1. to Look for the Critical Point and Phase Transition
New Ways
to Look for the Critical Point
and Phase Transition
Masayuki Asakawa
Department of Physics, Osaka University
in collaboration with C. Nonaka, B. Müller, and S.A. Bass
S. Ejiri and M. Kitazawa

2. CEP(critical end point)
QCD Phase Diagram
LHC T
RHIC
QGP (quark-gluon plasma)
CEP(critical end point)
MeV
crossover
1st order
Hadron Phase
order ?
chiral symmetry breaking
confinement
CSC (color superconductivity)
5-10r0 mB

3. Punchlines: Quantities to Look at
Quantities not subject to Critical Slowing Down and Final State Interactions
Quantities sensitive to CEP or Phase Transition
Conserved Quantities unlike correlation lengths, (usual) fluctuations...etc.
What to see? How to see?
Question

4. CEP = 2nd order phase transition, but...
Divergence of Fluctuation
Correlation Length
Specific Heat ?
Critical Opalescence?
CEP =
2nd Order Phase Transition Point
2SC
CFL
even if the system goes right through
the critical end point…
Subject to Final State Interactions
If expansion is adiabatic
AND
no final state int.

5. Critical Slowing Down and Final State Int.
Furthermore,
critical slowing down limits the size of fluctuation, correlation length !
Time Evolution along given
isentropic trajectories (nB/s : fixed)
x and xeq
big difference
almost no enhancement
1-dim Bjorken Flow
Nonaka and M.A. (2004)
Model H (Hohenberg and Halperin RMP49(77)435)

6. Locally Conserved Charges
Fluctuations of Locally Conserved Quantities
Net Baryon Number
Net Electric Charge
Net Strangeness
Energy ...
Net Electric Charge
Hadron Gas Phase
~2/3 of hadrons carry
electric charge
QGP Phase
only ~ 1/2 of d.o.f., i.e.
quarks and anti-quarks
carry electric charge
Heinz, Müller, and M.A., PRL (2000)
Also, Jeon and Koch, PRL (2000)

7. Charge Fluctuation at RHIC
D-measure
Predictions
QGP phase
Hadron phase
( : hadron resonance gas)
Experimental Value
(STAR)
(PHENIX)
Białas, PLB(2002)
Nonaka, Müller, Bass, M.A., PRC (2005)
Well-Explained by Quark Recombination

8. Odd Power Fluctuation Moments
We are in need of observables that are not subject to final state interactions
Fluctuation of Conserved Charges: not subject to final state interactions
Usually fluctuations such as
have been considered
even power
One of exceptions: Stephanov (2008) in conjunction with CEP
Usual Fluctuations such as
: positive definite
Absolute values carry information of states
Asakawa, Heinz, Müller, Jeon, Koch
On the other hand,
Odd power fluctuations :
NOT positive definite
In general, do not vanish (exception, )
Sign also carry information of states

9. Physical Meaning of 3rd Fluc. MomentcB
: Baryon number susceptibility
in general, has a peak along phase transition
changes the sign at QCD
phase boundary !
In the Language of fluctuation moments:
more information than
usual fluctuation

10. (Hopefully) More Easily Measured Moments
Third Moment of Electric Charge Fluctuation cB
cI/9
iso-vector susceptibility
nonsingular when Isospin-symm.
Hatta and Stephanov 2002

11. Mixed Moments T m Energy is also a conserved charge and mesurable !
E : total energy in a subvolume
Regions with Negative
fluctuation moments
Result with the standard
NJL parameters (Nf=2)

12. More and More 3rd Moments
These quantities are expressed concisely
as derivatives of “specific heat” at constant T m
“specific heat” at constant
diverges at critical end point
peaks on phase transition line

13. Comparison of Various Moments
2-flavor NJL
with standard
parameters
Far Side
G=5.5GeV-2
mq=5.5MeV
L=631MeV
Near Side
Different moments have different regions with negative moments
By comparing the signs of various moments,
possible to pin down the origin of moments
Negative m3(EEE) region extends to T-axis (in this particular model)
Sign of m3(EEE) may be used to estimate heat conductivity

14. Principles to Look for Other Observables
We are in need of observables that are not subject to final state interactions
After Freezeout, no effect of final state interactions
Chemical Freezeout
usually assumed
momentum independent
but this is not right
chemical freezeout time:
pT (or bT) dependent
Larger pT (or bT),
earlier ch. freezeout
Principle I

15. Emission Time Distribution
Larger bT, earlier emission
No CEP effect (UrQMD)

16. Principle II (T,mB) (r,h) Universality:
QCD CEP belongs to the same universality class as 3d Ising Model
(T,mB) (r,h)
QCD Critical End Point
3d Ising Model End Point
same universality class
h : external magnetic field
Further Assumptions
Size of Critical Region
No general universality
Lattice calculation: not yet V limit
Effective Model Results ?
need to be treated as an input, at the moment

17. Singular Part + Non-singular Part
Matching between Hadronic and QGP EOS
Entropy Density consists of Singular and Non-Singular Parts
Only Singular Part shows universal behavior
Requirement:
reproduce both the singular behavior and known asymptotic limits
Matched Entropy Density
Hadron Phase (excluded volume model)
QGP phase
crit
Bcrit
Dimensionless Quantity: Sc
D: related to extent of critical region

18. Isentropic Trajectories
In each volume element, Entropy (S) and Baryon Number (NB) are conserved,
as long as entropy production can be ignored (= when viscosities are small)
Isentropic Trajectories (nB/s = const.)
An Example
CEP
Near CEP s and nB change rapidly
isentropic trajectories show
non-trivial behavior
Bag Model EOS case

19. Consequence FO, CO QCP FO, CO QCP ratio
For a given chemical freezeout point,
prepare three isentropic trajectories:
w/ and w/o CEP
Along isentropic trajectory:
FO, CO
QCP
Principle I
As a function of pT(bT):
FO, CO
QCP
Bag Model EOS
(w/o CEP, usual hydro input)
ratio
: near CEP steeper

20. Evolution along Isentropic Trajectory
As a function of pT(bT):
FO, CO
QCP
with CEP
steeper spectra at high PT

21. Final State Interaction and Critical Slowing Down
Summary
Final State Interaction and Critical Slowing Down
Usual observables
such as large fluctuation, critical opalescence, ...etc. do not work
Conserved Charges and Higher Moments:
i) Third Fluctuation Moments of Conserved Charges
take negative values in regions on the FAR SIDE of Phase Transition
　　　(more information!)
ii) Different Moments have different regions with negative moments
By comparing different moments, possible to pin down the origin
of hot matter, get an idea about heat conductivity...etc.
Ratio : Two Principles
i) Chemical Freezeout is pT(bT) dependent
ii) Isentropic Trajectory behaves non-trivially near CEP (focusing)
ratio behaves non-monotonously near CEP
Information on the QCD critical point:
such as location, size of critical region, existence...