This document discusses using lattice QCD and experimental data to search for the QCD critical point. It proposes a 4-step method: 1) Assume the fireball created in heavy ion collisions is described by temperature and chemical potential. 2) Expand the grand partition function in a fugacity expansion involving the canonical partition functions Zn. 3) Determine Zn from lattice QCD calculations and experimental measurements of particle ratios. 4) With Zn known as a function of temperature, calculate observables at any chemical potential to search for signatures of the critical point.

1. How to use
Lattice and Experimental data
for QCD Critical Point Search
CPOD 2016,Wrocław, Poland
May 30th - June 4th, 2016
1/31
V.Bornyakov, D.Boyda, V.Goy, A.Molochkov,
A.Nakamura, A.Nikolaev, V.Zakharov
R.Fukuda, S.Oka, A.Suzuki,
Y.Taniguchi
K.Nagata

2. Volume 149B,number4,5 PHYSICSLETTERS 20 December1984
BEHAVIOR OF QUARKS AND GLUONS AT FINITE TEMPERATUREAND DENSITY IN SU(2) QCD
Atsushi NAKAMURA 1
INFN, Laboratori Nazionali di Frascati, CP 13, 00044 Frascati, Rome, Italy
Received9 August 1984
Wehaverun a computer simulationin SU(2)lattice gaugetheory on a 83 × 2 lattice includingdynamicalquark loops.
No rapidvariationis observedin the valueof the Polyakovline, whilethe energydensitiesof quark and gluonshowa strong
indication of a secondorderphasetransition around T ~ 250 MeV.In order to reducefinite sizeeffects,the resultsare
comparedwith those of a free gas on a lattice of the samesize.The quark and gluonenergydensitiesovershootthe freegas
valuesat hightemperature. The effect of the chemicalpotential is alsostudied. The behaviorsof the energydensitiesand
of the number density are fax fromthe free gasease.
It has been conjectured that systems of quarks and
gluons at high temperature and density show a com-
pletely different behavior from those at zero temper-
ature and normal density [1-3]. Above some temper-
ature and/or chemical potential, quarks and gluons
are expected to be liberated in a deconfined quark-
gluon plasma.
Monte Carlo (MC) studies of SU(2) Yang-Mills
theory in the absence of dynamical quarks by
McLerran and Svetitsky [4] and by Kuti, Polonyi and
Szlachanyi [5] have given the first numerical evi-
dence for a second order transition from a conf'med
phase to a deconf'med one. Groups at the University
of Bielefeld and at the University of illinois have
performed MC simulations of the gluon matter at
finite temperature in detail; for SU(3) Yang-Mills
theory, they have observed a first order phase tran-
sition and ideal gas behavior ofgluons at high tem-
perature *a
Such studies of QCD in unusual environments are
done not only for a theorist's fun and amusement.
We hope that in high energy heavy ion collisions high
temperature and density matter might be produced in
a controlled experimental environment. To under-
ments, we may develop and study models of the
quark-gluon system. MC simulation of lattice QCD
probably provides the most fundamental informa-
tion for such an analysis. For the study of hadronic
matter, it is important to include quark loops in the
calculation since they play a crucial role in screening.
The phase transition observed in the pure gauge cal-
culation might be washed out by them [7,8]. In the
presence of quark fields, the Polyakov line is no more
a good order parameter for the confined and decon-
freed phases, mathematically because the presence of
quark fields breaks the symmetry under the center of
the gauge group, or physically because isolated heavy
quarks can survive due to the quark pair creation.
We will report here a MC study of the quark gluon
system with dynamical quarks. We simulate the finite
temperature and baryon number density plasma on
an N t X N 3 lattice. The temperature of the system is
given by T = 1/Nta t, where at(g) is the lattice distance
in the fourth direction. The action is composed of the
kinetic term of gauge variables and the fermion part:
S = S C + S v, Sr: = ~ A qJ .
We employ the Wilson form for the action [9]. The
Volume 149B, number 4,5 PHYSICS LETTERS 20
tential. The gluon energy density shown in fig. 4a in-
creases quickly when we increase the chemical po-
tential, i.e., the gluons are not independent of quark
matter density and exhibit behavior far from that of a
"free gas". However it falls suddenly at large chemical
potential. The quark energy density in fig. 4b increases
like a "free gas" but the value is much higher. At
these chemical potential regions, the free quark
gluon picture is not correct. There might be other
degrees of freedom. The number density, n, shown in
fig. 5 also overshoots the "free gas" values at large
chemical potential in a similar manner to the quark
energy density. To obtain the system with large
chemical potential, a higher density is required than
that estimated from the ideal gas equation.
This calculation was done at CERN and Frascati.
I am grateful to the theory divisions there for their
hospitality and to N. Oshima for his advise in the
numerical computation. I am indebted to the parti-
cipants and organizers of the Warsaw symposium and
of Zacopane summer school, 1984, for constructive
criticisfn, especially A. Bialas and L.D. McLerran for
valuable discussions and careful reading of the manu-
script.
R e.ferences
[2] J. Collins and M. Perry, Phys. Rev. Lett
[3] M.B.Kisslinger and P.D. Morley, Phys.
2765.
[4] L.D. McLerran and B. Svetitsky, Phys. L
195;Phys. Rev. D24 (1981) 450.
[5] J. Kuti, J. Polonyi and K. Szlachanyi,P
(1981) 199.
[6] H. Satz, Phys. Rep. 88 (1982) 349;
J.B. Kogut, Illinoispreprint ILL-(TH)-8
I. Montvay, DESY preprint 83-001.
[7] T. Banks and A. Ukawa, Nuel. Phys. B2
(1983) 145.
[8] P. Hasenfratz, F. Karsch and I.O. Stama
Lett. 133B (1983) 221.
[9] K. Wilson, in: New phenomena in subnu
(Eriee), ed. A. Zichichi (Plenum, New Y
[10] P. Hasenfratz and F. Karsch, Phys. Lett
308.
[11] J. Engels, F. Karsch and H. Satz, Phys. L
(1982) 398.
[12] J. Engels and F. Karsch, Phys. Lett. 125
[13] V. Azcoiti and A. Nakamura, Phys. Rev
255.
[14] V. Azcoiti, A. Cruz and A. Nakamura, F
LNF-84/25(P).
[15] F. Karsch, Nuel. Phys. B205[FS5] (198
[16] J. Engels, F. Karsch and H. Satz, Nuel. P
[FS5] (1982) 239.
[17] J.D. Stack, Phys. Rev. D27 (1983) 412.
[18] A. Martin, Phys. Lett. 100B (1981) 511
[19] T. Celik, J. Engels and H. Satz, Phys. Le
(1983) 427.
I am indebted to the participants
and organizers of Warsaw sym-
posimu and of Zacopane summer
school, 1984, for constructive
criticism, especially, A. Bialas and
L.D.McLerran for valuable discus-
sions and careful reading of the
manuscript.
2

