Chemical potentials and parity breaking: the Nambu-Jona-Lasinio model

We consider the "two flavour" Nambu--Jona-Lasinio model in the presence of a vector and an axial external chemical potentials and study the phase structure of the model at zero temperature. The Nambu--Jona-Lasinio model is often used as a toy replica of QCD and it is therefore interesting to explore the consequences of adding external vector and axial chemical potentials in this model, mostly motivated by claims that such external drivers could trigger a phase where parity could be broken in QCD. We are also motivated by some lattice analysis that attempt to understand the nature of the so-called Aoki phase using this simplified model. Analogies and differences with the expected behaviour in QCD are discussed and the limitations of the model are pointed out.