T

PB

Hadronic matter

Quark-Gluon Plasma

Chiral symmetry broken

Chiral symmetry restored

LHC

A-A collisions

x

Probing phase diagram of QCD with fluctuations of conserved charges

Confronting 2nd order fluctuations and correlations: ALICE LHC LQCD

Work done with: Pok M. Lo, B. Friman, O. Kaczmarek, C. Sasaki & K.R. P. Braun-Munzinger, A. Kalweit , J. Stachel, & K.R. K. Morita, B. Friman & K.R. Phys.Lett. B741 (2015) 178

LQC

D

cT

^

>

Krzysztof Redlich, Wroclaw Uni., EMMI

Interplay between deconfinement & chiral symmetry breaking in QCD: LQCD & chiral models

Phys.Lett. B747 (2015) 292 Phys .Rev. D90 (2014) 7, 074035

Polyakov loop on the lattice needs renormalization

� Introduce Polyakov loop:

� Renormalized ultraviolet divergence � Usually one takes as an order parameter

2 1/ V

c

c

0 deconfined T T

0 confined T<T

z � !­ ½° °® ¾

�° °¯ ¿

renren| | qFL e E�� ! o

NL c L� 2 / ( )i k NNc e Z NS �

HotQCD Coll.

To probe deconfinement : consider fluctuations

� Fluctuations of modulus of the Polyakov loop However, the Polyakov loop Thus, one can consider fluctuations of the real and the imaginary part of the Polyakov loop.

SU(3) pure gauge: LGT data

R IL L Li �

RF

IF

HotQCD Coll.

Fluctuations of the real and imaginary part of the renormalized Polyakov loop

� Imaginary part fluctuations � Real part fluctuations

Pok Man Lo, B. Friman, O. Kaczmarek, C. Sasaki & K.R. , PRD (2013)

Compare different Susceptibilities:

� Systematic

differences/simi-larities of the Polyakov lopp susceptibilities

� Consider their ratios!

Ratios of the Polyakov loop fluctuations as an excellent probe for deconfinement

� In the deconfined phase Indeed, in the real sector of Z(3)

Expand modulus and find:

2 2 2| | | | ( )RL L LG� ! � � ! | � !

Pok Man Lo, B. Friman, O. Kaczmarek, C. Sasaki & K.R. , PRD (2013) 1AR |

0 0withR R RL L L L LG| � � !

0 0with 0, thusI II IL L L LG| �

2 2( ) , ( )L LR R I IV L V LF G F G � ! � !

Expand the modulus, 2

2 20 2

0 0

( )| | (1 )2

R IR I

L LL L L LL LG G

� | � �

get in the leading order

thus A RF F|

LA

A LR

R FF

Ratios of the Polyakov loop fluctuations as an excellent probe for deconfinement

� In the confined phase Indeed, in the Z(3) symmetric phase, the probability distribution is Gaussian to the first approximation,

with the partition function consequently

Expand modulus and find:

Pok Man Lo, B. Friman, O. Kaczmarek, C. Sasaki & K.R. , PRD (2013) 0.43AR |

In the SU(2) case is in agreement with MC results

Thus and

LA

A LR

R FF

3 2 2[ ( )( )]R IVT T L LR IZ dL dL e D � ³

3 3

1 1,2 2R IT T

F FD D

3

1 (2 ),2 2A T

SFD

�

(3) (2 ) 0.4292

SUAR

S �

(2) 2(2 ) 0.363SUAR S

�

Ratio Imaginary/Real of Polyakov loop fluctuations

� In the confined phase for any symmetry breaking operator its average vanishes, thus

and thus

2 2 0LL L LF � ! � � !

