HHIQCD, YITP, Kyoto, March 4 2015

Canonical partition functions and Lee-Yang zeros in QCD
Keitaro Nagata(KEK)collaboration with
K. Kashiwa(YITP), A. Nakamura(Hiroshima),
S. M. Nishigaki(Shimane)
KN, S. Motoki, Y. Nakagawa, A. Nakamura, T. Saito [PTEP01A103(2012)]
A. Nakamura, KN [arXiv:1305.0760]
KN, K. Kashiwa, A. Nakamura, S. M. Nishigaki arXiv:1410.0783
HHIQCD, YITP, Kyoto, March 4, 2015
1

P(n)

n

A grand canonical partition function=> n-particle state with a probability P(n)

P(n)

n

A grand canonical partition function=> n-particle state with a probability P(n)What is the shape of this probability in QCD ?

Contents

1. Introduction : motivation
2. result 1
1. canonical partition function
2. Lee-Yang zeros
3. result 2 - High T QCD
4. summary

4Introduction

HHIQCD, YITP, Kyoto, March 4, 20155

Today, I would like to focus on
Beam energy scan
experiment : exp. data at finite
density
Developments of techniques
for finite density LQCD

6

BES experiments
Investigate phase diagram using HIC with different beam energy
Number of hadrons measured in heavy ion collision
success of thermal statistical models [e.g.Andronic, et., al, (2005)]
provide information at freeze out point (µB and T)

Nayak, Pramana, 79, 719(’12)
Cleymans et. al. PRC73, 034905(’06)
7BES experiments
Grand canonical ?
Due to the experimental setup, hadrons
can be observed for a limited region
seeing a part of fireballanalogous to grand canonical system

Agaarwal et.al. PRL105,
022302(’10), arXiv:1004.4959.

8

Canonical approacha method to study the probability distribution= probability for an n-particle state(references for canonical approach) !
Barbour, Davies, Sabeur, PLB215, 567(1988) 2^4, Barbour, Bell NPB372, 385(1992)., Barbour et. al.,arXiv:hep-lat/9705042 !
A. Hasenfratz, D. Toussaint, NPB371, 539(‘92) 2^4!
de Forcrand, Kratochvila NPB Proc. Suppl. 153, 62 (2006), Kratochvila, de Forcrand, 0509143, PoS
Lat2005. !
Ejiri, PRD78, 074507(2008) 16^3x4 !
Li, Meng, Alexandru, Liu, 0810.2349, PoS Lat(2008) , Li, 1002.4459, PoS, Lat(2009) , Li, Alexandru, Liu,
Meng 1005.4158, Phys.Rev. D82 (2010) 054502 , Li, Alexandru, Liu, PRD84, 071503, arXiv:
1103.3045!

9

Canonical approach
It can be applied to both theory and experiment.
probability to observe nparticle statecalculable in LQCD at µ=0

There may be an opportunity to compare theory with exp.
Practically, there are controversy
difficulty to measure neutron
non-equlibrium
10What we can learn from distribution
Shape of the distribution ~ signal for CEPHigher order moments of the distribution of conserved charges
are sensitive to the correlation length

S : skewness (asymmetry)kappa : kurtosis : sharpenessHatta, Stephanov, PRL91, 102003(2003),
Stephanov, PRL102, 032301(2009), Asakawa,
Ejiri, Kitazawa, PRL103, 262301 (2009) ,Stephanov PRL 107, 052301 (2011), etc

11What we can learn from the distribution
First order phase transition from Maxwell construction
Chemical potential ~ an energy to add one particle(Left) A.Li, PoS Lat09,(Right) de Forcrand &Kratochvila, NPB Proc.
Suppl. 153, 62 (2006),

12

What we can learn from the distribution
Canonical approach extends data at a given µ to wide range.
Accessible values of µ are limited both in theory and experiments
lattice QCD(Monte Carlo) is possible at µ=0
HIC data are obtained at chem. freeze-out{Zn} is given

Z(µ) for any µ

13What can we learn from the distribution
Canonical approach extends data at a given µ to wide range.

Application to experimental data of proton number
distribution [Nakamura, Nagata(2013)]
sqrt(s)=62.4

1

0.01

Zn
Z-n

Z-n

Zn

0.0001

1e-06

1e-08

1e-10
-20

-15

-10

-5

0

5

10

Net proton number

15

20

14

Extraction of Zn and µ/Tµ/T from the CP invariance vs thermal statistical15

n µ/TZ(np) e p1µ/T=0.0
0.9µ/T=1.0
0.8µ/T=2.0µ/T=2.5
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-15
-10

-5

, sqrtsNN=200

0
np

5

10

15

Z(np) enp µ/T, sqrtsNN=11.5
1µ/T=0.0
0.9µ/T=1.0
0.8µ/T=2.0µ/T=2.5
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-30
-20

-10

0
np

10

20

30
16

Studies of Zn
Higher order moments can be a signal
However, it is unclear if freeze out points in experiments hit
CEP
The data would contain information of finite density QCD even
if the CEP is achieved.
!

