Table Of ContentHydrodynamical evolution near the QCD critical end point
Chiho Nonaka
Department of Physics, Duke University, Durham, NC 27708, U.S.A.
Masayuki Asakawa
Department of Physics, Osaka University, Toyonaka 560-0043, Japan
(Dated: February 8, 2008)
Hydrodynamical calculations have been successful in describing global observables in ultrarela-
tivisticheavyioncollisions,whichaimtoobservetheproductionofthequark-gluonplasma. Onthe
otherhand,recently,alotofevidencethatthereexistsacriticalendpoint(CEP)intheQCDphase
diagram has been accumulating. Nevertheless, so far, no equation of state with the CEP has been
employed in hydrodynamical calculations. In this paper, we construct the equation of state with
theCEP on thebasis of theuniversality hypothesisand show that theCEP acts as an attractor of
5 isentropic trajectories. We also consider the time evolution in the case with the CEP and discuss
0 howtheCEPaffectsthefinalstateobservables,suchasthecorrelationlength,fluctuation,chemical
0
freezeout,kineticfreezeout,andsoon. Finally,wearguethattheanomalouslylowkineticfreezeout
2
temperatureat the BNL Relativistic Heavy Ion Collider suggests thepossibility of the existence of
n theCEP.
a
J
I. INTRODUCTION important point is that the existence of the CEP was
8
2 shownsemi-quantitativelybynon-perturbativeQCDcal-
The structure of the QCD phase diagram is one of culation. Theprecisedeterminationofthelocationofthe
2 CEP in the QCD phase diagram is a difficult problem.
the mostinterestingandimportanttopicsinnuclearand
v ThelocationsoftheCEPindifferentstudiesare,infact,
particle physics, and a lot of intensive studies have been
8
scatteredoverthe T (temperature)-µ (baryonchemical
carried out from both theoretical side and experimen- B
7
potential)plane[12]. Therefore,analysesfromtheexper-
0 tal side [1]. Recent theoretical studies have revealed the
imentalsidearealsoindispensableinordertounderstand
0 rich structure in the QCD phase diagram at finite den-
1 sity, which consists of not only the quark-gluon plasma the properties of the QCD critical end point.
4 (QGP) phase and hadron phase but also the two flavor A promising way to investigate the existence and lo-
0 colorsuperconductorphaseandcolor-flavorlockedphase cation of the QCD critical end point is to carry out ex-
/
h [2]. Furthermore, in the finite temperature direction, re- perimentsatvariouscollisionenergiesandcomparetheir
t cent lattice calculations suggestthe existence of mesonic results. However, we have to find the observables which
-
l bound states even above the deconfinement phase tran- are suitable for detecting the CEP [16, 17, 18, 19, 20]
c
u sition temperature [3, 4, 5, 6, 7, 8]. for that purpose. In this paper, we study the CEP from
n Inthispaper,wefocusonthecriticalendpoint(CEP) thepointofviewofphenomenologicalanalyseswhichcan
: in the QCD phase diagram, which is the terminating becomparedwithexperimentalresultseasilyanddiscuss
v
point of the first order phase transition 1. In partic- the indication of the existence of the CEPin experimen-
i
X ular, we will consider its effects on the observables in tal observables.
r ultrarelativistic heavy ion collisions in detail. Severalef- We divide the discussion into three steps. First, we
a
fective theory analyses have predicted the existence of construct equations of state that include the CEP. This
the CEP in the QCD phase diagram [10, 11, 12]. Since step is necessary in order to consider the hydrodynami-
Fodor and Katz presented a QCD phase diagram with calevolutionofthesystem. Theguidingprinciplehereis
the CEP for the first time with lattice calculation[13], a the theory of critical phenomena, in particular, the uni-
lotofremarkableprogressinfinite temperatureandden- versality hypothesis. Second, we consider the deviation
sity lattice calculation has been made [14, 15]. However, from the thermal equilibrium near the CEP. This is im-
unfortunately,itwillneedmoretimebeforewereachthe portantbecausei)thetypicaltimescaleiselongatednear
conclusive result on the existence of the CEP, its precise the CEP,andii)the systemproducedinultrarelativistic
position, and so on, because finite density lattice QCD heavyioncollisionscannotstayneartheCEPdue tothe
stillhasdifficulties inperformingaccuratecalculationon strong expansion. Finally, we discuss the possibility to
large lattices and at large chemical potential. Here, the observe the consequences of the CEP experimentally by
using the preceding results.
The structure of this paper is as follows. In Section
II we argue the general feature of the equation of state
1 Theterminology‘criticalendpoint’issometimesusedforadif- withtheCEPonthebasisoftheuniversalityhypothesis,
ferent meaning [9]. However, it is rather customary in hadron
the focusing effect in isentropic trajectories in the T-µ
physicstouseitforthe meaningweemployhere. Thus,weuse B
plane, and hydrodynamical expansion near the CEP. In
‘critical end point’ instead of ’critical point’ in this paper. We
thankM.Stephanov forinformingusofRef.[9]. SectionIII we investigatethe growthanddecreaseofthe
2
correlationlengthneartheCEPfromthepointofviewof G(h,r),
slowing-out-of-equilibrium. In Section IV we discuss the
consequences of the CEP in ultrarelativistic heavy ion G(h,r)=F(M,r)−Mh, (2)
collision experiments. Section V is devoted to summary.
