The chiral kinetic theory of Weyl fermions with collisions in the presence of weak electric and magnetic fields is derived from quantum field theories. It is found that the side-jump terms in the perturbative solution of Wigner functions play a significant role for the derivation. Moreover, such terms manifest the breaking of Lorentz symmetry for distribution functions. The Lorentz covariance of Wigner functions thus leads to modified Lorentz transformation associated with sidejump phenomena further influenced by background fields and collisions.Introduction.-Novel quantum transport processes in Weyl fermionic systems have been widely investigated, in particular for the so-called chiral magnetic and vortical effects such that charged currents are induced by magnetic and vortical fields [1-3]. These effects associated with quantum anomaly have been studied from different theoretical approaches including relativistic hydrodynamics [4][5][6][7], lattice simulations [8][9][10][11][12], and gauge/gravity duality [13][14][15]. These effects might be (in-)directly observed in heavy ion collisions [16] and in condensed matter systems such as Weyl semimetals [17].From both theoretical and experimental perspectives, it is imperative to understand these anomalous effects in non-equilibrium conditions. One promising approach is kinetic theory, which can delineate non-equilibrium transport of a particle when the interaction and background fields are sufficiently weak. Nevertheless, it is hard to incorporate anomalous effects through the standard Boltzmann equations [6]. The chiral kinetic theory (CKT), which describes anomalous transport of Weyl fermions, has been thus developed from the path-integral [18], Hamiltonian [19], and local-equilibrium quantum kinetic approaches [20,21]. In such formalism, the effective velocity and forces for a single particle are modified by the Berry curvature Ω p = p/(2|p| 3 ), where p represents the spatial momentum of the particle, which originates from the Berry phase in an adiabatic process [22]. Further generalization to massive Dirac fermions can be found in Ref. [23]. In order to bridge the semi-classical approaches [18,19] and quantum field theories, the CKT is also derived from Wigner functions in the high-density effective theory [24] (see also Ref. [25] for relevant study of the on-shell effective field theory.)However, there still exist potential issues in the chiral kinetic equation. First, the field-theory derivation in Refs.[24] and [20,21] are subject to a predominant chemical potential and local equilibrium, respectively. The derivation for more general systems beyond local equilibrium is thus needed. Second, the non-manifestation of Lorentz invariance in the chiral kinetic equation has been recently discussed in Refs. [26,27] from the semi-

The second-order nonlinear responses of inviscid chiral fluids near local equilibrium are investigated by applying the chiral kinetic theory (CKT) incorporating side-jump effects. It is shown that the local equilibrium distribution function can be non-trivially introduced in a co-moving frame with respect to the fluid velocity when the quantum corrections in collisions are involved. For the study of anomalous transport, contributions from both quantum corrections in anomalous hydrodynamic equations of motion and those from the CKT and Wigner functions are considered under the relaxation-time (RT) approximation, which result in anomalous charge Hall currents propagating along the cross product of the background electric field and the temperature (or chemical-potential) gradient and of the temperature and chemical-potential gradients. On the other hand, the nonlinear quantum correction on the charge density vanishes in the classical RT approximation, which in fact satisfies the matching condition given by the anomalous equation obtained from the CKT.

We consider the formation of chiral density waves in Quarkyonic matter, which is a phase where cold, dense quarks experience confining forces. We model confinement following Gribov and Zwanziger, taking the gluon propagator, in Coulomb gauge and momentum space, as ∼ 1/( p 2 ) 2 . We assume that the number of colors, N c , is large, and that the quark chemical potential, µ, is much larger than renormalization mass scale, Λ QCD . To leading order in 1/N c and Λ QCD /µ, a gauge theory with N f flavors of massless quarks in 3 + 1 dimensions naturally reduces to a gauge theory in 1 + 1 dimensions, with an enlarged flavor symmetry of SU (2N f ). Through an anomalous chiral rotation, in two dimensions a Fermi sea of massless quarks maps directly onto the corresponding theory in vacuum. A chiral condensate forms locally, and varies with the spatial position, z, as ψ exp(2iµzγ 0 γ z )ψ . Following Schön and Thies, we term this two dimensional pion condensate a (Quarkyonic) chiral spiral. Massive quarks also exhibit chiral spirals, with the magnitude of the oscillations decreasing smoothly with increasing mass. The power law correlations of the WessZumino-Novikov-Witten model in 1 + 1 dimensions then generate strong infrared effects in 3 + 1 dimensions.

We show that charged vector mesons cannot be condensed by a magnetic field. Although some hadron models predict the charged vector meson condensation in a strong magnetic field, we prove, by means of the Vafa-Witten theorem, that this is not the case in QCD. We also perform the numerical analysis for the meson mass and condensation in lattice QCD. The lattice QCD data confirm no charged vector meson condensation in a magnetic field.Comment: 6 pages, 2 figures; some discussion improved; typos correcte