Transcript How to Locate the Critical Point?

Fluctuations and Correlations of Conserved
Charges in QCD Thermodynamics
Wei-jie Fu, ITP, CAS
16 Nov. 2010
AdS/CFT and Novel Approaches to Hadron and Heavy
Ion Physics, week 6
Outline
Introduction
 Fluctuations and Correlations at
Vanishing Chemical Potential
 Fluctuations and Correlations near the
QCD critical point
 Summary and Discussions

Introduction
Purposes of
BES at RHIC:


Looking for
evidence of a
Critical Point.
Looking for
evidence of a
first-order
phase
transition.
How to Locate the Critical Point?

Fluctuation Measures
M. Cheng et al. , Phys. Rev. D. 79, 074505
(2009).
The characteristic signature of the existence of a critical point is an increase of fluctuations.
<pT> Fluctuations
J. Adams et al. (STAR Collaboration), Phys. Rev. C. 72,
044902 (2005).
<pT> fluctuations are challenging as there are a number of effects that can swamp the signal,
for example the elliptic flow can cause a non-statistical fluctuation.
<pT> fluctuations increase with the collision energy, but scaled <pT> fluctuations are independent
of the energy.
K/πFluctuations
C. Alt et al. (NA49), PRC 79, 044910 (2009); B.I.Abelev et al. (STAR), PRL 103,
092301 (2009).
Solid line---UrQMD, dashed line---HSD
 Comparison between the statistical hadronization model results and the experimental data.
 When the light quark phase space occupancy is one, corresponding to equilibrium, the
calculations underestimate the experimental results at all energy.
 When the light quark phase space occupancy is varied to reproduce the excitation function of
K+/π+ yield ratios, SH correctly predicts at higher enegier.
 The right figure shows the comparison of the prediction of the transport model HSD and the
UrQMD models to the experimental data.
p/πand K/p Fluctuations
C. Alt et al. (NA49), PRC 79, 044910 (2009); M.M.Aggarwal, et al. (STAR),
arXiv:1007.2613;
T. Schuster (NA49), PoS CPOD2009,029(2009); J.Tian (STAR), SQM 2009.
 The left figure shows comparison of the predictions of the UrQMD model to the
experimental data for P/π. The predictions of the model is close to the experimental results.
 The correlation between strangeness S and baryon number B is sensitive to the state of
matter. V. Koch et al. , PRL 95, 182301 (2005).
Higher Moments of Net Proton
Multiplicity Distributions at RHIC
M. M. Aggarwal et al. (STAR Collaboration), Phys. Rev.
Lett. 105, 022302 (2010).
 The skewness (S) and kurtosis (k) of net proton multiplicity distributions are constant as
functions of the centrality.
 There is no evidence for a critical point in the QCD phase diagram for baryon chemical
potential below 200 MeV.
Why are Fluctuations of Conserved
Charges so important?

They provide information about the degrees of freedom---confined
hadrons or deconfined QGP, of strongly interacting matter.
S. Jeon et al. , PRL 85, 2076 (2000); V. Koch et al. , PRL 95, 182301 (2005);
B. Stokic et al. , PLB 673, 192 (2009); M. Cheng et al. , PRD 79, 074505 (2009)

Conserved charges are conserved through the evolution of the fire
ball, thus the fluctuations of conserved charges can be directly
measured by heavy-ion experiments.
M. M. Aggarwal et al. (STAR Collaboration), arXiv:1007.2613

Fluctuations are directly related to various generalized
susceptibilities, and these susceptibilities can be calculated by
Lattice QCD simulations and other theoretical methods. Therefore,
conserved charge fluctuations provide the first direct connection
between experimental observables and theoretical calculations.
M. Cheng et al. , PRD 79, 074505 (2009); M. M. Aggarwal et al.
(STAR Collaboration), arXiv:1007.2613
Fluctuations and Correlations

Denoting the ensemble average of the conserved charge number N ( X  B, Q, S )
with N X  . The deviation of a event multiplicity from its average value is
given by  N  N   N  . Then we can relate fluctuations and correlations
with the generalized susceptibilities.
X
X
X
X
W. Fu et al. , PRD 81, 014028 (2010); W. Fu et al. , PRD 82, 074013 (2010)
Theoretical framework---2+1 Flavor
PNJL Model

