Exploring the phase diagram of quantum chromo-dynamics (QCD) at finite temperature and real chemical potential is one of the most interesting and important subjects in particle and nuclear physics. Lattice-QCD (LQCD) simulations are a powerful method to investigate the QCD thermodynamics at zero chemical potential. At finite chemical potential, however, LQCD suffers from the so-called sign problem which restricts the applicability of LQCD to the region of small real chemical potential and high temperatures (see Ref. [1]). Therefore, model calculations (admittedly with substantial ambiguities [2]) are used to investigate the phase structure at moderate and large.

A promising strategy for studying the QCD phase diagram at finite is the imaginary-chemical-potential matching approach [3]. It is similar to the usual imaginary chemical potential approach for LQCD [4-7]: LQCD data at finite imaginary chemical potential are extrapolated to the region by using an analytic function. In the imaginary-chemical-potential matching approach we extract some important restrictions for the model design from the region. This allows us to extend the model to the real-chemical-potential region more realistically (cf. Ref. [8]). The important point is that the region encodes almost all information of the region. This fact can be understood through a Fourier transformation of the grand-canonical partition function, , in terms of in the case that the baryon number is a good quantum number:

Here, is the canonical partition function with real quark numbers (N_q). Moreover, QCD possesses the so-called Roberge-Weiss (RW) periodicity [9]: thermodynamical quantities have a periodicity of along the -axis. This periodicity is described by invariance under the extended symmetry [10]

with integer, where is the Polyakov loop and its conjugate. The RW periodicity enables us to determine which interactions are relevant and how strong the couplings are by comparing model results with LQCD data at finite. Note, that can be absorbed in the boundary angle of the temporal direction of the quark field. From this viewpoint, quarks are fermions at and these become boson-like at . Therefore, the dual quark condensate was proposed [11] as an order parameter for the chiral and the deconfinement phase transition.

At another characteristic property of QCD, the so-called RW transition, arises: this RW transition can be related to charge-conjugation or symmetry breaking [12]. symmetry is explicitly broken at finite, but it is not explicitly broken at because of RW periodicity (see, e. g., Ref. [13]). On the RW transition line, -odd quantities can have a finite value, but they vanish for temperatures below the RW end point [9]. Therefore, we can interpret -odd quantities as order parameters of spontaneous -symmetry breaking.

The Polyakov-loop-extended Nambu-Jona-Lasinio (PNJL) model is a promising approach as it preserves the RW periodicity in the same way as QCD. In the present study we extend the nonlocal version of the two-flavor PNJL model from Refs. [14,15] by introducing a nonderivative vector-current interaction between quarks both at imaginary and real chemical potentials.

This paper is organized as follows: In Section 2 we introduce the nonlocal PNJL model that is used in our calculations. In particular, we describe in detail the treatment of the vector-type interaction in the nonlocal framework. We show how this approach can be extended to imaginary chemical potentials. Section 3 presents the results of our calculations. Section 4 closes this work with a discussion and a summary.

Figure 1 displays the -dependence of the chiral order parameter and the real part of. Here we show the results of model A with and and that of model B with and at, in both cases with.In all cases the transitions are crossovers. The transition temperatures for the chiral and deconfinement crossovers almost coincide. This property comes from the entanglement of the chiral and deconfinement transitions through the nonlocality distribution functions.

Figure 2 shows the T-dependence of at . From the right figure we see that there is a first-order RW end point in the case of which turns into second-order for. The LQCD data [25,26] suggest that the order of the RW end point is first-order at sufficiently small. From this perspective, is not a suitable choice, but this situation can be modified as shown below.

In the previous figures we have ignored the vectorcurrent interaction. As a consequence, the ratio exceeds the LQCD prediction [4,7]. Choosing G_v = 0.4 G in model A with T₀ = 240 MeV, this ratio becomes [8]. Model B with G_v = 0.4 G and b = 0.01 leads to. These values are close to the LQCD result. Figure 3 shows again the T-dependence of the imaginary part of the modified Polyakov loop. One observes that the transition at becomes first-order when introducing the vector-current interaction in model B with. Henceforth, we only refer to the results of model A as both models lead to almost identical results.

Finally, we study the -dependence of the position of these critical point in the phase diagram. In Figure 4, the circles and triangles represent the positions of the critical end points for the nonlocal version of the PNJL model, respectively. Results are presented for

and, respectively. In the local PNJL model, the critical end point disappears or shifts to very small when considering a realistic range for the vector coupling strength. In the nonlocal PNJL model, the location of the critical point has a less pronounced dependence on temperature, at least for small. This behavior can be traced to the weakening of the NJL interaction by the nonlocality distribution. The downward trajectory of the critical point becomes very steep, however, once reaches values of 0.4 and beyond (see Figure 4). Around , the canonical ratio corresponding to an effective quark-quark interaction induced by color-octet (gluon-exchange) currents, the critical points tends to disappear altogether and the first-order phase transition turns into a continuous crossover.

In this study we have investigated the impact of a (nonderivative) vector-current interaction in the nonlocal PNJL model at real and imaginary chemical potentials. The presence of the vector-current interaction makes the transition at the Roberge-Weiss end point more pronounced. The RW end point becomes first-order, consistent with recent LQCD simulations.

The location of the critical point in the phase diagram for real chemical potentials is highly sensitive to the vector coupling strength. In the nonlocal PNJL model used here, the critical point tends to be eliminated in favor of a continuous crossover once the ratio of vectorto-scalar couplings is increased toward and beyond, the value characteristic of an effective gluon-exchange interaction between quarks. Qualitatively similar tendencies are found in recent related work [27] and in a -flavor study using the local PNJL model [28] in which the disappearance of the chiral first-order transition turning to a crossover is indicated already at values.

K. K. thanks H. Kouno and M. Yahiro for fruitful discussions. K. K. is supported by RIKEN Special Postdoctoral Researchers Program. T. H. acknowledges the kind hospitality at Brookhaven National Laboratory during his stay. This work is supported in part by BMBF, by the Excellence Cluster “Origin and Structure of the Universe”, and by DFG and NSFC through CRC 110.