kek2014
TRANSCRIPT
Towards Understanding QCD Phase Diagram
Lattice and RHIC Experiments
Atsushi Nakamura in Collaboration with K.Nagata
Lattice QCD at finite temperature and density20 Jan. 2014 KEK
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14年1月19日日曜日
2
QCD Phase Diagram
From Wiki-Pedia “QCD matter”
Very Exciting !Now it’s time for
Lattice QCD.
But how do you reveal it?
Please No Sales-Talk !
14年1月19日日曜日
A few years ago,
Nagata and I were looking for a Reduction formula for Wilson fermions in a finite density QCD project:
4
det �̃ = det�
Reduction Matrix Original Fermion Matrix
Z(µ) =�
DU det �(µ)e�SG
Rank(det �̃) < Rank (det �)Nagata and Nakamura Phys. Rev. D82 094027 (arXiv:1009.2149)
14年1月19日日曜日
5
Obtained formula has the form of the fugacity expansion
� � eµ/T Fugacity
Z(µ) =�
DU�
n
cn�ne�SG
=�
n
Zn�n
Zn : Canonical Partition Function
14年1月19日日曜日
We assumethe Fireballs created in High Energy Nuclear Collisons are described as
a Statistical System.with (chemical Potential)and T (Temperature)
µ
Grand Canonicalpartition functionZ(µ, T )
All QCD Phase Information is in Z(µ, T )
14年1月19日日曜日
If
Tr e�(H�µN̂)/T
=�
n
�n|e�(H�µN̂)/T |n�
=�
n
�n|e�H/T |n� eµn/T
=�
n
Zn(T )�n
Z(µ, T ) =
�� � eµ/T
�
Fugacity
Z(µ, T )
9 /40
Zn(T )
14年1月19日日曜日
Partition Function is Sum of the Probabilities with n ...If I know , then I have Zn.�
Experiment
10 /40number
14年1月19日日曜日
How can we extract from ?Pn = Zn�n
� unknown
Z+n = Z�n
We require (Particle-AntiParticle Symmetry)
Experiment
Zn
Observables in Experiments
14年1月19日日曜日
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� = 1.0 � = 1.2
� = 1.4 � = 1.5
� = 1.534Final Value/40
Zn Zn
ZnZn
Z�n
Z�n
Z�n
Z�n
nn
n n
From Experiment
14年1月19日日曜日
1e-07
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
-20 -15 -10 -5 0 5 10 15 20
sqrt(s)=62.4PnP-n
PP n-n
n 1e-10
1e-08
1e-06
0.0001
0.01
1
-20 -15 -10 -5 0 5 10 15 20
Net proton number
ZnZ-n
nZZ-n
sqrt(s)=62.4
�Fit
Experiment
Z+n = Z�nDemand
14年1月19日日曜日
Fitted are very consistent with those by Freeze-out Analysis.
�
0
2
4
6
8
10
12
0 50 100 150 200
!
sNN1/2
Chemical Freeze-Out
x This work
J.Cleymans, H.Oeschler, K.Redlich and S.WheatonPhys. Rev. C73, 034905 (2006)
Freeze-out
14年1月19日日曜日
�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
15 /40
14年1月19日日曜日
Now we have Zn of RHIC data (sqrt(s)= 10.5,19.6, 27, 39, 62.4, 200 GeV)
µ/T
T
Wao ! We can calculate at any density !This includes all QCD Phase information !
(� � eµ/T )14年1月19日日曜日
What happens ?
if we increase these points 15%if we dropthese points
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1 1.1 1.2 1.3 1.4 1.5 1.6
freeze-out point
/T
42
=19.6 GeV
0.35
0.4
0.45
0.5
0.55
0.6
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5/T
Number Susceptibility
freeze-out point
=27 GeV
Number Susceptibilityps = 27GeV
Usually we consider
(only) here.
