#  Locating critical point of QCD phase transition basing on finite-size
scaling

Chen Lizhu  Institute of Particle Physics, Hua-Zhong Normal University, Wuhan
430079, China  X.S. Chen  Institute of Theoretial Physics, Chinese Academy of
Sciences, Beijing 100190, China  Wu Yuanfang  Institute of Particle Physics,
Hua-Zhong Normal University, Wuhan 430079, China  Key Laboratory of Quark $\&$
Lepton Physics (Huazhong Normal University), Ministry of Education, China

###### Abstract

It is argued that in relativistic heavy ion collisions, due to limited size of
the formed matter, the reliable criterion of critical point is finite-size
scaling, rather than non-monotonous behavior of observable. How to locate
critical point by finite-size scaling is proposed. The data of $p_{\rm t}$
correlation from RHIC/STAR are analyzed. Critical points are likely observed
around $\sqrt{s}=62$ and $200$ GeV. They could be, respectively, the
transition of deconfinement and chiral symmetry restoration predicted by
lattice-QCD. Further confirmation with other observable and energies is
suggested.

###### pacs:

12.38.Mh, 25.75.Nq, 25.75.Gz

Lattice-QCD simulations have shown that the transition of deconfinement in
quantum chromodynamics (QCD) at vanishing baryon chemical potential $\mu_{\rm
B}$ is crossover lattice-1 . There has been much speculation that the
crossover becomes a true first-order phase transition for larger values of
$\mu_{\rm B}$. This suggests that the QCD phase diagram can exhibit a critical
endpoint where the line of first order transition matches that of second order
or analytical crossover 1st .

Chiral symmetry restoration is another QCD originated phase transition. It has
been shown that the transition for $\mu_{\rm B}=0$ is crossover c-crossover .
So there could also be a chiral critical endpoint in phase diagram. But it is
unclear if the critical temperature of chiral symmetry restoration is above
karsch-h , or equal to karsch-s , or below quarkyonic  that of the
deconfinement.

Locating the critical endpoints of QCD phase transitions by lattice
calculation is still a formidable challenge. But if the critical endpoint is
in the region accessible to current relativistic heavy ion collisions, it
should be discovered experimentally.

Most of the current signatures for finding the critical point are focused on
the anomalous, or non-monotonous, behavior of the observable at various
incident energies ebye-t . The argument is that in infinite system, the
correlation length $\xi$ diverges when approaching the critical point. The
contribution of this singularity to the observable is supposed to be
proportional to $\xi^{2}$. However, the data from RHIC and SPS in more than a
decade accumulation show no sign of anomalous behavior as a function of
$\sqrt{s}$ ebye-e .

In relativistic heavy ion collisions, two nuclei move with relativistic
velocity and collide as two contracted pancakes. More central collision makes
overlapped area larger. It is just because the large number of strongly
interacting nucleons in more central nuclear collisions make the transition
between hadron and quark-gluon plasma possible. The centrality (or the system
size) dependence of the observable is noticeable starprc .

Due to the finite size of system, no divergence can be practically observed at
critical point. The physical quantities, which are divergent in infinite
system, become finite and have a maximum, i.e., so called non-monotonous
behavior. However, the position of the maximum changes with system size and
deviates from the true critical point.

The appearance of non-monotonous behavior is not always associated with
critical point. Taking one-dimensional Ising model as an example, there is no
critical point in this model, but its specific heat in a finite system has
non-monotonous behavior.

Moreover, the absence of non-monotonous behavior does not mean no critical
point. The physical quantities like order parameter, which are finite in
infinite system, have a monotonous behavior near critical point in a finite
system chen1996 . Therefore, non-monotonous behavior is not a reliable
criterion for the critical point of finite system.

An effective identification of critical point of finite system is the finite-
size scaling, which was proposed from phenomenological fss-1  and
renormalization-group fss-RG  theories, and was approved by the Monte Carlo
results of finite systems in different universal classes fss-2 .

In this letter, we first propose how to locate critical point by finite-size
scaling. Then the data of $p_{\rm t}$ correlation at 6 centralities and 4
incident energies from RHIC/STAR are analyzed. The behavior of critical point
is likely observed around $\sqrt{s}=62$ and $200$ GeV. Finally, we suggest how
to confirm the findings and precisely locate the critical point in coming
experimental study.

