presentation file version 2

1. Towards Understanding
QCD Phase Diagram
Lattice and RHIC Experiments
Atsushi Nakamura
in Collaboration with K.Nagata
Lattice QCD at finite temperature and density
20 Jan. 2014 KEK
1 /38
14年1月20日月曜日

2. QCD Phase Diagram
Very Exciting !
Now it’s time for
Lattice QCD.
But how do
you reveal it?
Please
No Sales-Talk !
14年1月20日月曜日
From Wiki-Pedia “QCD matter”
2

3. Canonical Partition Functions
Experiments
Lattice QCD
Simulations
3 / 38
14年1月20日月曜日

4. A few years ago,
Z(µ) =
DU det
(µ)e
SG
Nagata and I were looking for a Reduction formula
for Wilson fermions in a finite density QCD project:
det ˜ = det
Reduction Matrix
Original Fermion Matrix
Rank(det ˜ ) < Rank (det )
Nagata and Nakamura
Phys. Rev. D82 094027 (arXiv:1009.2149)
4
14年1月20日月曜日

5. Obtained formula has the form of the fugacity
expansion
Z(µ) =
DU
cn
n
e
SG
n
=
Zn
n
n
µ/T
e
Fugacity
Zn : Canonical Partition Function
5
14年1月20日月曜日

6. Z(µ, T )
Zn (T )
Z(µ, T ) = Tr e
=
If
n|e
ˆ
(H µN )/T
ˆ
(H µN )/T
n
=
n|e
H/T
n
=
Zn (T )
n
µn/T
|n e
µ/T
e
Fugacity
6 /38
14年1月20日月曜日
n
|n

7. RHIC
(Relativistic Heavy Ion Collider)
7
14年1月20日月曜日

8. Multiplicity Distribution of RHIC
Wao,
Multiplicity !
Interesting !
It is almost Zn
14年1月20日月曜日

9. We assume
the Fireballs created in High Energy
Nuclear Collisons are described as
a Statistical System.
with µ (chemical Potential)
and T (Temperature)
Z(µ, T )
Grand Canonical
partition function
All QCD Phase
Information is
in Z(µ, T )
14年1月20日月曜日

10. Experiment
Partition Function is
Sum of the Probabilities
with n ...
If I know , then I have Zn.
number
10 /38
14年1月20日月曜日

11. How can we extract Zn
n
from Pn = Zn ?
Observables in
Experiments
Experiment
unknown
We require (Particle-AntiParticle Symmetry)
Z+n = Z
14年1月20日月曜日
n

12. From
Experiment
= 1.0
Z
Zn
n
= 1.4
Z
= 1.2
= 1.5
Zn
Z
n
n
Zn
n
n
12 /38
14年1月20日月曜日
Zn
n
07:12-08:05
n
n
Z
Final Value
= 1.534

13. Experiment
Demand Z+n = Z n
sqrt(s)=62.4
1
Fit
0.1
0.01
0.001
P-n
sqrt(s)=62.4
1
Pn
P-n
Pn
0.01
Zn
Z-n
Z-n
Zn
0.0001
0.0001
1e-06
1e-05
1e-08
1e-06
1e-07
-20
14年1月20日月曜日
-15
-10
-5
n
0
5
10
15
20
1e-10
-20
-15
-10
-5
0
5
10
Net proton number
15
20

14. Fitted
are very consistent with
those by Freeze-out Analysis.
12
Chemical Freeze-Out
x This work
Freeze-out
10
ξ
8
6
4
2
0
µ/T
e
14年1月20日月曜日
0
50
100
1/2
sNN
150
200
J.Cleymans,
H.Oeschler,
K.Redlich and
S.Wheaton
Phys. Rev. C73,
034905 (2006)