14年1月19日日曜日
0.35
0.4
0.45
0.5
0.55
0.6
1 1.1 1.2 1.3 1.4 1.5 1.6 1.7µ/T
Number Susceptibility, sNN1/2=19.6
freeze-out point
0.35
0.4
0.45
0.5
0.55
0.6
0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5µ/T
Number Susceptibility, sNN1/2=27
freeze-out point
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4µ/T
Number Susceptibility, sNN1/2=39
freeze-out point
0.38 0.4
0.42 0.44 0.46 0.48
0.5 0.52 0.54 0.56 0.58
0.6
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1µ/T
Number Susceptibility, sNN1/2=62.4
freeze-out point
0.4
0.45
0.5
0.55
0.6
0.65
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1µ/T
Number Susceptibility, sNN1/2=200
freeze-out point
Susceptivility
14年1月19日日曜日
-1
-0.5
0
0.5
1
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4µ/T
R42, sNN1/2=19.6
freeze-out point
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 1µ/T
R42, sNN1/2=27
freeze-out point
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 1µ/T
R42, sNN1/2=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 0.65µ/T
R42, sNN1/2=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 0.45µ/T
R42, sNN1/2=200
freeze-out point
Kurtosis
14年1月19日日曜日
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Multiplicity tells us
Not only Freeze-out points
Information of wider regions
Nmax →largeWider
14年1月19日日曜日
Lee-Yang ZerosZeros of in Complex Fugacity Plane.
Z(↵k) = 0
Great Idea to investigate a Statistical System
�xxxxxx
Phase Transition
Z(�)
25 /40
(1952)
14年1月19日日曜日
26
Lee-Yang Zeros
Non-trivial to obtain.But once they are got, it is easy to figure out the Free-energy
Lee-Yang zeros
F: Free-energy
: zeros�k
2-d Electro-Magnetic
F: Potential
:Point charge�k
Z(�, T ) = e�F/T
14年1月19日日曜日
cut Baum-Kuchen (cBK) Algorithm
1
1
Number ofZeros in
Contour C
A Coutour is cut into four pieces
if there are zeros inside.50 - 100 number
of significant digits
i ( )
and
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14年1月19日日曜日
Lee-Yang Zeros: RHIC Experiments
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Im[! B
]
Re[!B]
sNN1/2=19.6
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Im[! B
]
Re[!B]
sNN1/2=27
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Im[! B
]
Re[!B]
sNN1/2=39
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Im[! B
]
Re[!B]
sNN1/2=62.4
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Im[! B
]
Re[!B]
sNN1/2=200
Experiment
30/40
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
cos(t), cos(t),sin(t)’plotXY11.5’
14年1月19日日曜日
Lee-Yang Zeros Lattice QCD
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Im[!]
Re[!]
T/Tc=0.99
� = 1.85
T/Tc � 0.99
Z(�) =�
m
Z3m � �3m
Zn = 0 n �= 3munless
� = ei�
� =2�
3Periodicity
( )
32 /40
14年1月19日日曜日
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Im[!]
Re[!]
T/Tc=0.99
-1
-0.5
0
0.5
1
-1 -0.5 0 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)
� = 1.85
� = 1.87
� = 1.89
T/Tc � 0.99
T/Tc � 1.01
T/Tc � 1.04
33 /40
14年1月19日日曜日
Lee-Yang Zeros Lattice QCD
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Im[!]
Re[!]
T/Tc=1.20⇠ = eµ/T
The Unit Cirle in
Imaginary
⇠µ
Roberge-WeiseTransition !
T/Tc � 1.20
34 /40
(� � eµ/T )
14年1月19日日曜日
-1 -0.5 0 0.5 1
Im[ q
]
Re[ ]
-1
-0.5
0
0.5
1
q
q
10 43
8 43
T/Tc=1.08
-1 -0.5 0 0.5 1Re[ B]
B
10 43
�q �B = �q3
35 /40
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A Message to Experimentalists
In the Lee-Yang Diagram constructed from your multiplicity,
Zeroshere �q�B
No Roberge-Weise Transition T � TRW� 1.2Tc
Your Temperature
36 /40
14年1月19日日曜日
Lee-Yang Zeros: RHIC Experiments
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Im[! B
]
Re[!B]
sNN1/2=19.6
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Im[! B
]
Re[!B]
sNN1/2=27
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Im[! B
]
Re[!B]
sNN1/2=39
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Im[! B
]
Re[!B]
sNN1/2=62.4
-1
-0.5
0
0.5
1
-1 -0.5 0 0.5 1
Im[! B
]
Re[!B]
sNN1/2=200
37/40
14年1月19日日曜日
Effects of NmaxKim’s Model
38
Z(µq) = I0 + (⇠3q + ⇠�3q )I1 +(⇠6q + ⇠�6
q )I2 + · · ·Ik :Modified Bessel
In Confinement
Lesson from the Model
Nmax Large
Lee Yang ZerosLarge regions|µ|
It should be so!14年1月19日日曜日
Find Rooms where No Criminal.
