Manuel Calderón de la Barca Sánchez

Intro. Relativistic Heavy Ion Collisions

The QCD Phase Transition

!   What happens to strongly interacting matter !   at high temperature? !   at high density?

!   Key Features: !   Intrinsic size of hadrons

!   Hadron radius: !   Need a Volume to exist. !   è Implies a limiting density: !   where

! Pomeranchuk, Doklady Akad. Nauk. SSSR 78 (1951) 2

!   Resonances !   Exponential hadron spectrum:

!   First appearance: Statistical Bootstrap Model, R. Hagedorn: Nuovo Cim. Suppl. 3, 147 (1965) 2, 16

!   Hadron Thermodynamics: Limiting temperature of Hadronic Matter. !  

!   Can we go beyond nc and Tc?

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rh ≈ 1 fmVh ≈

43πrh

3

nc ≈1Vh

= 0.23 1/fm3 ≈ 1.5nNM nNM = 0.16 1/fm3

ρ(m) ~ exp(m /T )

Tc ≈ 150−200 MeV

! Deconfinement transition: Wear your colors proudly! ! Hadronic Matter: Colorless constituents of hadronic size !  Quark-Gluon Plasma: Colored constituents, pointlike.

! Deconfinement: è Color Conductor Transition in QCD !   Chiral transition: Lose the weights that bind you!

!  Shift in the effective mass of constituent quarks. !  At T=0 in the vacuum: quarks are “dressed” with gluons

!   Constituent quarks !   Bare quark mass: , becomes constituent quark mass:

!   In a hot QGP: Dressing melts. !  Since , Lagrangian has chiral symmetry for

!   : Chiral symmetry is spontaneously broken. !   : Chiral symmetry restoration.

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mq ≈ 0 Mq ≈ 300 MeV

Mq → 0

Lchiral

QCD ~ mψψ mq ≈ 0

Mq ≈ 300 MeV

Mq → 0

!  Chiral: “Handedness” !  Dirac Fields can be decomposed into “right” or “left” handed projections

!   where

!  For massless particles: chirality is the same as helicity ! Helicity:

!   Right Handed: direction of motion equals direction of spin.

!  For massive particles: chirality ≠ helicity !  Must rely on definition of ψL and ψR

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ψL =1−γ 5

2ψ, and ψR =

1+γ 5

2ψ γ 5 ≡ iγ 0γ1γ 2γ 3 =

0 0 1 00 0 0 11 0 0 00 1 0 0

#

$

%%%%

&

'

((((

!   When ψL and ψR transform independently !   QCD with two massless quarks

!   In terms of left and right handed spinors:

!   Define: so we can write:

!   This is invariant under !  a rotation of qR by any 2x2 unitary matrix !  a rotation of qL by any 2x2 unitary matrix !  This symmetry is called chiral symmetry

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L = uiγ µ ∂µu+diγ

µ ∂µ d +Lgluons

L = uLiγ

µ ∂µuL +uRiγµ ∂µuR +dLiγ

µ ∂µ dL +dRiγµ ∂µ dR +Lgluons

q = ud

!

"#

$

%&

L = qRiγ

µ ∂µ qR +qLiγµ ∂µ qL +Lgluons

!  For two massless quarks, the QCD vacuum breaks the exact SU(2)RxSU(2)L symmetry. !  Quark condensate: at low temperature !  Goldstone bosons which correspond to the three broken generators: pions. !   They are massive: QCD gives mass to pions, protons.

!   at large T : order parameter. ! Pions would be approximately massless at large T.

!  In the real world, QCD vacuum only has an approximate SU(2)RxSU(2)L symmetry. ! Pions have small, but non-zero mass, at large T.

!   Changes in the spectral properties of ρ meson?