whereF(M,r)isthefreeenergydensity. Settingthefree
energy density,
II. EQUATION OF STATE WITH THE
CRITICAL END POINT F(M,r)=h M r2−αg(θ), (3)
0 0
Inthissection,weconstructrealisticequationsofstate togetherwiththe relationh=(∂F/∂M) ,we obtainthe
r
including the CEP. The equation of state with the CEP differential equation for g(θ),
consists of two parts, the singular part and non-singular
part. We assume that the CEP in QCD belongs to the h˜(θ)(1−θ2+2βθ2)=2(2−α)θg(θ)+(1−θ2)g′(θ), (4)
same universality class as that in the three dimensional
Ising model on the basis of the universality hypothesis. whereacriticalexponentαis0.11[21]. Substitutingh˜(θ)
After mapping the variables and the equation of state
into Eq. (4), we obtain
near the CEP in the three dimensional Ising model onto
those in QCD, we match the singular entropy density g(θ)=g(1)+c (1−θ2)+c (1−θ2)2+c (1−θ2)3, (5)
1 2 3
near the CEP with the non-singular QGP and hadron
phase entropy densities which are known away from the
where g(1) is determined by Eqs. (1) and (4),
CEP. From this procedure we determine the behavior of
theentropydensitywhichincludesboththesingularpart
β
andnon-singularpartinalargeregionintheT-µ plane. g(1)= h˜(1), (6)
B 2−α
Finally,fromtheentropydensity,weextractthebehavior
of other thermodynamical quantities such as the baryon
and the coefficients c , c , and c are given by
1 2 3
number density, pressure, and energy density, and show
that in the T-µ plane the CEP acts as an attractor 1 1
B
c = − {(1−2β)(1+a+b)−2β(a+2b)},
of isentropic trajectories, nB/s = const., with nB and 1 2α−1
s being the baryon number density and entropy density, 1
respectively. c2 = − {2βb−(1−2β)(a+2b)},
2α
1
c = − b(1−2β), (7)
3
2(α+1)
A. Singular Part of Equation of State
where a=−0.76201and b=0.00804 are the coefficients
In the three dimensional Ising model, the magnetiza- oftheθ3andθ5termsinh˜(θ),respectively[21]. Actually,
tionM (theorderparameter)isafunctionofthereduced there is freedom to add the function C(1 − θ2)2−α to
temperature r =(T −Tc)/Tc and the external magnetic g(θ),withC beingaconstant. However,wewillnottake
field h with Tc being the critical temperature. The CEP into account this part with a free parameter, because,
is located at the origin (r,h) = (0,0). At r < 0 the whereas the universality tells us about only the critical
order of the phase transition is first and at r > 0 it is behavior near the phase transition, this part does not
crossover. A useful form of the equation of state of the show a singular behavior.
three-dimensionalIsingmodelis giventhroughthe para- Differentiating the Gibbs free energy by the tempera-
metric representation by the two variables R and θ, ture, we obtain the singular part of the entropy density
s near the QCD critical end point,
M = M Rβθ, c
0
h = h Rβδh˜(θ)
0 (1) ∂G
r == Rh0(R1β−δ(θθ2)−0.7(R62≥010θ,3−+10.1.05048≤04θθ5≤),1.154), sc = −(cid:18)∂T(cid:19)µB
∂G ∂h ∂G ∂r
where M0 and h0 are normalization constants and crit- = − ∂h ∂T − ∂r ∂T, (8)
ical exponents β and δ are 0.326 and 4.80, respectively (cid:18) (cid:19)r (cid:18) (cid:19)h
[21]. We set the normalization constants M0 and h0 by where ∂G and ∂G are, respectively, given by
imposing M(r=−1,h=+0) = 1 and M(r=0,h=1) = 1, ∂h r ∂r h
which assures that the critical magnetization behaves as (cid:0) (cid:1) (cid:0) (cid:1)
∂G ∂F(M,r) ∂M
M(r=0,h) ∝ sgn(h)|h|1/δ and M(r,h = +0) ∝ |r|β = − h−M
∂h ∂h ∂h
(r <0) around the origin. (cid:18) (cid:19)r (cid:18) (cid:19)r (cid:18) (cid:19)r
In order to determine the singular part of the en- = −M, (9)
tropydensity,westartfromtheGibbsfreeenergydensity
3
∂G ∂F(M,r) ∂M
= − h
∂r ∂r ∂r
(cid:18) (cid:19)h (cid:18) (cid:19)h (cid:18) (cid:19)h
h M R1−α
= 0 0 × (2−α)h˜′(θ)g(θ)−βθh˜′(θ)h˜(θ)−βδh˜(θ)g′(θ)+βδh˜2(θ) . (10)
2βδθh˜(θ)+(1−θ2)h˜′(θ)
n o
∆
T
µ crit
∆
critical region
B
T crit Sc 1
0
h −1
r
1
0.5
CEP 180 0 CEP −1 700
140 −0.5
500
100
T [MeV] 60 100 300 µB[MeV]
µ
B
FIG. 2: Dimensionless variable Sc(T,µB) as a func-
FIG. 1: Sketch of the r-h axes (three dimensional Ising tion of T and µB. The CEP is located at (T,µB) =
model) mapped onto the T-µB plane (QCD). The r axis is (154.7 MeV,367.8 MeV) (thetriangle in the contourplot).
tangential to the QCD phase boundary at the CEP. We set
theh direction perpendicular to ther direction.
