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Physics Letters B

www.elsevier.com/locate/physletb

Signature of QCD critical point:

Anomalous transverse velocity dependence of antiproton-proton ratio

Xiao-Feng Luo *, Ming Shao, Cheng Li, Hong-Fang Chen

Department of Modern Physics, University of Science and Technology of China, Hefei, Anhui 230026, China

ABSTRACT

We formulate the QCD critical point focusing effect on transverse velocity (ft ) dependence of antiproton to proton (p/p) ratio, which was recently proposed by Asakawa 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 ft dependence of pi/p ratio for three gedanken focused isentropic trajectories with different focusing degree on QCD phase diagram. Finally, we obtained an observable anomaly in ft dependence of p/p ratio, which can be employed as a signature of QCD critical point.

24.10.Jv © 2009 Elsevier B.V. All rights reserved.

24.10.Lx

25.70.Pq

Keywords: QCD critical point Heavy ion collisions UrQMD

THERMINATOR QGP

Isentropic trajectory

ARTICLE INFO

Article history:

Received 8 December 2008

Received in revised form 31 January 2009

Accepted 25 February 2009

Available online 28 February 2009

Editor: L. Alvarez-Gaume

PACS: 25.75.Ld

1. Introduction

In recent years, ultra-relativistic high energy heavy ion collisions (HICs) experiments, such as SPS/CERN sNN ~ 10A GeV) and RHIC/BNL (VsNw ~ 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) [1-4]. Lattice-QCD calculation predicts that the phase transition between hadronic and QGP phases at vanishing baryon chemical potential fiB is crossover transition whereas at higher fiB 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

* Corresponding author.

E-mail address: xfluo@lbl.gov (X.-F. Luo).

0370-2693/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2009.02.044

experimental programs have planed to search for the QCP, such as the energy scan program of RHIC/BNL sNN = 5 ~ 39A GeV) [8-10], the light ion program of NA61 experiment at SPS/CERN (^5NÑ = 5 ~ 17.3A GeV) [11,12] and also the CBM experiment at FAIR/GSI (^sNn < 8.5A GeV) [13-15]. Many experimental observables have been also proposed to be the QCP signatures, such as dynamical fluctuations in K/n 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 nb ratio (s/nb) to be constant [19]. A trajectory with s/nb = 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 /)t dependence of p/p ra-

Phase Boundary ■ ■ ■ , ■ ; . QCD Critical Polnt-

......*;::rB

/ ■'**

/ / .••*

; P .** • Trajectory 1 -

J' Chemical Freeze Out ■ Trajectory 2 _

"......... ▼ Trajectory 3 _ .......-

400 500

Fig. 1. Three gedanken isentropic trajectories on the QCD phase diagram with different focusing degrees. AH trajectories meet at the same QCD critical point (550,159) MeV and chemical freeze out point (400,145) MeV on the phase diagram.

tio [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 /)t dependence of p/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 /)t dependence of p/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 relativis-tic 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 relativis-tic heavy-ion collisions into hadrons. The model has successfully been applied to reproduce the experimental results from SIS/GSI to SPS/CERN energies [23]. The THERMINATOR 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 fiB, etc.

2. 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 (^B, T) plane and respectively labeled as trajectory 1, 2 and 3 in Fig. 1. The location of the QCP (j^c, Tc) = (550,149) MeV chosen here is the same as the Ref. [21], and the chemical freeze out point (^ch, Tch) = (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 (^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/Ltot. For each isentropic trajectory, the L represents the path length along the

Fig. 2. The normalized time evolution of the proton (upper panel) and antiproton (lower panel) numbers along the three gedanken isentropic trajectories.

trajectory from the considered point to the QCP and Ltot is the length along the trajectory from the chemical freeze out point to QCP. The colliding system is evolving from the QCP along the isen-tropic trajectory to the chemical freeze out point and assumed to be thermodynamical equilibrium. Then, the fiB 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, in which the dash lines are the corresponding 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 of p/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 fiB/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 ft dependence of p/p ratio and also the corresponding mechanism are still ambiguous. In qualitative analysis, particles with large ft 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 (ft) dependences of the average emission time ((temjssjon)) for antiproton and proton in Pb + Pb 40A GeV 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 /)t-t anti-correlation during the cooling down of the colliding system. The general fit-t anti-correlation pattern cannot be reproduced by hydrodynamic inspired model for their particular particle freeze out mechanism. The significant increase of p/p ratio with the normalized time t

Fig. 3. The normalized time evolution of the antiproton to proton ratio for three gedanken isentropic trajectories.