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ψ ψ ≠ 0

ψ ψ → 0

! Diquark Matter ! Deconfined quarks : attractive interaction

!  Can form colored, bosonic, “diquark” pairs !  QCD Equivalent of QED

Cooper pairs

!  Form QCD quark condensate !   Color superconductor

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!  Simplest form of Confined Matter: !  Pion Gas:

!  Simplest form of Deconfined Matter: !  Ideal, weakly coupled, Quark-Gluon Plasma !   !   B: Bag Pressure

!  QCD Vacuum: ideal color dielectric !   Expels color field !   Color charge is confined !   Bag pressure:

!  Difference btw physical vacuum & quark-gluon ground state

!  B1/4 ~ 200 MeV (Quarkonium Spectroscopy) 4/8/12 Phy 224C 8

Pπ =π 2

903T 4 ≈

13T 4

PQGP =π 2

902×8+ 7

8[2×2×2× 3]

#

$%

&

'(T 4 − B ≈ 4T 4 − B

!   Compare: !  

!   Phase transition from Hadronic Matter at low T to QGP at high T.

!   Critical Temperature:

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Pπ (T ), PQGP (T ) vs. T

Increasing T

Pπ = PQGP ⇒3π 2

90Tc

4 =37π 2

90Tc

4 − B⇒ T 4c =

4517π 2 B⇒ Tc ≈ 150 MeV

!  Pion Gas: !  QGP: !  1st Order phase transition,

by construction: !  At Tc, energy density changes abruptly

!  Discontinuous jump !   Latent heat of

deconfinement

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π =

π 2

10T 4

QGP =

37π 2

30T 4 + B

!  Compare energy density and pressure !  Ideal Gas:

!  “Interaction Measure”:

!  Shows that for QGP is strongly interacting (not ideal gas)

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= 3P

Δ ≡− 3PT 4 =

0, T <Tc4BT 4 , T ≥Tc

%

&'

('

Tc ≤T ≤ 2− 3Tc

!  Partition Function: Z(T,V) !  Thermodynamic quantities:

!  For 2, 2+1 flavors: !  

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= (T 2 /V ) ∂lnZ

∂T"

#$

%

&'V

P =T ∂lnZ∂V

"

#$

%

&'T

Tc 175 MeV, c ~ 1 GeV/fm3

! Deconfinement Transition !   Polyakov Loop :

!   free energy of pair for

!   Chiral Transition !  Chiral Condensate:

!   Measures dynamically generated (constituent) quark mass

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mq →∞

L(T )=Tr U4 ( j,

n)j=0

Nt−1

∏#

$%%

&

'((= lim

r→∞e−FQQ (r ,T )2T

FQQ QQ r→∞

L(T )= 0, T <TL confinement≠ 0, T >TL deconfinement

"#$

%$

χ(T )≡ ψψ ~ Mq

χ(T )≠ 0, T <Tχ chiral symmetry broken

= 0, T >Tχ chiral symmetry restored

#$%

&%

mq → 0

! Polyakov Loop and Chiral Condensate !   In the appropriate limits: both transitions fall in the same

universality class as the Z(3) Potts model !   In the real world, symmetries are not exact

!  L(T) and χ(T) still indicate a rapid change !   Both transitions occur at about the same temperature

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F. Karsch and E. Laermann: Phys. Rev. D 50, 6954 (1994)

!  Interaction measure: !  Peaks above Tc

!  Two regimes of QGP !  Strongly coupled QGP

! sQGP:

!  Weakly coupled QGP ! wQGP:

!  Interaction measure in real QCD: !  interpreted as arising due to Bag pressure B.

!  M. Asakawa and T. Hatsuda: Nuc Phy A 610, 470c (1996)

!   or colored “resonance” states above Tc. !   E. Shuryak and I. Zahed: Phys. Rev. C 70, 021901 (2004) 4/9/12 Phy 224C 15

Tc ≤T ≤ (2−2.5)Tc

T ≥ 2.5Tc

!  At µ~0: rapid cross-over !  No discontinuity, no thermal

singularity !  Strictly speaking:

!  Phases are not distinct !  No phase transition between them

!  Lattice: rapid transition. !  Something is changing, but how to better define/understand it?

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!   Making pudding, boiling egg !   Treated using “percolation”

!  Formation of clusters !  Correspond to critical behavior

!   Geometric (not thermodynamic) quantities diverge !   Cluster size

!   Cannot be obtained from partition function

!   Two-dimensional disk percolation !  distributed lilies randomly on surface of a pond (overlap ok) !   small disks, area a !  Pond, area !  When can ant walk across?

