1

Recent LGT results and their implications in Heavy Ion Phenomenology

quark-gluon plasma

hadron gas

colorsuperconductor

Bμ

Equation of state in LGT and freeze-out critical conditons

Charge fluctuations below and at chiral phase transition : LGT and Chiral Effective Model results how to surch for TCP in hevy ion collisions?

T

K. Redlich

m=0O(4) 2nd order

TCP Z(2), 2nd order

1st order

Crossover m=0

2

Chemical freeze-out and its characteristics

3

s

T

[ ]s GeV measures of the effective number of degrees of freedom

3

s

T .const≈ :expected for if there are leading quasiparticles

m g T≈ ⋅ since

3( , )m

s f TT T

μ≈ ⋅

Cleymans et al.. , A.Tawfik

withq Tμ <

J. Cleymans & K.R/ 1E N GeV≈

3

Deconfinement is density driven - (percolation)

(2) fSU

cT

lines of constant energy density in HRG

Hadron resonance gas partition function

provides a good description of m and depen. of deconfinement temperature

A. Peikert, et al..

LGT result shows: strong dependence of on and , howeverfor and for all

hadrons

+ gluballs

hadrons(3) fSU

qm fN

3( ) (0.6 0.3) /c GeV fmTε ≈ ±2,3fN = qm

condition for deconfinement

percolation deconfinement

H. Satz

1.18.p

h

n constV

≈ =

30.6c

GeV

fmε ≈3

10.55Pn fm

≈

1.09E

GeVN

< >≈

< >

fN

0.8hR fm;

4

Phase boundary of the fixed energy density versus chemical freezeout

30.6c

GeV

fmε ;

Splitting of the chemical freeze-out and the phase boundary surface appears when the densities of mesons and baryons are comparable?

particles production processes

0.77m GeVπ =0.14m GeVπ =

LGT (Allton et al..)

1meson

baryon≈

MesonDominated

BaryonDominated

(6 8)NNs GeV> −

Z. Fodor et al..

: QGP hadronization

(6 8)NNs GeV< − : Hadronic rescattering

R. Gavai, S. Gupta

5

Scaling properties of particle production yields

H. Oeschler, et al..,

A. Andronic, P. Braun-Munzinger &K.R.

RHIC

SPS

Scalings appear due to the strangeness exchange production processes in the hadronic fireball,

6

Scaling relations of production

K YNπ ++ +€

[ ] [ ]

[ ] [ ]N

K

Kκ

− +

+ ≈Ξ

NKπ −+Λ +€ K π− −+Λ Ξ +€

and K−Ξ

1

[ ] [ ]

[ ] [ ]NKκπ

− ≈⋅Λ⋅

2

[ ] [ ]

[ ] [ ]

K

πκ

−

−

ΛΞ

⋅≈

⋅[ ] [ ]

[ ] [ ]Nκ

−

≈Ξ ΛΛ

However, thus and

[ ] [ ]K +Λ ≈

3

[ ] [ ]

[ ] [ ]N

K

K

πκ−

+ ≈

N Kπ ++ Λ+€

P. Koch, B. Muller & J. Rafelski

C. M. Ko

H. Oeschler et al.. A. Andronic, P.Braun-Munzinger & K.R..

7

EQS and evolution path of fixed entropy/baryon

V.D. Toneev , J. Cleymans , E.G. Nikonov , K. Redlich , A.A. Shanenko , (2001,2006)

Mixed phase model L. Bravina et al.. (2001)UrQMD

S. Ejiri, F. Karsch, E. Laermann& C. Schmidt, (2006) LGT 2-flavor

8

QCD at non-vanishing chemical potential C. Allton, M.Doring,S. Ejiri, S.Hands, O. Kaczmarek, F. Karsch, E. Laermann & K. Redlich (2005)

3

3 2 2I q

I

0

ln|

! ( ) ( ) n

n

nc

N Z

n N T Tτ

μσ μ μ − =

∂=

∂ur

2

40

2 ( )( , )

n

n

q qn

P T

T Tc T

μ μ∞

=

Δ ⎛ ⎞= ⎜ ⎟

⎝ ⎠∑

2

2 2 4( / )q

q

P

T T T

χμ∂ Δ

=∂

6[( / ) ]qO Tμ

( , )( , , ) det ( ) S V TZ V T DA M eμ μ −=∫

Taylor expansion of

)/(P TμΔ

( ) (0)P P PμΔ = −

3 4( / )q

q

n P

T T Tμ∂ Δ

=∂

complex fermion determinant

0.05Q

T

μ≈0.5q

T

μ≈

Expansion valid for :

interesting region for heavy ions

SPS RHIC LHC

Problem: (0.18)

