Signature of QCD critical point: Anomalous transverse velocity dependence of antiproton-proton ratio

We formulate the QCD critical point focusing effect on transverse velocity ($\beta_{t}$) dependence of antiproton to proton ($\bar{p}/p$) ratio, which was recently proposed by Asakawa {\it et al.} as an experimental signature of QCD critical point in high energy heavy ion collisions (HICs). For quantitative analysis, Ultra-relativistic Quantum Molecular Dynamics (UrQMD) transport model and THERMal heavy-IoN generATOR (THERMINATOR) are applied to calculate the corresponding $\beta_{t}$ dependence of $\bar{p}/p$ ratio for three gedanken focused isentropic trajectories with different focusing degree on QCD phase diagram. Finally, we obtained an observable anomaly in $\beta_{t}$ dependence of $\bar{p}/p$ ratio, which can be employed as a signature of QCD critical point.


I. INTRODUCTION
In recent years, ultra-relativistic high energy heavy ion collisions (HICs) experiments, such as SPS/CERN ( √ s NN ∼ 10A GeV) and RHIC/BNL ( √ s NN ∼ 200A GeV), have been aimed to search for the new form of matter, which is composed of deconfinement free quarks and gluons, and thus called Quark Gluon Plasma (QGP) [1,2,3,4]. Lattice-QCD calculation predicts that the phase transition between hadronic and QGP phases at vanishing baryon chemical potential µ B is crossover transition whereas at higher µ B the transition may become first-order [5,6]. However, there are many uncertainties for Lattice-QCD or model calculation to determine the first-order phase transition boundary as well as the location of corresponding end point, the so-called QCD Critical Point (QCP) [7]. Thus, the location and even the existence of the QCP are still open questions. The uncertainty of theoretical calculation drives us to explore the properties of hot QCD matter at higher net baryon density and find experimental evidence of the QCP. Recently, several experimental programs have planed to search for the QCP, such as the energy scan program of RHIC/BNL ( √ s NN = 5 ∼ 39A GeV) [8,9,10], the light ion program of NA61 experiment at SPS/CERN ( √ s NN = 5 ∼ 17.3A GeV) [11,12] and also the CBM experiment at FAIR/GSI ( √ s NN ≤ 8.5A GeV) [13,14,15]. Many experimental observables have been also proposed to be the QCP signatures, such as dynamical fluctuations in K/π ratio [16], two experimental observables correlation [17], high order moment of transverse momentum [18], etc..
In the hydrodynamical description of relativistic heavy ion collisions, the expansion of the central fireball should be regarded as isentropic for the negligible entropy production in the latter evolution, which can be represented as entropy density s to baryon density n b ratio (s/n b ) to * Corresponding author: xfluo@lbl.gov be constant [19]. A trajectory with s/n b = constant for a given colliding system on the QCD phase diagram is called isentropic trajectory. Recently, the critical singular properties of QCP have been implemented in equation of state (EOS) for hydrodynamical description of HICs [19,20] by Asakawa, et al.. They pointed out that when the isentropic trajectory passes through the vicinity of the QCP, it may be deformed-the so-called "QCP focusing effect" [19]. They also argued that the QCP focusing effect may result in an observable anomaly in the β t dependence ofp/p ratio [21], which can be employed as a robust signature of the QCP. However, whether the focusing effect can effectively result in an observable anomaly in the β t dependence ofp/p ratio has not been worked out yet; neither has the corresponding mechanism. In this letter, we will formulate the QCP focusing effect on the β t dependence ofp/p ratio and apply the UrQMD and THERMINATOR model to calculate the dependence patterns for three gedanken focused isentropic trajectories with different focusing degrees on the QCD phase diagram.
The UrQMD model [22] used here is based on the quark, di-quark, string and hadronic degrees of freedom and relativistic Boltzmann transport dynamics. It includes 50 different baryon species(nucleon, hyperon and their resonances up to 2.11 GeV) and 25 different meson species. It is usually used to describe the freeze-out and breakup of the fireball produced in relativistic heavyion collisions into hadrons. The model has successfully been applied to reproduce the experimental results from SIS/GSI to SPS/CERN energies [23]. The THERMINA-TOR model [24] is a Monte-Carlo event generator designed for studying of particle production in relativistic heavy ion collisions from SPS to LHC energies. It implements thermal models of particle production with single freeze out. The input parameters are those thermodynamical parameters at freeze out, such as temperature T , baryon chemical potential µ B , etc..