It is important to study the shape of the distribution theoretically.
find properties sensitive to the shapeclarify its physical meaning17

Lattice QCD simulations

HHIQCD, YITP, Kyoto, March 4, 201518

Lattice simulations
Canonical approach extends data at a given µ to wide range

How do we obtain Zn ? : reduction formula + reweighing

Reduction formula[Gibbs	  ('86).	
  Hasenfratz,	
  Toussaint('92).	
  Adams(’03,	  ‘04),	
  Borici('04).	
  KN&AN(‘10),	
  Alexandru	
  &Wenger(’10)]

We use a reweighing in µ

[Barbour,	
  et.	
  al.	  ('91).]19

Lattice simulations
gauge configurations are generated at µ = 0 and use
reweighingvolume : 8^3x4, 10^3x4
mass: mps/mv~0.8!

action : clover-improved Wilson fermion +renormalization
improved gauge# of statistics : 400 (20 trajectory-intervals, 3000 therm.)

20

Result - Zn
Lines :
Gaussian for T/Tpc = 1.35 and 1,
exp( - a|n| ) for T/Tpc = 0.93.

[KN, S. Motoki, Y. Nakagawa, A.
Nakamura, T. saito, PTEP(2012). ]volume : 8^3x4, 10^3x4, mass

: mps/mv~0.8

gauge configurations are generated at µ = 0 and use reweighing
action : clover-improved Wilson fermion +renormalization improved gauge# of statistics : 400 (20 trajectory-intervals, 3000 therm.)

21

Result - Zn exp(µ/T) at high T
If Zn is the Gaussian, then the baryon number distribution is
also Gaussian.

22

Result - Zn exp(µ/T) at low T
Increasing µ at low T, a non-trivial shape change has been
observed
ZN exp( NB µ/T)/Z
B

0.3
0.25
0.2
0.15

µB/T=0.6µB/T=1.2µB/T=1.8µB/T=2.1µB/T=2.7

T/Tc=0.93

T/Tc = 0.93 !
µB/T = 1.8 : right tail!
µB/T = 2.1 : flat!
µB/T = 2.7 : left tail

0.1
0.05
0
-20 -15 -10 -5

0
NB

5

10 15 20

23

Baryon number distribution & fluctuations
ZN exp( NB µ/T)/Z

m

B

0.3
0.25
0.2
0.15

µB/T=0.6µB/T=1.2µB/T=1.8µB/T=2.1µB/T=2.7h( N ) i =T/Tc=0.93

2

0.1

0
NB

5

10 15 20

1.5
1

g m2

Sm

0.5
0
-0.5
-1
-1.5

T/Tc=0.93
HRG
0

0.5

1

1.5
µB/T

2

2.5

3

3
2
1
0
-1
-2
-3
-4
-5

( N )m ZnB enB µB /T /ZGCnB

2= h( N ) iS=h( N )3 i=h( N )4 i

0.05
0-20 -15 -10 -5

X

3

3

4

T/Tc=0.93
HRG
0

0.5

1

1.5
µB/T

2

2.5

3

24

Lee-Yang zeros : from CPF to Phase transition
Lee-Yang zero Theorem :zeros of the partition function control the analyticity of the freeenergy [Lee & Yang 1952]

25

Result-Lee-Yang zeros[Nakamura, Nagata(2013)]

26

Result-Lee-Yang zeros[Nakamura, Nagata(2013)]

27

Result-Lee-Yang zeros[Nakamura, Nagata(2013)]

Lee-Yang zero distribution catches the difference
between Tc and TRW

28

Result - 2

HHIQCD, YITP, Kyoto, March 4, 201529

Questions and Subtleties in the calculation
Lee-Yang zeros are sensitive to Zn; We need careful analysis
There remains some questions
error bars in Zn : statistical stability
truncation of the polynomial : convergence

We focus on high T region, and perform
analytic calculation
reanalysis of lattice data
30

Analytic calculation of Zn of high T QCD
Properties of high T QCD leads to Gaussian Zn
Zn can be obtained from the F.T.

Use properties of high T QCD

Nagata, Nakamura, JHEP(2012)

also use the saddle point approximation

31

Analytical calculation of LY zeros of high T QCD
Gaussian Zn + RW periodicity leads to Lee-Yang zeros as

32

Analytical calculation of LY zeros of high T QCD
Gaussian Zn + RW periodicity leads to Lee-Yang zeros as

High T QCD

Theta function

33

Alternative calculation
Cancellation of two types of free energy allows Z=0
Cancellation of free energies [Biskup et al(’01)]

Approximate solution for c2>>c4

It is also possible to solve it in the presence of c4
fI and fIII, and fII and fIII.