LL R IF F F � R IF F

� In deconfined phase the ratio of and its value is model dependent

/ 0I RF F z

Pok Man Lo, B. Friman, O. Kaczmarek, C. Sasaki & K.R. , PRD (2013)

LILR

FF

Ratio Imaginary/Real of Polyakov loop fluctuations

� Due to Z(3) symmetry breaking in confined phase, the fluctuations of transverse Polyakov loop strongly

suppressed,

� In deconfined phase

Pok Man Lo, B. Friman, O. Kaczmarek, C. Sasaki & K.R. , PRD (2013)

LILR

FF

Due to Z(3) symmetry breaking in deconfined phase, the fluctuations of transverse Polyakov loop strongly suppressed,

1I

R

FF

��

RL

IL

Deconfinement phase transition in a large quark mass limit: effective approach

� Effective partition function where

a

lndet[ ]( , ) fVU L L QZ dLdL e E �� �� ³

�T T

Start with QCD thermodynamic potential Employ the background field approach Consider a uniform background Consider effective gluon potential with parameters fixed to reproduce a pure LQCD thermodynamic as: C. Sasaki &K.R. Phys.Rev. D86 (2012) 014007

cl quantumA A gAP P P| �

4A AP P PG

( , )U L L�

PL and heavy quark coupling

� Effective potential

� Tree level result

where Compare with LGT:

3ln det[ ] [ , ; ]f q qQ VT U L L M� �

[ / ]q eff q RU h M T L �

G f RG e fU U h Lo �

qM T!!

22( / ) ( / )eff f q qh N M T K M T|

3(2 )(2 )(2 ( ))NLGTeff f ch N N N NW

W WN

G. Green & F. Karsch (83)

The critical point of the 2nd order transition

Pok Man Lo, et al. Phys. Rev. D (2014)

Susceptibility at the critical point

� Divergent longitudinal susceptibility at the critical point for decondinement

=1.48 GeV

Pok Man Lo, et al. Phys. Rev. D (2014)

Critical masses and temperature values

� Different values then in the matrix model by K. Kashiwa, R. Pisarski and V. Skokov,

Phys. Rev. D85 (2012)

Pok Man Lo, et al. Phys. Rev. D (2014)

LGT C. Alexandrou et al. (99) 1 1.4fNcM GeV |

Polyakov loop and fluctuations in QCD

� Smooth behavior for the Polyakov loop and fluctuations difficult to determine where is “deconfinement”

The inflection point at 0.22decT GeV|

HOT QCD Coll

The influence of fermions on the Polyakov loop susceptibility ratio

� Z(3) symmetry broken, however ratios still showing deconfinement

� Dynamical quarks imply

smoothening of the susceptibilities ratio, between the limiting values as in the SU(3) pure gauge theory

Pok Man Lo, B. Friman, O. Kaczmarek, C. Sasaki & K.R.

LA

A LR

R FF

� Change of the slope in the

narrow temperature range signals color deconfinement

/ pcT T

this work

The influence of fermions on ratios of the Polyakov loop susceptibilities

� Z(3) symmetry broken, however ratios still showing the transition

� Change of the slopes at fixed T

Pok Man Lo, B. Friman, O. Kaczmarek, C. Sasaki & K.R.

LA

A LR

R FF

LRLI

FF

Fluctuations of net baryon number sensitive to deconfinement in QCD

� Kurtosis measures the squared of the baryon number carried by leading particles in a medium S. Ejiri, F. Karsch & K.R. (06)

2

2

2 4

119

PC

BPC

B T T

T TBF

FN V

�§ ·¨ ¸ | ¨ ¸!¨ ¸© ¹

Pok Man Lo, B. Friman, (013) O. Kaczmarek, C. Sasaki & K.R. S. Ejiri, F. Karsch & K.R. (06)

316 4u lattice with p4 fermion action

0.77GeV, =0BmS P|

S. Ejiri, F. Karsch & K.R. (06)

( , ) ( )cosh( / )Bq BP T F BT TPP

� HRG factorization of pressure: 4

2

B

B

FF 200 MeVpcT |

2 770fN m MeVS |

4( / )( / )

nBn n

B

P TT

FP

w w

Kurtosis of net quark number density in PQM model V. Skokov, B. Friman &K.R.

� For the assymptotic value

2

24 2

( ) 2 3 3cosh

3qq f q q

c

qP T N mK

T T Tm

N TP§ · § ·

| ¨ ¸ ¨ ¸© ¹ © ¹

due to „confinement” properties

� Smooth change with a very weak dependence on the pion mass

cT T�9

4 2/ 9c c

� For

cT T!!

