Magnetic Catalysis in Quantum Electrodynamics

We derive the (Wilsonian) low energy effective Lagrangian for Quantum Electrodynamics under external constant magnetic field by integrating out all electrons except those in the lowest Landau level. We find the one-loop effective Lagrangian contains a marginal four-Fermi interaction with anomalous dimension, (ln 2) 2 2π e 4. Renormalization group analysis shows that the four-Fermi interaction will break chiral symmetry in QED if the external magnetic field is extremely strong, B > 1042 gauss, or if the Landau gap, √ |eB| > 6.5×1010 GeV. 11.30.Rd, 12.20.-m, 11.10.Gh, 12.20.Ds Typeset using REVTEX 1 Recently, it has been shown that external (constant) magnetic field acts as catalysis of dynamical symmetry breaking in quantum electrodynamics with or without a nonrenormalizable four-Fermi interaction [1–3]. It is then generalized to QCD under external chromo-magnetic field [4]. The essense of this magnetic catalysis is that electrons of energy much less than the Landau gap (E ≪ √ |eB|) are effectively 1 + 1 dimensional, since the quantum fluctuations perpendicular to the external magnetic field are suppressed by E √ |eB| . Due to this dimensional reduction, the critical coupling for dynamical symmetry breaking becomes zero. Namely, dynamical symmetry breaking occurs for any arbitrarily weak attraction. This has been shown either by calculating the vacuum energy or by solving the Schwinger-Dyson equations for the fermion two-point function. On the other hand, dynamical symmetry breaking is believed to occur when particles interact strongly as in QCD. Indeed, it is found that four-Fermi interaction of electrons in the lowest Landau level (LLL) are marginal and the β-function of four-Fermi coupling is negative for attractive interaction, leading to strong attraction at low energy [3]. Thus, the result of Schwinger-Dyson analysis can be understood in terms of renormalization group (RG). But, it was unclear how the weak electromagnetic interaction leads to dynamical symmetry breaking when the four-Fermi interaction is absent, as shown in the SchwingerDyson analysis [1–3], since the Coulomb interaction of electrons remains weak at low energy [3]. In this paper, we attempt to understand the dynamical symmetry breaking in pure QED under external magnetic field in terms of RG analysis. In this attempt, we find that, if one integrates out electrons in the higher Landau levels, there will be a new low-energy effective operator for electron-photon coupling at tree level, among others, and this operator will generate a four-Fermi interation at one-loop, which becomes strong in infrared region. Magnetic catalysis is a very interesting phenomenon and has potential applications in astrophysics such as the cooling process of neutron stars or particle interactions in early universe under premodial magnetic field [5]. In order for the magnetic catalysis to operate, the external magnetic field has to be strong enough so that the average energy of charged particles is much smaller than the Landau gap. Namely, the gap has to be bigger than the 2 rest mass energy, √ |eB| ≫ m, or, in the early universe, the temperature if the particles are relativistic, √ |eB| ≫ T . Therefore the electrons will be catalyzed only when the external magnetic fields are stronger than a critical field, B > Bc, where Bc = m 2 e/|e| ≃ 10 gauss (G). If electrons are massless, Bc = 0 and therefore for any weak external magnetic field they will be catalyzed to get a dynamical mass, mdyn ≃ √ |eB|e−1/g, where g is the coupling at the cut-off scale, √ |eB|. But, the effect is relevant only at distance larger than 1/mdyn, which is enormously large for weak field. In this paper, we consider electrons of energy larger than the rest mass energy but smaller than the Landau gap, m < E < √ |eB|, and neglect the electron mass. As derived by Schwinger [6], the electron propagator in a constant external magnetic field is given as S(x, y) = S̃(x− y) exp [ ie 2 (x− y)A μ (x+ y) ] , (1) with the Fourier transform of S̃, S̃(k) = ie−k 2 ⊥ /|eB| ∞