# 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 ${\displaystyle A^{\mu }}$ is described by

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

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

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

where the fermion mass ${\displaystyle m}$ can be positive or zero. And the interaction term is

${\displaystyle qA^{\mu }({\bar {\psi }}\gamma ^{\mu }\psi )}$

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

${\displaystyle {\alpha \over 2}(\partial ^{\mu }A^{\mu })^{2}}$

for ${\displaystyle \alpha \geq 0}$

There is a remarkable difference between the case ${\displaystyle \alpha >0}$ and the case ${\displaystyle \alpha =0}$: 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 (${\displaystyle m=0}$), the model is exactly solvable. One solution was found, for ${\displaystyle \alpha =1}$, by Thirring and Wess [1] using a method introduced by Johnson for the Thirring model; and, for ${\displaystyle \alpha =0}$, two different solutions were given by Brown[2] and Sommerfield.[3] Subsequently Hagen[4] showed (for ${\displaystyle \alpha =0}$, but it turns out to be true for ${\displaystyle \alpha \geq 0}$) that there is a one parameter family of solutions.

## References

1. ^ Thirring, WE; Wess, JE (1964). "Solution of a field theoretical model in one space one time dimensions". Annals of Physics. 27 (2): 331–337. Bibcode:1964AnPhy..27..331T. doi:10.1016/0003-4916(64)90234-9.
2. ^ Brown, LS (1963). "Gauge invariance and Mass in a Two-Dimensional Model". Il Nuovo Cimento. 29 (3): 617–643. Bibcode:1963NCim...29..617B. doi:10.1007/BF02827786.
3. ^ Sommerfield, CM (1964). "On the definition of currents and the action principle in field theories of one spatial dimension". Annals of Physics. 26 (1): 1–43. Bibcode:1964AnPhy..26....1S. doi:10.1016/0003-4916(64)90273-8.
4. ^ Hagen, CR (1967). "Current definition and mass renormalization in a Model Field Theory". Il Nuovo Cimento A. 51 (4): 1033–1052. Bibcode:1967NCimA..51.1033H. doi:10.1007/BF02721770.