The Target is in other Room.
Not here ! Then, ..
Hunting the QCD Phase Transition Regions
Lower Bound14年1月19日日曜日
0.15
0.16
0.17Te
mpe
ratu
re (G
eV)
0 0.1 0.2 0.3 0.4 0.5Chemical Potential (GeV)!
200
62.439
27
19.6
Freeze-out PointLower bound determined by susceptivility
Phase Transition Regions estimated byLee-Yang Zero distribution
Lower bound determined by negative Kurtosis
14年1月19日日曜日
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.
42
14年1月19日日曜日
Canonical Partition Functionsis a Bridge
between Two Approachesto Study QCD Phase
Lattice QCD Simulaitions
Experiments
44 /40
14年1月19日日曜日
Lattice QCD Canonical Approach
Miller and RedlichPhys. Rev. D35 (1987) 2524
A.Hasenfratz and ToussaintNucl. Phys.B371 (1992) 539
Barbour and BellNucl. Phys. B372 (1992) 385
Engels, Kaczmarek, Karsch and LaermannNucl.Phys. B558 (1999) 307
deForcrand and KratochvilaNucl. Phys. B (P.S.) 153 (2006) 62 (hep-lat/0602024)
A.Li, Meng, Alexandru, K-F. LiuPoS LAT2008:032 and 178
Phys.Rev. D82(2010) 054502, D84 (2011) 071503
Danzer and GattringerarXiv:1204-1020 Europian Journal
Lattice
45/40
14年1月19日日曜日
Lattice: How to Calculate
Fugacity ExpansionNagata and A. Nakamura, Phys. Rev. D82 (2010) 094027
Lattice
46 /40
Alexandru and Wenger, Phys.Rev.D83 (2011) 034502
14年1月19日日曜日
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.
47
14年1月19日日曜日
ZGC(µ) =�
Zn�n
�DU(
�an�n)(det�(0))Nf e�SG
=�
DU(�
an�n)(det�(0))Nf e�SG
=�
n
�n
�DUan(det�(0))Nf e�SG
Lattice
48
=
/40
14年1月19日日曜日
from lattice QCDZn
1e-160
1e-140
1e-120
1e-100
1e-80
1e-60
1e-40
1e-20
1
-60 -40 -20 0 20 40 60m=3n
·=Q�����RULJ·
1e-60
1e-50
1e-40
1e-30
1e-20
1e-10
1
-40 -30 -20 -10 0 10 20 30 40
m=3n
'Zn1950-orig'
1e-90
1e-80
1e-70
1e-60
1e-50
1e-40
1e-30
1e-20
1e-10
1
-50 -40 -30 -20 -10 0 10 20 30 40 50
m=3n
'Zn1850-orig'
1e-40
1e-35
1e-30
1e-25
1e-20
1e-15
1e-10
1e-05
1
-30 -20 -10 0 10 20 30
m=3n
'Zn2000-orig'
1e-40
1e-35
1e-30
1e-25
1e-20
1e-15
1e-10
1e-05
1
-30 -20 -10 0 10 20 30
m=3n
'Zn1900-orig'
1e-20
1e-19
1e-18
1e-17
1e-16
15 16 17 18 19 20m=3n
'Zn1850-orig'
Im(Zn) are used as an error
Lattice
49 /40
14年1月19日日曜日
A Strange Fact
There are Lee-Yang Zeros on the unit circle.
Theoretically, a bit annoying.
Phenomenologically, very natural
50 /40
14年1月19日日曜日
51
Z(µ) =� �
f
det �(mf , µf )e�SG
is REAL det �(m,µ)
if is pure Imaginary.µ
On the unit circle in complex plane.�
(� = eµ/T )
14年1月19日日曜日
52
det �(m,µ) is REAL and Positive,
if is pure Imaginary and m is sufficiently large.
µ
Z(µN ) =�
det �(Nucleon)e�SG > 0
Lee-Yang zeros on the unit circle tell usthat Nucleon is a composite.
14年1月19日日曜日
53
Current lattice QCD simulations assumes
mu = md
Z(µN ) =�
(det�(mq, µq))2 � det �(ms, µs) · · · e�SG
Z(µN ) can not take zero.
14年1月19日日曜日