Probing the QCD phase structure with higher order baryon number susceptibilities within the NJL model

  • Conserved charge fluctuations can be used to probe the phase structure of the strongly interacting nuclear matter in relativistic heavy-ion collisions. To obtain the characteristic signatures of the conserved charge fluctuations for the QCD phase transition, we perform a detailed study on the susceptibilities of dense quark matter up to 8th order by using an effective QCD based model. We studied two cases, one with the QCD critical end point (CEP) and one without due 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 show similar sign change pattern, but with relatively smaller magnitude comparing to 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.
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Wenkai Fan, Xiaofeng Luo and Hongshi Zong. Probing the QCD phase structure with higher order baryon number susceptibilities within the NJL model[J]. Chinese Physics C. doi: 10.1088/1674-1137/43/3/033103
Wenkai Fan, Xiaofeng Luo and Hongshi Zong. Probing the QCD phase structure with higher order baryon number susceptibilities within the NJL model[J]. Chinese Physics C.  doi: 10.1088/1674-1137/43/3/033103 shu
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Probing the QCD phase structure with higher order baryon number susceptibilities within the NJL model

  • 1. Kuang Yaming Honors School, Nanjing University, Nanjing 210093, China
  • 2. Key Laboratory of Quark & Lepton Physics (MOE) and Institute of Particle Physics, Central China Normal University, Wuhan 430079, China
  • 3. Department of Physics, Nanjing University, Nanjing 210093, China

Abstract: Conserved charge fluctuations can be used to probe the phase structure of the strongly interacting nuclear matter in relativistic heavy-ion collisions. To obtain the characteristic signatures of the conserved charge fluctuations for the QCD phase transition, we perform a detailed study on the susceptibilities of dense quark matter up to 8th order by using an effective QCD based model. We studied two cases, one with the QCD critical end point (CEP) and one without due 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 show similar sign change pattern, but with relatively smaller magnitude comparing to 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.

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    1.   Introduction
    • Exploring the phase structure of strongly interacting nuclear matter is one of the main goals of heavy-ion collision experiments. Due 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 [1]. Lattice QCD calculations show that at small baryon chemical potential and high temperature this transition is a smooth crossover. However, it can not give accurate predictions when chemical potential becomes larger due to the sign problem of lattice QCD [2]. A first-order phase transition is expected at high baryon chemical potential from various model calculations [3-6]. The end point of this possible first-order phase boundary towards the crossover region is called the QCD critical end point (CEP). The location of CEP in the QCD phase diagram or even its existence has not been confirmed yet [7-9].

      It has long been predicted that the fluctuations (susceptibilities) of conserved charges, such as baryon number, electric charge and strangeness, are sensitive to the QCD phase transition. The experimental measurements of the fluctuations of conserved quantities have been performed in the beam energy scan (BES) program by the STAR experiment at the Relativistic Heavy-Ion Collider (RHIC). Interestingly, the STAR experiment observed a non-monotonic energy dependence of the fourth order ( $ \kappa\sigma^{2} $ ) net-proton fluctuations in the most central Au+Au collisions [10-15]. Furthermore, this non-monotonic behavior cannot be described by various transport models [13, 16, 17]. The STAR experiment has also performed the measurements of the sixth order cumulants of net-proton multiplicity distributions in Au+Au collisions at 200 GeV [18]. This is to search for the signature of crossover at small baryon chemical potential values. To investigate the contribution of the possible criticality physics to the conserved charges fluctuations, we adopt an effective quark model, the Nambu-Jona-Lasinio (NJL) model [19, 20], to calculate the various fluctuations up to 8th order in two cases, one with the CEP and one without. By comparing the difference between the two cases, one can see how the various baryon number susceptibilities change sign and magnitude in the QCD phase diagram as a function of temperature and baryon chemical potential. This can provide us valuable guidance to search for signatures of the QCD phase transition and/or critical point in heavy-ion collisions with baryon number fluctuations.

      Previous work has studied these quantities up to fourth order [21-28]. We choose the NJL model as a representative since other effective models like the Polyakov-loop improved NJL model [29, 30], linear $ \sigma $ model [31], the Polyakov-Quark-Meson (PQM) model [32-34], the Gross-Neveu (GN) model [22] or Dyson-Schwinger equations [35-40] all share similar phase structure with the NJL model and we are mainly interested in the qualitative behavior of the susceptibilities (the location of the CEP in the NJL model tends to be at higher $ \mu_B $ and lower T though).

      The chemical potential of u, d quarks are almost the same in experiments [41], so we set them to be equal throughout this work. The chemical potential of the strange quark is smaller, but due to the large mass of s quark, it does not vary the phase diagram much, thus having little influence on the susceptibilities [28]. Throughout our calculation, we assume that the fire-ball is near thermal equilibrium at freeze-out, though critical slowing of dynamics would be important if the fire-ball passes the CEP [42, 43]. Additionally, changes in expansion dynamics and interactions that produce variations in particle spectra and acceptance independent of critical phenomena may blur the signal [44]. However, these non-equilibrium effects do not affect the purpose of our present study: to find the difference of higher-order susceptibilities between critical and non-critical phase transition.

