Thirring–Wess model

Definition

The Lagrangian density is made of three terms:

the free vector field A μ is described by

for F μ ν = ∂ μ A ν − ∂ ν A μ and the boson mass μ must be strictly positive; the free fermion field ψ is described by

where the fermion mass m can be positive or zero. And the interaction term is

Although not required to define the massive vector field, there can be also a gauge-fixing term

for α ≥ 0

There is a remarkable difference between the case α > 0 and the case α = 0 : the latter requires a field renormalization to absorb divergences of the two point correlation.

History

This model was introduced by Thirring and Wess as a version of the Schwinger model with a vector mass term in the Lagrangian .

When the fermion is massless ( m = 0 ), the model is exactly solvable. One solution was found, for α = 1 , by Thirring and Wess using a method introduced by Johnson for the Thirring model; and, for α = 0 , two different solutions were given by Brown and Sommerfield. Subsequently Hagen showed (for α = 0 , but it turns out to be true for α ≥ 0 ) that there is a one parameter family of solutions.