4

( )qqP TT

4 22/ 6 /c c S

4 2/c c

4 2 2

2

1 1 7[ ]2 6 180f cN N

T TP P S

S§ · § ·� �¨ ¸ ¨ ¸© ¹ © ¹

Polyakov loop susceptibility ratios still away from the continuum limit:

� The renormalization of the Polyakov loop susceptibilities is still not well described:

Still strong dependence on in the presence of quarks.

NW

Ch. Schmidt et al.

The phase diagram at finite quark masses

CP Asakawa-Yazaki Stephanov et al., Hatta & Ikeda

At the CP: Divergence of Fluctuations, Correlation Length and Specific Heat The existence of the CP in QCD has not been established yet

� To identify chiral crossover consider fluctuations of

lnT ZV m

\\ w� !

w

2

2

lnT ZV m\\F w

w

S. Borsayi et al. Wuppertal-Budapest Coll. A. Bazavov et al. HotQCD Coll.

The interplay between deconfinement and the chiral crossover

� The change of properties of observables which are sensitive to deconfinement and chiral transition appear in the same narrow temperature range

.

23

� Due to the expected O(4) scaling in QCD the free energy:

� Consider generalized susceptibilities of the net-quark number

� Since for , are well described by the search for deviations (in particular for larger n) from HRG to quantify the contributions of , i.e. the O(4) criticality

11 (2 ) /( , , ) ( , )Sq IRF T b b t bFF hD EG QP P�� � �

Quark fluctuations and O(4) universality class

4( ) ( / )

( / )

n

nB

nB

P TcTP

w w

( )s

(2 )/ ( ) ( )n nn d h fc zD EG� �r

pcT T�

with

( )nRc HRG

( )nSc

( )( )nR

nSc c�

S. Ejiri, F. Karsch & K.R. Phys. Lett. B633, (2006) 275 M. Asakawa, S. Ejiri and M. Kitazawa, Phys. Rev. Lett. 103 (2009) 262301 V. Skokov, B. Stokic, B. Friman &K.R. Phys. Rev. C82 (2010) 015206 F. Karsch & K. R. Phys.Lett. B695 (2011) 136 B. Friman, et al. . Phys.Lett. B708 (2012) 179, Nucl.Phys. A880 (2012) 48

Momentum cut dependence of the cumulant ratio of the electric charge fluctuations

� Strong dependence of the electric charge kurtosis on the low momentum cut, thus

� direct comparisons of LQCD and STAR data with low momentum cut contain unknown systematic errors

Consider fluctuations and correlations of conserved charges to be compared with LQCD

� They are quantified by susceptibilities: If denotes pressure, then

� Susceptibility is connected with variance

� If probability distribution of then

( , , , )B Q SP T P P P2

2 2

( )( )

N

N

PTF

Pw

w

2

2

( )NM

N M

PTF

P Pw

w w

2 22 3

1 ( )N N NT VTF

� ! � � !

,q qN N N� � , ( , , ),N M B S Q / ,TP P 4/P P T

P( )N NP( )n n

NN N N� ! ¦

Excellent probe of: � QCD criticality A. Asakawa at. al. S. Ejiri et al.,… M. Stephanov et al., K. Rajagopal et al. B. Frimann et al. � freezeout

conditions in HIC F. Karsch & S. Mukherjee et al., P. Braun-Munzinger et al.,,

Consider special case:

� Charge and anti-charge uncorrelated and Poisson distributed, then � the Skellam distribution

� Then the susceptibility

/2

P( ) (2 )exp[ ( )]N

N q q qq

qq

NN N N N

NN I � �

�

§ · � �¨ ¸¨ ¸© ¹

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

2 3

1 ( )Nq qT VT

N NF�� ! � � !