    2.   Model setup
    • 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 to QCD in the low-energy limit, assuming that gluon degrees of freedom can be frozen into effective point-like interactions between quarks. The strength of this model is that it can be designed to incorporate all global symmetries of QCD and enables one to “see” 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 weakness of the model is that it does not have the color confinement property of QCD [19, 20]. Though in the last section of this paper we will also adopt the recent EPNJL model which incorporate confinement by including the Polyakov loop [45, 46].

      The Lagrangian density we adopt is the 3-flavor NJL model with scalar and vector interactions, along with the t'Hooft interaction which breaks the U(1)A symmetry:

      $\begin{split} {\cal L} =& \bar \psi (i\not \partial - {m_0})\psi + {G_S}\left[{(\bar \psi {\lambda _i}\psi )^2} + {(\bar \psi i{\gamma _5}{\lambda _i}\psi )^2}\right]\\& - {G_V}\left[{(\bar \psi {\gamma _\mu }{\lambda _i}\psi )^2} + {(\bar \psi {\gamma _\mu }{\gamma _5}{\lambda _i}\psi )^2}\right]\\& - K({\rm det}\Big[\bar \psi (1 + {\gamma _5})\psi \Big] + {\rm det}\Big[\bar \psi (1 - {\gamma _5})\psi \Big]), \end{split}$

      (1)

      where $ m_0=diag(m_{u0},m_{d0},m_{s0}) $ is the bare mass matrix. $ \lambda_i $ are the Gell-Mann matrices and the determinant is taken in flavor space. The model's parameters are taken from Ref. [47] which are determined by fitting to the mass and decay constant of various mesons. The bare quark masses are $ m_{u0}=m_{d0}=5\ {\rm MeV}, m_{s0}=136\ {\rm MeV} $ . The 3-momentum cutoff $ \Lambda=631\ {\rm MeV} $ , and $ G_S=1.83/\Lambda^2, K= $ $ 9.29/\Lambda^5$ . At finite temperature and chemical potential, we have one additional term $ \mu_0 \psi^{\dagger}\psi $ , where $ \mu_0 $ is the bare chemical potential matrix. After performing mean-field approximation, the following equations hold:

      $\left\{ \begin{array}{l} {m_i} = {m_{i0}} - 4{G_S}\langle {{\bar q}_i}{q_i}\rangle + 2K\langle {{\bar q}_m}{q_m}\rangle \langle {{\bar q}_n}{q_n}\rangle (i \ne m \ne n),\\ {\mu _i} = {\mu _{i0}} - 4{G_V}\langle q_i^†{q_i}\rangle , \end{array} \right.$

      (2)

      with $ \langle\Theta\rangle=\displaystyle\frac{{\rm Tr} (\Theta {\rm e}^{-\beta(\mathcal{H}-\mu_i \mathcal{N}_i)})}{{\rm Tr}( {\rm e}^{-\beta(\mathcal{H}-\mu_i\mathcal{N}_i)})} $ being the grand canonical ensemble average, and i=u, d, s. These are a set of self-consistent equations which relates the bare and effective mass/chemical potential of the quarks.

      The various susceptibilities are defined as:

      $\frac{{\partial \langle q_i^†{q_i}\rangle }}{{\partial {\mu _j}}} = {\chi _{ij}},\ \ \frac{{{\partial ^2}\langle q_i^†{q_i}\rangle }}{{\partial {\mu _j}\partial {\mu _k}}} = {\chi _{ijk}},\ \ \frac{{{\partial ^3}\langle q_i^†{q_i}\rangle }}{{\partial {\mu _j}\partial {\mu _k}\partial {\mu _p}}} = {\chi _{ijkp}},...$

      (3)

      We calculate these susceptibilities using symbolical differentiation which prevents truncation and rounding errors brought by numerical differentiation. Furthermore, we change the base from {u, d, s} at quark level to the conserved charges {B, Q, S} by using:

      $\left\{ \begin{array}{l} {\mu _u} = \displaystyle\frac{1}{3}({\mu _B} + 2{\mu _Q}),\\ {\mu _d} = \displaystyle\frac{1}{3}({\mu _B} - {\mu _Q}),\\ {\mu _s} = \displaystyle\frac{1}{3}({\mu _B} - {\mu _Q} - 3{\mu _S}). \end{array} \right.$

      (4)
    3.   Comparison between two cases with and without the CEP
    • Two cases are considered in this work: one with GV=0 which has a CEP and has been studied extensively, another one with GV=0.5GS (given by renormalization-group analysis [48, 49]) which remains crossover transition throughout the phase diagram [7]. As can be seen from Fig. 1, the two cases have very similar phase structure at low $ \mu_B $ .

      Figure 1.  The phase diagram of the order parameter mu (the constituent up quark mass) in MeV. (a) GV=0 case where a CEP is present. The thick line shows the boundary of the first-order phase transition (b) GV=0.5GS case where there is no critical behavior.

