QCD Phase Diagram and Critical Point

Lecture at 5th CBM India Collaboration

Meeting, BHU, Varanasi, India

December 28, 2009

Reviews: C.P. Singh, Phys. Rep. 236,147(1993)

Int. J. Mod Phys. A7, 7185 (1992)

Outlines :Outlines :

1. Introduction1. Introduction

2. History of QCD Phase Transition.2. History of QCD Phase Transition.

3. QCD phase transition and Critical 3. QCD phase transition and Critical Point.Point.

4. Summary.4. Summary.

What happens to a matter at extremely large What happens to a matter at extremely large temperature and/or densitytemperature and/or density

1010 50102010 3010 401010101

Present Universe Atomic matter

3

33

30.14 /

4 4 1.23

A

A

AGeV fm

R

3

3

0.5 /43

pp

p

mGeV fm

r

31 /GeV fm

Nuclear matter

Energy density

Nucleons/ 3cm

Comparepr(if = 0.8 fm)

2

22

2

2

22

ln)233(

12

4)(

)(

4

1)(

Q

Q

QN

gQ

GGfgGGG

GTigD

GGmDiL

QCD

QCDf

ss

cbabcsaaa

ajkasjkjk

aakjkjkjQCD

QCD – Lagrangian density

Confinement

Asymptotic freedom0

s

s

Continued….

QCD Two Important features = 0.2 GeV/cQCD Two Important features = 0.2 GeV/c

(1)(1) ConfinementConfinement → Infra red slavery→ Infra red slavery

or or

(2)(2) Asymptotic freedomAsymptotic freedom

QCD, the theory of strongly interacting matter, predicts that, above a QCD, the theory of strongly interacting matter, predicts that, above a critical energy density, hadrons, the constituents or normal nuclear critical energy density, hadrons, the constituents or normal nuclear matter, decompose into a plasma of quarks and gluons, the Quark Gluon matter, decompose into a plasma of quarks and gluons, the Quark Gluon

PlasmaPlasma, , QGP. SuchQGP. Such a state existed shortly after the Big Bang and may a state existed shortly after the Big Bang and may exist as well in the neutron stars. Colliding heavy ions at extreme exist as well in the neutron stars. Colliding heavy ions at extreme energies is the only way to study QGP on Earth.energies is the only way to study QGP on Earth.

1r

( )s r 2 2Q

2Q 2( ) 0s Q

( ) 0s r

0r

QCD QCD

QED QED

Debye screenedDebye screened

confiningconfining

Bound Bound

statestate screenedscreened

rr

ss rV r

r

exp 1 expsd

d d

r rr

r r r

( )V rr

r

expd

r

r

sV r

At large T and large ρ

r D is small ( r D < r) and thus

Hadrons→ melt down

Hadrons→ QGP

P

T

Tri Critical point

Ice

Steam

Critical PointWater

Cross-over region

Phase Diagram of H2O

C2

C1

C2 ( TC= 273.16 0K , PC= 600 N / m2 )

C1 ( TC= 647 0K , PC= 2.21x !07 N / m2 )

Ist order Discontinuity in S, V

II order Discontinuity in CP, KTP

P

P

T

GTC

T

GS

2

2

TT

T

P

G

TK

P

GV

2

21

QCD predicts a colour deconfining Phase QCD predicts a colour deconfining Phase transitiontransition

QGP- Quarks and gluons are coloured matter.

- Force increases with separation.

- Large density matter.

Hadrons- Hadrons are colour insulators.

- Force decreases with separation.

- Dilute and low density matter.

HISTORY OF QCD PHASE TRANSITION

- Baym (1982)

Order of Phase transitionOrder of Phase transitionF = Free energy of the systemF = Free energy of the system

discontinuous then it is nth order phase discontinuous then it is nth order phase

transition.transition.

11stst order order = Latent Heat = Latent Heat

QGPQGP

11stst order order

HGHG

SYMMETRIC 2 SYMMETRIC 2ndnd order order

Broken symmetryBroken symmetry

TT

n

n

F

T

FT

T

F

F TE TV V

O

4CT 4T

Phase transition between H.G. (Pion Phase transition between H.G. (Pion Gas) QGPGas) QGP

1st order phase transition1st order phase transitionMaxwell’s construction Maxwell’s construction

IfIf

PP

TT -B-B

( , ) ( , )c c Q c cP T P T

0

243

90P T

2437

90QP T B

QP

cT

P1

40.72cT B

140MeV

TWO QCD VACUA

Bag Model:

Perturbative QCD Perturbative QCD vacuum vacuum

((High T) High T) Real or PhysicalReal or Physical QCD vacuum QCD vacuum (Low T) (Low T)

Confinement pressure B Confinement pressure B

Stress arising due to K.E. of quarks Stress arising due to K.E. of quarks

0qm

3

0 0

30 0

00

0

4( )

3

0

( ) 4

4

3( )

4

CE R R B

RE

RE R V B

V R

E RB

V

3

30

175 /

0.7 /

B MeV fm

GeV fm

QGP in two situationsQGP in two situations

Temperature T Measures the mean energy of the system.