The main points of finite-size scaling can be described as the following. An
observable $Q$ of finite system is a function of temperature $T$ and system
size $L$. When $L$ is much larger than the microscopic length scale and $T$ is
in the vicinity of critical point $T_{c}$, the observable $Q(T,L)$ can be
written in a finite-size scaling form fss-1 ; fss-RG ; fss-2 ,

$Q(T,L)=L^{\lambda/\nu}F_{Q}(tL^{1/\nu}).$ (1)

$t=(T-T_{c})/T_{c}$ is the reduced temperature and $\lambda$ is the critical
exponent of the observable. $\nu$ is the critical exponent of the correlation
length $\xi=\xi_{0}t^{-\nu}$.

Finite-size scaling not only characterizes the scaling behavior of
thermodynamic quantities of finite system near critical point, but also
provides criterion for locating the critical point. At critical point
$T=T_{c}$, the finite-size scaling function $F_{Q}$ in Eq. (1) becomes

$F_{Q}(0)=Q(T_{c},L)L^{-\lambda/\nu},$ (2)

which is constant and independent of system size $L$. In the plot of
$Q(T,L)L^{-\lambda/\nu}$ vs $T$, the critical point [$T_{c}$, $F_{Q}(0)$] is a
fixed point, where all curves of different system sizes converges to.
Reversely, the appearance of fixed point indicates the existence of a critical
point.

If the critical exponent $\lambda=0$, like Binder cumulant ratio binder1981 ,
the fixed point can be obtained directly from the temperature dependence of
this observable at different system sizes. This is why Binder cumulant ratio
has been used very widely in determining critical point of finite-size system.

If the critical exponent $\lambda\neq 0$ and is unknown, the fixed point can
be found by investigating the temperature dependence of $Q(T,L)L^{-a}$ at
different system sizes. When a fixed point is observed at a certain parameter
$a_{0}$, it indicates the existence of a critical point and the parameter
$a_{0}$ is related to the ratio of critical exponents, i.e.,
$\lambda/\nu=a_{0}$.

The critical point can also be found directly from the system size dependence
of the observable. Taking logarithm in the both sides of Eq. (1), it becomes

$\ln Q(T,L)=\lambda/\nu\ln L+\ln F_{Q}(tL^{1/\nu}).$ (3)

At critical point $t=0$, the second term of Eq. (3) becomes a constant and
$\ln Q(T_{c},L)$ becomes a straight line with respect to $\ln L$. If system is
away from the critical point, the second term of Eq. (3) is no longer a
constant. It gives an additional size dependent contribution to the observable
and makes $\ln Q(T,L)$ deviate from the straight line with respect to $\ln L$.

It is found recently that the finite-size scaling holds not only for
thermodynamic quantities like order-parameter, susceptibility, and so on, but
also for various cluster sizes liangsheng  and their fluctuations lizhu-ising
. Therefore, the finite-size scaling of various critical related observable
could be used to identify critical point and its critical exponents.

In relativistic heavy ion collision, correlation and fluctuation of final
state particles is regarded as critical related observable lattice-corr .
Although much attention have been drawn in measuring them, but influence of
system size has been neglected. The available data for system size study is
very few. The $p_{\rm t}$ correlation at Au+ Au collisions from RHIC/STAR
starprc  is the only data which can be used for the analysis, where the
centrality dependence of $p_{\rm t}$ correlation at 4 incident energies are
well presented starprc . But the errors of the data at $\sqrt{s}=20$ GeV are
much larger than that at other collision energies. The $p_{\rm t}$ correlation
is defined as

$\displaystyle P(\sqrt{s},L)=\frac{1}{N_{\rm e}}\sum\limits_{k=1}^{N_{\rm
e}}\frac{\sum\limits_{i=1}^{N_{k}}\sum\limits_{j=1,i\not=j}^{N_{k}}(p_{{\rm
t},i}-\langle p_{\rm t}\rangle)(p_{{\rm t},j}-\langle p_{\rm
t}\rangle)}{N_{k}(N_{k}-1)}.$ (4)

$N_{\rm e}$ is the number of event, $p_{{\rm t},i}$ is the transverse-momentum
of the $i$th particle in each event, and $N_{k}$ is the number of particles in
the $k$th event. $\langle\ldots\rangle$ is the average over event sample.

