Smallx Theory Overview

1
Small-x Theory Overview

• Yuri Kovchegov
• The Ohio State University
• Columbus, OH

2
Outline

• A brief review of Saturation/Color Glass Physics
• Hadron production in p(d)A collisions going from
  mid- to forward rapidity at RHIC, transition from
  Cronin enhancement to suppression.
• Open questions what needs to be measured?

• Di-leptons in p(d)A.
• Back-to-back jet correlations.
• Is suppression tied to fragmentation region?
• Open charm production in p(d)A.

3
Nuclear/Hadronic Wave Function

• Imagine an UR nucleus
• or hadron with valence
• quarks and sea gluons
• and quarks. Define
• Bjorken (Feynman) x as

p
k
4
Rest frame of the hadron/nucleus

• Longitudinal coherence length of a gluon is
• such that for small enough xBj we get
• with R the nuclear radius.
• (e.g. for x10-3 get lcoh100 fm)

5
Color Charge Density

• Small-x gluon sees the whole nucleus coherently
  
• in the longitudinal direction! It sees many
  color charges which
• form a net effective color charge Q g (
  charges)1/2, such
• that Q2 g2 charges (random walk). Define color
  charge
• density
• such that for a large nucleus (A1)
• Nuclear small-x wave function is
  perturbative!!!

McLerran Venugopalan 93-94
6
McLerran-Venugopalan Model

• The density of partons in the nucleus (number of
  partons per
• unit transverse area) is given by the scale m2
  A/pR2.
• This scale is large, m ?QCD , so that the
  strong coupling
• constant is small, aS (m)
• Leading gluon field is classical! To find the
  classical gluon field
• Aµ of the nucleus one has to solve the Yang-Mills
  equations,
• with the nucleus as a source
• of color charge

Yu. K. 96 J. Jalilian-Marian et al, 96
7
Shadowing Ratio

• Defining the shadowing ratio
• for the unintegrated gluon
• distributions

Enhancement (anti-shadowing), most glue are here
we plot it for the distribution found
Shadowing! (small k ? small x)
? Gluons are redistributed from low to high pT.
8
DIS in the Classical Approximation

• The DIS process in the rest frame of the target
  is shown below.
• It factorizes into

with rapidity Yln(1/x)
9
DIS in the Classical Approximation
The dipole-nucleus amplitude in the classical
approximation is
A.H. Mueller, 90
Black disk limit,
Color transparency
1/QS
10
Quantum Evolution

• As energy increases
• the higher Fock states
• including gluons on top
• of the quark-antiquark
• pair become important.
• They generate a
• cascade of gluons.

These extra gluons bring in powers of aS ln s,
such that when aS 1 this
parameter is aS ln s 1.
11
BFKL Equation
Balitsky, Fadin, Kuraev, Lipatov 78
Start with N particles in the protons wave
function. As we increase the energy a new
particle can be emitted by either one of the N
particles. The number of newly emitted particles
is proportional to N.
12
Nonlinear Equation
At very high energy parton recombination becomes
important. Partons not only split into more
partons, but also recombine. Recombination
reduces the number of partons in the wave
function.

Yu. K. 99 (large NC QCD) I. Balitsky 96
(effective lagrangian)
13
Nonlinear Equation Saturation

Gluon recombination tries to reduce the number of
gluons in the wave function. At very high energy
recombination begins to compensate gluon
splitting. Gluon density reaches a limit and does
not grow anymore. So do total DIS cross
sections. Unitarity is restored!
14
Nonlinear Evolution at Work
Proton

• First partons are produced
• overlapping each other, all of them
• about the same size.
• When some critical density is
• reached no more partons of given
• size can fit in the wave function.
• The proton starts producing smaller
• partons to fit them in.
• The story repeats itself for smaller
• partons. The picture is similar to
• Fermi statistics.This way some
• critical density is never exceeded.

Color Glass Condensate
15
Phase Diagram of High Energy QCD
Saturation physics allows us to study regions of
high parton density in the small coupling
regime, where calculations are still under
control!
QS
(or pT2)
Transition to saturation region is characterized
by the saturation scale
We are making real progress in understanding
proton/nuclear structure!
16
What Happens to Gluon Distributions?

• With the onset of evolution, as energy/rapidity
  increases, the shadowing ratio starts to decrease

Anti-shadowing
Suppression! A new phenomenon.
Shadowing
(Toy model picture)
17
What Happens to Gluon Distributions?

