Thirring–Wess model

Not to be confused with Thirring 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 $\alpha \geq 0$

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 $\alpha \geq 0$ ) that there is a one parameter family of solutions.

References

↑ 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.
↑ Brown, LS (1963). "Gauge invariance and Mass in a Two-Dimensional Model". Il Nuovo Cimento. 29 (3): 617–643. doi:10.1007/BF02827786.
↑ 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.
↑ 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.

External links

Quantum field theories

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Four-fermion interactions	Thirring model Gross–Neveu model Nambu–Jona-Lasinio model

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