      In order to relate our calculation with experiments and other model calculation, we consider the following ratios defined as:

      ${m_n}(X) = \frac{{{T^n}\chi _X^{(n + 2)}}}{{\chi _X^{(2)}}},\ \ \ n = 1,2,3,...$

      (5)

      where x=B, Q, S. These ratios are then independent of the volume of the system. The signs of these ratios with x=B are shown in Fig. 2 and 3. Red regions are of positive value, and blue regions are of negative value. The yellow regions represent values very close to 0.

      Near the phase boundary (the crossover line and the possible 1st order phase transition line), there is really not much difference of the signs of the signals (the moments). The negative regions in Fig. 2 (a) and (b) are the same as predicted in Refs. [24] and [25] by universal analysis. However, a similar sign pattern also shows up in the case with no CEP (see Fig. 3). If we measure data away from the phase boundary, we may not be able to tell whether the CEP is present by only analyzing the sign of the data. The most significant difference between the two cases lies at the low T (large $ \mu_B $ ) part of the phase boundary. Especially across the first-order phase transition line (denoted by crosses in Fig. 2), the signal changes sign for only 0 (even order) or 1 (odd order) time. Within the no CEP case, however, the signal changes sign more and more times across the boundary as the order becomes higher. If we are able to measure enough data points across this part of the phase boundary, we may be able to tell whether the phase transition is first order or crossover.

      Next, we want to study the magnitude of the various susceptibilities. In Fig. 4, we plot the region where the magnitude of $ m_n(B) $ is greater than 5n or 10n for both cases. The case with the CEP has a large area where large signals are expected while the case with no CEP has only a small area. Besides, due to criticality, the maximum magnitude of signals of the case with CEP can be even larger if we look close enough around the CEP (in Fig. 4, the region where $ m_n(B)>10^n $ is still sizable). However, by only analyzing the magnitude of the signal in current experiments, we can not yet tell whether the CEP is present. Critical phenomenon or a very rapid crossover phase transition can both give large susceptibilities. However, we can perform finite size scaling analysis to identify the nature of the phase transition. The magnitude of the crossover transition signals will not depend on the volume of the system, while the critical signals will show power law dependence.

      Figure 4.  (color online) Region where the magnitude of ${ m_n(B)>5^n }$ (yellow crosses and red triangles) or 10n (blue squares and gray diamonds) for the two cases. The left regions belong to the GV=0 case and the right regions belong to GV=0.5GS case.

      In Fig. 5, we show various baryon number susceptibilities with fixed $ \mu_B $ , where Tc is the critical temperature which is around 170 MeV and is the transition temperature at $ \mu_B=0 $ . Again, we can see that the susceptibilities of the two cases have very similar trend. And the case with CEP will give larger susceptibilities than the case without the CEP. The sign changes are also more clearly shown here compared to Fig. 2 and 3.

      Figure 5.  (color online) Susceptibilities of the two cases with different ${ \mu_B }$ . (a) ${ \mu_B=0,\ G_V=0} $ (b) $ {\mu_B=0.6\ GeV, \ G_V=0 }$ (c) ${ \mu_B=0,\ G_V=0.5G_S }$ (d) ${ \mu_B=0.6\ GeV, \ G_V=0.5G_S} $

      Figure 2.  (color online) Sign of mn of baryon number for the GV=0 case. Red region represents positive value while blue zone represents negative value. The dashed line is the crossover line while the crosses represents the first--order phase transition curve. The negative region are also enclosed by a solid green line and filled with stripes for illustration purpose.

      Figure 3.  (color online) Sign of mn of baryon number for the GV = 0.5GS case. Red region represents positive value while blue zone represents negative value. The dashed line is the crossover line.

      More importantly, we expect our conclusion will still hold for more complicated and realistic model calculations. In Fig. 6, we have calculated the baryon number moments mn(B) of the EPNJL model, which contains the confinement effect by introducing the Polyakov loop [45, 46]. We have calculated the susceptibilities up to the 4th order at present as this calculation is much more slow. The point is, the sign pattern of the moments is very similar to the NJL result.

      Figure 6.  (color online) m1(B) and m2(B) from an EPNJL calculation. Similar sign pattern as the NJL result is present. The CEP is denoted by a black dot. Parameters are taken from Ref. [46].

    4.   Conclusion and outlook
    • In this paper, we studied various baryon number susceptibilities up to the 8th order within the NJL model. We find that higher order susceptibilities are very sensitive and can carry lots of information about the phase transition. The flip of 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 close enough to the CEP. On the other hand, a rapid crossover transition can also yield large signals and give a similar sign pattern as approaching to the phase boundary. The two cases both agree with the phase structure given by lattice calculation at low chemical potential[50]. Also, the sign patterns agree with current experimental data qualitatively [13] (the comparison between the case with CEP and experiment data is done in Ref. [28]). If we only focus on the sign of the baryon number fluctuations, we may not be able to tell whether the CEP is present. We need to move across the phase boundary (the possible first-order transition line) for more information and perform finite size scaling analysis to identify the nature of the phase transition. For future experimental measurements, it should be meaningful to use different kinds of ions (with different freeze-out curves) for collision in order to search a larger region on the phase diagram [51, 52]. If there is a CEP and all the data points measured are inside the phase boundary, we should expect a rapid increase in the magnitude of the data as we approach the phase boundary.

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