Gauge independent approach to chiral symmetry breaking in a strong magnetic field

Abstract

The gauge independence of the dynamical fermion mass generated through chiral symmetry breaking in QED in a strong, constant external magnetic field is critically examined. We show that the bare vertex approximation, in which the vertex corrections are ignored, is a consistent truncation of the Schwinger-Dyson equations in the lowest Landau level approximation. The dynamical fermion mass, obtained as the solution of the truncated Schwinger-Dyson equations evaluated on the fermion mass shell, is shown to be manifestly gauge independent. By establishing a direct correspondence between the truncated Schwinger-Dyson equations and the 2PI (two-particle-irreducible) effective action truncated at the lowest nontrivial order in the loop expansion as well as in the 1/N f expansion (N f is the number of fermion flavors), we argue that in a strong magnetic field the dynamical fermion mass can be reliably calculated in the bare vertex approximation.

Chiral transition in a magnetic field and at finite baryon density

“…This effect is now well established, see e. g. [63,[70][71][72][73][74][75]. In the present case, chiral symmetry at vanishing magnetic field B corresponds to choosing m 2 > 0 1 .…”

Effect of discrete quark momenta on the Goldstone mode in a magnetic field

“…The neutral mesons in magnetized NJL model are investigated in vacuum and at finite temperature by taking different methods like the assumption of four-momentum independent meson polarizations [14], magnetic field independent regularization scheme [18,19], and derivative expansion [20,21] and Φ-derivable approach [22]. In this paper, we focus on how the quark dimension reduction in magnetic field affects the neutral meson properties at finite temperature and density in the Pauli-Villars regularization scheme.The SU(2) NJL model is defined through the Lagrangian density [13][14][15][16][17] (1) where the covariant derivative D µ = ∂ µ − iQA µ couples quarks to the external magnetic field B = (0, 0, B) in z-direction, Q = diag(Q u , Q d ) = diag(2e/3, −e/3) and µ = diag(µ u , µ d ) = diag(µ B /3, µ B /3) are electric charge and quark chemical potential matrices in flavor space with µ B being baryon chemical potential, G is the coupling constant in scalar and pseudo-scalar channels, and m 0 is the current quark mass characterizing the explicit chiral symmetry breaking.Taking the Leung-Ritus-Wang method [23][24][25][26][27], the quark condensate ψ ψ or the dynamical quark mass m q = m 0 − G ψ ψ at mean field level is controlled by the gap equationwith J 1 = N c f,n α n |Q f B|/(2π) dp z /(2π)J 0 /(2E f ) and, where N c = 3 is the number of colors which is trivial in the NJL model, α n = 2 − δ n0 is the spin degeneracy, and E ± f = E f ± µ B /3 are the quark energies with E f = p 2 z + 2n|Q f B| + m 2 q . Note that, the quark three-momentum integration in vacuum is reduced to a summation over Landau energy levels plus a onedimensional momentum integration in magnetic field.…”

“…Taking the Leung-Ritus-Wang method [23][24][25][26][27], the quark condensate ψ ψ or the dynamical quark mass m q = m 0 − G ψ ψ at mean field level is controlled by the gap equation [1][2][3][4][5][6][7] …”

Non-perturbative Euler–Heisenberg Lagrangian and paraelectricity in magnetized massless QED

“…It is then justified the approximation J nn ′ (q ⊥ ) ≃ n!δ nn ′ . Hence, the electron self-energy (17) in the strong-field approximation is given bŷ…”