qqN N� !{ =>

Consider special case: particles carrying

� The probability distribution

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

qq SS ��� !{

1, 2, 3q r r r

1, 2, 3q r r r

Fluctuations Correlations

Variance at 200 GeV AA central coll. at RHIC

� Consistent with Skellam distribution

� Consider ratio of cumulants in in the whole momentum range:

P. Braun-Munzinger, et al. Nucl. Phys. A880 (2012) 48)

2 1.022 0.016p pV

� ! � � ! r 1

3

1.076 0.035FF

r

STAR Collaboration data in central coll. 200 GeV

2

6.18 0.14 0.4 0.8ti Vp

Gp

p enV r � �

�

7.67 1.86 0.0 tp p i

ppn e

pG V�

� r � � f

Particle production yields in Heavy Ion Collisions at LHC

Paolo Giubellino & Jürgen Schukraft for ALICE Collaboration

A. Kalweit

Use ALICE data to quantify the 2nd order correlations and fluctuations of conserved charges

Constructing net charge fluctuations and correlation from ALICE data

� Net baryon number susceptibility

� Net strangeness

� Charge-strangeness correlation

002 3

1 ( )B p parNT VTF � �� �6| � � /�6 � � � ; � ; � : �6

0 0

0 002 3

1 ( 4 4 9

)( )S LK K K K

SSK K par

T VT

M M M MM

F

� �

�

o o o

�

o

� ��

� * �

| � � / �6 � � � ; � ; � :

* �

6

*

�

�*

6

* *0 0

0

2

*

3

1 ( 2 3

( ) ( ))K K K

Q

K

S

K K

K parT VT

KM M

F

M� � � �o o o

�

o

� �

� * �* � * �*

| � ; � : �

constructed from ALICE particle yields

� use also from pBe at

� Net baryon fluctuations

� Net strangeness fluctuations

� Charge-Strangeness corr.

� Ratios is volume independent

, ,B S QSF F F

0 / 0.2786 / 25s GeV

2 3

1 (203.7 11.4)B

T VTF

| r

2 3

1 (504.2 24)S

T VTF

| r

2 3

1 (178 17)QS

T VTF

| r

0.404 0.028B

S

FF

r 1.14 0.13B

QS

FF

rand

Compare the ratio with LQCD data:

Phys.Rev.Lett. 113 (2014) and HotQCD Coll. A. Bazavov et al.

A. Bazavov, H.-T. Ding, P. Hegde, O. Kaczmarek, F. Karsch, E. Laermann, Y. Maezawa and S. Mukherjee

Phys.Rev. D86 (2012) 034509

� Is there a temperature where calculated ratios from ALICE data agree with LQCD?

Baryon number, strangeness and Q-S correlations

� There is a very good agreement, within systematic uncertainties, between extracted susceptibilities from ALICE data and LQCD at the chiral crossover

� How unique is the determination of the temperature at which such agreement holds?

Compare ratios at the chiral crossover

Consider T-dependent LQCD ratios and compare with ALICE data

� The LQCD susceptibilities ratios are weakly T-dependent for � We can reject for saturation of at the

LHC, and can fix T to be in the range , however � LQCD => for thermodynamics cannot be anymore

described by the hadronic degrees of freedom

cT Tt

0.15T GeVd ,B S QSandF F F0.15 0.21T GeV� d

0.163T GeV!

35

Baryon-Strangeness Correlations

2

( )( )( )

BSBS

S

B SCS

FG GG F

� ! � �

� !

� Excellent observable to fix temperature

� Data fix only the lower limit since e.g.

not included

Consider Volker Koch 05

Extract the volume by comparing data with LQCD

� Since thus

� All volumes, should be equal at a given temperature if originating from the same source, thus

3 2

203.7 11.4)/ )

((B

LQCDB

V TT TF F

r 3 2

504.2 24.2)/ )

((S

LQCDB

V TT TF F

r

3 2

178 17( )( / )QS

B LQCD

V TT TF F

r

2 22

3

( )( / ) LHN LQCD

C

N

N NTV T

F � ! � � !

150T MeV!