• At very high energy anti-shadowing disappears

Suppression everywhere!
(Toy model picture)
18
Why Suppression?
In general one can write
Without quantum evolution g1 and there is no
suppression
Quantum evolution leads to g1/2, such that
leading to

• Kharzeev, Levin, McLerran,
• hep-ph/0210332

19
Hadron Production
Lets start with gluon production, it will have
all the essential features, and quark production
will become clear after that.
20
How to Calculate Observables

• Start by finding the classical field of the
  McLerran-Venugopalan model.
• Continue by including the quantum corrections of
  the nonlinear evolution equation.
• Works for structure functions of hadrons and
  nuclei, as well as for gluon production in
  various hadronic collisions. Let us consider pA
  collisions first.

21
Gluon Production in Proton-Nucleus Collisions
(pA) Classical Field
To find the gluon production cross section in pA
one has to solve the same classical
Yang-Mills equations for two sources proton
and nucleus.
This classical field has been found by Yu. K.,
A.H. Mueller in 98
22
Gluon Production in pA McLerran-Venugopalan
model
The diagrams one has to resum are shown here
they resum powers of
Yu. K., A.H. Mueller, hep-ph/9802440
23
McLerran-Venugopalan model Cronin Effect
To understand how the gluon production in pA is
different from independent superposition of A
proton-proton (pp) collisions one constructs the
quantity
Enhancement (Cronin Effect)
We can plot it for the quasi-classical cross
section calculated before (Y.K., A. M. 98)
Kharzeev Yu. K. Tuchin 03
Classical gluon production leads to Cronin
effect! Nucleus pushes gluons to higher
transverse momentum!
(see also Kopeliovich et al, 02 Baier et al,
03 Accardi and Gyulassy, 03)
24
Understanding Cronin Effect
It is easier to understand this enhancement if we
note that, amazingly, one can rewrite gluon
production cross section in a kT factorized form
in terms of gluon distributions
Since for nuclear unintegrated gluon distribution
we had enhancement at high pT, it leads to
enhancement in RpA .
25
Proof of Cronin Effect

• Plotting a curve is not a proof of
• Cronin effect one has to trust the
• plotting routine.
• To prove that Cronin effect actually
• does take place one has to study the
• behavior of RpA at large kT
• (cf. Dumitru, Gelis, Jalilian-Marian,
• quark production, 02-03)

Note the sign!
RpA approaches 1 from above at high pT ? there is
an enhancement!
26
Cronin Effect
The position of the Cronin maximum is given by
kT QS A1/6 as QS2
A1/3. Using the formula above we see that the
height of the Cronin peak is RpA (kTQS)
ln QS ln A.

• The height and position of the Cronin maximum are
  increasing functions of centrality (A)!

27
Including Quantum Evolution

• To understand the energy dependence of particle
  production
• in pA one needs to include quantum evolution
  resumming
• graphs like this one. It resums powers of a
  ln 1/x a Y.
• 
  (Yu. K., K. Tuchin, 01)

28
Including Quantum Evolution

• Amazingly enough, gluon production cross section
• reduces to kT factorization expression (Yu. K.
  Tuchin, 01)

29
Our Prediction

• Our analysis shows that as
• energy/rapidity increases the
• height of the Cronin peak
• decreases. Cronin maximum
• gets progressively lower and
• eventually disappears.
• Corresponding RpA levels
• off at roughly at

Toy Model!
RpA
energy / rapidity increases
(Kharzeev, Levin, McLerran, 02)
k / QS
D. Kharzeev, Yu. K., K. Tuchin, hep-ph/0307037
(see also numerical simulations by Albacete,
Armesto, Kovner, Salgado, Wiedemann,
hep-ph/0307179 and Baier, Kovner, Wiedemann
hep-ph/0305265 v2.)

• At high energy / rapidity RpA at the Cronin peak
  becomes a decreasing
• function of both energy and centrality.

30
Understanding Suppression
Again, using the kT-factorization formula
And remembering that the unintegrated nuclear
gluon distribution at high (forward) rapidity y
is suppressed at all pT, we conclude that RpA is
also suppressed.
31
Other Predictions

• Color Glass Condensate /
• Saturation physics predictions
• are in sharp contrast with other
• models.
• The prediction presented here
• uses a Glauber-like model for
• dipole amplitude with energy
• dependence in the exponent.

figure from I. Vitev, nucl-th/0302002, see also
a review by M. Gyulassy, I. Vitev, X.-N. Wang,
B.-W. Zhang, nucl-th/0302077
32
RdAu at different rapidities
RdAu
RCP central to peripheral ratio
Most recent data from BRAHMS Collaboration
nucl-ex/0403005
Our prediction of suppression was confirmed!
33
Our Model
RdAu
pT
RCP
pT
from D. Kharzeev, Yu. K., K. Tuchin,
hep-ph/0405045, where we construct a model based
on above physics add valence quark contribution
34
Our Model
We can even make a prediction for LHC
Dashed line is for mid-rapidity pA run at LHC,
the solid line is for h3.2 dAu at RHIC.
Rd(p)Au
pT
from D. Kharzeev, Yu. K., K. Tuchin,
hep-ph/0405045
35
Future Experimental Tests
36
Di-lepton Production

• To calculate hadron production one always needs
  to
• convolute quark and gluon production cross
  sections with the
• fragmentation functions. Since fragmentation
  functions are
• impossible to calculate and are poorly known in
  general, they
• introduce a big theoretical uncertainty.
• Di-lepton production involves no fragmentation
  functions.
• It is, therefore, a much cleaner probe of the
  collision
• dynamics.
• Theoretical calculation for di-lepton
  production in dAu is
• pretty straightforward.