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A a

!  For constituents with intrinsic scale è formation of infinite connected clusters

!  S(n):average cluster size !  increases with increasing density

!   n=N/A

! Suddendly, at n=nc, clusters are large enough to span pond !   S~A

!  Limit: !   diverge at

!  Geometric form of critical behavior 4/9/12 Phy 224C 18

N→∞, A→∞, n = constant

S(n), and dS(n)dn

n→ nc

!  2-D disks : !  68 % of space covered, 32 % empty !  when an ant can cross, a ship cannot, and vice versa

!   But Only in 2-D

!  3-D spheres (Hadrons: ) !   :

!   29 % of space covered, 71 % empty !   both cluster and empty space are connected

!   : !   71% of space covered, 29% empty !   connected vacuum disappears !   Assume medium is ideal gas of hadrons and

their resonances: 4/9/12 Phy 224C 19

nc =1.13 / πr2

nc 0.34 / (4πr3 / 3)= 0.16 fm−3

nc 1.24 / (4πr3 / 3)= 0.56 fm−3

rh 0.8 fm

nres (Tc )= nc ⇒ Tc 170 MeV

Hadron Percolation

Vacuum Percolation

!   At high temperatures and/or densities, strongly interacting matter becomes a QGP; !  How can we probe its properties and its behavior as function of temperature and density?

!  Given a volume of strongly interacting matter and an energy source, how can we determine its state at different temperatures?

!   Possible probes: !  Hadron Radiation !  EM Radiation ! Quarkonium Dissociation !  Parton energy loss

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!   Emission of light hadrons !  made of (u,d,s) quarks !   scale ~ 1 fm ~ 1/(200 MeV) !  Cannot exist in hot interior

!   Formed at transition surface btw QGP and vacuum

!   Light hadrons are born at T=Tc !   independent of initial temperature

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!   Carry information about hadronization stage

!  Experimental observable: !   Same relative abundances, independent of initial energy density

!  Does it not carry any information about pre-hadronic medium?

!   Medium is not contained !  Can expand freely !  Hydrodynamic flow

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!   Radial Flow !  Boosts hadron momenta

!   Elliptic Flow !  Non-spherical initial state in peripheral collisions !  Pressure gradients are different in different directions:

!   Azimuthal anisotropy

!   Final configuration of system depends on hydrodynamic time-evolution !  Can provide information about Equation of State of hot QGP

!   Quark-gluon Interactions !   Quark-antiquark annihilation

!  photons and dileptons

!   Key: Do not carry color !  Leave medium ~ without further interaction

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!  Provide information at time of their production !   Probe of Hot QGP

!  Difficulty: They are produced throughout system evolution !   Hadron gas can also produce electromagnetic radiation

!  Experimental goal: disentangle hot thermal radiation from QGP versus other sources

! Hadronic/EM Radiation: produced by medium. !   Other possibilities: External probes

! Quarkonium !   bound states !  Smaller than other hadrons

!  

!  Binding energies: !  ~0.5-1 GeV > Tc

!  Can survive above Tc in QGP

! Charmonia:

!  Different quarkonia: !  melt at different Temperatures

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cc ,bb

rQQ rh 1 fm

J /ψ(1S)→ rJ /ψ 0.2 fmχc(1P)→ rχ 0.3 fmψ '(2S)→ r $ψ 0.4 fm

!   Shoot an energetic parton (q or g) through QGP !  Measure energy of outgoing beam

!   Attenuation or “Quenching” !  Depends on medium density, ρ. !   Increases with temperature

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!   How do we get these external probes? !   “Hard probes”

! Quarkonia, charm, beauty, jets, energetic photons & dileptons

!  Formed early in the collision: τ~1/Q !  Hard scale: pQCD is applicable !  Rate can be compared to pp, pA data

!  Thermodynamics of the Strong Interaction !  There is a transition around 160-190 MeV !  Color deconfinement !  Chiral symmetry restoration !  latent heat increases energy density !  The plasma near Tc is strongly coupled.

!  Experimentally, we can probe it using: ! Hadronic and electromagnetic radiation ! Quarkonia !  Jets

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