Supported in part by the National Natural Science Foundation of China (11475085, 11535005, 11690030, 11575069, 11221504), and the MoST of China 973-Project ( 2015CB856901)

^{3}and published under licence by Chinese Physical Society and the Institute of High Energy Physics of the Chinese Academy of Sciences and the Institute of Modern Physics of the Chinese Academy of Sciences and IOP Publishing Ltd

Conserved charge fluctuations can be used to probe the phase structure of strongly interacting nuclear matter in relativistic heavy-ion collisions. To obtain the characteristic signatures of the conserved charge fluctuations for the quantum chromodynamics (QCD) phase transition, we study the susceptibilities of dense quark matter up to eighth order in detail, using an effective QCD-based model. We studied two cases, one with the QCD critical end point (CEP) and one without owing to an additional vector interaction term. The higher order susceptibilities display rich structures near the CEP and show sign changes as well as large fluctuations. These can provide us information about the presence and location of the CEP. Furthermore, we find that the case without the CEP also shows a similar sign change pattern, but with a relatively smaller magnitude compared with the case with the CEP. Finally, we conclude that higher order susceptibilities of conserved charge can be used to probe the QCD phase structures in heavy-ion collisions.

Article funded by SCOAP^{3}

Exploring the phase structure of strongly interacting nuclear matter is one of the main goals of heavy-ion collision experiments. Owing to the asymptotic freedom nature of quantum chromodynamics (QCD), nuclear matter is expected to undergo a phase transition from hadrons to a quark-gluon plasma (QGP) phase [

It has been predicted that the fluctuations (susceptibilities) of conserved charges, such as baryon number, electric charge, and strangeness, are sensitive to QCD phase transition. The experimental measurements of the fluctuations of conserved quantities were performed in the beam energy scan (BES) program by the STAR experiment at the Relativistic Heavy-Ion Collider (RHIC). The STAR experiment observed a nonmonotonic energy dependence of the fourth order (

In previous studies, these quantities were investigated up to the fourth order [

The chemical potential of

Nambu-Jona-Lasinio (NJL) model is an effective Lagrangian of quarks with local four-point/six-point interactions. This model might serve as a suitable approximation for QCD in the low-energy limit, assuming that gluon degrees of freedom can be frozen into effective point-like interactions between quarks. An advantage of this model is that it can be designed to incorporate all global symmetries of QCD and enables one to “see” the dynamical symmetry breaking mechanisms at work. It offers a simple scheme to study spontaneous chiral symmetry breaking and its manifestations in hadron physics, such as dynamical quark mass generation, the appearance of a quark pair condensate, and the role of pions as Goldstone bosons. The disadvantage of the model is that it does not have the color confinement property of QCD [

The Lagrangian density we adopt is the three-flavor NJL model with scalar and vector interactions, along with the t'Hooft interaction that breaks the _{A} symmetry:

where

with

The various susceptibilities are defined as:

We calculated these susceptibilities using symbolical differentiation, which prevents truncation and rounding errors due to numerical differentiation. Furthermore, we changed the base from {

Two cases are considered in this work: _{V}_{V}_{S}

Phase diagram of the order parameter _{u}_{V}_{V}_{S}

To relate our calculation with experiments and other model calculation, we consider the following ratios:

where

(color online) Sign of _{n}_{V}

(color online) Sign of _{n}_{V}_{S}

Near the phase boundary (the crossover line and the possible first-order phase transition line), the difference in the signs of the signals (the moments) is not significant. The negative regions in

Next, we want to study the magnitude of the various susceptibilities. In ^{n} or 10^{n} for both cases. For the CEP case, has a large area where large signals are expected while the case with no CEP has only a small area. Besides, owing to criticality, the maximum magnitude of signals of the case with CEP can be even greater if we carefully observe the region around the CEP (in

(color online) Region where the magnitude of
^{n} (blue squares and gray diamonds) for the two cases. The left regions indicates the _{V}_{V}_{S}

_{c} is the critical temperature, which is approximately 170 MeV and is the transition temperature at

(color online) Susceptibilities of the two cases with different

We expect our conclusion will still apply to more complicated and realistic model calculations. In _{n}

(color online) _{1}(_{2}(

In this study, we investigated various baryon number susceptibilities up to the eighth order within the NJL model. We find that higher order susceptibilities are very sensitive and carry a large amount of information about the phase transition. The flip in the sign is closely related to the location of the phase transition, and the magnitudes of higher-order signals are generally larger. It should be very beneficial to measure higher-order fluctuations in experiments. The case with the CEP will give very large signals if we can come sufficiently close to the CEP. However, a rapid crossover transition can also yield large signals and give a similar sign pattern as it approaches the phase boundary. The two cases both agree with the phase structure given by the lattice calculation at low chemical potential [