/ 1q Tμ ≤

0.005q

T

μ≈

/ 0.7m mπ ρ ;3

3 0q

ln|

! ( )n

n

n

N Z

n N Tc τ

μσ μ =

∂=

∂ur

( ) I I I2 4

2

q6

4

qI2

2 12 30T T

c cT

cμ μχ

μ⎛ ⎞ ⎛ ⎞

= + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

L

9

Quark-fluctuations and chiral symmetery restoration

Free energy: expected 2-order transition in 3-d, O(4)

universality class:

( , , ) ( )Analitic q I SingularFF tF T μ μ= +

2 ,( ) ~ 0F t t and smallα α− <

Net Quark

Fluctuations

2( )qNδ⎧

< >≈⎨⎩

1

0 .

0 0

0

q

qt

t fo

for cusp struct t

r vanishes tα

α μ

μ−

−

≠

= →

→

{4( )qNδ< >≈(2 ) 0

. 0

0

0

q

q

t

t fo

fo

r cusp struc

r diverg

t

es t

tα

α μ

μ− −

−

≠

=

→

→

2 (( 2) 2)q diverges at Z nd order p tN oinδ< >

(2)Z

(4)O

1st order

T

μ

2fN =

10

Derivatives of pressure and susceptibilities

Difference between and is small at μ=0. As pQCD Large spike in : for as in pQCD

( ) I I I2 4

2

q6

4

qI2

2 12 30T T

c cT

cμ μχ

μ⎛ ⎞ ⎛ ⎞

= + + +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

L( )2 4

q2 4 6

q

22 12 30q c c

T Tc

T

χ μ μμ

⎛ ⎞ ⎛ ⎞= + + +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠L

6 0c <cT T>4cqχ Iχ

3( )O g

3( )O g

Q Q Qδ = −< >1 2

2 (2!) ( )c Qδ−≈ < >

1 4 2 24 (4!) ( ( ) 3 ( ) )c Q Qδ δ−≈ < > − < >

Interestingin Heavy IonCollisions

11

Ratios of cumulants reflect carriers of the net-quark and charge

( ) ( )4 2 0

ln

( // |

)q Q q Q x

n

n

nx

Zw di hd d t

T μμ =∂

=∂

S. Ejiri, F. Karsch & K.R. (2006)

cT T< 44,2

2

( )9

( )

q

qq d T

d TR = =

(3) (2)

(3) (2)

(4)

4,2 (4)

4 3

4

27

3 9Q F

FG FR

G F+ +=

+ +HRG

12

The Hadron Resonance Gas model and net-quark fluctuations

2

2 2

31

QGPq q

fNT T

χ μπ

⎡ ⎤⎛ ⎞= +⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦( ) /~ Nm Te μ−

Allton et al.. O(4) S. Ejiri, F. Karsch & K.R , O(6)

HRG provides good description of LGT

charge fluctuations in confined phase

13

factorization on the Lattice as in HRG/Tμ

/ cT T

HRG-partition function:

cosh(3( / )) qmeson BP F T TP μ= +

T

1 ( )3 sinh(3 / )Bq qF Tn T Tμ−=

1 ( )9 cosh(3 / )q qBT T TF Tχ μ−=

fixed ratios of baryonic observables independent of

q

T

μ=

T

14

Charge Fluctuations in the NJL model Ch. Sasaki, B. Friman & K.R.

( ) ( )

2 25

2 23

( ) [( ) ( ) ]

( ) ( )

S

S

NJ

q IV

L

VV

L i m G

G

i

Gμ μ

ψ ψ ψψ ψ τγψ

ψγ ψ ψτγ ψ ψμ ψ τ ψμψ+ +

= ∂− + +

− − + +

r

r

Generic structure of the phase diagram as expecetd in different chiral models: see eg. Y. Hatta & T. Ikeda; M. Stephanov, K. Rajagopal,..

: quark and isovector

chemical potentials

,q Iμ μ

/

15

Quark and isovector susceptibilities along the critical curve

Non-monotonic behavior of the net quark susceptibility in the vicinity to TCP

Crude estimate of the critical region

in using chemical freeze-out

condition gives: NNs

( ) 0.7NN TCPs GeVΔ ≈

( )TCPTΔ

TΔ

Ch. Sasaki et al..

Ch. Sasaki et al..

16

Conclusions

For the chemical freeze-out appears in the near vicinity to deconfinement

Phenomenological partition function of the Hadron Resonance Gas provides a good approximation of the reqular part of the QCD partition function in the hadronic phase

To identify the chiral end-point experimentally one needs to search for non-monotonic in dependence of charge fluctuations

(6 8)NNs GeV> −

NNs