Collision energy is the controllable condition. Here we let it play the role
of temperature in the analysis of finite-size scaling. The size of the formed
matter is mainly limited by the size of overlapping transverse region, which
is proportional to the number of participant nucleons and is quantified as
centrality. So the initial mean size of the formed matter can be approximately
estimated by the square root of participants, $\sqrt{N_{\rm part}}$. We choose
dimensionless (or relative) size,

$\displaystyle L=\sqrt{N_{\rm part}}/\sqrt{2N_{\rm A}},$ (5)

as scaled mean size of initial system, where $N_{\rm A}$ is the number of
nucleons of incident nucleus. The system size at transition should be a
monotonically increasing function of $L$. The position of critical point is
insensitive to the concrete form of this function, but only the critical
exponents changes with it. As the first step, the system size at transition is
assumed to be proportional to $L$.

In the case of a few critical points, the finite-size scaling of $p_{\rm t}$
correlation in the vicinity of each critical collision energy
$\sqrt{s_{c,i}}(i=1,2,...)$ can be written as

$\displaystyle
P(\sqrt{s},L)=L^{\lambda_{i}/\nu_{i}}F_{P,i}[e_{i}L^{1/\nu_{i}}].$ (6)

$e_{i}=(\sqrt{s}-\sqrt{s_{c,i}})/\sqrt{s_{c,i}}$ is the reduced collision
energy at $i$th critical point, which is unknown in priori. $\lambda_{i}$ is
the $i$th critical exponent of $p_{\rm t}$ correlation. In the following, we
demonstrate how to locate the critical point by the data of $p_{\rm t}$
correlation from RHIC/STAR.

Figure 1:  (a) The energy dependence of $p_{\rm t}$ correlations at different
sizes $L$ (or centralities). Data come from RHIC/STAR starprc . (b), (c) and
(d) are $p_{\rm t}$ correlation multiplied by the factor, $L^{-a}$, with
$-a=1.0$, 2.09 and 4, respectively

.

Firstly, we change the centrality dependence of $p_{\rm t}$ correlation at
different collision energies in Ref. starprc  to the collision energy
dependence at different sizes (or centralities). The results are shown in Fig.
1(a). Since in the most peripheral collisions, the size of the formed matter
is too small to be inside the asymptotic region of finite-size scaling, we
choose six centralities at mid-central and central collisions to do the
analysis. The sizes corresponding to the 6 centralities are indicated in the
legend of Fig. 1(a). It is clear that at a given collision energy, the
correlation strength increases with the decrease of system size. The influence
of finite size is obvious.

If critical collision energy of QCD phase transition is in the range of
incident energy at RHIC, the behavior of fixed point should be observable. So
we multiply $P(\sqrt{s},L)$ by a size factor $L^{-a}$ with different $a$ to
see how it changes with the system size $L$. Varying $-a$ from small to large,
it is interesting to see that at collision energy $\sqrt{s}=62$ GeV, all
points of different sizes move firstly toward each other, then well converge
at $-a_{0,1}=2.09$, and finally move again apart from each other. The
corresponding steps and typical $a$ values are presented in Fig. 1(b), (c),
and (d) respectively, where the errors in each sub-figures come from the
measure of $P(\sqrt{s},L)$ only, and the errors of $N_{\rm part}$ are not
included.

At $\sqrt{s}=200$ GeV, the points of different sizes show the same behavior
and best converge at $-a_{0,2}=2.08$. While in the whole process, the points
of different sizes at energies $\sqrt{s}=20$ (or 130) GeV never move close to
each other as those at $\sqrt{s}=62$ (or 200) GeV do. So there are likely two
fixed points around $\sqrt{s}=62$ and 200 GeV.

In order to confirm the position of fixed points, we study the $\ln L$
dependence of $\ln P(\sqrt{s},L)$ for four incident energies, respectively. A
parabola fit, $c_{2}(\ln L)^{2}+c_{1}\ln L+c_{0}$, is used at each collision
energy. The better straight-line behavior results in smaller $|c_{2}|$ and
larger ratio of $|c_{1}/c_{2}|$. The fit parameters, $c_{2}$ and $c_{1}$, for
4 collision energies are listed in Tab. 1. It shows that the better straight-
line behavior happen to be at $\sqrt{s}=62$ and 200 GeV, which are the same
collision energies of fixed points found above. The data at these two energies
can be well fitted, respectively, by the straight lines with slopes $a_{0,1}$
and $a_{0,2}$ obtained above by the fixed points. The results are shown in
Fig. 2(a). While, the data at $\sqrt{s}=20$ and 130 GeV are better fitted by
parabola as shown in Fig. 2(b).