Baryon chemical potential measures the mean number of baryons in the system.

Heat Heat ---------------> ---------------> dense indense in mesonsmesons T > 200 MeV T > 200 MeV Early UniverseEarly Universe ( ( second after Big Bang)second after Big Bang) Baryons MesonsBaryons Mesons

K

B

610

121016.1100 TMeVTKB

Compress

940B MeV

dense in Baryons ( Core of neutron star)

Chiral Symmetry Restoration

Global flavour rotations

matrix

So rotation group is

Chiral group

2 (2) (2) Flavour + rotations

R R R

L L L

U

U

f fU N N

( ) ( )R f L fU N U N

(1) (1) ( ) ( )V A f fU U SU N SU N

SUSU fN 5

Noether Current

if 0

LL

RR

5

5

2

2

0

0

a a

a as

a

a

V

A

V

A C

m

00000

2

)1(2

)1(

5,

5,

RLLR

LR

LR

Chiral Symmetry Restoring Phase Transition

This is non zero(250 MeV)3

0m Helicity of the quark is fixed (gluon interaction

does not change the helicity ).

BL & BR both are good quantum numbers.

0m Quark can exist in both the helicities.

B= BL + BR is a good quantum number.

chiral symmetry is broken ( left and right quarks are not

Independent ).

00000 RLLR The vacuum contains qq pairs

≈ (250 MeV)3

= Rate at which quarks flip their helicity.

Order parameter => 0 T > TC

= large T < TC

Confinement => 0)/exp( TFL qT < TC

= 1 T > TC

Susceptibilities 2

2 LLL

22 q

Courtesy : K. Fukushima, Feb 2008, QM08

Courtesy : K. Fukushima, Feb 2008, QM08

Courtesy : K. Fukushima, Feb 2008, QM08

LATTICE GAUGE THEORYLATTICE GAUGE THEORY- A numerical simulation of finite temperature QCD.- Entire range of strong interaction thermal dynamics.

Main FeaturesMain Features1. We make space-time discrete and lattice if points with

finite spacing a 0 gives continuous limit. We have lattice of sites. and V = T = 1. Divergences due to small (infra-red) and large (ultra-

violet) momenta disappear.2. = plaquette = square in space-time grid

3N N 3( )N a

1( )N a

U

matter fields as site variables

= Link variable between adjacent sites

Wilson Loop Order Parameter

, U

exp ( )i iiga A x

1

3

0exp ( , , )

V

z d d dU d d xL A

1 ln1

V

Z

VT

ln

T

ZP T

V

exp FL T

Results:Results:

• No quarks Only gluons : 1st order

• 3 or 4 massless quarks : 1st order

• 2 massless quarks Continuous

• 2 massless, non zero Continuous

• Now calculations started with on the lattice Action

sm

G QS S S B

QCD Phase Diagram Reference :QCD Phase Boundary and Critical Point in a Bag Model Calculation

C. P. Singh, P. K. Srivastava, S. K. Tiwari, Physical Review D (Accepted)

QGP Equation Of State -

BTT

TTTTP

Bs

BBSB

BQGP

2/3222

2

2/3

43

222

42

42242

]9

2

3

8[

23

8

]81

1

9

2

9

11[

1629

1

90

37

where1

2

22)]

622.15089.0[ln(

29

12

TBs

Here we have used B1/4=216 MeV and Λ =100 MeV.

Hadron Gas Equation Of State -The Grand canonical partition function using full statistics and including excluded

volume correction in a thermodynamically consistent manner -

]1)[exp(

1

6ln

022

4

20

TEmk

dkkdV

T

gZ

iii

VNV

V

iexi

jjj

i

Where g i is the degeneracy factor of ith species of baryon, E is the energy of the particle

V0i is the eigen volume of one ith species of baryon and is the total volume

occupied

0j

jjVN

We can write above equation as -

])[exp(

1

6

)1(ln

022

2

2

0

ii

i

ii

iij

jexj

exi

T

Emk

dkk

T

gI

IVnVZ

Where

)exp(Ti

i

and Is the fugacity of the particle, nexi is the number density of jth type

of baryons after excluded volume correction.

