Paul Balduf#paperOfTheDay "Large N solution of generalized Gross-Neveu model with two coupling constants" from 2009.
The Gross-Neveu model is a #quantumFieldTheory of fermionic fields with a 4-fermion interaction vertex. Such theory would be non-renormalizable in 4-dimensional nature, but the Gross-Neveu model is usually considered in 2 dimensions. Its interaction term is of the form (psibar*psi)^2, where psibar is the Hermitian conjugate spinor to psi (i.e. this interaction vertex has 2 incoming and 2 outgoing fermions). The theory has a global discrete symmetry psi -> gamma_5*psi, which effectively exchanges psi and psibar (gamma_5 is the 5th Dirac matrix).
On the other hand, one can augment this model with a second interaction term of the form (psibar*i*gamma_5*psi)^2. If both terms appear with the same coupling constant, the full model (assuming that there is no mass term for the fermion) has a continuous symmetry psi -> exp(i*alpha*gamma_5)psi. This is called "chiral symmetry", and the continuous parameter alpha gives rise to a massless Goldstone boson. The so-defined model is called Nambu-Jona-Lasinio model.
The present paper investigates the transition between GN and NJL model, that is, starting from only the Gross-Neveu interaction term, and gradually adding the other term until both of them are equally strong and the NJL model is recovered. Indeed, this results in a continuous interpolation between the two models. One can also view this as a version of chiral symmetry breaking "through interaction", which is different from the more familiar version of chiral symmetry breaking where the fermions simply get a non-zero mass term.
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.80.125038
The Gross-Neveu model is a #quantumFieldTheory of fermionic fields with a 4-fermion interaction vertex. Such theory would be non-renormalizable in 4-dimensional nature, but the Gross-Neveu model is usually considered in 2 dimensions. Its interaction term is of the form (psibar*psi)^2, where psibar is the Hermitian conjugate spinor to psi (i.e. this interaction vertex has 2 incoming and 2 outgoing fermions). The theory has a global discrete symmetry psi -> gamma_5*psi, which effectively exchanges psi and psibar (gamma_5 is the 5th Dirac matrix).
On the other hand, one can augment this model with a second interaction term of the form (psibar*i*gamma_5*psi)^2. If both terms appear with the same coupling constant, the full model (assuming that there is no mass term for the fermion) has a continuous symmetry psi -> exp(i*alpha*gamma_5)psi. This is called "chiral symmetry", and the continuous parameter alpha gives rise to a massless Goldstone boson. The so-defined model is called Nambu-Jona-Lasinio model.
The present paper investigates the transition between GN and NJL model, that is, starting from only the Gross-Neveu interaction term, and gradually adding the other term until both of them are equally strong and the NJL model is recovered. Indeed, this results in a continuous interpolation between the two models. One can also view this as a version of chiral symmetry breaking "through interaction", which is different from the more familiar version of chiral symmetry breaking where the fermions simply get a non-zero mass term.
https://journals.aps.org/prd/abstract/10.1103/PhysRevD.80.125038
Stephen Brooks 🦆