Inhomogeneous phases in the 3+1-dimensional Nambu-Jona-Lasinio model and their dependence on the regularization scheme

In this work we study the $3+1$-dimensional Nambu-Jona-Lasinio (NJL) model in the mean field-approximation. We carry out calculations using five different regularization schemes (two continuum and three lattice regularization schemes) with particular focus on inhomogeneous phases and condensates. The regularization schemes lead to drastically different inhomogeneous regions. We provide evidence that inhomogeneous condensates appear for all regularization schemes almost exclusively at values of the chemical potential and with wave numbers, which are of the order of or even larger than the corresponding regulators. This can be interpreted as indication that inhomogeneous phases in the $3+1$-dimensional NJL model are rather artifacts of the regularization and not a consequence of the NJL Lagrangian and its symmetries.