Four-fermion interactions

In quantum field theory, fermions are described by anticommuting spinor fields. A four-fermion interaction describes a local interaction between four fermionic fields at a point in spacetime. A theory involving such an interaction might be an effective field theory or it might be fundamental.

In four spacetime dimensions, such theories are not renormalisable.^{[citation needed]}

Relativistic models

Some examples are the following:

• Fermi's theory of the weak interaction.^[1] The interaction term has a V − A (vector minus axial) form.
• The Gross–Neveu model.^[2] This is a four-fermi theory of Dirac fermions without chiral symmetry and as such, it may or may not be massive.
• The Thirring model.^[3] This is a four-fermi theory of fermions with a vector coupling.
• The Nambu–Jona-Lasinio model.^[4] This is a four-fermi theory of Dirac fermions with chiral symmetry and as such, it has no bare mass.

Nonrelativistic models

A nonrelativistic example is the BCS theory at large length scales with the phonons integrated out so that the force between two dressed electrons is approximated by a contact term.^{[citation needed]}

See also

• Oblique correction
• Peskin–Takeuchi parameter

References

1. Zee 2010, p. 170.
2. Thies 2014.
3. Kondo 1995.
4. Klevansky 1992.

• Klevansky, S. P. (1992). "The Nambu—Jona-Lasinio model of quantum chromodynamics". Reviews of Modern Physics. 64 (3): 649–708. doi:10.1103/revmodphys.64.649.
• Kondo, Kei-ichi (1995). "Thirring model as a gauge theory". Nuclear Physics B. 450 (1–2): 251–266. arXiv:hep-th/9502070. doi:10.1016/0550-3213(95)00316-K.
• Thies, Michael (2014). "Integrable Gross-Neveu models with fermion-fermion and fermion-antifermion pairing". Physical Review D. 90 (10): 105017. arXiv:1408.5506. doi:10.1103/PhysRevD.90.105017.
• Zee, Anthony (2010). Quantum Field Theory in a Nutshell (2nd ed.). Princeton University Press. ISBN 9780691140346.