15. Zn from RHIC data
s = 19.6GeV
s = 27GeV
s = 39GeV
1
0.01
1
'Zn_27'
'Zn_19.6'
0.0001
0.01
0.0001
1e-06
1e-06
1e-08
1e-08
1e-10
1e-10
1e-12
'Zn_39'
0.01
0.0001
1e-06
Experiment
1e-12
1e-08
1e-10
1e-12
1e-14
1e-16
1e-18
-25
-20
-15
-10
-5
0
5
10
15
20
25
1e-14
-25
-20
-15
-10
-5
0
5
10
15
25
1e-14
-25
-20
-15
-10
-5
0
5
s = 200GeV
s = 62.4GeV
0.1
20
1
'Zn_62.4'
'Zn_200'
0.01
0.1
0.001
0.01
0.0001
0.001
1e-05
1e-06
0.0001
1e-07
1e-05
1e-08
1e-06
1e-09
1e-10
-20
-15
-10
-5
0
5
10
15
20
15 /38
14年1月20日月曜日
1e-07
-15
-10
-5
0
5
10
15
10
15
20
25

16. Now we have Zn of RHIC data
(sqrt(s)= 10.5,19.6, 27, 39, 62.4, 200 GeV)
Wao ! We can calculate
at any density !
This includes all QCD Phase
information !
T
µ/T
14年1月20日月曜日
(
µ/T
e
)

17. Do not forget that your n is finite !
I need cooling down
14年1月20日月曜日

18. Moments
k
14年1月20日月曜日
⌘
✓
@
T
@µ
◆k
k
log Z

19. What happens ?
0.6
Number Susceptibility
p
if we increase
these points 15%
if we drop
these points
s = 27GeV
Number Susceptibility
Usually we
=27 GeV
consider
(only) here.
0.55
0.5
freeze-out point
1
4
0.8
2
0.6
0.4
0.45
=19.6 GeV
0.2
0.4
0.35
1.2
freeze-out point
0.7
14年1月20日月曜日
0.8
0.9
1
1.1
/T
1.2
1.3
0
1.4
1.5
-0.2
1
1.1
1.2
1.3
/T
1.4
1.5
1.6

20. Susceptivility
Number Susceptibility, sNN1/2=19.6
0.6
Number Susceptibility, sNN1/2=27
0.6
freeze-out point
0.55
0.55
0.5
0.5
0.45
0.45
0.4
0.35
freeze-out point
0.4
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7 0.35
µ/T
0.7
0.8
0.9
Number Susceptibility, sNN1/2=39
0.65
freeze-out point
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.5
0.6
0.7
0.8
0.9
1
1.1
µ/T 0.65
1
1.1 1.2 1.3 1.4 1.5
µ/T
Number Susceptibility, sNN1/2=62.4
0.6
freeze-out point
0.58
0.56
0.54
0.52
0.5
0.48
0.46
0.44
0.42
0.4
0.38
1/2
=200
1.2 Number Susceptibility, sNN 0.3 0.4 0.5 0.6 0.7
1.3 1.4
µ/T
freeze-out point
0.6
0.55
0.5
0.45
0.4
14年1月20日月曜日
0.1
0.2
0.3
0.4
0.5 0.6
µ/T
0.7
0.8
0.9
1
0.8
0.9
1
1.1

21. Kurtosis
R42, sNN1/2=19.6
R42, sNN1/2=27
1.5
freeze-out point
1
freeze-out point
1.4
1.3
0.5
1.2
1.1
0
1
Usually we
consider (only) here.
-0.5
-1
0.9
1
1.05
1.1
1.15
1.2
µ/T
1.25
1.3
0.8
0.7
1.35
1.4
0.6
R42, sNN1/2=39
2.2
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.86
0.88
0.9
0.92 0.94
µ/T
0.96
0.98
1
R42, sNN1/2=62.4
1.1
freeze-out point
R42, sNN1/2=200
1.2
freeze-out point
1
1
0.9
freeze-out point
0.8
0.8
0.6
0.7
0.4
0.6
0.2
0.5
0.6
0.7
0.8
µ/T
14年1月20日月曜日
0.9
1
0.5
0.35
0.4
0.45
0.5
µ/T
0.55
0.6
0.65
0.1
0.15
0.2
0.25
0.3
µ/T
0.35
0.4
0.45

