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

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. 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.

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