Initial State of Heavy-Ion Collisions
1. Why is this interesting?
2. Light-cone kinematics.
3. Single nucleus: saturation and Color Glass
Condensate.
4. CGC meltdown: a linear treatment.
5. Lattice formulation.
6. Observables.
7. Where do we go from here? (potential uses &
shortcomings).
Why is this interesting?
Historically, QCD has mostly been studied in 2 regimes:
1. High energy (>> QCD ), weak coupling, weak fields: asymptotic
freedom, perturbative treatment of scattering.
2. Low energy (< QCD), strong coupling: symmetry breaking, hadron
spectrum, nonperturbative methods (lattice QCD).
Now RHIC, and even more so future accelerators, allow us to study a
novel regime:
3. High energy (>> QCD ), weak coupling, strong fields: classical
physics, possible evolution toward quark-gluon plasma.
The grand scheme
Partons (predominantly gluons) carrying small fractions (x) of
longitudinal momentum in the nuclear wave function are very
numerous and form a coherent state (a likeness of a condensate).
Large field magnitudes allow application of classical physics.
Large field magnitudes mean that nonlinearities are very important.
In a collision of heavy ultra-relativistic nuclei, gluons are liberated
(radiated) from the condenstate. This particle formation process
leads to a state far from thermal equilibrium, which may or may
not evolve into quark-gluon plasma.
Light-cone notation and kinematics
T
x
,
+
x
Z
v T wT
2
m2
1 pT
2
p
2
(thought of as the
light-cone “energy”,
so x is the light-cone
“time”)
-
y
ln
*
y
2
&
)('
p ,
%
1
"
$# "
p
"
p ,
!
Under boosts: p
1 2p
ln
2 m2
t
1 p
y ln
2 p
Momentum-space rapidity:
pT
2p p
2
m
In particular, since
p
v w
v
Invariant product:
v w v w
1
v0 v3
2
Call the beam axis z. Then
If a projectile is a composite object (a hadron, a nucleus),
Feynman's x of a parton is its share of the longitudinal momentum:
xP
(hence x is boost invariant).
<
@
M
mT
=
@
ln
A
A
=
ln x
?
X is related to rapidity:
Y
@>
p
1
ln
2
p
<
y
=
.
/
.
p
please work it out!
x
;
dN
P
d p
:
6
pT x
p
:
7
Q
2
8
d pT
2
6
2
4
21
0
xG x , Q
2
3
5
9
A parton distribution function (integrated)
DF is measured in DIS
3
Deep Inelastic Scattering in the Infinite Momentum Frame
If the struck parton absorbs
momentum Q from the lepton
m
2
F
C
xP Q
Q Photon
2
E D
B
Electron
4-
0
Struck parton
G
and x 2 M 2
2
Q2
2P Q
K
JI
x
Q
H
2
2
With P M
Hadron
xG(Q²,x)
Q²=200(Gev²)
20
N
M
X
These measurements reveal the smallx problem: the low-x multiplicity
threatens to violate
the Froissart
1
xG C ln
bound
x
L
5
The unitarity bound requires that xG(x,Q²) level off at some x, Q². But this is
not achieved perturbatively!
1
x
Z[
]
S R
T
xG Q² , x
Q
O
DGLAP
P
O
BFKL
exp
\
X W
Y
U
V
xG Q² , x
ln x
O
Saturation
O
PT is not up to the task because low-x matter is strongly interacting: as the
constituents proliferate, they begin to overlap (percolate), leading to
Classical physics
Both emerge naturally from
Decreasing x
McLerran – Venugopalan Model
The valence partons are confined to the longitudinal extent of the boosted
nucleus:
2R
b
`
x
2 RM
P
For the wee partons, the longitudinal extent is bounded from below by the
uncertainty relation
1
1
x
,
p
xP
g
g
e
h
f
d
O
c
a
_
^
O
The low-x (sea, wee) and the high-x (valence) components of the nuclear WF
behave differently. In the IM (lab) frame
so for x << 1/RM the wee partons “see” the valence ones as pointlike.
l
2p
j
1
p
m
k
x
j
i
On the other hand, the valence partons have much longer lifetimes:
pT
2
so they are seen by the wee partons as static.
Fast partons are static sources of color charge:
|
Nc A
The transverse density of the fast partons
o
RA
1 3
with R A A
2
QCD
Charge sources belong to different nucleons and add up incoherently
Q²
2
g²
C
n
F
QCD
squared charge per unit areaS
for a heavy
T
nucleus.