B. Thermodynamical Quantities
Forquantitativeconstructionofequationsofstatewith
Note that T in Eq. (8) is the temperature on the QCD
the CEP, we fix the relation between the scales in (r,h)
side. InEqs.(9)and(10),weusethedifferentialequation
and(T,µ )variables,whichprovidesthesizeofthecrit-
Eq. (4) and the following relations, B
icalregionaroundtheCEPintheT-µ plane,asfollows:
B
∆r =1(r-h plane)↔∆µ (T-µ plane)and∆h=1
Bcrit B
(r-h plane)↔∆T (T-µ plane)(Fig.1). ∆µ and
∂R 1 2θ crit B Bcrit
= , ∆T give, respectively, approximate extensions of the
(cid:18)∂h(cid:19)r h0Rβδ−12βδθh˜(θ)+(1−θ2)h˜′(θ) critcirciatlregionin the µ andT directions when the loca-
B
∂θ 1 1−θ2 tion of the CEP is close to the T axis and, as a result,
= ,
(cid:18)∂h(cid:19)r h0Rβδ 2βδθh˜(θ)+(1−θ2)h˜′(θ) the r axisisapproximatelyparallelto theµB axis,asre-
cent lattice calculations suggest [13, 14, 15]. In order to
∂R h˜′(θ)
= , connect the equations of state in the singular regionand
(cid:18)∂r(cid:19)h 2βδθh˜(θ)+(1−θ2)h˜′(θ) the non-singular region smoothly, we define the dimen-
∂θ βδ h˜(θ) sionless variable Sc(T,µB) for the singular part of the
= − . (11) entropydensity s , whichhas the dimension [energy]−1,
∂r R 2βδθh˜(θ)+(1−θ2)h˜′(θ) c
(cid:18) (cid:19)h
S (T,µ )=A(∆T ,∆µ )s (T,µ ), (12)
c B crit Bcrit c B
Now we map the r-h plane in the three dimensional where A(∆T ,∆µ ) = ∆T2 +∆µ2 ×D and
crit Bcrit crit Bcrit
Ising model onto the T-µ plane in QCD in order to D is adimensionless constant. The extensionofthe crit-
B
p
determine ∂h/∂T and ∂r/∂T in Eq. (8). The CEP in ical domain around the CEP is specified by the param-
the three dimensionalIsing Model, which is the originin eters ∆T , ∆µ , and D. In Fig. 2, an example of
crit Bcrit
the r-h plane, is mapped to the CEP in QCD, (T,µ )= S is shown as a function of T and µ . The CEP is
B c B
(T ,µ ). The r axis is tangential to the first order located at (T,µ )=(157.4 MeV, 367.8 MeV) (the trian-
E BE B
phase transition line at the CEP [22]. However, there gle in the contour plot). We use the same parameters
is no general rule about how the h axis is mapped in (∆T ,∆µ ,D) as those in Fig. 4 (left). S is ob-
crit Bcrit c
the T-µ plane. Here, for simplicity, we set the h axis tained by solving numerically the implicit relations de-
B
perpendicular to the r axis. In Fig.1, at r <0 the order rived in the preceding subsection. It is clearly seen that
ofthephasetransitionisfirstandatr>0itiscrossover. the order of the phase transition changes from crossover
to first order at the CEP. The contour lines, S =±0.5,
c
4
give a rough idea about the size of the phase transition the temperature direction. Using the dimensionless vari-
region. At lower µ below the critical chemical poten- ableS (T,µ ),wedefinetheentropydensityintheT-µ
B c B B
tial, the phase transition is smeared and the effect of plane,
the phase transition is observed in a larger domain in
1 1
s(T,µ )= (1−tanh[S (T,µ )])s (T,µ )+ (1+tanh[S (T,µ )])s (T,µ ), (13)
B c B H B c B Q B
2 2
where s ands are the entropy densities in the hadron In the first order phase transition region, in order to
H Q
phase and QGP phase away from the CEP, respectively. take into account the discontinuity in the entropy den-
This entropy density includes both singular and non- sity and baryon number density on the phase boundary,
singular contributions, and more importantly, gives the it is necessary to add the following term to Eq. (16) for
correct critical exponents near the QCD critical end T >T (µ ),
c B
point. s is calculated from the equation of state of the
H
hadronphaseintheexcludedvolumeapproximation[23], ∂Tc(µB)
(s(T (µ )+0,µ )−s(T (µ )−0,µ )), (17)
c B B c B B
∂µ
(cid:12) B (cid:12)
P(T,{µ }) = Pideal(T,µ −V P(T,{µ })) (cid:12) (cid:12)
Bi i Bi 0 Bi (cid:12) (cid:12)
i wh(cid:12)ere Tc(µB(cid:12)) is the temperature of the first order phase
X
transitionatµ . ThepressureP isobtainedbyintegrat-
= Pideal(T,µ˜ ), (14) B
Bi ing the entropy density with regard to the temperature,
i
X
whereP isthepressure,Piideal isthepressureoftheideal T
′ ′
gas of particle i, µ is the baryon chemical potential of P(T,µ )= s(T ,µ )dT +P(0,µ ). (18)
Bi B B B
particle i, and V0 is the hadron volume common to all Z0
hadronspecies. Non-strangeresonanceswith mass upto
Finally, from the thermodynamical relation the energy
2GeVareincludedinthe sumandthe radiusofhadrons
density E is obtained as
is fixed at 0.7 fm. We obtain s from the equation of
Q
state of the QGP phase in the Bag model, E =Ts−P −µ n . (19)
B B
(32+21N )π2 N µ 2
P(T,µ ) = f T4+ f B T2 Here we make a comment on the choice of the param-
B
180 2 3 eters ∆T , ∆µ , and D. The linear mapping from
crit Bcrit
+Nf µB 4−B, (cid:16) (cid:17) (15) (r,h)to(T,µB),strictlyspeaking,holdsonlyintheprox-
4π2 3 imityoftheCEP,wherethe thermodynamicalquantities
(cid:16) (cid:17) comply with the universality. We use the linear map-
wherethenumberoftheflavorsN is2andthebagcon-
f ping throughout the critical region. In order to avoid
stant B is (220MeV)4. In Eq. (13) the relative strength
artifacts due to this simplification, ∆T , ∆µ , and
of the singularity also depends on the distance between crit Bcrit
D need to be appropriately chosen so that thermody-
the QCD critical end point, where S = 0, and the
c namical constraints are not violated. In particular, the
s = s line. If the CEP is located near the s = s
Q H Q H following thermodynamical inequalities [22, 24, 25] need
line, less singularity is realized. However, in the follow-
to be observed,
ingcalculations,weassumethattheCEPisonthephase
transition line in the bag plus excluded volume model, ∂S
> 0,
where the phase transition is always of strong first order
∂T
as shown later. Thus, s is substantially larger than s (cid:18) (cid:19)V,NB
Q H
∂P
at the CEP and the singularity is not suppressed.
> 0,
Once we construct the entropy density, we can calcu- ∂V
(cid:18) (cid:19)T,NB
late the other thermodynamical quantities such as the
∂µ
B
baryon number density, pressure, and energy density. > 0, (20)
∂N
The baryon number density n is given by (cid:18) B(cid:19)T,V
B
whereS isthe entropy,N isthe baryonnumber,andV
∂P B
n (T,µ ) = is the volume of the system.
