Thirring–Wess model

The Thirring–Wess model or Vector Meson model is an exactly solvable quantum field theory describing the interaction of a Dirac field with a vector field in dimension two.

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 [1] using a method introduced by Johnson for the Thirring model; and, for \( \alpha = 0 \) , two different solutions were given by Brown[2] and Sommerfield.[3] Subsequently Hagen [4] showed (for \( \alpha = 0 \), but it turns out to be true for \( \alpha \ge 0 \) ) that there is a one parameter family of solutions.
References

^ Thirring, W; Wess J (1964). "Solution of a field theoretical model in one space one time dimensions". Annals Phys. 27: 331–337.
^ Brown, L (1963). "Gauge invariance and Mass in a Two-Dimensional Model". N.Cimento. 29.
^ Sommerfield, C (1964). Annals Phys. 26.
^ Hagen, C (1967). "Current definition and mass renormalization in a Model Field Theory". N. Cimento A 51.