Thirring–Wess model

Definition

The Lagrangian density is made of three terms:

the free vector field A^\mu is described by

: {(F^{\mu\nu})^2 \over 4} +{\mu^2\over 2} (A^\mu)^2

for F^{\mu\nu}= \partial^\mu A^\nu - \partial^\nu A^\mu and the boson mass \mu must be strictly positive; the free fermion field \psi is described by

: \overline{\psi}(i\partial!!!/-m)\psi

where the fermion mass m can be positive or zero. And the interaction term is : qA^\mu(\bar\psi\gamma^\mu\psi)

Although not required to define the massive vector field, there can be also a gauge-fixing term : {\alpha\over 2} (\partial^\mu A^\mu)^2 for \alpha \ge 0

There is a remarkable difference between the case \alpha 0 and the case \alpha = 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 \alpha = 1 , by Thirring and Wess using a method introduced by Johnson for the Thirring model; and, for \alpha = 0 , two different solutions were given by Brown

References