22. Multiplicity tells us
Not only Freeze
-out points
Information of
wider regions
Nmax →large
Wider
14年1月20日月曜日
22

23. Lee-Yang Zeros
23
14年1月20日月曜日

24. Lee-Yang Zeros
(1952)
Zeros of Z( ) in Complex Fugacity Plane.
Z(↵k ) = 0
Great Idea to investigate
a Statistical System
x
x
x
x
x
x
24 /38
14年1月20日月曜日
Phase Transition

25. Lee-Yang Zeros
Non-trivial to obtain.
But once they are got, it is easy to figure
out the Free-energy
Z( , T ) = e
F/T
Lee-Yang zeros
2-d Electro-Magnetic
F: Free-energy
F: Potential
k :Point charge
k : zeros
25
14年1月20日月曜日

26. ’out’
20
15
10
5
0
-5
-10
1
-1
0.5
-0.5
0
0
0.5
F( ) =
1 -1
log(
k
26
14年1月20日月曜日
-0.5
k)

27. cut Baum-Kuchen (cBK) Algorithm
and
1
1
i
50 - 100 number
of significant digits
14年1月20日月曜日
27/38
(
Number of
Zeros in
Contour C
)
A Coutour is cut into
four pieces
if there are zeros inside.

28. Is this my
Original ?
I donot
think so.
Let us wait
until someone
claims.
28
14年1月20日月曜日

29. Is this my
Original ?
I donot
think so.
Let us wait
until someone
claims.
28
14年1月20日月曜日
It’s me !
Riemann (1826 - 1866)

30. Experiment
Lee-Yang Zeros: RHIC Experiments
cos(t), cos(t),sin(t)
’plotXY11.5’
1
1/2
=19.6
0
-1
-0.5
0
sNN
1
0.5
1/2
-0.5
-1
-1
-1
1
-1
=39
-0.5
0
Re[ξB]
sNN
1
0.5
1/2
1
-1
=62.4
0
sNN
1/2
0.5
1
=200
0
-0.5
-0.5
-0.5
-1
-1
-1
-1
14年1月20日月曜日
0
Re[ξB]
0.5
Im[ξB]
0
-0.5
1
0.5
Im[ξB]
0.5
=27
0
-0.5
-0.5
1/2
0.5
Im[ξB]
Im[ξB]
0
sNN
1
0.5
0.5
Im[ξB]
sNN
1
-0.5
0
Re[ξB]
0.5
1
-1
-0.5
29/40
0
Re[ξB]
0.5
1
-1
-0.5
0
Re[ξB]
0.5
1

31. sNN
1
1/2
=200
Im[ξB]
0.5
0
-0.5
-1
-1
14年1月20日月曜日
-0.5
0
Re[ξB]
30
0.5
1

32. Lee-Yang Zeros
Lattice QCD
Z( ) =
1
Z3m
m
T/Tc=0.99
Zn = 0
Im[ξ]
0.5
0
Periodicity
-0.5
-1
-1
-0.5
0
Re[ξ]
0.5
1
= 1.85
T /Tc
0.99
31 /38
14年1月20日月曜日
3m
unless
2
=
3
(
n = 3m
= ei
)

33. T/Tc=0.99
1
Im[ξ]
0.5
0
= 1.85
0.99
= 1.87
T /Tc
1.01
= 1.89
-0.5
T /Tc
T /Tc
1.04
-1
-1
-0.5
0
Re[ξ]
0.5
1
'./plotXY_B1870'
cos(t), sin(t)
1
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1
'./plotXY_B1890'
cos(t), sin(t)
1
0.5
0
-0.5
-1
-1
14年1月20日月曜日
-0.5
0
0.5
1
32 /38

34. Lee-Yang Zeros
Lattice QCD
1
T/Tc=1.20
⇠=e
Im[ξ]
0.5
⇠
Imaginary µ
The Unit Cirle in
0
-0.5
-1
-1
-0.5
0
Re[ξ]
T /Tc
0.5
Roberge-Weise
Transition !
1
1.20
(
33 /38
14年1月20日月曜日
µ/T
µ/T
e
)