3. Message of Talk
To determine the Confinement/Deconfinement
transition line is very hard.
But a non-standard method, Canonical Approach,
may make it possible.
T
µ3

4. Why difficult ?
Experimentally, measurements are done within
the confinement region, i.e., we measure hadrons.
Theoretically, the first-principle calculation, lattice
QCD suffers from
Sign problem.
4

5. The Message consists
of 4 Steps
5 /31

6. Step 1
Let us assume a Fireball created in Heavy Ion
Collisions is described by (Chemical Potential and
(Temperature).
µ T
T
µ
P.Braun-Munziger, K.Redlich
and J. Stachel
Quark Gluon Plasma 3
Chap.10
6

7. Z(µ, T)
T
µ
This means the system is described by
the Grand Partition function
7

8. Step 2
is expanded as Fugacity ExpansionZ(µ, T)
Z(µ, T) =
X
n
Zn(T)(eµ/T
)n
Canonical Partition
Function
Fugacity
8

9. Step 3
Z(µ, T) =
X
n
Zn(T)(eµ/T
)n
can be determined by
Lattice QCD and
Experiments
9

10. Z(µ, T) =
X
n
Zn(T)(eµ/T
)n
Step 4
After you get , you can see information
at any µ/T T
µ
does not
depend on !µ
Zn
10 /31

11. Let us see details of
these steps.
11

12. Step 1
Temperature and Chemical
Potential at each
Cleymans, Oeschler, Redlich and
Wheaton
Phys. Rev. C73, (2006) 034905.
p
sNN
µB
p
sNN GeV
T
12

13. Step2
Fugacity Expansion
Z(µ, T) =
X
n
Zn(T)(eµ/T
)n
Tr e (H µ ˆN)/T(Left Hand Side)=
If
=
n
n|e (H µ ˆN)/T
|n
=
n
n|e H/T
|n eµn/T
Zn(T)
13

14. Alternative Proof of
Fugacity Expansion
Z(µ, T) =
Z
DU(det )Nf
e SG
Diagonalize Q
Z(µ, T) =
+2NcNf N3
sX
2NcNf N3
s
Zn⇠n
14

15. Step 3
How to determine
I. Experimentally
STAR@RHIC
Z(µ, T) =
X
n
Zn(eµ/T
)n
P20 =
Z20(eµ/T
)20
Z15

16. Pn =
Zn⇠n
Z
P n =
Z n⇠ n
Z
Zn = Z n
(CP-invariance, or particle anti-particle symmetry)
Experimantal Data
Pn/P n = ⇠2n
Now is determined.⇠
⇠ ⌘ eµ/T
Zn
Z
= Pn/⇠n
16 /31

17. Fitted are consistent with those
by Freeze-out Analysis ?
x This work
J.Cleymans,
H.Oeschler,
K.Redlich and
S.Wheaton
Phys. Rev. C73,
034905 (2006)
Freeze-out
0
2
4
6
8
10
12
0 50 100 150 200
ξ
sNN
1/2
Chemical Freeze-Out
⇠
p
s GeV
17 /31
⇠ = eµ/T