'22 )1()1( iii

iiiiexi IR

RIIRn

Using the basic thermodynamical relation between number density and partition function

We can write as -

Where i

iexi VnR 0

Is the fractional volume occupied. We can write R in an opera-

-tor equation - RRR

Where 0

0

1 R

RR

with20'000iiiiii VIVnR

And the operator

i iiiiVnR 00

01

1

Using Neumann iteration method, we get -

RRRR 2

Solving this equation numerically, we can get the total pressure of hadron gas after

Excluded volume correction is - i

mesoni

iii

exHG PIRTP )1(

To draw the Phase diagram we uses the Gibbs’ equilibrium condition of

phase transition - ),(),( ccQGPccexHG TPTP

New and Interesting Features :

(1) By Maxwell construction, it gives first order phase transition : PH (Tc,µC) = PQ (Tc,µC)

(2) In cross over region, PQ > PH

(3) End point of first order line is critical point

(4) Our EOS for HG is thermodynamically consistent nB=∂PH/ ∂µB

(5) Freeze out curve from HG description.

(6) We have used full quantum statistics so we cover entire (T, µB) plane

(7) We have used QGP EOS in which perturbative corrections have been added and

non perturbative term includes Bag constant B0

Chemical Freeze out points :By fitting the hadron multiplicities at different energy

√SNN

(GeV)

2.7 3.32 3.84 4.32 4.84 6.3 12.3 17.3 130 200

T (MeV)

70 87.1 98.1 106.4 113.6 128.0 150.4 156 163.3 163.5

µB

(MeV)

760 684 630 588.5 548.7 461.1 278.4 209 31.7 20.7

Color Flavour Locked ( CFL) Phase :

30

0

16.00

9220

fmnn

BEMMeV

B

B

Fermi degenerate matter:

3

2

2/322

2

3

03

2

1

332

42

fm

Mkdkkn COF

k

BCF

MeVCO 1200so For quarks

Neutron star densities

3CO

Fk

0105 nAt Fermi surface -> qq interaction is weakly attractive ( Single gluon exchange)

Results into BCS pairing instability diquark condensate <qq> ≠ 0

2 Δ = energy gap between highest occupied and lowest vacant one particle state

Analogous to BCS cooper pair condensate in Superconductor

2 Δ

EF

E

k

K. Rajagopal

Δ 10 -100 MeV

Sm3

),(),( kkqq baabSC

a, b = 1,2

α, β =1,..3

Not colour neutral≠ 0

Colour Superconductivity

T no true phase transition between SC and QGP

If Sm Color Flavour Locked phase

0),(),( kkqq baabii

iCFL

Gluon mass ~ Δ

Superfluidity Chiral Symmetry is broken, B is not a good Q. No.

Critical point predicted by our

new model

Chemical freeze out points

Critical point by Our new Model

LTE04 LR04 LTE03LR01

NJL/inst

NJL

NJL/I

NJL/II

Critical point by different Lattice Model

Critical point by different NJL Models

New Findings :

(1)We show the presence of cross-over region and precise location of critical

point

TC= 160 MeV, µC= 156 MeV

(2) Entire conjectured phase boundary has been reproduced

(3) We get a first order deconfining phase transition

(4) Chemical Freeze out curve lies in close proximity to the critical point

Steps in space –time picture of nucleus-nucleus Steps in space –time picture of nucleus-nucleus collisions:collisions:

Pre equilibrium stagePre equilibrium stage At (z,t) =(0,0), nuclei collides and pass At (z,t) =(0,0), nuclei collides and pass through each other, nucleons interact with each other.through each other, nucleons interact with each other.

Formation stageFormation stage Quarks and gluons (qq,gg) are produced in Quarks and gluons (qq,gg) are produced in the central region a large amount of energy is deposited. the central region a large amount of energy is deposited.

EqulibrationEqulibration Due to parton interaction plasma evolves from Due to parton interaction plasma evolves from formation stage to a thermalized QGP. formation stage to a thermalized QGP.

HadronizationHadronization Thermalized plasma expands and cools until Thermalized plasma expands and cools until hadronization takes place and mesons and baryons are hadronization takes place and mesons and baryons are created.created.

Freeze-outFreeze-out When temperature falls further, the hadrons no When temperature falls further, the hadrons no longer interact and they stream out of the collision regionlonger interact and they stream out of the collision region

towards the detectors.towards the detectors.

