B

Hadronic matter

Quark-Gluon Plasma

Chiral symmetrybroken

x

Exploring QCD Phase Diagram in Heavy Ion Collisions

Krzysztof Redlich University of Wroclaw and EMMI/GSI

QCD phase boundary and freezeout in HIC

Cumulants and probability distributions of conserved charges as Probe for the Chiral phase transition:

theoretical expectations and recent STAR data at RHIC

-CEP?

?

AA collisions

2

– probing the response of a thermal medium to an external field, i.e. variation of one of its external control parameters:

(generalized) response functions == (generalized) susceptibilities

pressure:

thermal fluctuations density fluctuations condensate fluctuations

generalized susceptibilities:generalized susceptibilities:

energy density net charge number order parameter

Bulk thermodynamics and critical behaviour

, )i 4

1( / )~ |

iii

N P T

Mean particle yields

3

Susceptibilities of net charge number

– The generalized susceptibilities probing fluctuations of net -charge number in a system and its critical properties

pressure:

particle number density quark number susceptibility 4th order cumulant

net-charge q susceptibilities

31 1 ,q N

VT 2 2

32 1 ( )q N N

VT 3

34 41 ( ( ) 3 ( ) )q N N

VT

q qN N N N N N expressed by and central moment

( )nq

(2)q

(4)q

4

Only 3-parameters needed to fix all particle yields

Tests of equlibration of 1st “moments”: particle yields

resonance dominance: Rolf Hagedorn partition function

22ln ( , ) ( ) ( , )2

iQB WT

i ii hadrons

VT sZ T d e ds s K F m sT

Breit-Wigner res.

Re .[ ( , ) ( , )]KB BK

ti

h si

thiiV T TnN n

particle yield thermal density BR thermal density of resonances

Chemical freezeout and the QCD chiral crossover

A. Andronic et al., Nucl.Phys.A837:65-86,2010. O(4) universality

2

0 0.0061 6c

c

TT T

HRG model

Chiral

/ ( )T f s

crossover

Thermal origin of particle production: with respect to HRG partition function

Chiral crosover Temperature from LGT

HotQCD Coll. (QM’12)

154 9cT MeV

Chemical freezeout and the QCD chiral crossover

A. Andronic et al., Nucl.Phys.A837:65-86,2010. O(4) universality

2

0 0.0061 6c

c

TT T

HRG model

Chiral

/ ( )T f s

crossover

Is there a memory that the system has passed through a region of QCD chiral transition ?

What is the nature of this transition?

Chiral crosover Temperature from LGT

HotQCD Coll. (QM’12)

154 9cT MeV

QCD phase diagram and the O(4) criticality

In QCD the quark masses are finite: the diagram has to be modified

Expected phase diagram in the chiral limit, for massless u and d quarks:

Pisarki & Wilczek conjecture

TCP: Rajagopal, Shuryak, Stephanov Y. Hatta & Y. Ikeda

TCP

The phase diagram at finite quark masses

The u,d quark masses are small

Is there a remnant of the O(4) criticality at the QCD crossover line?

CPAsakawa-YazakiStephanov et al., Hatta & Ikeda

At the CP: Divergence of Fluctuations, Correlation Length and Specific Heat

The phase diagram at finite quark masses

Can the QCD crossover line appear in the O(4) critical region?It has been confirmed in LQCD calculations

TCP

CP

LQCD results: BNL-Bielefeld

Critical region

Phys. Rev. D83, 014504 (2011)Phys. Rev. D80, 094505 (2009)

10

singular

critical behavior controlled by two relevant fields: t, h

Close to the chiral limit, thermodynamics in the vicinity of the QCD transition(s) is controlled by a universal scaling function

K. G. Wilson,Nobel prize, 1982

Bulk Thermodynamics and Critical Behavior

non-universal scalescontrol parameter for amountof chiral symmetry breaking

regular

O(4) scaling and magnetic equation of state

Phase transition encoded in the magnetic equation of state

Pm

11/

/ ,( )sf z z tmm

pseudo-critical line

1/m

F. Karsch et al

universal scaling function common forall models belonging to the O(4) universality class: known from spin modelsJ. Engels & F. Karsch (2012)

mz

QCD chiral crossover transition in the critical region of the O(4) 2nd order

12

Find a HIC observable which is sensitive to the O(4) criticality

Consider generalized susceptibilities of net-quark number

Search for deviations from the HRG results, which for quantifies the regular part

