Discrete Anomaly and Dynamical Mass in 2+1 dimensional U(1)V × U(1)A Model

We note that in (2+1)-dimensional gauge theories with even number of massless fermions, there is anomalous Z2 symmetry if theory is regularized in a parity-invariant way. We then consider a parity invariant U(1)V × U(1)A model, which induces a mutual Chern-Simons term in the effective action due to Z2 anomaly. The effect of the discrete anomaly is studied in the induced spin and in the dynamical fermion mass. PACS numbers: 12.50.Lr, 11.15.Pg, 11.30.Hv, 12.20.Ds E-mail address: [email protected] Permanant address The symmetry of classical lagrangians often breaks down upon quantization. A well-known example is the axial anomaly in quantum electrodynamics [1], where any gauge invariant regularization necessarily breaks the axial symmetry. On the other hand the irreducible spinor representation of Lorentz group in odd dimensions does not have γ5-like object. Namely, there is no matrix anti-commuting with all γ matrices in odd dimensions. For instance, in three dimensions the irreducible spinor representation is two dimensional and the product of all γ-matrices Γ0Γ1Γ2 = 1. Therefore there is no axial anomaly in odd diemsions. But, a discrete symmetry might be anomalous in odd dimensional gauge theories due to the incompatibility of the gauge-invariant regulator with the discrete symmetry. The anomalous discrete symmetry is realized as an induced quantum number for the vacuum [2]. Redlich [3] has shown that parity is anomalous in (2+1)-dimensional SU(N) gauge theories since the parity invariant regularization results in an effective action, which is not invariant under large gauge transformations, because Π3 (SU(N)) = Z for N ≥ 2, and thus one needs a parity-violating Chern-Simons term to recover the gauge invariance in the effective action. For the abelian case, parity is anomalous in perturbation theory [3] and for time-independent gauge fields the parity anomaly can be understood as the (1+1)D axial anomaly [4]. However, when the number of fermions is even, one can find a parity-preserving Pauli-Villars regulator of four-component fermions [5, 6]. Then, parity is no longer anomalous and the Chern-Simons term is not induced in the effective Lagrangian. In this paper, we note that for even number of two-component fermions there is another anomalous discrete symmetry, which is not parity, and we study the effect of this discrete anomaly in U(1)V ×U(1)A model. This model itself is also interesting since it might be realized in parity-invariant planar superconductivity [7].