37
Di-lepton Production
from J. Jalilian-Marian, hep-ph/0402014
M2 is the photons invariant mass, kT and qT are
total and relative transverse momenta of the
lepton pair.

• The photon does not interact (while everything
  else is just
• like for gluons) theoretical calculations are
  simpler! They were
• first performed by Kopeliovich, Schafer, and
  Tarasov in 98.

38
Di-lepton Production
Again one should get suppression at forward
rapidities at RHIC.
Here we plot Rp(d)A integrated over kT and qT ,
for both pA and dA, for y2.2 as a function of M.
from J. Jalilian-Marian, hep-ph/0402014
39
Di-lepton Production
The suppression at forward rapidities at RHIC can
also be seen as a function of kT
RpA as a function of kT for M2GeV for y1.5
(short-dashed) and y3 (dashed), as well as for
M4GeV and y3 (lower solid line).
from Baier, Mueller, Schiff, hep-ph/0403201
40
Di-lepton Production
and as a function of centrality
RpA as a function of A for kT5 GeV, M2GeV and
y3 for the saturation model (solid curve) and
for analytical estimate RpA A-0.124.
from Baier, Mueller, Schiff, hep-ph/0403201
41
Back-to-back Correlations
Saturation and small-x evolution effects may also
deplete back-to-back correlations of jets.
Kharzeev, Levin and McLerran came up with the
model shown below (see also Yu.K., Tuchin 02)
which leads to suppression of B2B jets at
mid-rapidity dAu (vs pp)
42
Back-to-back Correlations
and at forward rapidity
from Kharzeev, Levin, McLerran, hep-ph/0403271
Warning only a model, for exact analytical
calculations see J. Jalilian-Marian and Yu.K.,
04.
43
Back-to-back Correlations
An interesting process to look at is when one jet
is at forward rapidity, while the other one is at
mid-rapidity
The evolution between the jets makes the
correlations disappear
from Kharzeev, Levin, McLerran, hep-ph/0403271
44
Back-to-back Correlations

• Disappearance of back-to-back correlations in
  dAu collisions predicted by KLM seems to be
  observed in preliminary STAR data. (from the
  contribution of Ogawa to DIS2004 proceedings)

45
Back-to-back Correlations

• The observed data shows much less correlations
  for dAu than predicted by models like HIJING

46
Is Suppression due to Proximity to Fragmentation
Region?
Kopeliovich et al suggest that observed
suppression may not be due to small-x evolution
in the nuclear wave function, but is due to
large-x effects in the deuteron.
To resolve this issue one would ideally want to
have dAu collisions at higher energy, where we
could test the same small-x region of the nuclear
wave function for which x of the deuteron is not
that large anymore CGC would expect
suppression, Kopeliovich no suppression.
Alternatively one can run dAu at lower energy
large-x effects would stay the same, nuclear x
would increase CGC would expect less
suppression, Kopeliovich would expect the same
amount of suppression.
47
Open Charm Production
Kharzeev and Tuchin model open charm production
in the saturation/Color Glass formalism by the
following model
It results in the spectra which fall off slower
than PYTHIA
Caution only a model, for exact calculation see
Tuchin 04 and Blaizot et al, 04.
48
Open Charm Production
Similar suppression applies to heavy flavors, in
particular to open charm. The figure below
demonstrates that in going from mid-rapidity to
forward rapidity open charm production
should start scaling slower than Ncoll , which
indicates suppression.
Charmed Meson Yield
from D. Kharzeev and K. Tuchin, hep-ph\0310358
Ncoll
49
Summary

• A lot of interesting new information about
  saturation/Color Glass dynamics can be found by
  studying dAu collisions at RHIC II.
• Things to look at are
• RpA of hadrons in forward direction
  (suppression?).
• RpA of open charm and J/y (forward suppression?).
• Di-leptons cleaner signal, expect suppression in
  forward direction as well.
• Various back-to-back correlations broadening,
  disappearance...

50
Summary

• Understanding nuclear and hadronic wave functions
  reveals interesting physics of strong gluonic
  fields and complicated interactions.