18. s = 19.6GeV s = 27GeV s = 39GeV
s = 62.4GeV s = 200GeV
from RHIC dataZn
1e-18
1e-16
1e-14
1e-12
1e-10
1e-08
1e-06
0.0001
0.01
-25 -20 -15 -10 -5 0 5 10 15 20 25
'Zn_19.6'
1e-14
1e-12
1e-10
1e-08
1e-06
0.0001
0.01
1
-25 -20 -15 -10 -5 0 5 10 15 20 25
'Zn_27'
1e-14
1e-12
1e-10
1e-08
1e-06
0.0001
0.01
1
-25 -20 -15 -10 -5 0 5 10 15 20 25
'Zn_39'
1e-10
1e-09
1e-08
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
-20 -15 -10 -5 0 5 10 15 20
'Zn_62.4'
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
-15 -10 -5 0 5 10 15
'Zn_200'
Experiment
Can I see
Difference?
Yes,You Can !
We will see it.
18/31

19. Step 3
How to determine
II. Lattice QCD
(1) Glasgow method
Z(µ, T) =
Z
DU(det (µ))Nf
e SG
=
X
Zn(eµ/T
)n
19

20. II. Lattice QCD
(2) Hasenfratz-Toussant
A.Hasenfratz and Toussant, 1992
µIf is pure imaginary, real.det
Zn =
Z
d✓
2⇡
ei✓n
Z(✓ ⌘
Imµ
T
, T)
It looks great, but it did not work.
Numerically unstable in Fourier
Transformation
20 /31

21. 21
Big Cancellation in Fourier Transformation !
✓integration Multi-Precision (50 - 100)

22. V. Bornyakov, D. Boyda, M. Chernodub,V. Goy, A. Molochkov,
A. Nikolaev and V. I. Zakharov
Now in FEFU,Vladivostok,
Zn are being produced
at many imaginary µ
22

23. Step 4
What kind of Physics from Zn ?
Z(µ, T) =
X
n
Zn(T)(eµ/T
)n
T
µ
Experimental Point
Determine here.
Then see QCD Phase
at higher density !
Zn(T)
23

24. Moments k
k ⌘
✓
T
@
@µ
◆k
log Z
We determine Zn
at some T and µ
µ/T
T
We predict
at any /T for
fixed T.
µ
k
24/31

25. Lattice
They look
similar.
Can I see
Difference?
Different
above and
below Tc
25

26. 26
µ
µ
T < Tc ( = 0.9, 1.1)
µ
µ
µ
µ
( = 1.3, 1.5)
( = 1.7, 1.9)T > Tc
T  Tc
µ/T
T
Pessure
Tc
P(µ/T) P(0)
T4
Zn Collaboration
(Taniguchi, Oka, AN)
/31

27. µ
µ
µ
Number Density
µ/T
T
Tc
27/31
T/Tc = 3.62 T/Tc = 1.77
T/Tc = 0.83 T/Tc = 0.72 T/Tc = 0.65

28. 28
µ
T > Tc
µ
hNqi(2)
c/(VT3
)
T < Tc
Second Cummulant
/31

29. Then how RHIC data look like?
i.e.,We construct from RHIC data
and calculate the Moments using
Z(µ, T) =
X
n
Zn(T)(eµ/T
)n
Zn
at arbitrary values of µ/T T
µ
We construct Zn
and calculate moments
on
on .
29

30. 0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
2.2
0.5 0.6 0.7 0.8 0.9 1
µ/T
R42, sNN
1/2
=39
freeze-out point
0.5
0.6
0.7
0.8
0.9
1
1.1
0.35 0.4 0.45 0.5 0.55 0.6 0.65
µ/T
R42, sNN
1/2
=62.4
freeze-out point
0.2
0.4
0.6
0.8
1
1.2
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
µ/T
R42, sNN
1/2
=200
freeze-out point
Kurtosis
p
s = 62.4
p
s = 39 p
s = 200
RHIC Data 4
2
as a function of
µ
T
µ/Tµ/Tµ/T
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
0.86 0.88 0.9 0.92 0.94 0.96 0.98 1
µ/T
R42, sNN
1/2
=27
freeze-out point
p
s = 27
µ/T
-1
-0.5
0
0.5
1
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4
µ/T
R42, sNN
1/2
=19.6
freeze-out point
p
s = 19.6
µ/T
p
s = 11.5
30/31

31. Summary
I introduced recent activity for Critical Point Study
at Far East (Vladivostok and Japan).
Now Zn are evaluated from data at many
imaginary chemical potential values.
Baryon number distribution is hard to measure in
experiment. Proton number gives us a lot of hints
which suggest very interesting goal.
We are preparing Net Charge and Strangeness in
lattice QCD canonical approach.
31