II. FORMULATION OF THE QCD CRITICAL POINT FOCUSING EFFECT
To quantitatively describe the QCP focusing effect, three gedanken isentropic trajectories with different focusing degrees, passing through the common QCP and chemical freeze out point are constructed on the (µ B , T ) plane and respectively labeled as Trajectory 1, 2 and 3 in Fig.1. The location of the QCP (µ c , T c )=(550,149) MeV chosen here is the same as the Ref. [21], and the chemical freeze out point (µ ch , T ch )=(440,145) MeV is the statistical model fit result to the data of Pb+Pb 40A GeV fixed target reactions at SPS/CERN [25]. Besides the start and end points, three other (µ B , T ) points are also sampled for every trajectory. As the actual evolution time scale of the isentropic trajectory on the QCD phase diagram is unknown, the normalized time is used and defined by the normalized path length, t = L/L tot . For each isentropic trajectory, the L represents the path length along the trajectory from the considered point to the QCP and L tot is the length along the trajectory from the chemical freeze out point to QCP. The colliding system is evolving from the QCP along the isentropic trajectory to the chemical freeze out point and assumed to be thermodynamical equilibrium. Then, the µ B and T of every sampling point are used as the input thermodynamical parameters for THERMINATOR model to calculate the corresponding antiproton and proton numbers along the corresponding isentropic trajectory on QCD phase diagram.
The normalized time dependence of the antiproton and proton numbers at the sampled points for the three gedanken isentropic trajectories are respectively shown in Fig.2 3rd-order polynomial fitting results. It is found in Fig.2 that QCP focusing effect results in a sharp decrease of proton number and a non-monotonous increase of antiproton number along the focused isentropic trajectory. We can also see large discrepancy among the antiproton numbers of the three trajectories, whereas there is little discrepancy for the proton number. This means the antiproton is much more sensitive to the focusing degree than proton. In Fig. 3, the time evolution ofp/p ratio along the three gedanken isentropic trajectories and the corresponding 3rd-order polynomial fitting dash lines are shown. The antiproton to proton number ratio (p/p) increases monotonously from QCP (t = 0) to chemical freeze out point (t = 1), which can be simply explained as the decreasing value of µ B /T along each focused isentropic trajectory [21].
Although we have obtained three gedanken focused isentropic trajectories, the quantitative description of the QCP focusing effect on β t dependence ofp/p ratio and also the corresponding mechanism are still ambiguous. In qualitative analysis, particles with large β t would be emitted earlier from the fireball created in HICs, since the mean free path generally grows with increasing hadron momentum and becomes comparable with the fireball size [21]. In Fig. 4, the transverse velocity ( β t ) dependences of the average emission time ( < t emission > ) for antiproton and proton in Pb+Pb 40 AGeV fixed target reactions are calculated by UrQMD. It is implied that the antiproton and proton, which are dynamically emitted from the fireball, would show strong β t − t anticorrelation during the cooling down of the colliding system. The general β t − t anti-correlation pattern cannot be reproduced by hydrodynamic inspired model for their particular particle freeze out mechanism. The significant increase ofp/p ratio with the normalized time t shown in Fig. 3, together with the strong β t -t anti-correlation indicates the experimental observable, β t dependence of p/p ratio, should be enhanced in the lower β t region and suppressed in the higher β t region [21]. The formulation of the QCP focusing effect on β t dependence ofp/p ratio is based on two simple assumptions: one is that the numbers of antiproton and proton evolve independently along the isentropic trajectories and the other is that the numbers of antiproton and proton emitted at time t are proportional to their corresponding total number (p(t), p(t)). The two assumptions can be formulated as:p ,where the Np(β t , t) and N p (β t , t) are the twodimensional β t − t distributions of the antiproton and proton, respectively. The two β t − t distributions indicate antiproton and proton are emitted from the colliding system with a finite probability for certain emission time t and transverse velocity β t . The k 1 and k 2 are the two unknown constant coefficients. In addition to equ.(1) and (2), we also have the boundary condition: , which means the numbers of antiproton and proton at chemical freeze out point (t = 1) are equal to their corresponding sum of emitted numbers along the isentropic trajectory.