B B
∂µ
B Figure 3 indicates the entropy density and the baryon
= T ∂s(T′,µB)dT′+n (0,µ ).(16) number density as a function of T and µB. We can see
∂µ B B that in the high chemical potential region the order of
Z0 B
5
the phase transition is first and that in the low chemical Two comments are in order here. First, as we ex-
potentialregionasmoothphasetransitionoccurs,which plained, the universality does not tell us about the sizes
reflects the existence of the CEP. In this calculation the of∆T ,∆µ ,D,andsoforth. IfDislarger,thesize
crit Bcrit
CEP is located at (T,µ ) = (154.7 MeV, 367.8 MeV). of the critical region becomes smaller and the behavior
B
Since the universality does not provide the information of the isentropic trajectories approaches that in Fig. 5,
on the location of the CEP, it is treated as a parame- which can be seen from Eqs. (12) and (13). Namely in
ter throughoutthis paper as well as the values of ∆T , the case of larger D, the contour lines of S (for exam-
crit c
∆µ , and D. Note, however, that the local singular ple, S = −1,−0.5,0.5 and 1 in Fig. 2) are closer to the
Bcrit c
behaviorofthermodynamicalquantitiesaroundtheCEP S = 0 line, i.e., the tangential line to the QCD phase
c
is fixed by the universality hypothesis and that accord- boundary at the CEP. Then, in the limit of D → ∞,
ingly the local features such as the focusing, which we all contour lines shrink to one line, which is identical to
discuss below, are not affected by this ambiguity. the QCD phase boundary as long as the curvature of
the phase boundary can be ignored. This can be seen
from Eq. (13). Therefore, the bag plus excluded volume
s[fm − 3 ] nB[fm − 3 ] model canbe consideredas anextreme case of the equa-
18 0.6
tion of state with the CEP. Second, the focusing on the
0.4
10 right hand side of the CEP was first discussed in Ref.
0.2
[16]. Our numerical result indicates that the CEP at-
0
0 tracts isentropic trajectories not only on the right hand
160 160
T [MeV80] 0 200µB[Me4V0]0 600 T [Me8V0] 0 200µB[M40e0V] 600 shiadned(sfiirdset(ocrrdoesrsosviedre)sidoef)tohfethCeECPEbPu.tTahlusso,oifnthtehecrlietfit-
calregionis largeenough,the fine-tuning ofthe collision
FIG. 3: Entropy density (left) and the baryon number energy is not necessary to hit the vicinity of the CEP
density (right) as a function of T and µB. (TE,µBE) = not only on the low energy side [16] but also on the high
(154.7 MeV, 367.8 MeV). Thevaluesof(∆Tcrit,∆µBcrit,D) energy side.
are thesame as in Fig. 4 (left).
V] 2000 (200,145) 0 (200,461) V] 200
jecFtoirrsitesw,ei.ein.,vceostnitgoautreltihneesboehfanvio/rs.ofWthheeniseennttrrooppyicptrrao-- [MeT 160 0.29 (155,361) 0.34 (152,490) [MeT 160
B
duction can be ignored, the entropy and baryon number 120 0.4 (119,227) 120
areconservedineachvolumeelementand,therefore,the
80 0.4 (144,481) 80
temperature and chemical potential in a given volume
element change along the contour lines specified by the 100 200 30µ0B[MeV 4]00 500 600 100 300µB[MeV] 500 700
initial condition.
FIG.4: IsentropictrajectoriesinthecaseswiththeCEP.The
Figure 4 shows the isentropic trajectories in the T-µ
B CEP is located at (TE,µBE) = (154.7 MeV, 367.8 MeV)
plane. The values of (∆T , ∆µ , D) for the Fig. 4
crit Bcrit (left) and (TE,µBE) = (143.7 MeV, 652.0 MeV) (right).
(left) and Fig. 4 (right) are (100 MeV, 200 MeV, 0.15) The values of nB/s on the trajectories are 0.01, 0.02, 0.03,
and (100 MeV, 200 MeV, 0.4), respectively. In both and 0.04 (left) and 0.01, 0.02, 0.03, 0.04, and 0.05 (right)
cases, the trajectories are focused to the CEP. This ‘fo- from left to right. L/Ltotal(T,µB) is shown for some points
cusing’ effect is more clearly observed in the case of on two trajectories in theleft figure.
(T ,µ ) = (143.7 MeV, 652 MeV) (Fig. 4 (right)).
E BE
Thus, the CEP acts as an attractor of isentropic trajec- Next we argue the change of the square of the sound
tories. Figure 5 shows isentropic trajectories in the bag velocity c2,
s
plusexcludedvolumemodel,whichiscurrentlyemployed
in most of hydrodynamical calculations [26]. The order ∂P
c2 = , (21)
of the phase transition is always first in this case. There s ∂E
(cid:18) (cid:19)nB/s
is no focusing effect on the isentropic trajectories in this
case. Instead, the trajectories are just shifted to the left along the isentropic trajectories, which gives us the in-
on the phase transition line. This implies that the hy- formation on how the equation of state changes in the
drodynamicalevolutioninthe casewiththe CEPis very course of the hydrodynamical evolution. Figure 6 indi-
different from the one in the case with the equation of cates the sound velocity as a function of L/L , where
total
state given by the bag plus excluded volume model. Be- L is the path length to a point along the isentropic tra-
cause of this attractor character of the CEP, it is not jectory with a given n /s from a reference point on the
B
needed to fine-tune the collision energy to make the sys- same isentropic trajectoryonthe T-µ plane, and L
B total
tem pass near the CEP,which is pointed out in Ref. [16] istheonetoanotherreferencepointalongthetrajectory.