1(21 /)( ),( , , ) SingularAnalytic q I bP T b hb tP P

0(4) : .2O

Quark fluctuations and O(4) universality class

To probe O(4) crossover consider fluctuations of net-baryon and electric charge: particularly their higher order cumulants with

F. Karsch & K. R. Phys.Lett. B695 (2011) 136

B. Friman, V. Skokov et al, P. Braun-Munzinger et al. Phys.Lett. B708 (2012) 179

Nucl.Phys. A880 (2012) 48

4( ) ( / )

( / )

n

nB

nB

P TcT

( ) (2 /2)/ ( /2) ( 0)nr

nnc d h f z and n even

( ) (2 )/ ( ) ( ) 0nn nrc d h f z

cT T( )nrc 6n

or compare HIC data directly to the LGT results, S. Mukheriee QM^12 for BNL lattice group

Effective chiral models Renormalisation Group Approach

coupling with meson fileds PQM chiral model FRG thermodynamics of PQM model:

Nambu-Jona-Lasinio model PNJL chiral model

1/3

0

i *4

nt ( , ),[ ](( ) )T

qV

S d d x iq V U Lq q q LA q q

the SU(2)xSU(2) invariant quark interactions described through:

K. Fukushima; C. Ratti & W. Weise; B. Friman , C. Sasaki ., ….

B.-J. Schaefer, J.M. Pawlowski & J. Wambach; B. Friman, V. Skokov, ...

int ( , )V q q

*( , )U L L the invariant Polyakov loop potential (Get potential from YM theory, C. Sasaki &K.R. Phys.Rev. D86, (2012); Parametrized LGT data: Pok Man Lo, B. Friman, O. Kaczmarek &K.R.)

(3)Z

B. Friman, V. Skokov, B. Stokic & K.R.

fields

Including quantum fluctuations: FRG approach

FRG flow equation (C. Wetterich 93)J. Berges, D. Litim, B. Friman, J. Pawlowski, B. J. Schafer, J. Wambach, ….

start at classical action and includequantum fluctuations successively by lowering k

Regulator function suppressesparticle propagation with momentum Lower than k

0lim(( ) ), ( /) k kk

T T VV

k k kk R

k-dependentfull propagator

B. Stokic, V. Skokov, B. Friman, K.R.

FRG for quark-meson model

•LO derivative expansion (J. Berges, D. Jungnicket, C. Wetterich) (η small) •Optimized regulators (D. Litim, J.P. Blaizot et al., B. Stokic, V. Skokov et al.)

•Thermodynamic potential: B.J. Schaefer, J. Wambach, B. Friman et al.

,

11 2 1 2( , : ) [3 4 ]q a qk k o k f c

q

n nn nT N NE E E

Non-linearity through self-consistent determination of disp. rel. 2 2

i iE k M with ' ' ''2 2 2 2

0, 0,2 2k k kk q kM M M g

and 2k k h 0,

'/ |

kk k with

Employed Taylor expansion around minim

Get Potential Ignore flow of mesonic field get Mean Field result

Essential to include fermionic vacuum fluctuations:

Solving the flow equation with approximations:

0( ), ,) (kT T

E. Nakano et al.

Deviations of the ratios of odd and even order cumulants from their asymptotic, low T-value: are increasing with and the cumulant order Properties essential in HIC to discriminate the phase change by measuring baryon number fluctuations !

4 2 3 1/ / 9c c c c /T

4,2 4 2/R c cRatios of cumulants at finite density: PQM +FRG

HRG

B. Friman, F. Karsch, V. Skokov &K.R. Eur.Phys.J. C71 (2011) 1694

HRGHRG

STAR data on the first four moments of net baryon number

Deviations from the HRG

Data qualitatively consistent with the change of these ratios due to the contribution of the O(4) singular part to the free energy

4)2

(

(2)B

B

(3)

(2) ,B

B

S

HR

G

, 1| |p p

p pHRG HRG

N N

N NS

Kurtosis saturates near the O(4) phase boundary

The energy dependence of measured kurtosis consistent with expectations due to contribution of the O(4) criticality. Can that be also seen in the higher moments?

