Towards Understanding
QCD Phase Diagram
Lattice and RHIC Experiments

Atsushi Nakamura
in Collaboration with K.Nagata
Latt...
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2...
Canonical Partition Functions
Experiments

Lattice QCD
Simulations

3 / 38
14年1月20日月曜日
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 finit...
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 : ...
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
...
RHIC
(Relativistic Heavy Ion Collider)

7
14年1月20日月曜日
Multiplicity Distribution of RHIC

Wao,
Multiplicity !

Interesting !

It is almost Zn

14年1月20日月曜日
We assume
the Fireballs created in High Energy
Nuclear Collisons are described as
a Statistical System.
with µ (chemical P...
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...
How can we extract Zn
n
from Pn = Zn ?
Observables in
Experiments

Experiment

unknown

We require (Particle-AntiParticle ...
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

...
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
...
Fitted
are very consistent with
those by Freeze-out Analysis.

12

Chemical Freeze-Out

x This work
Freeze-out

10

ξ

8
6...
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

1...
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 inclu...
Do not forget that your n is finite !

I need cooling down

14年1月20日月曜日
Moments

k

14年1月20日月曜日

⌘

✓

@
T
@µ

◆k

k

log Z
What happens ?

0.6

Number Susceptibility

p

if we increase
these points 15%
if we drop
these points
s = 27GeV

Number S...
Susceptivility
Number Susceptibility, sNN1/2=19.6
0.6

Number Susceptibility, sNN1/2=27
0.6

freeze-out point

0.55

0.55
...
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...
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
Lee-Yang Zeros

23
14年1月20日月曜日
Lee-Yang Zeros

(1952)

Zeros of Z( ) in Complex Fugacity Plane.

Z(↵k ) = 0
Great Idea to investigate
a Statistical Syste...
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

L...
’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)
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
Zero...
Is this my
Original ?

I donot
think so.
Let us wait
until someone
claims.

28
14年1月20日月曜日
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 - 186...
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....
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
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

...
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...
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]

...
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...
A Message to Experimentalists
In the Lee-Yang Diagram constructed
from your multiplicity,
Zeros
here

No Roberge-Weise
Tra...
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...
Effects of Nmax
Kim’s Model
In Confinement
Z(µq ) = I0 +

3
(⇠q

+ ⇠q

3

6
+(⇠q + ⇠q 6 )I2 + · · ·
)I1
Ik :Modified Bessel
...
Summary
Grand-Partition functions, Z(µ, T ) , provide us the QCD
phase information, which can be constructed from Zn .
Lat...
Backup Slide

39
14年1月20日月曜日
BES(Beem Energy Scan)

14年1月20日月曜日
Hunting the QCD Phase
Transition Regions
Find Rooms where No Criminal.
The Target is in other Room.

Not here ! Then, ..

...
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 suscepti...
Other Messages
Net proton multiplicity is Not a conserved
quantity.
Baryon multiplicity is perfect
Can you estimate Baryon...
You have a Big Chance to find
QCD phase Transition !

44 /40
14年1月20日月曜日
Canonical Partition Functions
is a Bridge
between Two Approaches
to Study QCD Phase
Lattice QCD
Simulaitions

Experiments
...
Lattice QCD
Canonical Approach
Miller and Redlich
Phys. Rev. D35 (1987) 2524

A.Hasenfratz and Toussaint
Nucl. Phys.B371 (...
Lattice: How to Calculate

Fugacity Expansion
Nagata and A. Nakamura,
Phys. Rev. D82 (2010) 094027
Alexandru and Wenger,
P...
Four Excuses why not Baryon
Multiplicities
1. This is a formulation. Let’s wait until
Experimentalists measure Baryon mult...
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

N...
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-6...
A Strange Fact
There are Lee-Yang Zeros on
the unit circle.
Theoretically, a bit annoying.
Phenomenologically, very
natura...
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
...
det

(m, µ) is REAL and Positive,

if µ is pure Imaginary
and m is sufficiently large.

Z(µN ) =

det (Nucleon)e

SG

>0

L...
Current lattice QCD simulations assumes
mu = m d
Z(µN ) =

2

(det (mq , µq ))

det (ms , µs ) · · · e

Z(µN ) can not tak...
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
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Kek2014 v2

  1. 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. 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. 3. Canonical Partition Functions Experiments Lattice QCD Simulations 3 / 38 14年1月20日月曜日
  4. 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. 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. 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. 7. RHIC (Relativistic Heavy Ion Collider) 7 14年1月20日月曜日
  8. 8. Multiplicity Distribution of RHIC Wao, Multiplicity ! Interesting ! It is almost Zn 14年1月20日月曜日
  9. 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. 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. 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. 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. 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. 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. 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. 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. 17. Do not forget that your n is finite ! I need cooling down 14年1月20日月曜日
  18. 18. Moments k 14年1月20日月曜日 ⌘ ✓ @ T @µ ◆k k log Z
  19. 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. 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. 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. 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. 23. Lee-Yang Zeros 23 14年1月20日月曜日
  24. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 40. Backup Slide 39 14年1月20日月曜日
  41. 41. BES(Beem Energy Scan) 14年1月20日月曜日
  42. 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. 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. 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. 45. You have a Big Chance to find QCD phase Transition ! 44 /40 14年1月20日月曜日
  46. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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

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