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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  using a method introduced by Johnson for the Thirring model; and, for $$\alpha = 0$$ , two different solutions were given by Brown and Sommerfield. Subsequently Hagen  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.

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