Date post: | 31-Dec-2015 |
Category: |
Documents |
Upload: | ariel-moss |
View: | 28 times |
Download: | 0 times |
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.