for the first time. In other words, it is expected that For example, on the insentropic trajectory n /s = 0.01
B
the effect of the CEP, if any, changes only slowly as the in Fig. 4 (left), L/L =0, which is the starting point
total
collision energy is changed. of n /s trajectory, corresponds to the point (T,µ ) =
B B
6
200 III. SLOWING OUT OF EQUILIBRIUM
V]
e
M
[
T 160 The equilibriumcorrelationlengthandfluctuationbe-
comelargeneartheCEPandtheydivergeattheCEP.At
120 the same time, the typical time scale becomes long near
theCEP,whichmakesthesystemtakelongtimetoreach
80 the thermal equilibrium near the CEP. Therefore, in the
time evolution in ultrarelativistic heavy ion collisions,
100 200 300 400 500 600 neartheCEPslowing-out-of-equilibriumoccursandnon-
µB[MeV]
equilibrium dynamics has to be taken into account. We
assume that the thermal equilibrium is established soon
FIG. 5: Isentropic trajectories (solid lines) in the bag plus
after collisions and that the quark-gluon plasma follows
excluded volume model. The values of nB/s on thetrajecto-
riesare0.01,0.02,0.03,0.04,and0.05fromlefttoright. The Bjorken’sscalingsolution[27]forhydrodynamicalevolu-
dashed line stands for the phase boundary. The order of the tion. The initial temperature and proper time are set to
phase transition is always first in this case. 200 MeV and 1 fm/c, respectively 2. Before discussing
the correlation length in out-of-equilibrium time evolu-
tion,wecalculatetheequilibriumcorrelationlengthnear
(200 MeV,145 MeV) and L/L = 0.29 corresponds the CEP in the r-h plane. Using Widom’s scaling law
total
to the point (T,µB) = (155 MeV,361 MeV). At about [28], the equilibrium correlation length ξeq is given by
L/L = 0.25∼ 0.35 the hadronization process occurs
total
|r|
and at about L/Ltotal = 0.35 ∼ 0.5 kinetic freezeout ξ2 (r,M)=a2M−2ν/βg , (22)
takes place. We can see clear difference between the be- eq |M|1/β
(cid:18) (cid:19)
havior of the sound velocities on n /s = 0.01 and 0.03
B where ν =0.63is a criticalexponentof the three dimen-
trajectories. In the case of n /s = 0.01, the sound ve-
B sional Ising model [21]. a is a constant with the dimen-
locity changes smoothly along the trajectory, which re-
sion of length and fixed to 1 fm. There is not so large
flectsthe occurrenceofasmoothphasetransitionatthis
ambiguity in the determination of a, since ξ is of the
n /svalue,andittakesitsminimumatL/L =0.29. eq
B total orderof1fmatT ∼200MeVThefunctiong(x)isgiven
On the other hand, the sound velocity at n /s = 0.03
B by the ǫ expansion to order ǫ2 [28],
changes suddenly at L/L = 0.34, due to the first
total
order phase transition. The hydrodynamical expansion g(x) = g (x)
ǫ
along various n /s paths differs in how the effect of the ǫ
B = 6−2νz 1− [(5+6ln3)z−6(1+z)lnz]
phase transition appears because of the existence of the 36
CEP.Ina realcollisionsystem, nB/s changesfromposi- 1n+2z2 z 1
tion to position. As a result, the time evolution of such +ǫ2 ln2z+ z− (1−ln3)
72 18 2
a system is described as a superposition of trajectories (cid:20) (cid:18) (cid:18) (cid:19)
1 47 56
with differentnB/s’s. Thus, the systemmay notexpand − 16z2− z− lnz
uniformly and this effect may appear in physicalobserv- 216 3 3
(cid:18) (cid:19)(cid:19)
ablesrelatedtoexpansionsuchasthe collectiveflowand 1 101 2
Hanbury Brown-Twiss (HBT) radii if the collision pa- + + I +6ln23+4ln3−10 z2
216 6 3
rameters are appropriate, i.e., isentropic trajectories in (cid:18) (cid:19)
1 44 137 8
the system pass through and/or near the CEP. − 6ln23+ ln3+ + I z ,
216 3 9 3
(cid:18) (cid:19) (cid:21)(cid:27)
2C0S.3 nBs= 0.01 20C.S3 (23)
where z ≡ 2 , I ≡ 1 ln[x(1−x)]dx ∼ −2.344, and
1+x/3 0 1−x(1−x)
0.2
0.2 ǫ=4−d with d being the dimension of the space, 3. At
R
0.1 (T , µ B )=(155,361) 0.1( T , µ B )=(152, 490) nBs= 0.03 blaerguesexd,ia.en.d,atrhoeuansdymthpetcortoicssfoovremragnivdenCEinP,Rgeǫf(.x[)29c]a,nnot
0
(2000, 145) 0.1 (1601.,2 2L90)Lto0ta.3l (1109.4, 277) (T 0 , .µ 5 B )(2000, 461) (1605.,2 457)L (L14t0o4t.,a4 4l81) (909,. 6486)(T , µ B ) 1 2ν
g(x)=g (x)= , (24)
large
3+x
(cid:18) (cid:19)
FIG. 6: Square of the sound velocity c2s as a function of
L/Ltotal along nB/s = 0.01 (left) and nB/s = 0.03 (right)
isentropic trajectories. The CEP is located at (TE,µBE) = 2 In Bjorken’s one-dimensional hydrodynamical evolution, the
(154.7 MeV, 367.8 MeV)inbothcases. Theparametersare non-equilibrium correlation length is not sensitive to the choice
thesame as in Figs. 2, 3, and 4 (left). oftheinitialconditions. Instead, aswediscusslater,the reduc-
tionofthetimescaleduetothetransverseexpansionisexpected
tohavemoreeffectonit.
7
shouldbeused. Wesmoothlyconnectg (x)andg (x) Γ[m (τ)] is given as
ǫ large σ
aroundx=5,i.e.,inmostpartofthecriticalregiong(x)
A
is given by Eq. (23). Γ[m (τ)]= (m (τ)ξ )z, (26)
σ σ 0
ξ
Figure 7 shows the correlation length in equilibrium 0
and the isentropic trajectories at some n /s’s in the r-h
B wherewetakeξ =ξ (T =200MeV),andz isauniver-
0 eq
plane. In Fig. 7 the CEP is located at the origin, and
salexponentofModelHin[30],i.e.,z ∼3[29,31]. Aisa
the first order phase transition occurs at r < 0 and the
dimensionlessnon-universalparameterandweuseA=1
crossover phase transition takes place at r > 0. The
as in Ref. [29], though there is considerable uncertainty
correlation length in equilibrium ξ is divergent at the
eq in the parameter A. However, fortunately, the relation
origin and the region where ξ is large spreads out in
eq between ξ and ξ does not depend on the parameter
eq
the smooth phase transition region, r > 0. In Fig. 7
choice of A so much at the kinetic freezeout point. The
solid lines stand for the isentropic trajectories in the
non-equilibrium correlation length approaches the equi-
caseof(T ,µ )=(154.7 MeV, 367.8 MeV)(left)and
E BE librium correlation length as A increases. At A=100, ξ
(T ,µ ) = (143.7 MeV, 652.0 MeV) (right). Note
E BE isalmostequaltoξ . Ontheotherhand,atsmallAthe
eq
that this r-h plane is actually placed in the T-µ plane
B effect of slowing-out-of-equilibrium appears strongly. At
with the tilt given by the slope of the tangential line of
thesametime,however,themaximumvalueofξbecomes
the firstorderphase transitionline atthe CEPasshown
small as A decreases. As a result, the absolute value of
in Fig. 1. The value of n /s is determined not only by
B the difference between ξ and ξ at the kinetic freezeout
eq
r and h but also by the non-critical component of the
temperature remains almostunchanged for a wide range
equation of state. This is the reason why the behavior
of A. Indeed, we have checked that the following discus-
ofthe trajectoriesrelativetothe CEPchangesaccording
sion about non-equilibrium correlation length holds for
to the location of the CEP in the T-µ plane. Thus,
B 0.1.A.10.