Analytical calculation of LY zeros of high T QCD

angularradial35

Analytical calculation of LY zeros of high T QCD

angular

1

Jacobi Theta"jtzerofinite.dat"

radial

l=0

Im[j]

0.5

l=1
0

-0.5

l=2

-1
-1

-0.5

0
Re[j]

0.5

1

36

Analytical calculation of LY zeros of high T QCD

angular

1

Jacobi Theta"jtzerofinite.dat"

radial

l=0

l=0

T

Im[j]

0.5

l=1

D

2 /3

C

( / 3, TRW )

l=1

0

T pc

A

-0.5

l=2

-1
-1

-0.5

0
Re[j]

0.5

1

I

/T

37

Lee-Yang zeros : Ising vs free fermion gasa)Isingb) high T QCDc) high T QCD

Reanalysis of Lattice data & analytic result
We reanalyze previous lattice data.
errors of Zn : bootstrap analysis (1000 BS samples.)estimate the convergence
finite size scaling

1
1e-10

a

Ns=10
Ns=8

Zn

B

1e-20
1e-30
1e-40
1e-50
-40 -30 -20 -10

c
b
0
nB

10 20 30 40a) Ns=10 n0=37b) Ns=8 n0=32c) Ns=8 n0=19

Reanalysis of Lee-Yang zeros for high T QCD
Analytic and lattice calculation are consistent !
0.2
0.1

∠ξ

0 1 2

0

0

0.8

1

-0.1

c) Ns=8, n0=19b) Ns=8, n0=32a) Ns=10, n0=37
exp(-3(2l+1)/(4c2))

0.7
0.6
VT3

Ns=8, n0=32
Ns=8, n0=19
Ns=10, n0=37|Re[j]|

0.3

0.5
0.4
0.3
0.2
0.1

-0.2

0

-0.3
-1

-0.9 -0.8 -0.7 -0.6 -0.5

0

1

2

3

4

l

Solid Line :Lee-Yang zeros near the unit circle are statistically stableb)-c) : convergence oka)-c) : volume scaling consistent with the analytic one.
underestimation : saddle point approximation5

implication
1
1e-10

Theta
function

B

1e-20
Zn

c2-dominance
RW periodicity

Ns=10
Ns=8

1e-30
1e-40
1e-50
-40 -30 -20 -10

10nB

Ns=8, n0=19
Ns=10, n0=37

10 20 30 40

j

0.5
0
-0.5
-1

Completion of
deconfinement

Gaussian behavior

-1

-0.5

0

0.5

Roberge-Weissphase transition1

Summary
1
1
1e-10

Ns=10
Ns=8

j

0.5
0

B

1e-20
Zn

Ns=8, n0=19
Ns=10, n0=37

1e-30

-0.5

1e-40
1e-50
-40 -30 -20 -10

0
nB

-1

10 20 30 40

-1

-0.5

0

0.5

1

y
r
a
in

m
i
l
re

p

42

Summary
Canonical approachprovides a way to extend data obtained at a certain µ/T
to other values of µ/T.
can be applied not only to lattice QCD but also toexperimental data.
We showed that the net baryon number follows
Gaussian distribution at high temperatures
Gaussian shape is an indication of RW phasetransition(high T)
Lee-Yang zeros are sensitive to a shape of the canonicalpartition functions.

43

Applications to Beam energy scan
experiment

HHIQCD, YITP, Kyoto, March 4, 201544

Heavy ion collisions
arXiv:1201.4264

Number of hadrons
fixed at a time when inelastic process ceases
chemical freeze-out!

Number of hadrons provide information at the freezeout time45

Hadron yields and thermal statistical model

Andronic, et., al, (2005)46

Hadron yields and thermal statistical model

T
,
µ
)
s
~
r
y
e
t
g
r
e
e
m
n
a
e
r
a
n
p
o
i
t
s
i
u
l
l
o
co ezeerf(Andronic, et., al, (2005)

47

Energy dependence of freeze-out parameters

d
e
t
a
c
o
l
e
re
n
i
a
l
ts -out
n
i
o
e
p
z
t
e
u
e
r
o
f
e
~
z
e
e
e
n
i
r
l
F
a
on

Cleymans et. al.

48

Energy dependence of freeze-out parameters

Cleymans et. al.

49

Fluctuation
Aggarwal et al. STAR, !
PRL105, 022302(’10)

Probability distribution for some hadrons are obtained.
This is used to obtain event-by-event fluctuation at afreeze-out point.