O
O
n
n
Estimate of the charge magnitude:
~
w v
x
u
Note that this current is not
covariantly conserved and must be
modified
rT
}
s
p
p
a
{zy
x
t
r
r
a
q
J
MV make a (reasonable) simplifying assumption (to be relaxed later): the charge
density is a Gaussian
white
noise (noteabthe locality, hence gauge invariance)
a
b
xT
yT
delta² y T x T
S
Describe slow partons using classical Yang-Mills EOM:
J
classical physics! D F
strong fields
Strong sources
¡
rT , x
²
¦
¦
,
®
rT , x W
¤£¢
¥
¤£¢
d xF F
S A,
i
d²x T dx Tr
gN C
4
¥§
EOM follow from the classical effective action
v
¬
ª©¨
ig
dx A r T , x
u
«
«±
¯
P exp
«
W u ,v r T , x
ª¬
ª©¨
°
where
»
, x traceless
º
0 ; A , Ai
Å
Ä
Â
x
static (independent of
É
U
Ì
Æ
É
Æ
Ç
ÊË
i
x , xT
Ê
È
Ê
È
There is still a lot of gauge freedom left!
i
F
0
A
x
,
x
U x , xT
Note that
ij
i
T
g
È
·
rT W
É
Ã
Â
Solution: insist on F ij 0 ; A
À
¿¾½
¼
,
º
·
Wx
»
x
¶
µ
g
´
³
J
¸
Á
¹
Show that this action gives EOM with a covariantly conserved current
).
Ï
Ó
Ñ
Ò
2
Õß
à
Ø
ÝÜÛ
Þ
i
.
satisfied byU I
Ai 0
Î
0
Ñ
A
x , xT
Ø Ù
×
T
å
á
i
0
, so we transform from CG to LCG by
æ
Ä
In the light-cone gauge (LCG) A
(transverse electric).
âã
Fi
The only non-vanishing field is
ä
Ú
Ô
Ô
A
Õ
Then
Ö
In the covariant gauge (CG)
Ï Î
Ð
Í
This leaves two independent degrees of freedom : U and
A
î
ð
x
éé
ç
z , xT
è
ï
è
dz
ñ
ig
è
P exp
ì ç
í
ê é
ë
è
ç
V x , xT
ô
ò
ôú
ò
ô
ò
ó
÷
õ ô
ò
ó
ùøô
ò
ò
õ ô
ö
÷
ó
ó
ò
ô
û
so in LCG U=V. In particular, for an infinitely thin (boosted) source
i
V x , xT
x V xT
Aj x , xT
x V xT jV xT
g
Now use this machinery to compute observables!
Mother of all observables: gluon distribution function.
þ
d³N
,
d²k T dk
P
ý
x k
þ
Q² k² T x
ý
d²k T d k
ÿþ
ý
xG x , Q²
d³N
d²k T dk
is the gluon k-space density.
a k ,kT a k ,kT
where
Work in LCG, then GDF will have a simple gauge-invariant interpretation:
â
x,y ,
' y U
' x U x,y Fj
.
k
I
in LGC for a suitably chosen
(
-
/.**
dz A z
)
P exp i g
)
+*
( )
,
#
d³x d³y e
Fj
"
k
U x,y
with
But
Fj k Fj
&%$
²
!
2
i x y k
Q² k² T Tr F j k F j
d²k T
xG x , Q²
ik A j k ,
Fj k
since in LCG
d kT
2
:
4
s
NC
rT ²
s
4
r T ²ln
;
5
2
2
<2
8
ik T r T
7
6
T
3
²
14
5
2
2
C
3
0
2 1
2
2
9
R² N in 1Gaussian
d²r
Tour
de force
integration:
d N
1
e
1 exp
N
4 =
>
ü
?
C
QCD
rT
Important points:
@
N
1. There is an infrared divergence cut off at
QCD
D E
kT
s
NC
H
2. At hard momenta
BC
A
(result not valid for softer momenta)
G
F
d²N d²k T
kT ²
kT
P
\
0.3
N
M
L
Z
^
_
X
[ZY
X
Z
Q0
2
QS x Q0 x 0 x
1 GeV ; x 0 3 10 4 ;
`Z
2
Y]
W
How can the saturation scale be
estimated?
HERA data are well described by
viewing a photon as a quarkantiquark pair multiple scattering
off the target (a dipole). From the
analysis of the dipole X-section
NC
U V
S R
T
Q
s
N
O
QCD
3. At soft momenta
2
R² N C 1
d²N d²k T 2
ln
g² N C
KC
K
JIA
S
D E
K
?
ΛQCD
s
NC
kT
which means:
(i) soft fields are strong if the coupling is
weak;
(ii) there is saturation (very slow growth of
multiplicity at soft momenta).
This is the promised Color Glass Condensate
An AA collision as seen by MV model
b
a
Source
Source
x A
0
d
c
x A
ab
Work in the gauge
(this is equivalent to LCG for each of the
single-source solutions).
f
2
Y
xT
X
l onY
i
j
mi k
kX
f
Y
ef
X
x
t
l t
u rs
m
p
q
Af
from EOM in the vicinity of z=t=0.
