Anomalous Behavior in the Massive Schwinger Model

  • Yutaka Hosotani, Ramón Rodŕıguez
  • Published 1996


We evaluate the chiral condensate and Polyakov loop in two-dimensional QED with a fermion of an arbitrary mass (m). We find discontinuous m dependence in the chiral condensate and anomalous temperature dependence in Polyakov loops when the vacuum angle θ∼π and m=O(e). These nonperturbative phenomena are due to the bifurcation process in the solutions to the vacuum eigenvalue equation. e-mail: e-mail: The Schwinger model, QED in two dimensions has been a preferred theoretical laboratory for the study of physical phenomena such as chiral symmetry, gauge symmetry, anomalies, and confinement. [1]-[7] In a nontrivial topology it allows us to inquire about finite volume and temperature effects while keeping the computations infrared safe. [8][24] Results at finite temperature (T ) can be obtained by Wick rotating the solution on a circle S of circumference L and replacing L by T−1. The theory is exactly solvable with massless fermions, but not with massive fermions. The effect of a small fermion mass (m/e ≪ 1) in the one flavor case is minor other than necessiating the θ vacuum.[4, 6, 10] The opposite limit of weak coupling, or heavy fermions, has been analized by Coleman. [5, 6] In this work we investigate physical quantities such as chiral condensate and Polyakov loop with no restriction on values of the parameters of the system. The effect of nonvanishing fermion masses has been investigated in lattice gauge theory and light cone quantization methods as well. [25]-[27] The Lagrangian of the system is given by L = − 4 FμνF μν + ψ − γ(i∂μ − eAμ)ψ −m(M +M †) M = ψ −1 2 (1− γ)ψ, (1) where γ = (σ1, iσ2) and ψ T a = (ψ a +, ψ a −). We study the model on a circle of circumference L and boundary conditions Aμ(t, x+ L) = Aμ(t, x) ψa(t, x+ L) = −ψa(t, x) . (2) The only physical degree of freedom associated with gauge fields is the Wilson line phase ΘW(t): [9, 10, 11] eW = exp { ie ∫ L 0 dxA1(t, x) } . (3) In the Matsubara formalism of finite temperature field theory boson and fermion fields are periodic and anti-periodic in imaginary time (τ), respectively. Mathematically, the model at finite temperature T = β−1 is obtained from the model defined on a circle by Wick rotation and replacement L → β, it → x and x ↔ τ . The Polyakov loop of a charge 2 q in the finite temperature theory corresponds to the Wilson line phase: Pq(x) = exp { iq ∫ β 0 dτ A0(τ, x) }

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Cite this paper

@inproceedings{Hosotani1996AnomalousBI, title={Anomalous Behavior in the Massive Schwinger Model}, author={Yutaka Hosotani and Ram{\'o}n Rodŕıguez}, year={1996} }