the strength of focusing effect of the CEP is determined
Figure 8 shows the correlation length as a function
also by the non-singular part. On the other hand, we
of L/L . The dashed and solid lines stand for the
total
use the universality hypothesis to calculate ξ and, as a
eq correlation lengths in equilibrium at n /s = 0.008 and
B
result, it is a function of only r and h. The trajectories
n /s=0.01, respectively. The parameters are the same
B
areattractedtotheCEP,whichmakesthelengthofeach
asforFig.4(left). Thethinandthicklinesareξ andξ,
eq
trajectory in the critical region, where ξ is large, long.
eq respectively. Themaximumvalueofξ alongtheformer
eq
This behavior is very different from the assumption in
trajectoryislargerthanthatalongthelatter,becausethe
usualschematic analyses,for example,Fig. 1 inRef. [29]
formerapproachesthe CEPmorecloselythanthelatter.
byBerdnikovandRajagopal. Themaximumvalueofξ
eq The non-equilibrium correlation length ξ is smaller than
depends on the value of n /s.
B ξ at the beginning. Then, ξ becomes larger than ξ
eq eq
later. These are both due to the critical slowing down
h h around the CEP, as pointed out in Ref. [29]. However,
0.2 TE=154.7 MeV 0.2 0.035 TE=143.7 MeV the differencebecomessmallbythe time thesystemgets
0.01
0.008 0.02 tothekineticfreezeoutpoint. Ifthetransverseexpansion
0 0
is takeninto account,the time scale inthe hadronphase
CEP CEP
−0.2 crossover 1st order ξeq>>(f12m..50) −0.2 crossover 1st order ξeq>>(f12m..50) becomes much shorter, but ξeq is already small in the
>>32..05 >>32..05 hadron phase and the difference is expected to remain
0.2 0 −0.2 r 0.2 0 −0.2 r small. Figure 9 shows the cube of the correlation length
as a function of the temperature for n /s=0.008,0.01,
FIG. 7: Correlation length in equilibrium and the nB/s B
and 0.015. The parameters are the same as for Fig. 8.
contour lines in the r-h plane. The solid lines are trajec-
The fluctuation of the sigma field scales approximately
tories with fixed nB/s values 0.008 and 0.01 in the case of
as ∝ξ3. The singularpartof the fluctuationof intensive
(TE,µBE) = (154.7 MeV, 367.8 MeV) (left) and 0.02 and
0.035 in the case of (TE,µBE) = (143.7 MeV, 652.0 MeV) thermalquantitiesofpionsnearthe CEPis proportional
(right). to ξ2 [16]. The general relation between ξ and observed
fluctuations still needs to be clarified. These relations,
however, suggest that there is room for the observation
Next we calculate the non-equilibrium correlation
of some enhancement in fluctuations when the system
length ξ(τ) as a function of the proper time τ by using
passes the vicinity of the CEP even if the enhancement
the rate equation given in Ref. [29],
of ξ is small.
d 1
m (τ)=−Γ[m (τ)] m (τ)− , (25)
σ σ σ
dτ ξ (τ) IV. CONSEQUENCES OF THE CRITICAL END
(cid:18) eq (cid:19)
POINT IN EXPERIMENTS
where 1/m (τ) = ξ(τ) and Γ[m (τ)] is the parameter
σ σ
which represents the rate of slowing-out-of-equilibrium. Many experimental analyses have been carried out in
From the theory of dynamical critical phenomena, search of the evidence of the CEP in the QCD phase
8
12 τ[fm/c] ξ(0.01) V] 200
m] 3 8 ξξeq((00..00018)) [Me 10 fm−3
[f 4 ξeq(0.008) T 160 5 fm−3
ξ 2 0 0.2 0.4
or L/Ltotal 120
q
e
ξ 1
80
0 100 200 300 400 500 600
0 0.1 0.2 0.3 0.4 L/Ltotal µB[MeV]
200 ~170 ~160 ~153 ~120 T [MeV]
FIG.10: Isentropictrajectories andentropydensitycontour
FIG. 8: Equilibrium correlation length ξeq (thin lines) and linesin thecase with theCEP. Theparameters are thesame
non-equilibriumcorrelation lengthξ (thicklines)ontheisen- as for Fig. 3 and the left figure of Fig. 4. The solid lines are
tropictrajectorieswithnB/s=0.008and0.01,togetherwith isentropic trajectories (nB/s= 0.01, 0.015, 0.02, 0.025, 0.03,
τ as functions of L/Ltotal (inlet). z = 3 was used in the 0.035, and 0.04 from left to right) as in Fig. 4. The dashed
calculation. lines are entropy density contour lines, which are shown at
every 1 fm−3.