35. q
1
B
=
T/T =1.08
c
B
q
q
3
Im[ q]
0.5
0
-0.5
3
-1
10 4
3
8 4
-1
14年1月20日月曜日
3
-0.5
0
Re[ q]
0.5
1
10 4
-1
34 /38
-0.5
0
Re[
B]
0.5
1

36. A Message to Experimentalists
In the Lee-Yang Diagram constructed
from your multiplicity,
Zeros
here
No Roberge-Weise
Transition
14年1月20日月曜日
q
B
Your Temperature
35 /38
T
TRW
1.2Tc

37. Lee-Yang Zeros: RHIC Experiments
sNN
1
1/2
=19.6
1/2
=27
0
=39
0
-0.5
-0.5
-0.5
-1
-1
-1
-1
-0.5
0
Re[ξB]
0.5
1
-1
sNN
1
1/2
-0.5
0
Re[ξB]
=62.4
1
-1
sNN
1/2
-0.5
=200
0.5
Im[ξB]
Im[ξB]
0.5
1
0.5
0
0
-0.5
-0.5
-1
-1
-1
14年1月20日月曜日
1/2
0.5
Im[ξB]
0
sNN
1
0.5
Im[ξB]
0.5
Im[ξB]
sNN
1
-0.5
0
Re[ξB]
0.5
1
36/38
-1
-0.5
0
Re[ξB]
0.5
1
0
Re[ξB]
0.5
1

38. Effects of Nmax
Kim’s Model
In Confinement
Z(µq ) = I0 +
3
(⇠q
+ ⇠q
3
6
+(⇠q + ⇠q 6 )I2 + · · ·
)I1
Ik :Modified Bessel
Lesson from the
Model
Nmax
Large
Lee Yang Zeros
Large |µ| regions
It should be so!
14年1月20日月曜日
37

39. Summary
Grand-Partition functions, Z(µ, T ) , provide us the QCD
phase information, which can be constructed from Zn .
Lattice QCD can calculate Zn
But we need much more works to obtain reliable
Experiments provide us the multiplicities
We can calculate Zn from them.
Present data are those of net-proton, which are not conserved quantities.
Either correction, or ask experimentalists to measure net-baryon
Charge multiplicity is a conserved quantity, and another probe.
Large Nmax are wanted, but even finite Nmax data give us the lower bound.
Lee-Yang zeros provide us a new tool of the QCD phase
study.
They are sensitive to the data, i.e., they teach us which regions are important.
38
14年1月20日月曜日

40. Backup Slide
39
14年1月20日月曜日

41. BES(Beem Energy Scan)
14年1月20日月曜日

42. Hunting the QCD Phase
Transition Regions
Find Rooms where No Criminal.
The Target is in other Room.
Not here ! Then, ..
14年1月20日月曜日
Lower Bound

43. 0.17
Temperature (GeV)
200
62.4
39
27
0.16
0.150
14年1月20日月曜日
19.6
Freeze-out Point
Lower bound determined by susceptivility
Lower bound determined by negative Kurtosis
Phase Transition Regions estimated by
Lee-Yang Zero distribution
0.1
0.2
0.3
Chemical Potential
0.4
(GeV)
0.5

44. Other Messages
Net proton multiplicity is Not a conserved
quantity.
Baryon multiplicity is perfect
Can you estimate Baryon multiplicity from that of
Proton ?
Another conserved quantity is the Charge
multiplicity. It should work as well.
43
14年1月20日月曜日

45. You have a Big Chance to find
QCD phase Transition !
44 /40
14年1月20日月曜日

46. Canonical Partition Functions
is a Bridge
between Two Approaches
to Study QCD Phase
Lattice QCD
Simulaitions
Experiments
45 /40
14年1月20日月曜日