2x x
In particular,
Af
x
xT , ; A f
T xT ,
j
y
|Y
X
{}
|Y
}
{X
o~
z
j
y
Obtain
r
w m s
t
rvus
p
r
m s l
xwt
t
rs
l t p r
s
2
++
+
+Z
-
h
xT ; J 2
1
J2 ,
p
1
J1
Given the structure of the source, seek solution in
the form
A A1
x
x
A2
x
x
AF
x
x
p
3
x
k
m X
onY
i
j
il k
where
J1
T
-
D F
e
Central
region
g
Solve the classical EOM, now with 2 sources:
Note:
is dictated by the choice of gauge.
only through , but not the space-time rapidity
y
2. the forward light-cone fields depend on x
1. The form of A
x
1
ln
2
x
Now plug A
p
into the classical EOM. The step functions give rise to singularities at
The solution is boost-invariant!
0.
T
A 2 A1
0
ig
A1 A 2
2
A1 A 2 ;
0
Require their cancellation to obtain (you are welcome to work this out!):
This is a nonabelian magic: an overlap of two pure gauges in the FLC is nontrivial.
the solution contains all powers of
T
§
¦
¥
,
¢
¢
1,2
2
hence all powers of
¤
;
£
V 1,2 exp ig
¡
;
1,2
V
T
i
V
g
A1,2
Now solve the EOM in the FLC. Recall that the single-nucleus transverse fields are of the form
1,2
1,2
.
How do you linearize?
kT
¨
©
g2
Perturbation expansion in
And the dimension is reduced to 2+1 by
is an (SU(N) adjoint) scalar field.
Ì
Í
2
Æ
Å Ä Â
T
Å
Ä Â
Ê
T
1
0.
Ê
Ë
1
ÉÈÇ
Å
A 0;
Æ
Ä ÅÂ
Ä Ã
 Á
The linearized EOM:
2
·
0.
½
À
A
.
»
A
¼
0
¾ ¸
¿ »
¹ º
¹¸
In this language x A x A
boost invariance. Now
,
¶
xT ,
A more convenient choice of coordinates in the FLC of the collision is
³
±
³
±
²
µ´
®
¯
°
«
¬
ª
Keep the lowest nontrivial (2nd) order in
both in the initial conditions and in the EOM. Then
(i) the EOM linearize upon a ( - independent) gauge choice Tj A fj 0 (WWIO!) ;
(ii) the expansion involves only powers of
T ;
2
g
s
(iii) the theory is classical, hence g can be eliminated by A A g ;
This gives, e.g., for the transverse modes
4
ÐÔ
ÕÖ
Ñ
cos k T
Ï
Î
Î
A0 k T J 0 k T
A0 k T
×
Ñ
Ð
2
ä ã
Üâ
äåÝ ã
Üâ
à
so the particle number at late times
A0 k T
ßâ
Ú
Ù
kT A kT ,
2
(at late times)
2
Û
Ï
Ø
kT
d d d xT ,
Û
áàß
ÞÝÜ
Û
N kT
d x
â
The space-time measure now is
Ù
4
Ó
ÐÑ
Ï
Ñ
Ò/Ð Ñ
Ï
Î
Ï
A kT ,
(per mode per unit rapidity). Unfortunately, to this order, this is IR-divergent mode by mode!
í
2
1
² p² p k ²
ï
ç
d²p T
æ
î
1
kT ²
ìç
ì
è
s
æ
ç ëêé
æ
ç
4
ST N C N C ² 1
g²
2
æ
N kT
4
ø
ñ
÷
ö
ô
ð
óòñ
ð
ö
õ
So all momenta are involved, and our low-density expansion is bound to fail. Cutting the
divergence by hand,
4
k
4
s
N kT ST N C N C ² 1
ln T
g²
2 kT
s
but we must do better than that if we want real numbers.
?
Implement the initial conditions and solve the EOM numerically to all orders in the charge density.
To this end, reformulate it on a 2d lattice (discretize only the transverse plane).
UP
j,y U
j A z , At
j,x U
x ,y
í
n
U j , n At , j
U
S exp ir
r 0
r
Variation wrt A has the usual meaning. Variation wrt U means D S U
t
n
At , j
ý
t
U j , n At , j
U j ,n
t
n
U j ,n
ý
2
t
1
2
U j ,n
Tr U P
2
At , j
ý
i At , j
1 2
At , j
2
e.g. , M t , jn
1
NC
þ
1
ÿ
n
M z , jn
þ
M t , jn
Tr
û
1
NC
j
1
2
Tr F zt
2 NC
ý
dz dt
S
ú
û
î
é
ü
ù
Consider IC first. Work with z, t or x . Vary the lattice action (U are SU(N) unitary matrices).