3
ξ3(0.015)
2.5
ξ3(0.01) asanattractoroftheisentropictrajectories. Thus,ifthe
2 ξ3(0.008) size of the critical domain is large enough, as stressed in
3m ] 1.5 ξξ33eeqq((00..00115)) Rsaerfy.[t1o6]fifnoer-tthueneritghhethcoalnlidsisoindeeonfertghyeCtoEmP,aikteisthneotsnysetceems-
[f ξ3eq(0.008)
3 approach the CEP closely enough. However, such phys-
ξ 1
ical quantities that show the critical behavior near the
0.5 CEP are hadronic observables, and are subject to final
state interactions in the hadronphase. It has been often
0 argued that if the system passes near the CEP, kinetic
200 180 160 140 120 100
freezeout takes place near the CEP [16]. It is based on
T [MeV]
theexpectationthat,insuchacase,thesystemstayslong
neartheCEP,wherethephasetransitionissecondorder,
FIG. 9: Cube of the equilibrium correlation length (thin and that when the system starts to leave away from the
lines) as a function of the temperature and the cube of the
CEP, the entropy density or particle density is already
non-equilibrium correlation length (thick lines) on the isen-
small enough so that kinetic freezeout takes place. To
tropic trajectories with nB/s = 0.008, 0.01, and 0.015. The
the contrary, Fig. 3 does not show a sharp drop of the
parameters are the same as for Fig. 8. The isentropic lines
entropy density near the CEP. To see this more clearly,
with nB/s = 0.008, 0.01, and 0.015 pass left of, almost
through, and right of theCEP, respectively. weshowthecontourplotoftheentropydensityinFig.10
forthesameparametersasinFig.3. Theentropydensity
gives a semi-quantitative measure of the whereabouts of
diagram. Since the correlationlengthandfluctuationdi- kinetic freezeout. Alternatively, we could use the energy
verge at the CEP in thermal equilibrium, it has been density as a measure as well. Figure 10 shows that the
expected that some enhancement of the fluctuation, for contourlines of the entropydensity are not focused near
instance, is observed if the collision energy of nuclei is the CEPin contrastto the isentropic trajectories. Thus,
properly adjusted so that the system goes right through it is notlikely that kinetic freezeouttakes place near the
the CEP. Generally, the higher the collision energy is, CEP. The reason can be traced back to the fact that at
the smaller chemical potential region is explored in the the CEP the entropy density is continuous and bound,
T-µ plane. Thus, it has also been expected that the although its derivative along the h axis diverges. If the
B
observables related to the critical behavior around the system does not freeze out near the CEP, the dilution
CEP such as fluctuations show non-monotonic behavior of the critical behaviors in the hadron phase needs to be
as a function of the collision energy. We discuss the ob- carefully considered.
servability of such behavior in ultrarelativistic heavy ion Empiricallyithasbeenknownthatchemicalfreezeout
collisionsonthe basis ofourfindings inthe previoussec- takes place just below the theoretically expected phase
tions. transition line. Since the isentropic trajectories are fo-
It has been naively expected that the collision energy cused around the CEP, it would be tempting to expect
needs to be carefully adjusted so that the system goes thatthe chemicalfreezeoutpointsarefocusedasthe col-
right through the CEP and its existence can be con- lision energy is varied. However, so far, only the free
firmed. In section II, we have shown that the CEP acts resonance gas model has been used in the analysis of
9
the chemical freezeout point. If the critical region is so 200
large that the focusing is realized in a large region, the V]
e
effect of the interactions on hadrons can by no means [M 160 10fm−3
be ignored. Then, the hadrons at chemical freezeout are T
1.0fm−3
at most quasi-particles which are quantum-mechanically
120
different from the hadrons in the vacuum [32] and it is 0.5fm−3
necessary to consider the time evolution of the quasi-
80
particlestatestorelatethechemicalfreezeoutpointsand
the observed particle number ratios. This is beyond the 100 200 300 400 500 600
scope of this paper and we would like to leave this as a µB[MeV]
future problem.
FIG.11: Isentropictrajectories andentropydensitycontour
The entropy density in Fig. 3 shows the following fea-
lines in the case without the CEP. The parameters are the
ture. Onthe left handside ofthe CEP,the entropyden-
same as in Fig. 5. The solid lines are isentropic trajectories
sity changes smoothly, reflecting the fact that the phase (nB/s= 0.01, 0.02, 0.03, 0.04, and 0.05 from left toright) as
transition here is crossover. The closer to the T axis, in Fig. 5. The dashed lines are the entropy density contour
the less rapid the change is. On the other hand, on the lines. Thecontourlinesareshownatevery0.1(1)fm−3inthe
righthandsideoftheCEP,thephasetransitionisoffirst hadron(QGP)phasesbecausetheincreaserateoftheentropy
order, and the jump in the entropy density increases as density in the QGP phase is much larger than that in the
the chemical potential increases. As a result, i) the sep- hadron phase dueto thestrong first order phase transition.
aration of the contour lines of the entropy density and
energy density gets larger as the distance from the CEP
becomes larger on the left hand side of the CEP, ii) the not increased as the expansion rate compared to SPS.
contour lines below the crossover are flatter than those This can be naturally understood if the separation of
in the hadron phase in the case with a first order phase the entropy density (∼ particle density) or energy den-
transition (See Figs. 3 and 10). This feature is not due sity contour lines is larger at kinetic freezeout at RHIC
to the universality. Instead, it is a general consequence than at SPS. This implies that the CEP exists in the
of the existence of the CEP.In fact, as shownin Fig. 11, QCD phase diagram and, furthermore, that the RHIC
thisfeatureisnotobservedinthecaseofthebagplusex- isentropic trajectory passes on the left hand side of the
cluded volume model. Using this feature of the entropy CEP. To confirm this tendency, it is necessary to obtain
density, we discuss the behavior of kinetic freezeout as a thekineticfreezeouttemperaturesatseveralcollisionen-
function of the collision energy. Kinetic freezeout takes ergies between the SPS and current RHIC energies.
place when the mean (elastic) collision rate and the ex- Inthe previoussection,we haveshownthat the differ-
pansion rate of the system become comparable. The ex- ence between the thermal correlation length ξeq and the
pansion rate in central collisions at RHIC is known to actual correlation length ξ is small at kinetic freezeout
be considerablylargerthanthatatSPS[33]. The elastic as well as ξ itself. As explained in Ref. [29], precisely
collisionratecanbe estimatedfromthe elastic crosssec- speaking,Eq. (26) is valid only near the CEP.Neverthe-
tions for pions and baryons and their densities. Due to less, the physical reason why ξeq and ξ are so close to
the chiral symmetry and the effect of the ∆, the elastic each other at kinetic freezeout, is general and holds also
π-N crosssectionisconsiderablylargerthanthatofπ-π. inthiscase;finalstateinteractionstendtowashoutnon-
On the other hand, in Pb+Pbcollisions at 158 AGeV at equilibrium effects. Since the kinetic freezeout points at
SPS,(p+p¯)/(π++π−)∼0.09[34,35]anditis∼0.09at the Super Proton Synchrotron (SPS) at CERN and the
RHIC [36]. Thus, at the same particle number density, Relativistic Heavy Ion Collider (RHIC) at Brookhaven
the collision rate at SPS is almost the same as that at National Laboratory are far below the phase transition
RHIC. From these considerations, it would be natural if line, the effect of final state interactions is expected to
the kinetic freezeout temperature in central collisions at dominate.
