How to use Lattice and Experimental data for QCD Critical Point Search

CPOD 2016, Wrocław, Poland May 30th - June 4th, 2016

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V.Bornyakov, D.Boyda, V.Goy, A.Molochkov, A.Nakamura, A.Nikolaev, V.Zakharov

R.Fukuda, S.Oka, A.Suzuki, Y.Taniguchi

K.Nagata

Volume 149B, number 4,5 PHYSICS LETTERS 20 December 1984

BEHAVIOR OF QUARKS AND GLUONS AT FINITE TEMPERATURE AND DENSITY IN SU(2) QCD

Atsushi NAKAMURA 1 INFN, Laboratori Nazionali di Frascati, CP 13, 00044 Frasca ti, Rome, Italy

Received 9 August 1984

We have run a computer simulation in SU(2) lattice gauge theory on a 83 × 2 lattice including dynamical quark loops. No rapid variation is observed in the value of the Polyakov line, while the energy densities of quark and gluon show a strong indication of a second order phase transition around T ~ 250 MeV. In order to reduce finite size effects, the results are compared with those of a free gas on a lattice of the same size. The quark and gluon energy densities overshoot the free gas values at high temperature. The effect of the chemical potential is also studied. The behaviors of the energy densities and of the number density are fax from the free gas ease.

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- stand the data which might arise from such experi-

! Fujukai Foundation fellow. ,1 See ref. [6] and references therein.

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 matrix A has the form

391

Volume 149B, number 4,5 PHYSICS LETTERS 20 December 1984

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 o f 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 l ike 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 hospitali ty and to N. Oshima for his advise in the numerical computat ion. 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 o f the manu- script.

R e.ferences

[1] N. Cabibbo and G. Parisi, Phys. Lett. 59B (1975) 67.

[2] J. Collins and M. Perry, Phys. Rev. Lett. 34 (1975) 135. [3] M.B. Kisslinger and P.D. Morley, Phys. Rev. D13 (1976)

2765. [4] L.D. McLerran and B. Svetitsky, Phys. Lett. 98B (1981)

195;Phys. Rev. D24 (1981) 450. [5] J. Kuti, J. Polonyi and K. Szlachanyi, Phys. Lett. 98B

(1981) 199. [6] H. Satz, Phys. Rep. 88 (1982) 349;

J.B. Kogut, Illinois preprint ILL-(TH)-83-45; I. Montvay, DESY preprint 83-001.

[7] T. Banks and A. Ukawa, Nuel. Phys. B225 [FS9] (1983) 145.

[8] P. Hasenfratz, F. Karsch and I.O. Stamateseu, Phys. Lett. 133B (1983) 221.

[9] K. Wilson, in: New phenomena in subnuelear physics (Eriee), ed. A. Zichichi (Plenum, New York, 1977).

[10] P. Hasenfratz and F. Karsch, Phys. Lett. 125B (1983) 308.

[11] J. Engels, F. Karsch and H. Satz, Phys. Lett. 133B (1982) 398.

[12] J. Engels and F. Karsch, Phys. Lett. 125B (1983) 481. [13] V. Azcoiti and A. Nakamura, Phys. Rev. D27 (1983)

255. [14] V. Azcoiti, A. Cruz and A. Nakamura, Frascati preprint

LNF-84/25(P). [15] F. Karsch, Nuel. Phys. B205[FS5] (1982) 285. [ 16] J. Engels, F. Karsch and H. Satz, Nuel. Phys. B205

[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. Lett. 133B

(1983) 427. [20] F. Fucito and S. Solomon, Phys. Lett. 140B (1984)

387. [21] P.V. Gavai, M. Lev and B. Petersson, Phys. Lett. 140B

(1984) 397.

395

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.

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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

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

The Message consists of 4 Steps

5 /31

Step 1Let 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

Z(µ, T )

T

µ

This means the system is described by the Grand Partition function

7

Step 2 is expanded as Fugacity ExpansionZ(µ, T )

Z(µ, T ) =X

n

Zn(T )(eµ/T )n

Canonical Partition Function

Fugacity

8

Step 3Z(µ, T ) =

X

n

Zn(T )(eµ/T )n

can be determined by Lattice QCD and Experiments

9

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 notdepend on !µZn

10 /31

Let us see details of these steps.

11

Step 1 Temperature and Chemical

Potential at each

Cleymans, Oeschler, Redlich and Wheaton Phys. Rev. C73, (2006) 034905.

psNN

µB

psNN GeV

T

12

Step2 Fugacity Expansion

Z(µ, T ) =X

n

Zn(T )(eµ/T )n

Tr e�(H�µN̂)/T(Left Hand Side)=

If=

�

n

�n|e�(H�µN̂)/T |n�

=�

n

�n|e�H/T |n� eµn/T

Zn(T )13

Alternative Proof of Fugacity Expansion

Z(µ, T ) =

ZDU(det�)Nf e�SG

Diagonalize Q

Z(µ, T ) =

+2NcNfN3sX

�2NcNfN3s

Zn⇠n

14

Step 3 How to determine

I. Experimentally

STAR@RHIC

Z(µ, T ) =X

n

Zn(eµ/T )n

P20 =Z20(eµ/T )20

Z15

Pn =Zn⇠n

ZP�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

Fitted are consistent with those by Freeze-out Analysis ?

x This work

J.Cleymans, H.Oeschler, K.Redlich and S.WheatonPhys. Rev. C73, 034905 (2006)

Freeze-out

0

2

4

6

8

10

12

0 50 100 150 200

ξ

sNN1/2

Chemical Freeze-Out

⇠

ps GeV

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⇠ = eµ/T

�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.

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Step 3 How to determine

II. Lattice QCD

(1) Glasgow method

Z(µ, T ) =

ZDU(det�(µ))Nf e�SG

=X

Zn(eµ/T )n

19

II. Lattice QCD

(2) Hasenfratz-Toussant

A.Hasenfratz and Toussant, 1992

µIf is pure imaginary, real.det�

Zn =

Zd✓

2⇡ei✓nZ(✓ ⌘ Imµ

T, T )

It looks great, but it did not work. Numerically unstable in Fourier Transformation

20 /31

21

Big Cancellation in Fourier Transformation !

✓integration Multi-Precision (50 - 100)

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

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

Moments �k

�k ⌘✓T

@

@µ

◆k

logZ

We determine Zn at some T and µ

µ/T

T

We predictat any /T for fixed T.

µ�k

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Lattice

They look similar.

Can I see Difference?

Different above and below Tc

25

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

PessureTc

P (µ/T )� P (0)

T 4

Zn Collaboration (Taniguchi, Oka, AN)

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µ

µ

µ

Number Density

µ/T

T

Tc

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T/Tc = 3.62 T/Tc = 1.77

T/Tc = 0.83 T/Tc = 0.72 T/Tc = 0.65

28

µ

T > Tc

µ

hNqi(

2) c/(VT

3)

T < Tc

Second Cummulant

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Then how RHIC data look like?

i.e., We construct from RHIC dataand 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 Znand calculate moments

on

on .29

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

ps = 62.4

ps = 39 p

s = 200

RHIC Data �4

�2as 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 1µ/T

R42, sNN1/2=27

freeze-out point

ps = 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 1.4µ/T

R42, sNN1/2=19.6

freeze-out point

ps = 19.6

µ/T

ps = 11.5

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SummaryI 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.

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