(Lie derivative). Derive from here Lagrangean EOM, and, just as in the continuum, obtain the mathing
conditions on the FLC of the collision event.
U2 U
0;
n
*
h.c.
'
%(
'
U2
(
I U1
'
U
j
n ,n
+
j ,n
)
' %$
h.c.
'
'
(
)
I
%(
U1
'
Tr
*
,
h.c.
I
U2 U
&%$ #
i
4 NC
U1
"
!
Tr
We then obtain the IC for the lattice fields (WWIO):
.
Note that the IC for U is implicit. There is an explicit expression for SU(2), but we could not find one for
higher N (you are welcome to try). U can always be found numerically (by relaxation).
5
34
D
2
P
1
BT
01
01
ET
6
First, in the continuum
d xT
0
2
0
H
-
.
/
Now move on to the EOM. These follow from the hamiltonian in the Schwinger gauge (per unit
rapidity).
1
1 2
1
2
2
2
2
2
n
NC
n
>
j
U j ,n
:
=
>
U j ,n
=
) . On the lattice
D
8
<
Tr U P ;
=
1
;
BT
:
;
8
d xT
8
9
7
(color indices2 suppressed. P is1 the2 conjugate
momentum of
1
j
@
E
@
@
@
B
?
?
BA
E j , nthat
, E l ,the
f
E j , n ; fields
E j , nare
, Unot
i jl momenta
m lattice
jl mn
l , m conjugate
mn U j , n of the rsp. A's:
Note, however,
chromo-electric
D
D
AE
D
D
G
C
CF
P,
NOL M
JK
I
Local charge conservation is expressed by the Gauss' constraint, D E
respected by the dynamics.
0,
Here is, then, the game plan:
j
(Gaussian random, independent for each nucleus and for
2
, compute the respective
For each nucleus, obtain the gauge functions by solving
gauge transformations V j exp i j
and from those the (pure-gauge) transverse fields
U j ,n V j V j n
V
Z
H
\ \
H
Use the matching conditions to obtain the initial values for the fields inside the light cone.
Solve the hamiltonian equations of motion, subject to Gauss' constraint.
H
[
W
Y
X
H
UTS
Q
R
H
P
Generate the color charge distribution
each lattice site j). More on that later.
Compute “observables”.
H
Tricks of the trade:
Use FFT to solve the Laplace eqns; use suitably modified leapfrog to respect Gauss' constraint
Relation to continuum physics
Nuclear radius R (6.5fm for Au)
Lattice spacing a (the reason you have not seen it is that other quantities are measured in units of a)
^
H
H
1.4 GeV @ RHIC
Saturation scale
H
]
We have the following dimensional input parameters at our disposal:
S
H
_
Confinement scale QCD , or other suitable color neutrality scale (disregard this for now, we will
come back to it later)
a
a 0, R a
, a S 0,
As
but R
expected to behave in the continuum limit as
d
Therefore, a quantity of dimension d is
R .
h
g
f
f
f
d
S
e
`
b`
`
c
const.
S
S
Now on to the results. First, consider (our proper-time Hamitltonian) energy
density.
How can all this information be put to a good use?
H
An obvious way of viewing these results is:
At some point, as the system expands,
classical approximation loses validity.
Use the classical configuration at some
earlier time as an initial state for, e.g.,
transport eqns (later in this school).
But perhaps the initial (classical) state can
be observed directly? Could there a
similarity between the initial gluon and
the final hadron configuratons (partonhadron duality [KLN])?
However, it is not clear yet whether the
recent dA data from RHIC (supposedly
more sensitive to the initial state) favor
this possibility.
H
Recent reviews of CGC: E. Iancu & R. Venugopalan, hep-ph/0303204; E. Iancu, A. Leonidov, L.
McLerran, hep-ph/0202270.
H
Light-cone physics (kinematics & much more): S.J. Brodsky, H.C. Pauli, S.S. Pinsky, Phys.Rep. 301
(1998) 299.
General reference on heavy-ion collisions: C.Y. Wong, Introduction to Heavy-Ion Collisions, World
Scientific (ISBN 981-02-0264-4).
H
H
Bibliography
Lattice field theory classic: M. Creutz, Quarks, Gluons and Lattices, Cambridge University Press
(many editions).
H
CGC – Color Glass Condensate
H
GDF – Gluon Distribution Function
H
DIS – Deep Inelastic Scattering
H
IMF – Infinite Momentum Frame
H
LCG – Light-cone Gauge
H
(Known) abbreviations
WWIO – Welcome to Work It Out
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