The Lagrangian density for the 2+1 flavor PNJL model is given as

The thermodynamical potential density is
C. Ratti et al. , PRD 73, 014019 (2006);
W. Fu et al. , PRD 77, 014006 (2008)
Fluctuations of Light Quarks and
Strange Quarks
W. Fu, Y. Liu, and Y. Wu, PRD 81, 014028 (2010); M. Cheng et al. , PRD 79, 074505 (2009)
 The quadratic fluctuations increase monotonously with
the increase of the temperature, and the fluctuations of
the heavier strange quarks are suppressed relative to
those of the light quarks.
 We find a plateau at the critical temperature in the
curve of the ratio of strange and light quark quadratic
fluctuations.
 Our calculations are consistent with the lattice
calculations, except the temperature corresponding to
strange quark quartic peak is relatively larger than that in
lattice simulations.
Ratio of the Quartic to Quadratic
Fluctuations
W. Fu, Y. Liu, and Y. Wu, PRD 81, 014028 (2010); M. Cheng et al. , PRD 79, 074505 (2009)
 The ratio is believed to be a valuable probe of
the deconfinement and chiral phase transitions.
S. Ejiri et al. , PLB 633, 275 (2006); S. Ejiri
et al. , NPA 774, 837 (2006); B. Stokic et al.,
PLB 673, 192 (2009).
 At high temperature, the system approaches
the Stefan-Boltzmann limit, and the pressure
can be easily obtained as
 However, the temperature –driven
deconfinement transition is a continuous
crossover. Assuming
 At low temperature, the Polyakov-loop
approaches zero, meaning the one- and twoquark states are suppressed, only permitting
three-quark states. Using Boltzmann
approximation, the pressure is
Fluctuations of Baryon Number at
Vanishing Chemical Potentials
 The singular behavior of the baryon number
fluctuations is expected to be controlled by the
universal O(4) symmetry group at vanishing
chemical potential and vanishing light quark
mass. And the baryon number fluctuations are
expected to scale like,
 Our results are consistent with the lattice
calculations, except for some quantitative
differences at high temperature, because of
the current quark mass effect.
 The ratio of the quartic to quadratic baryon
number fluctuations approaches 1 at low
temperature. We find a pronounced cusp in the
ratio at the critical temperature. The cusp is a
signal of the phase transition.
K. Karsch, PoS CPOD07 026 (2007);
B.Stokic et al. , PLB 673, 192 (2009)
Fluctuations of Electric Charge and
Strangeness
W. Fu, Y. Liu, and Y. Wu, PRD 81, 014028 (2010)
 There is a prominent cusp at the critical temperature in the
 The fluctuations of electric charge are well
consistent with the lattice results.
ratio of the quartic to quadratic electric charge fluctuations. At
low temperature, The ratio is a rising function.
 The sixth-order fluctuations of strangeness are qualitatively
consistent with the lattice calculations. But there are some
quantitative differences.
Second-order Correlations of
Conserved Charges
 The correlations of conserved charges are well
consistent with the lattice results.
The correlation between baryon number and electric
charge decreases with the increase of the temperature,
and in the high temperature limit, this correlation
approaches zero, because the sum of quark electric
charges is zero.
The correlation between baryon number and strangeness
is a useful diagnostic of strongly interacting matter.
V. Koch et al. ,
PRL 95, 182301 (2005)
 An interesting thing is that both our calculations and
lattice simulations indicate there is a plateau at the
critical temperature in the correlation between electric
charge and strangeness .
 In summary, our effective model reproduces all the
things that obtained in lattice simulations. This is expected,
because what governs the critical behavior of the QCD
phase transition is the universality class of the chiral
symmetry, which is kept in this model.
W. Fu, Y. Liu, and Y. Wu, PRD 81, 014028 (2010); M. Cheng et al. , PRD 79, 074505 (2009)
Fluctuations of Baryon Number in
the QCD Phase Diagram
 The quadratic fluctuation
of baryon number has a peak
structure, and the peak
becomes sharper and
narrower while moving toward
the QCD critical point.
 The cubic fluctuation
changes its sign, which is
argued to be used to
distinguish the two sides of
the QCD phase boundary.
M. Asakawa et al. ,
PRL 103, 262301 (2009
All amplitudes grow rapidly
when moving toward the QCD
critical point and diverge
there.
 The chiral phase transition
line in contour plots is obvious,
and the region near the QCD
critical point also be manifest.
W. Fu et al. , PRD 82, 074013 (2010)
 Comparing to the quadratic fluctuation, higher-order
fluctuations are superior in the search for the QCD
critical point.
Fluctuations of Electric Charge in
the QCD Phase Diagram
 The structure of the
