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In this work we study the $3+1$-dimensional Nambu-Jona-Lasinio (NJL) model inthe mean field-approximation. We carry out calculations using five differentregularization schemes (two continuum and three lattice regularization schemes)with particular focus on inhomogeneous phases and condensates. Theregularization schemes lead to drastically different inhomogeneous regions. Weprovide evidence that inhomogeneous condensates appear for all regularizationschemes almost exclusively at values of the chemical potential and with wavenumbers, which are of the order of or even larger than the correspondingregulators. This can be interpreted as indication that inhomogeneous phases inthe $3+1$-dimensional NJL model are rather artifacts of the regularization andnot a consequence of the NJL Lagrangian and its symmetries.

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