B. Friman, et al. EPJC 71, (2011)

Deviations of the ratios from their asymptotic, low T-value, are increasing with the order of the cumulant

Ratio of higher order cumulants in PQM model B. Friman, V. Skokov &K.R. Phys.Rev. C83 (2011) 054904

Negative ratio!

STAR DATA Presented at QM’12

Lizhu Chen for STAR Coll.

V. Skokov, B. Friman & K.R., F. Karsch et al.

The HRG reference predicts:

6 2/c c

HRG

6 2/ 1c c O(4) singular part contribution: strong deviations from HRG: negative structure already at vanishing baryon density

Moments of the net conserved charges

Obtained as susceptibilities from Pressure

or since they are expressed as polynomials in the central moment N N N

4( / ) / ( / )nn B

nTc P T

Moments obtained from probability distributions

Moments obtained from probability distribution

Probability quantified by all cumulants

In statistical physics

2

0

( ) (1 [ ]2

)expdy iP yNN iy

[ ( , ) ( , ]( ) ) k

kkV p T y p T yy

( )k k

N

N N P N

Cumulants generating function:

)( ()N

C T

GC

Z NZ

P eN

Probability distribution of the net baryon number

For the net baryon number P(N) is described as Skellam distribution

P(N) for net baryon number N entirely given by measured mean number of baryons and antibaryons

In Skellam distribution all cummulants expressed by the net mean

and variance

BB

/2

( ) (2 )exp[ ( )]N

NB B BN B BB

P I

2 B B M B B

P. Braun-Munzinger, B. Friman, F. Karsch, V Skokov &K.R. Phys .Rev. C84 (2011) 064911 Nucl. Phys. A880 (2012) 48)

Probability distribution of net proton number STAR Coll. data at RHIC

STAR data

Do we also see the O(4) critical structure in these probability distributions ?

Thanks to Nu Xu and Xiofeng Luo

Influence of O(4) criticality on P(N)

Consider Landau model:

Scaling properties:

Mean Field 0

O(4) scaling 0.21

2 2 41 12 4bg t

21 | ( , ) |4

t T ( )t T

sin3 gnn c

( , )t T

Contribution of a sigular part to P(N)(

(,

), ) N

C T

GC

Z N T VZ

P eN

Get numerically from:

For MF broadening of P(N) For O(4) narrower P(N)

Take the ratio of which contains O(4) dynamics to Skellam distribution with the same Mean and Variance at different / pcT T

( )FRGP N

Ratios less than unity near the chiral critical point, indicating the contribution of the O(4) singular part to the thermodynamic pressure

0 K. Morita, B. Friman et al.

The influence of O(4) criticality on P(N) for 0

The influence of O(4) criticality on P(N) for Take the ratio of which contains O(4) dynamics to Skellam

distribution with the same Mean and Variance near ( )pcT ( )FRGP N

Asymmetric P(N) Near the ratios less

than unity for For sufficiently large

the for

0 K. Morita, B. Friman et al.

0

( )pcT N N

( )( ) / 1FRG Skellam NP N P

N N

0

The influence of O(4) criticality on P(N) for

0

K. Morita, B. Friman & K.R.

0

In central collisions the probability behaves as being influenced by the chiral transition

Centrality dependence of probability ratio

O(4) critical

Non- criticalbehavior

For less central collisions, the freezeout appears away the pseudocritical line, resulting in an absence of the O(4) critical structure in the probability ratio.

STAR analysis of freezeout K. Morita et al.

Cleymans & RedlichAndronic, Braun-Munzinger & Stachel

Energy dependence for different centralities

Ratios at central collisions show properties expected near O(4) chiral pseudocritical line

For less central collisions the critical structure is lost

Conclusions:

Hadron resonance gas provides reference for O(4) critical behavior in HIC and LGT results

Probability distributions and higher order cumulants are excellent probes of O(4) criticality in HIC

Observed deviations of the and by STAR from the HRG qualitatively expected due to the O(4) criticality

Deviations of the P(N) from the HRG Skellam distribution follows expectations of the O(4) criticality

3 2/ , 4 2/ , 6 2/

Present STAR data are consistent with expectations, that in central collisions the chemical freezeout appears near the O(4) pseudocritical line in QCD phase diagram