Nambu-Jona-Lasinio model

From Wikipedia, the free encyclopedia

In quantum field theory, the Nambu-Jona-Lasinio model is a theory of interacting Dirac fermions with chiral symmetry. It is an example of a four fermion interaction. This model is defined in a spacetime with an even number of dimensions.

Let's start with the one flavor case first. The Lagrangian density is

$\mathcal{L}=i\bar{\psi}\partial\!\!\!/\psi+\frac{\lambda}{4} \left [\left(\bar{\psi}\psi\right)\left(\bar{\psi}\psi\right)-\left(\bar{\psi}\gamma^5\psi\right)\left(\bar{\psi}\gamma^5 \psi\right)\right]=i\bar{\psi}_L\partial\!\!\!/\psi_L+i\bar{\psi}_R\partial\!\!\!/\psi_R+\lambda \left(\bar{\psi}_L \psi_R\right)\left(\bar{\psi}_R\psi_L \right)$

The global symmetry of this theory is U(1)_Q×U(1)_χ where Q is the ordinary charge of the Dirac fermion and χ is the chiral charge.

Note that there is no bare mass term because of the chiral symmetry. However, there will be a chiral condensate (but no confinement) leading to an effective mass term and a spontaneous symmetry breaking of the chiral symmetry, but not the charge symmetry.

Let's look at the case with N flavors now with the flavor indices represented by the Latin letters a, b, c...

$\mathcal{L}=i\bar{\psi}_a\partial\!\!\!/\psi^a+\frac{\lambda}{4N} \left [\left(\bar{\psi}_a\psi^b\right)\left(\bar{\psi}_b\psi^a\right)-\left(\bar{\psi}_a\gamma^5\psi^b\right)\left(\bar{\psi}_b\gamma^5 \psi^a\right)\right]=i\bar{\psi}_{La}\partial\!\!\!/\psi_L^a+i\bar{\psi}_{Ra}\partial\!\!\!/\psi_R^a+\frac{\lambda}{N} \left(\bar{\psi}_{La} \psi_R^b\right)\left(\bar{\psi}_{Rb}\psi_L^a \right)$

Once again, chiral symmetry forbids a bare mass term. And once again, we might have chiral condensates. The global symmetry here is SU(N)_L×SU(N)_R× U(1)_Q × U(1)_χ where SU(N)_L×SU(N)_R acting upon the left handed flavors and right handed flavors respectively is the chiral symmetry (in other words, there is no natural correspondence between the left handed and the right handed flavors), U(1)_Q is the Dirac charge, which is sometimes called the baryon number in some contexts and U(1)_χ is the axial charge. If a chiral condensate forms, then the chiral symmetry is spontaneously broken into a diagonal subgroup SU(N) since the condensate leads to a pairing of the left handed and the right handed flavors. The axial charge is also spontaneously broken.

The broken symmetries lead to massless pseudoscalar bosons which are sometimes called pions. See Goldstone boson

This model is sometimes used as a phenomenological model of quantum chromodynamics in the chiral limit. While it is able to model chiral symmetry breaking and chiral condensates, it does not model confinement. Also, the axial symmetry is broken spontaneously in this model, leading to a massless Goldstone boson unlike QCD, where it is broken anomalously.

Since the Nambu-Jona-Lasinio model is nonrenormalizable in four spacetime dimensions, this theory can only be an effective field theory which needs to be UV completed.