47. Lattice QCD
Canonical Approach
Miller and Redlich
Phys. Rev. D35 (1987) 2524
A.Hasenfratz and Toussaint
Nucl. Phys.B371 (1992) 539
Barbour and Bell
Nucl. Phys. B372 (1992) 385
Engels, Kaczmarek, Karsch and Laermann
Nucl.Phys. B558 (1999) 307
deForcrand and Kratochvila
Nucl. Phys. B (P.S.) 153 (2006) 62 (hep-lat/0602024)
A.Li, Meng, Alexandru, K-F. Liu
PoS LAT2008:032 and 178
Phys.Rev. D82(2010) 054502, D84 (2011) 071503
Danzer and Gattringer
arXiv:1204-1020 Europian Journal
46 /40
14年1月20日月曜日
Lattice

48. Lattice: How to Calculate
Fugacity Expansion
Nagata and A. Nakamura,
Phys. Rev. D82 (2010) 094027
Alexandru and Wenger,
Phys.Rev.D83 (2011) 034502
47 /40
14年1月20日月曜日
Lattice

49. Four Excuses why not Baryon
Multiplicities
1. This is a formulation. Let’s wait until
Experimentalists measure Baryon multiplicities
2. After the Freeze-out, the proton number is
essentially constant.
3. Expect the proton multiplicity is similar to the
baryon multiplicity
4. By some event generators or models, let us
calculate the proton and baryon multiplicity.
From that data, we can estimate the baryon
multiplicity.
48
14年1月20日月曜日

50. Lattice
=
DU (
an
=
DU (
=
n
n
)(det (0))
an
Zn
)(det (0))
DU an (det (0))
n
49 /40
14年1月20日月曜日
e
SG
Nf
Nf
n
ZGC (µ) =
n
Nf
e
e
SG
SG

51. Lattice
Zn from lattice QCD
1e-16
'Zn1850-orig'
1e-17
'Zn1850-orig'
1
1
1e-10
1e-20
1e-18
1e-20
1e-40
1e-30
1e-60
1e-40
1e-19
1e-80
1e-50
1e-100
1e-60
1e-120
1e-20
1e-70
15
16
17
18
19
20
m=3n
1e-140
1e-80
1e-160
-60
-40
-20
0
m=3n
20
40
60
1e-90
-50
-40
-30
-20
-10
0
10
20
30
40
Im(Zn ) are
used
as an error
50
m=3n
'Zn1900-orig'
'Zn1950-orig'
1
1
1e-05
'Zn2000-orig'
1
1e-05
1e-10
1e-10
1e-10
1e-20
1e-15
1e-15
1e-30
1e-20
1e-25
1e-20
1e-25
1e-40
1e-30
1e-30
1e-50
1e-35
1e-40
-30
1e-35
-20
-10
0
m=3n
10
20
30
1e-60
-40
-30
-20
-10
0
10
m=3n
50 /40
14年1月20日月曜日
20
30
40
1e-40
-30
-20
-10
0
m=3n
10
20
30

52. A Strange Fact
There are Lee-Yang Zeros on
the unit circle.
Theoretically, a bit annoying.
Phenomenologically, very
natural
51 /40
14年1月20日月曜日

53. Z(µ) =
det
(mf , µf )e
SG
f
det
(m, µ) is REAL
if µ is pure Imaginary.
On the unit circle in complex
plane.
( =e
µ/T
52
14年1月20日月曜日
)

54. det
(m, µ) is REAL and Positive,
if µ is pure Imaginary
and m is sufficiently large.
Z(µN ) =
det (Nucleon)e
SG
>0
Lee-Yang zeros on the unit circle tell us
that Nucleon is a composite.
53
14年1月20日月曜日

55. Current lattice QCD simulations assumes
mu = m d
Z(µN ) =
2
(det (mq , µq ))
det (ms , µs ) · · · e
Z(µN ) can not take zero.
54
14年1月20日月曜日
SG

56. mu > md
µp = 2µu + µd
Pure imaginary µp does not mean
µu and µd
are pure imaginary.
55
14年1月20日月曜日