Constraining the volume from HBT and percolation theory

� Some limitation on volume from Hanbury-Brown–Twiss: HBT Take ALICE data from pion interferometry Use 3D hydro to transfer : the volume of the homogeneity at the last interaction :the volume at the thermal freezeout : the volume at temperature

34800 640HBTV fm� r

HBTV

( )th thV T

ch thT T!( )chV T

100th VT Me|D. Adamova, et al (CERES) Phys Rev. Lett 90, 022301 (2003).

S.V. Akkelin, P. Braun-Munzinger & Y. Sinyukov Nucl. Phys. A710 (2002).

Thermal origin of particle yields with respect to HRG

Re .[ ( , ) ( , )]th sii

th

Ki K iN T TnnV P Po

� � ! *�¦

Rolf Hagedorn => the Hadron Resonace Gas (HRG): “uncorrelated” gas of hadrons and resonances

A. Andronic, Peter Braun-Munzinger, & Johanna Stachel, et al.

� Measured yields are reproduced with HRG at

156 MeVT |

22

1 ( / ) ( / )2 1

mdN TKdy

V T mj

�

Particle yields with no resonance decay contributions at the LHC:

Conclusions:

� there is thermalization in heavy ion collisions at the LHC and the 2nd order charge fluctuations and correlations are saturated at the chiral crossover temperature

From direct comparisons of 2nd order fluctuations and correlations constructed from ALICE data and LQCD results one concludes that: Skellam distribution, and its generalization, is a good approximation of the net charge probability distribution P(N) for small N<10. The chiral criticality sets in at larger N>10 and implies shrinking of the Skellam distribution.

� Ratios of the Polyakov loop and the Net-charge susceptibilities are excellent probes for deconfinement and/or the O(4) chiral crossover in QCD

� .

Particle density and percolation theory

� Density of particles at a given volume

� Total number of particles in HIC at LHC, ALICE

� Percolation theory: 3-dim system of objects of volume

exp

( )( )totalNn T

V T

0

1.22cn V take => =>

30 04 / 3V RS

0 0.82R fm| 30.52 [ ]cn fm�| 152 [ ]pcT MeV|

P. Castorina, H. Satz &K.R. Eur.Phys.J. C59 (2009)

What is the influence of O(4) criticality on P(N)?

� For the net baryon number use the Skellam distribution (HRG baseline)

as the reference for the non-critical behavior � Calculate P(N) in an effective chiral

model which exhibits O(4) scaling and compare to the Skellam distribtuion

/2

( ) (2 )exp[ ( )]N

NB B BN B BB

P I§ · � �¨ ¸© ¹

)( ()N

C T

GC

Z NZ

P eNP

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 iyS

SF �³

[ ( , ) ( , ]( ) ) k

kkV p T y p T yy E PF P F � � ¦

( )k k

NN N P N� ! ¦

Cumulants generating function:

)( ()N

C T

GC

Z NZ

P eNP

What is the influence of O(4) criticality on P(N)?

� For the net baryon number use the Skellam distribution (HRG baseline)

as the reference for the non-critical behavior � Calculate P(N) in an effective chiral

model which exhibits O(4) scaling and compare to the Skellam distribtuion

/2

( ) (2 )exp[ ( )]N

NB B BN B BB

P I§ · � �¨ ¸© ¹

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

� 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 crossover, indicating the contribution of the O(4) criticality to the thermodynamic pressure

K. Morita, B. Friman &K.R. (PQM model within renormalization group FRG)

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

0P

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 P( )FRGP N

� Asymmetric P(N) � Near the ratios less

than unity for

0P K. Morita, B. Friman et al.

0P z

( )pcT PN N! � !

0P z

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

0P

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

0P z

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

STAR DATA

Centrality dependence of probability ratio

O(4) critical

Non- critical behavior

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

K. Morita et al.

Cleymans & Redlich Andronic, Braun- Munzinger & Stachel

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 ? Efficiency uncorrected data!!

Thanks to Nu Xu and Xiofeng Luo

Effective Polyakov loop Potential from Y-M Lagrangian Chihiro Sasaki & K.R.

(Gross, Pisarski & Yaffe)

.