RHICisnoticeablyhigherthanthatatSPS,whiletheki- Similarargumentsholdalsoforfluctuationsexceptfor
netic freezeout temperature at RHIC is actually slightly the cube effect we argued in the previous section. The
lower than that at SPS [33]. Two comments are in or- event-by-event fluctuations of the mean transverse mo-
der here. First, the kinetic freezeout we are considering mentuminPb+Aucollisionsat40,80,and158AGeV/c
here is that of pions and kaons. The analysis in Ref. [33] were measured by the CERES Collaboration [38]. They
does not take into account the effect of resonance de- are slightly smaller than those at RHIC, and unusually
cays. Whenresonancedecaysaretakenintoaccount,the large fluctuation or non-monotonic behavior which may
kinetic freezeout temperature becomes even lower [37]. suggestthe existence of the CEP has not been observed.
Second, more precisely speaking, kinetic freezeout takes However, in order to understand experimental observa-
place when the mean collision rate and the product of tions, we have to take account of the CEP character as
the expansion rate and the typical scale of the system an attractorof isentropic trajectoriesin the T-µ plane.
B
become comparable. However, according to the results Due to the focusing effectofthe CEP,the isentropictra-
of the HBT interferometry, the typical size at RHIC is jectories of various initial collision energies are gathered
10
totheCEP,ifthecriticalregionaroundtheCEPislarge 1.0
enough. As a result, when each isentropic trajectory η0 1.1 1.05
passesneartheCEP,similarcorrelationlengths,fluctua- /
η
tions,andsoonareinduced,andtheydonotshowstrong
1
non-monotonicbehaviorasafunctionofthecollisionen-
ergy. In other words, the absence of the non-monotonic
behavior, which is shown by the CERES Collaboration,
does not necessarily imply the non-existence of the CEP 0.2
0.2 0
in the region probed by SPS and RHIC. More precise 0 −0.2 −0.2 h
measurement and analyses which take account of the fo- r
cusingandslowing-out-of-equilibriummayrevealthe ex-
istence of the CEP.
Recently, the fluctuation of (K+ + K−)/(π+ + π−) FIG.12: Shearviscosity η/η0 neartheCEP.Fortheparam-
eters used in thecalculation, see thetext.
and (p+p¯)/(π+ +π−) was measured and it was found
that the collision energy dependence of the fluctuation
of (K++K−)/(π++π−) is not described by a cascade
in experimentalobservables. Firstwe constructed realis-
model [34]. While this result is interesting, we have to
tic equations of state with the CEP on the basis of the
be carefulinassociatingthis resultwith the criticalphe-
universality hypothesis and discussed the hydrodynami-
nomena around the CEP. Since neither the numerator
cal expansion near the CEP. The behavior of isentropic
nordenominatorof(K++K−)/(π++π−)isaconserved
trajectoriesneartheQCDcriticalendpointisclearlydif-
quantity in the strong interaction, substantial modifica-
ferent fromthat in the bag plus excluded volume model,
tion is expected for both in the hadronphase as we have
which is usually used in hydrodynamical models. We
demonstratedforthecorrelationlength. Furthermore,as
found that the CEP acts as an attractor of isentropic
we discussed for the particle ratios, quasi-particle states
trajectories, n /s= const., in the T-µ plane, not only
B B
near the CEP are quantum-mechanically different from
on the right hand side of the CEP but also on the left
the ones in the vacuum, i.e., observed pions, kaons, pro-
handsideoftheCEP.Becauseofthefocusingoftheisen-
tons, and so on. To relate the fluctuation of the particle
tropic trajectories, the path of the system in the course
ratios to the critical phenomena, it is necessary to know
ofthetimeevolutionintheT-µ planeisremarkablydif-
B
not only the time evolution of the quasi-particle states
ferent from that in the usual bag plus excluded volume
butalsothe relationbetweenthequasi-particlequantum
model. This will be reflected in the final state observ-
states in medium and the particle states in the vacuum
ables that are sensitive to the equation of state, such as
[32].
the anisotropic flows (directed, elliptic, and higher order
Up to now, we have been considering the isentropic
Fouriercoefficients)[39,40]andHBTradii. Forthispur-
trajectories in the T-µ plane. Actually, the viscosity
B pose, full 3-D hydrodynamical calculations [41, 42] with
is known to diverge at the CEP and the entropy is gen-
realistic equations of state are indispensable [43].
erated if the system passes near the CEP. However, the
Next we argued the critical slowing down near the
singularityintheviscosityisknowntobeveryweak[31],
QCD critical end point. We compared the equilibrium
and non-equilibrium correlation lengths along two isen-
η ξ (119ǫ+O(ǫ2)) tropic trajectories which pass near the CEP. We found
= eq , (27) that the difference between them at kinetic freezeout is
η0 (cid:18)ξeq,0(cid:19) very small.
Furthermore, we considered the experimental observ-
where η is the shear viscosity, η is the shear viscosity
0 abilityofthe consequencesofthe CEP.Fluctuationsand
at the point where the equilibrium correlation length is
viscosities diverge at the CEP, but fluctuation which is
ξ , and ǫ is 4−d as before. In Fig. 12, η/η in the
eq,0 0 inducedbytheCEPwillfadebykineticfreezeoutforthe
r-h plane is shown. The small correction to the criti-
same reason as for the correlation length. This explains
cal exponent 1/19 is set to 0 for simplicity. Because of
thenon-observationofsharpnon-monotonicbehaviorsin
the small critical exponent, the diverging feature of the
fluctuations. However, the singular part of the fluctua-
viscosity is hardly seen. The effect of the entropy pro-
tions is approximately proportional to the cube of the
duction around the CEP is thus expected to be small,
correlation length and there is a possibility that more
and it will not show easily-recognizable non-monotonic
precise measurements in the future may reveal the ex-
changesinobservablesasthe collisionenergyischanged.
istence of the CEP. The singularity in the viscosity is
too weak to affect observables. Concerning the chemical
freezeout process, we need to consider the effect of the
V. SUMMARY interactions on hadrons for detailed discussion on parti-
cle number ratios,when ittakes placenear the CEP.We
Inthispaper,wediscussedthehydrodynamicalexpan- found that, at least now, the most promising way is to
sionneartheQCDcriticalendpointanditsconsequences trace the behavior of the collision energy dependence of