fluctuations of electric
charge is similar with that of
the baryon number
fluctuations.
 The peak in the quadratic
fluctuation of electric charge
grows less rapidly than that
in the quadratic fluctuation
of baryon number.
Employing the quadratic
fluctuations of electric
charge to search for the
critical point is not easy.
But the situations for the
higher-order fluctuations are
different. They are sensitive
to the singular properties of
the critical point.
W. Fu et al. , PRD 82, 074013 (2010)
Fluctuations of Strangeness
 The structure of the fluctuations
of strangeness is similar with those of
the baryon number and electric
charge fluctuations.
 But, the strangeness fluctuation is
less sensitive to the singular behavior
of the critical point than the baryon
number and electric charge
fluctuations.
W. Fu et al. , PRD 82, 074013 (2010)
Second-order Correlations of Conserved
Charges in the QCD Phase Diagram
 The common feature of the
second-order correlations is
that there is a peak
structure during the chiral
phase transition.
 In chiral symmetric phase,
i.e. in the SB limit, the
correlation between B and Q
is vanishing, while other
correlations have finite
values.
 The correlation between B
and Q is superior to the
other correlations to be used
to search the critical point.
W. Fu et al. , PRD 82, 074013 (2010)
Third-order Correlations between
Two Conserved Charge
W. Fu et al. , PRD 82, 074013 (2010)
 The common feature of the third-order correlations is that they change their signs at the chiral
phase transition.
 Comparing with the second-order correlations, the third-order correlations are much more
sensitive to the singular behavior of the critical point. The critical point is much more obvious.
 Among the six correlations, we find the former three correlations are better than the other ones.
Third-order Correlations among
Three Conserved Charge
W. Fu et al. , PRD 82, 074013 (2010)
 The correlation changes its sign from negative to positive during the chiral phase transition and
diverges at the QCD critical point.
 Only when the system is near the chiral phase transition, the correlation has nonvanishing value.
The critical point in the contour plot is very obvious, therefore, the third-order correlation among
three conserved charge is an ideal probe to search for the QCD critical point.
Fourth-order Correlations between
Two Conserved Charge
W. Fu et al. , arXiv:1010.0892
 The magnitudes of all the
fourth-order correlations
grow rapidly and oscillate
drastically at the phase
transition, and all the
correlations diverge at the
QCD critical point.
 The common feature of the
fourth-order correlations is
that there are three extrema.
 All the correlations except
for chiBS13 and chiQS13
vanish rapidly once the
system deviates from the
QCD phase transition.
 chiBS13 and chiQS13 still
have finite values at large
baryon chemical potential,
since the constituent mass of
strange quark can not be
neglected even in the chiral
symmetric phase.
Fourth-order Correlations between
Two Conserved Charge
 We observe that all the
fourth-order correlations are
vanishing in the chiral
symmetry broken phase.
 The chiral phase transition
line is distinct and we can
easily recognize the region
near around the QCD critical
point.
 Among all the correlations,
we find the former seven
correlations are superior to
the chiBS13 and chiQS13 to
be used to search for the
critical point.
W. Fu et al. , arXiv:1010.0892
Fourth-order Correlations among
Three Conserved Charge
W. Fu et al. , arXiv:1010.0892
 There are two minima and
one maximum in the curves of
chiBQS121 and chiBQS211
and two maxima and one
minimum in chiBQS112.
 The fourth-order
correlations among three
conserved charges approach
zero rapidly once the system
deviates from the chiral
phase transition.
 The fourth-order
correlations among three
conserved charges are quite
sensitive to the singular
structure related to the
critical point, and the critical
point is very obvious in these
contour plots, so the
correlations are excellent
probes to explore the critical
point.
Summary and Discussions



We study the fluctuations and correlations of conserved
charges, such as the baryon number, the electric charge, and
the strangeness, in a effective model---2+1 PNJL model. The
results are well consistent with the lattice simulations.
We find the higher-order fluctuations and correlations,
especially the correlations are superior to the second-order
ones to be used to search for the critical point.
We suggest that experimentalists at RHIC use the higher-order
correlations to explore the QCD critical point.
Acknowledge
My collaborators: Prof. Yu-xin Liu, Prof. Yue-liang Wu
Thank you for your attentions!