Bose-Einstein Condensation
and Thermalizationof the Quark-Gluon Plasma
YITP, Kyoto
August 24, 2011

Jean-Paul Blaizot, IPhT- Saclay
Collaboration withF. Gelis, J. Liao, L. McLerran, R. Venugopalan[arXiv: 1107.5296 v2]
WORK IN PROGRESS ! and SPECULATIVE....

The sQGP : a puzzle
Where is the apparent strongly
coupled character of the quark-gluon
plasma coming from ?

The strongly coupled quark-gluon plasma
Empirical evidence from RHIC (and LHC) data
- Strong opacity of matter (jet quenching, energy loss,...)- Collective behavior (elliptic flow, ...)- Small ratio of viscosity to entropy density- Small thermalization time, etc

Why this is puzzling- The coupling constant is not small, but not huge α s ∼ 0.3 ÷ 0.4- Strict perturbation does not work, but successful resummations exist- Understanding of early stages of HI collisions relies on weak coupling

Clue- «Strong coupling» behavior may appear at weak coupling, when manydegrees of freedom contribute coherently (e.g. collective phenomena, BCS,CGC, etc)

The over-populated quark-gluonplasmaHigh density partonic systems

Large occupation
numbers1xG(x, Q2 )∼22αsπR Q sThermodynamical considerations
Initial conditions(t0 ∼ 1/Q s )3Q
sn0 = n(τ = Q−1s )∼αs4Qs�0 = �(τ = Q−1s )∼αs

overpopulation parameter�0 /n0 ∼ Qs

n0 �0−3/4 ∼ 1/α1/4s

In equilibrated quark-gluon plasma�eq ∼ T 4

neq ∼ T 3

mismatch by a large factor (at weak coupling)

α−1/4sChemical potential does not help
Assume that the nmber of gluons is conservedfeq (k) ≡1eβ(ω k −µ) − 1

Maximum value of µ

growing function of µµ ≤ ω p=0 = m � 0

Screening mass
m20 ∼ αs

�d f0
∼ Q2sp dω pmeq ∼ α1/2 T ∼ α1/4 Qs

Maximum number of gluons that can be accommodated by aBE distribution
�
1d3 k3∼Tnmax =(2π)3 eβ(ω k −m) − 1

Formation of a Bose-Einstein condensate(when elastic processes dominate)
Most particles are in the BEC

�Q3s �1/41−αnc ∼αnc = n − ng

BEC contributes little to the energy density
Q3s 1/4
αs Qs ∼ α1/4 T 4 � �0nc m ∼αsEntropy considerationss∼�

ln f pps0 ∼ Q3sseq ∼ T 3 ∼ Q3s /α3/4

Note: overpopulation disappears at equilibriumng ∼ T 3ng � −3/4 ∼ 1

Note: inelastic processes inhibate the formation of a condensate

Kinetic evolution dominated byelastic collisions

Simple kinetic equation and two important scales
Non expanding plasma∂t f (k, X) = C k [ f ]A very schematic distribution functionf (p) ∼1for p < Λs ,αsInitially,f (p) ∼t ∼ 1/Qs

1 Λsfor Λs < p < Λ,
αs ω pf (p) ∼ 0 for Λ < pΛs ∼ Λ ∼ Qs

In small angle approximation��� ���2ΛΛ∂f �αsdf
�� ∼ s2 ∂ p p2+f (p)(1 + f (p))∂t colld p Λsp

ΛΛs
≡−αs

ΛΛ2s
≡
2αs

� ∞0

� ∞d p p2

dfdpd p p2 f (1 + f )0Note: when f ∼ 1/αs all dependence on coupling disappears
Note fixed point for BE distribution with T = Λs /αsThen, T ∼ Λ ∼ Λ s /α s (thermalization condition)

Simple estimates
Momentum integrals dominated by hard scale

1 2ng ∼ Λ Λ sαs�g∼Λng

1�g ∼ Λs Λ3αs��c ∼ nc m ∼ nc ΛΛ sn = nc + ng

m2 ∼ αs

�d p p2

From transport equationΛ
tscat = 2 ,ΛsNote: the collision time is independent of α sd f (p)∼ ΛΛsdω p

Thermalization
Time dependence fixed by 2 conditions

Λ
tscat ∼ 2 ∼ tΛs3

Λ s Λ ∼ constantThenΛs ∼ Q sng ∼ n0

� t �1/7o� t � 730t� � 17tΛ ∼ Qs
t0m ∼ Qs (t0 /t)1/7

ts ∼ Λ3 ∼ Q3s (t/t0 )3/7

�c � t0 �1/7
∼�gt

At thermalization (Λ s ∼ α s Λ)�

1 1tth ∼Qs αs� 74

3s ∼ Q3s /α3/4∼Ts

Inelastic processes

The rate of inelastic and elastic processes
are comparable
1t scat∼ αn+m−2s

�

Λs
αs

�n+m−2 �

1m2

�n+m−4

Λn+m−5m2 ∼ Λ s ΛΛtscat = 2 ,ΛsRecall that there is no ‘equilibrium’ condensate with inelastic
processes
However the formation of a transient condensate is possible.
Whether it occurs or not is an interesting, difficult, question.

Effect of the longitudinal expansion

Longitudinal expansion
Simple (boost invariant) expansion
�d f ��pz
� = C[ f ]∂t f − ∂ pz f =t
dt � pz t

� + PL∂t � +=0t

Note: mean field terms have been dropped. But they are important.
E.g. instabilities may lead/maintain momentum isotropy.
Instead, ASSUME

PL = δ �0 < δ < 1/3

� t �1+δ� t �(4+δ)/7Then�g (t) ∼ �(t0 )and0Λs ∼ Qst

���tth1∼t0
αs0t7
� 3−δ,

Λ ∼ Qs� t �(1+2δ)/70t

Bose-Einstein condensation....
or not ??
With particle number conserved,n∂t n + = 0 ,tn = n0n� −3/4 ∼nc ∼Q2s 1.
∼tαs t0� t �1/4 � t �−3δ/400tt

3 � ��
Qs t0

αs�t �t1−

� t �(−1+5δ)/7 �0t

If inelastic processes important, situation unclear....

Summary
The Quark-Gluon plasma formed in the early stages ofheavy ion collisions is strongly interacting with itself up toparametrically late times when the system thermalizes.
A transient BEC may form on the way to thermalization