Signatures of QGP:Signatures of QGP:1.1. HGHG Hot,dense hadron gas ( Background)Hot,dense hadron gas ( Background)

QGPQGP Quark Gluon matter with a collective Quark Gluon matter with a collective

behaviour.behaviour.

How to eliminate background contributions.How to eliminate background contributions.

1.1. Lack of proper understanding of ultra relativistic Lack of proper understanding of ultra relativistic Nuclear collisions:Nuclear collisions:

a. Thermal statistical modela. Thermal statistical model

b. Superposition of hadron-hadron scattering.b. Superposition of hadron-hadron scattering.

c. Transport theory Non equilibrium.c. Transport theory Non equilibrium.

Signals of QGPSignals of QGP1. 1. Dilepton production Thermometers Production rates and

momentum distribution of these particles depend on the momentum distribution of the quarks and antiquarks in the plasma. Background contributions are Drell-Yann processes.

Effects Mass and widths of resonances can shift in QGP ( e.g. , etc) 2. J/ suppression J/ is produced in pre equilibrium stage. c

and c cannot be thermally produced in QGP. J/ while passing through the deconfining QGP medium, dissociates into c c pair which separate from each other and there is less probability that they can combine with each other to form J/ after QGP.

A-A collisions J/ QGP J/ formation less likely

Pre-equilibrium cc separates

A-A J/ HG J/ less in number absorption or Rescattering

3.Strangeness enhancement3.Strangeness enhancement

• For Baryon dense matter For Baryon dense matter 300 MeV300 MeV

150 MeV150 MeV

3

3 12 2 2

16

(2 )

exp 1

s s

s

d pn n

p m

T

3

3

16

(2 )exp 1

u

q

d pn

p

T

q

sm

sn un

Reference : M. Mishra and C. P. Singh

Phys. Rev. C 78, 024910 (2008)

a)a) Baryon free QGPBaryon free QGP

T >> = T >> =

a)a) Lower thresholdLower threshold

ss 300 MeV (QGP)ss 300 MeV (QGP)

KK 1000 MeV (HG)KK 1000 MeV (HG)

d) gg ss facilitates strangeness in QGPd) gg ss facilitates strangeness in QGP

After hadronization of QGP, ratio of particles like etc isAfter hadronization of QGP, ratio of particles like etc is

large.large.

4. Jet Quenching4. Jet QuenchingIn p p collisions, back to back two jets are produced. But when jet In p p collisions, back to back two jets are produced. But when jet

pass through the dense quark medium, one jet is more quenched and pass through the dense quark medium, one jet is more quenched and

this effect is seen at RHIC.this effect is seen at RHIC.

sm sn un

, ,K

5. HBT Interferometry:5. HBT Interferometry: When all the interactions are stopped, still Bose Einstein When all the interactions are stopped, still Bose Einstein

attraction or Fermi Dirac repulsion between a pair of particles attraction or Fermi Dirac repulsion between a pair of particles always exist.always exist.

Interference in Interference in coincident detectorscoincident detectors

Correlation C( , ) = Correlation C( , ) =

== Fourier transform of Fourier transform of

freeze-out densityfreeze-out density ==

Large radii large volume at thermal freeze-outLarge radii large volume at thermal freeze-out

Signals of 1Signals of 1stst order phase transition order phase transition

1p

1p2 1 2

1 1 2 2

( , )

( ) ( )

N p p

N p N p221 ( )Q 2 2 / 21 Q Re

2p

2p

Experimental StatusExperimental Status1987-1999:1987-1999:Brookhaven National Lab : AGSBrookhaven National Lab : AGS Si, Au beams wereSi, Au beams were

accelerated to 14.6 GeV/Aaccelerated to 14.6 GeV/A

CERN SPSCERN SPS S and Pb beams upto 200 GeV/AS and Pb beams upto 200 GeV/A

Inference Inference Colliding nuclei are stopped FireballColliding nuclei are stopped Fireball

Large number of produced particles which Large number of produced particles which

cannot be obtained by simple superposition of cannot be obtained by simple superposition of

p-p collisions.p-p collisions.

20002000::BNL RHICBNL RHIC Collider Experiments (Au-Au at 200 GeV/A)Collider Experiments (Au-Au at 200 GeV/A)

1

2CM t LabE m E

SUMMARYSUMMARY

- JET QUENCHING has been observed .- JET QUENCHING has been observed .

- J/ suppression.- J/ suppression.

- Large enhancement of strangeness.- Large enhancement of strangeness.

- HBT a large freezeout volume.- HBT a large freezeout volume.

- A perfect liquid with zero viscosity.- A perfect liquid with zero viscosity.