50Extension of data to wide range of µ
The probability distribution of net baryon number(here we use the proton number as an approximation)µ/T is determined from CP invariance : Z(n)= Z(-n)
sqrt(s)=62.4

1

0.01

Zn
Z-n

Z-n

Zn

0.0001

1e-06

1e-08

1e-10
-20

-15

-10

-5

0

5

10

Net proton number

15

20

51

Extension of data to wide range of µµ/T obtained from CP invariance agree with thoseobtained from thermal statistical model for wide
range of collision energies.
52

n µ/TZ(np) e p1µ/T=0.0
0.9µ/T=1.0
0.8µ/T=2.0µ/T=2.5
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-15
-10

-5

, sqrtsNN=200

0
np

5

10

15

Z(np) enp µ/T, sqrtsNN=11.5
1µ/T=0.0
0.9µ/T=1.0
0.8µ/T=2.0µ/T=2.5
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-30
-20

-10

0
np

10

20

30
53

n µ/TZ(np) e p1µ/T=0.0
0.9µ/T=1.0
0.8µ/T=2.0µ/T=2.5
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-15
-10

-5

, sqrtsNN=200

0
np

5

10

15

Z(np) enp µ/T, sqrtsNN=11.5
1µ/T=0.0
0.9µ/T=1.0
0.8µ/T=2.0µ/T=2.5
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-30
-20

-10

0
np

10

20

30
54

RHIC DataµKurtosis as a function of
2
T
4

µ
T

Buck up slides

HHIQCD, YITP, Kyoto, March 4, 2015
56

How to achieve LY zeros ?
Calculation of Zn : truncation is inevitable57

Calculation of Lee-Yang zeros

58Lee-Yang零点を２通りの方法で計算します。
Zeros of the fugacity polynomial

Cancellation of the free energy

1. Cancellation of free-energy
Free energy is different in different RW phases
RW phases are distinguished by the argument of Pol.
The arg(Pol) is translated into A4.

This modifies the free energy as1. Cancellation of free-energy
Cancellation of two types of free energy allows Z=0
Cancellation of free energies [Biskup et al(’01)]

Approximate solution for c2>>c4

Possible to solve it with c4
fI and fIII, and fII and fIII.

2. Zeros of the fugacity polynomial
We show that the fugacity polynomial of high T QCD iswell approximated by a well-known polynomial.

We need Zn. : Fourier integral with the free energy as input(analytic)(We use the fugacity expansion in lattice simulation. )First, we derive Zn using the Fourier transformation.

Zn from the Fourier transformation

We use the quartic form of f(µ) as input.
f(µ) is obtained for real µ. On the other hand, the Fourier
integral requires complex µ.

Next, we use the RW periodicity

We decompose the integral into three domains.

Shift of θ leads to(This relation holds for any temperature, regardless of the RW
phase transition)

Now, we use the quartic expression of f(µ) and SPA

At T/Tc > 1.1~1.2, c2/c4 ~ 10. We use the saddle point
approximation.

4

Exact
SPA

We assume the Gaussian Zn
is valid for large n.
SPA is valid for small Re[µ]- (f (µ/T) - f(0))/T4

3.5
3
2.5
2
1.5
1
0.5
0
0

0.5
µ/T

1

If Zn is Gaussian, then Z(µ) is a Jacobi-theta function.

Theta function

Zeros of theta function

Method1

Lee-Yang zero distribution of theta function

1

Jacobi Theta"jtzerofinite.dat"

Zeros approaches to the RW
transition point as 1/V.
Spacing of zeros is a prediction

Im[j]

0.5

0

k

-0.5

-1
-1

-0.5

0
Re[j]

0.5

1

How to achieve LY zeros ?
Calculation of Zn : truncation is inevitable

Cauchy integral + recursive division +multi-precision arithmetic

68

Validity of SPA

4

Exact
SPA

- (f (µ/T) - f(0))/T4

3.5
3
2.5
2
1.5
1
0.5
0

0

0.5
µ/T

1

69

Lee-Yang zeros - 20 years

70

RW periodicity&phase transition
T

T pc

D

2 /3

C

( / 3, TRW )

A

I

0.1

0.2

µI=0.00µI=0.16µI=//12µI=0.28`=1.80Im[Pol]Im[Pol]

0.2

0
-0.1
-0.2
-0.2

0.1

/T

µI=0.00µI=0.16µI=//12µI=0.28`=1.95

0
-0.1

-0.1

0

0.1

Re[Pol]

0.2

0.3

-0.2
-0.2

-0.1

0

0.1

0.2

0.3

Re[Pol]71

Taylor coefficients of free energy72Lee-Yang zeros : from CPF to Phase transition
Lee-Yang zeros [Lee & Yang 1952]

Z(µ) = 0 is an origin of a thermodynamic non-analyticity

73

Convergence on the negative real axis
Infinite sum of higher order terms is bounded on
the negative real axis.

c.f. Leibnitz’s test for an alternating series

74