The critical endpoint (CEP) and the phase structure are studied in the Polyakov-loop extended Nambu-Jona-Lasinio model in which the scalar type eight-quark (σ 4 ) interaction and the vector type four-quark interaction are newly added. The σ 4 interaction largely shifts the CEP toward higher temperature and lower chemical potential, while the vector type interaction does oppositely.At zero chemical potential, the σ 4 interaction moves the pseudo-critical temperature of the chiral phase transition to the vicinity of that of the deconfinement phase transition.

The chiral phase transition and color superconductivity in an extended NJL model with eight-quark interactions are studied. The scalar-type nonlinear term hastens the chiral phase transition, the scalar-vector mixing term suppresses effects of the vector-type linear term and the scalar-diquark mixing term makes the coexisting phase wider.Quantum Chromodynamics (QCD) has non-perturbative properties. First-principle lattice QCD simulations are useful to study thermal systems at zero or small density [1,2]. At high density, however, lattice QCD is still not feasible due to the sign problem. Therefore, effective models are used in finite density region. One of the models is the Nambu-Jona-This model has the mechanism of spontaneous chiral symmetry breaking, but it has not the confinement mechanism. However, this model has been widely used [4,5] with the mean field approximation (MFA), for example, for analyses of the critical endpoint of chiral phase transition [6,7,8,9,10,11].As for the NJL model, only a few studies were done so far on roles of higher-order multiquark interactions [12,13], except for the case of the six-quark interaction coming from the

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