Table 1:  Parameters of parabola fits. $\sqrt{s}$(GeV) |  | 20 |  | 62 |  | 130 |  | 200
---|---|---|---|---|---|---|---|---
$|c_{2}|$ |  |  1.86$\pm$ 0.93 |  |  0.6 $\pm$ 0.09 |  |  1.56$\pm$ 0.41 |  |  0.77$\pm$ 0.1
$|c_{1}|$ |  |  3.9$\pm$0.89 |  |  2.59$\pm$ 0.09 |  |  3.43$\pm$ 0.41 |  |  2.74$\pm$0.1
Figure 2:  Double-log plots of $p_{\rm t}$ correlation with respect to size,
(a): straight-line fits with slopes $a_{0,1}$ and $a_{0,2}$ obtained by fixed
points, and (b): parabola fits.

The same analysis has also been applied to the $p_{\rm t}$ correlation
normalized by the average $p_{\rm t}$ over the whole sample starprc . The
analysis for normalized $p_{\rm t}$ correlation at $\sqrt{s}=62$ and $200$ GeV
show exactly the same behavior of fixed points and straight lines as what
$p_{\rm t}$ correlation demonstrates above. The critical exponents of
normalized $p_{\rm t}$ correlation (about 1.1) are smaller than that of
$p_{\rm t}$ correlation.

So the critical collision energies are most probably around $\sqrt{s}=62$ and
$200$ GeV, rather than near $\sqrt{s}=20$ and 130 GeV. The same analysis for
other critical related observable, such as the fluctuations of mean $p_{\rm
t}$ per event, the moments of multiplicity, the ratio of $K$ to $\pi$, and so
on, will be greatly helpful in confirming the observed results. Therefore, the
incident energy and centrality dependence of those observable are called for.

If there were additional collisions around $\sqrt{s}=62$ and $200$ GeV, we
could determine the finite-size scaling function defined in Eq. (6). This is
impossible at present since there are only two collision energies in addition
to the critical ones, and they could be outside of the asymptotic region where
finite-size scaling holds.

The findings of the two critical points may imply that deconfinement and
chiral symmetry restoration occur at different temperatures. Which one is at
the lower or higher temperature (energy) has to be confirmed finally from
theoretical calculation. Two critical collision energies, $\sqrt{s}=62$ and
$200$ GeV, are both within the range estimated by lattice calculation hTc .

The similar ratios of critical exponents at two critical points is consistent
with current theoretical estimation, which shows that all critical exponents
of the deconfinement transition, in the same university as the $3$-dimensional
Ising model 3d-ising , are very close to that of chiral symmetry restoration,
in the same university as the $3$-dimensional O$(4)$ model with spin symmetry
3d-o4 .

To the summary, we argue in this letter that finite-size effects of the formed
matter in relativistic heavy ion collisions is not negligible. The finite-size
scaling, rather than non-monotonous behavior of observable is a reliable
criterion of the existence of critical point. Then we propose how to locate
critical point by finite-size scaling. As an application, we analyze the data
of $p_{\rm t}$ correlation and its normalized one at 6 centralities and 4
incident energies from RHIC/STAR. Two fixed points, and therefore two critical
points, are likely observed around $\sqrt{s}=62$ and 200GeV. They could be,
respectively, related to the transition of deconfinement and chiral symmetry
restoration predicted by lattice-QCD. The ratios of critical exponents at
these two critical points are similar, in consistence with current theoretical
estimation.

The confirmation of this observation requires the efforts from both
theoretical and experimental sides. From experimental side, it is proposed to
get more and better data on other critical related observable at current
collision energies, and a few additional collisions around $\sqrt{s}=62$ and
200 GeV. Then we can more precisely determine the critical endpoints and
critical exponents.

The authors are grateful to Dr. Li Liangsheng, Prof. Liu Lianshou and Prof.
Dr. Hou Defu for very helpful discussions. This work is supported in part by
the NSFC of China with project No. 10835005 and MOE of China with project No.
IRT0624 and No. B08033.

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