Fig. 4. The average emission time (temission) dependence of the transverse velocity pt calculated by UrQMD for Pb + Pb 40 AGeV fixed target reactions at mid-rapidity.

shown in Fig. 3, together with the strong pt-t anti-correlation indicates the experimental observable, pt dependence of p/p ratio, should be enhanced in the lower pt region and suppressed in the higher pt region [21].

The formulation of the QCP focusing effect on pt dependence of p/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 (t) = ki x J Np (Pt, t) dPt, p(t) = k2 xj Np (Pt, t) dPt,

(1) (2)

where the Np (Pt, t) and Np (Pt, t) are the two-dimensional pt-t distributions of the antiproton and proton, respectively. The two Pt-t distributions indicate antiproton and proton are emitted from the colliding system with a finite probability for certain emission

time t and transverse velocity Pt. The k1 and k2 are the two unknown constant coefficients. In addition to Eqs. (1) and (2), we also have the boundary condition:

p(1) = k1 x J Np (Pt, 1) dPt = JJ Np (Pt, t) dPt dt,

p(1) = k2 x / Np (Pt, 1) dPt = / / Np (Pt, t) dPt dt

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 k1 and k2 can be determined by performing integral over the normalized time t on both sides of Eqs. (1) and (2), respectively as:

/J p(t) dt

/0 p(t) dt

/J/Np(Pt, t) dPt dt p(1) /J p(t) dt = /0 p(t) dt

/0/Np (Pt, t) dPt dt p(1) '

We introduce Dp(t) and Dp(t) as the numbers of antiproton and proton emitted at time t along the isentropic trajectory, respectively. Then, with Eqs. (1), (2), (5) and (6), we have:

Dp (t) =( Np (Pt, t) dPt = ^ = 1 p(t) x p(1), (7)

J k1 Û p(t) dt

f p(t) p(t) Dp (t) = j Np (Pt, t) dPt = ^ = — x p(1)'

k2 /0 p(t) dt

It is found that the Dp(t) and Dp(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, pt dependence of p/p ratio, can be calculated as:

p(Pt) = /0 Np (Pt, t) dt p(pt) /J Np (Pt, t) dt'

which depends strongly on the pt-t distributions of antiproton and proton. Although, the two pt-t distributions are not exactly known, the anti-correlation between pt and t is well known.

To quantitatively analyze and set the benchmark for the pt dependence of p/p ratio with the QCP focusing effect, we extract the pt-t distributions of antiproton (NU(pt, t)) and proton (Nu(fit, t)) from the Pb + Pb 40A GeV fixed target reactions implemented by UrQMD. The emission time t here has been normalized. With the two pt-t distributions, the corresponding pt dependence of p/p ratio is calculated by Eq. (9) and shown in Fig. 5. As QGP phase transition and QCP have not been implemented in UrQMD model, the monotonous increase pattern with pt 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 pt dependence of p/p ratio resulted from the QCP focusing effects, we modify the pt-t distributions of antiproton (NU (fit' t)) and proton (Njj(fit, t)), which has been calculated by UrQMD model. The t distributions in the N!p(fit' t) and NU(fit' t) are replaced by the distributions Dp(t) and Dp(t) derived from Eqs. (7) and (8), respectively. For the three gedanken focused isentropic trajectories in Fig. 1, the resulted pt-t distributions for antiproton and proton, which have been introduced in QCP focusing effect, can be calculated as:

Fig. 5. pt dependence of p/p ratio calculated by UrQMD for Pb + Pb 40 AGeV fixed target reactions at mid-rapidity.

Fig. 6. pt dependence of p/p ratio for three gedanken isentropic trajectories. The solid lines are used to guide eyes.

ocp NU (ßt, t)

N0CP(ßt, t) - p

fNUp (ßt, t) dßt

N0CP(ß t)_ NU (ßt,t) Np (ßt,t) - -

fNU (ßt, t) dßt

x Dp(t),

x Dp(t).