The two constants k 1 and k 2 can be determined by performing integral over the normalized time t on both sides of equ. (1) and (2), respectively as: We introduce Dp(t) and D p (t) as the numbers of antiproton and proton emitted at time t along the isentropic trajectory, respectively. Then, with equ. (1), (2), (5) and (6), we have: It is found that the Dp(t) and D p (t) are only determined by the normalized time t dependence of antiproton and proton numbers along the isentropic trajectory, respectively, which are calculated by THERMINATOR model in Fig. 2.
The experimental observable, β t dependence ofp/p ratio, can be calculated as: , which depends strongly on the β t − t distributions of antiproton and proton. Although, the two β t −t distributions are not exactly known, the anti-correlation between β t and t is well known.
To quantitatively analyze and set the benchmark for the β t dependence ofp/p ratio with the QCP focusing effect, we extract the β t − t distributions of antiproton (N Ū p (β t , t)) and proton (N U p (β t , t)) from the Pb+Pb 40 AGeV fixed target reactions implemented by UrQMD. The emission time t here has been normalized. With the two β t −t distributions, the corresponding β t dependence ofp/p ratio is calculated by equ. (9) and shown in Fig.5. As QGP phase transition and QCP have not been implemented in UrQMD model, the monotonous increase pattern with β t up to 0.7 in Fig. 5 is thought to be normal and without suffering from QCP focusing effect.
In order to obtain the anomalous β t dependence ofp/p ratio resulted from the QCP focusing effects, we modify the β t −t distributions of antiproton (N Ū p (β t , t)) and proton ( N U p (β t , t)), which has been calculated by UrQMD model. The t distributions in the N Ū p (β t , t) and N U p (β t , t) are replaced by the distributions Dp(t) and D p (t) derived from equ. (7) and (8), respectively. For the three gedanken focused isentropic trajectories in Fig. 1, the resulted β t − t distributions for antiproton and proton, which have been introduced in QCP focusing effect, can be calculated as: Consequently, the QCP focusing effect has been implemented in the β t − t distributions for both antiproton and proton through the equ. (10) and (11), respectively. Thus, the β t dependence ofp/p ratio with QCP focusing effect can be also calculated by equ. (9) with the modified β t − t distributions ( N QCP p (β t , t), N QCP p (β t , t) ). The results for the three focused gedanken isentropic trajectories on the QCD phase diagram are illustrated in Fig.  6. Comparing the results of UrQMD calculation in Fig.  5 and the results with QCP focusing effect in Fig. 6, we find that they demonstrate completely opposite dependence patterns, which indicates the QCP focusing effect could actually result in an observable anomaly in the β t dependence ofp/p ratio. In Fig.6, it is also noticed that the higher focused degree of isentropic trajectory leads to the steeper β t dependence ofp/p ratio. Obviously, the anomalous β t dependence ofp/p ratio can be employed as a sensitive QCP signature.

III. SUMMARY AND DISCUSSION
We have formulated the QCP focusing effect on β t dependence ofp/p ratio with some reasonable assumptions. The quantitative calculations for the three gedanken focused isentropic trajectories have been made with UrQMD and THERMINATOR models. In fact, the real β t − t correlation pattern for antiproton and proton of a colliding system in HICs cannot be measured experimentally, which directly determines the β t dependence ofp/p ratio. Therefore, the UrQMD model is used to provide β t − t correlation pattern and the THERMI-NATOR model is applied to calculate the normalized time dependence of the antiproton and proton numbers along the isentropic trajectories. Then, the anomalous β t dependence ofp/p ratios have been obtained for three gedanken focused isentropic trajectories, which means the QCP focusing effect can efficiently result in an observable anomaly in the β t dependence ofp/p ratio. We argue that when the isentropic trajectory of the colliding system on the QCD phase diagram is passing through the vicinity of the QCP and deformed by QCP focusing effect, an observable anomaly in β t dependence ofp/p ratio, enhanced in low β t and suppressed in high β t region, will be observed. This anomaly may also be reflected in the p T spectra of antiproton and/or proton. The existing antiproton p T spectrum for Pb+Pb 40 AGeV fixed target collisions measured by NA49 collaboration at SPS/CERN exhibits a steeper exponential slope [21,26]. Finally, we propose that it may be helpful to extract the anomalous structures from the p T spectrum of antiproton and/or proton by performing inverse Laplace transform on the spectrum [27]. The excitation function of structure variable, which should be predefined, can be useful to search for the QCP.