30.Qc, 12.20.Ds Asymptotic properties of the QED diagrams and operators in strong magnetic fields B ≫ B e , B e = m 2 e /e ≃ 4.41 · 10 13 G (e is the elementary charge) are of conceptual interest both from the standpoint of the searches of the perturbation theory applicability borders, and also in view of possible applications in astrophysics and in cosmology of the early Universe. The investigations of this type are being performed by many authors during a rather long time. For example, a history of calculations of the electron mass operator in a strong magnetic field lasts more than 30 years already. However, as we show in this Letter, it is too early to put the final point in the problem.One-loop contribution into the electron mass operator in a strong magnetic field was obtained for the first time by Jancovici [1] in the leading (double) log approximation. Later on, in the papers by Loskutov and Skobelev [2, 3] the attempts were performed to calculate the two-loop contribution and to summarize all the manyloop contributions in the same approximation. A correct formula for the electron mass operator in this approximation was obtained recently in the paper [4]. However, the double log approximation becomes invalid in asymptotically strong magnetic fields [5], because of the crucial influence of the strong magnetic field on the virtual photon polarization operator. This influence provides an appearance of the effective photon mass, m 2 γ = (2α/π)eB, which replaces the electron mass in one of the two logarithms. A correct expression for the electron mass operator in the leading (single) log approximation was obtained recently in our paper [6] by the summation of the rainbow Feynman diagrams, in the form:where m 0 is the electron mass without field, γ E = 0.577 . . . is the Euler constant, α R is the electromagnetic coupling constant renormalized by the fieldHowever, the formula (1) has a restricted area of applicability with respect to the parameter η 0 = α 2π ln eB m 2 0

Improved ring potential of QED at finite temperature and in the presence of weak and strong magnetic fields

“…with Π(q 2 ⊥ , q 2 ) having the form [30] Π(q 11) where N f is the number of fermion flavors. Here, H(z) is defined by…”

“…To compare this result with the ring potential in the leading static limit, (III.29) will be evaluated in the high temperature expansion mβ → 0. This can be determined from the behavior of Bessel functions in this limit 30) and the Bessel function identities [39,40] …”

“…Since the LLL fermions couple only to the longitudinal (0, 3) components of the photon fields, no polarization effects are present in the transverse (1, 2) components of D µν (q). Therefore, the photon propagator in the LLL approximation including the one-loop correction is given by the Feynman-like covariant propagator [20,22] i D µν (q) = g µν q 2 + q 2 Π q 2 , q 2 ⊥ , (II.10) with Π(q 2 ⊥ , q 2 ) having the form [30] Π(q 2 ⊥ , q…”

“…III.D, they are indeed different. 30 In this section, we will determine the full dynamical mass and critical temperature using the ring potential in the static limit. In the next section, these quantities are calculated using the ring potential in the strong limit.…”