(10) (11)

Consequently, the QCP focusing effect has been implemented in the /)t-t distributions for both antiproton and proton through Eqs. (10) and (11), respectively. Thus, the /)t dependence of p/p ratio with QCP focusing effect can be also calculated by Eq. (9) with the modified &-t distributions (NQCP(fit, t), NQcp(fit, 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 /)t dependence of p/p ratio. In Fig. 6, it is also noticed that the higher focused degree of isentropic trajectory leads to the steeper /)t de-

pendence of p/p ratio. Obviously, the anomalous fit dependence of p/ p ratio can be employed as a sensitive QCP signature.

3. Summary and discussion

We have formulated the QCP focusing effect on /)t dependence of p/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 /)t-t correlation pattern for antiproton and proton of a colliding system in HICs cannot be measured experimentally, which directly determines the /)t dependence of p/p ratio. Therefore, the UrQMD model is used to provide /)t-t correlation pattern and the THERMINATOR model is applied to calculate the normalized time dependence of the antiproton and proton numbers along the isentropic trajectories. Then, the anomalous /)t dependence of p/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 /)t dependence of p/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 /)t dependence of p/p ratio, enhanced in low /)t and suppressed in high /)t region, will be observed. This anomaly may also be reflected in the pT spectra of antiproton and/or proton. The existing antiproton pT 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.

Acknowledgements

This work is supported by the Nation Natural Science Foundation of China (10835005, 10675111, 10775131).

References

[1 [2 [3 [4 [5 [6 [7

[8 [9 [10 [11 [12 [13 [14 [15

[16 [17 [18 [19

[20 [21

[22 [23 [24 [25 [26 [27

P. Braun-Munzinger, J. Wambach, arXiv:0801.4256. N. Xu, et al., Nucl. Phys. A 751 (2005) 109. J. Adams, et al., Nucl. Phys. A 757 (2005) 102. K. Adcox, et al., Nucl. Phys. A 757 (2005) 184.

F. Karsch, hep-lat/0601013.

Z. Fodor, S.D. Katz, JHEP 0203 (2002) 014.

M. Stephanov, Acta Phys. Pol. B 35 (2004) 2939;

M. Stephanov, hep-lat/0701002.

P. Sorensen, nucl-ex/0701028.

R.A. Lacey, et al., Phys. Rev. Lett. 98 (2007) 092301.

G.S.F. Stephans, J. Phys. G: Nucl. Part. Phys. 35 (2008) 044050. A. Laszlo, et al., NA61 Collaboration, arXiv:0709.1867.

M. Gazdzicki, nucl-ex/0512034.

J.M. Heuser, CBM Collaboration, J. Phys. G: Nucl. Part. Phys. 35 (2008) 4. P. Senger, Acta Phys. Hungarica a-Heavy Ion Phys. 22 (2005) 363. S. Chattopadhyay, CBM Collaboration, J. Phys. G: Nucl. Part. Phys. 35 (2008) 104027.

M.A. Stephanov, K. Rajagopal, E.V. Shuryak, Phys. Rev. D 60 (1999) 114028. X.F. Luo, et al., Phys. Rev. C 78 (R) (2008) 031901. J.X. Du, et al., arXiv:0810.1989.

C. Nonaka, M. Asakawa, Phys. Rev. C 71 (2005) 044904.

C. Nonaka, M. Asakawa, S.A. Bass, J. Phys. G: Nucl. Part. Phys. 35 (2008) 104099.

M. Asakawa, et al., Phys. Rev. Lett. 101 (2008) 122302.

S.A. Bass, et al., Prog. Part. Nucl. Phys. 41 (1998) 255.

L.A. Winckelmann, et al., nucl-th/9610033.

A. Kisiel, et al., Comput. Phys. Commun. 174 (2006) 669.

P. Braun-Munzinger, K. Redlich, J. Stachel, nucl-th/0304013.

C. Alt, et al., NA49 Collaboration, Phys. Rev. C 73 (2006) 044910.

E. Schnedermann, Z. Phys. C 64 (1994) 85.