Pomeron loop equations and phenomenological consequences

1
Pomeron loop equations and phenomenological
consequences
Cyrille Marquet
RIKEN BNL Research Center
ECT workshop, January 2007
2
Contents

• The B-JIMWLK equations- scattering off a dense
  target
• The dipole model equations- scattering off a
  dilute target
• The Pomeron loop equations- combining dense and
  dilute evolution- stochasticity in the QCD
  evolution
• Phenomenological consequences- diffusive
  scaling- implications for deep inelastic
  scattering- implications for particle production

3
Introduction
x parton longitudinal momentum fraction
kT parton transverse momentum
transverse view of the hadron
Regime of interest
weak coupling regime
effective coupling
? dense system of partons mainly gluons (small-x
gluons)
high-energy scattering processes are sensitive
to the small-x gluons
the dilute/dense separation is caracterized by
the saturation scale Qs(x)
4
The B-JIMWLK equationsscattering off a dense
target
5
Effective description of the hadron
McLerran and Venugopalan (1994)
the numerous small-x gluons are responsible for a
large color field which can be treated as a
classical field
light-cone gauge
6
The JIMWLK equation
Jalilian-Marian, Iancu, McLerran, Weigert,
Leonidov, Kovner
a functional equation for the rapidity evolution
of
the JIMWLK equation gives evolution of the hadron
wavefunction for large enough Y
study the high-energy scattering of simple
projectiles (dipoles) off this dense hadron
7
Dipoles as test projectiles
the dipole
u quark space transverse coordinate v
antiquark space transverse coordinate
scattering amplitude off the dense target
scattering of the quark
JIMWLK equation ? evolution equation for the
dipole correlators
8
An hierarchy of equations
for dipoles scattering off a dense target
Balitsky (1996)
an hierarchy of equation involving correlators
with more and more dipoles
in the large Nc limit, the hierarchy is
restricted to dipoles
general structure
BFKL
saturation
9
Something is missing
frame invariance requires that H is invariant
under the following transformation (dense-dilute
duality)
color charge
color field
Kovner and Lublinsky (2005)
10
The dipole model equationsscattering off a
dilute target
11
The dipole model
in the large Nc limit, the emission cascade of
soft gluons is a dipole cascade
N-1 gluons emitted at transverse coordinates
? N dipoles
ansatz for the wavefuntion of a dilute hadron
dipole creation operator
Iancu and Mueller (2004) Mueller, Shoshi and Wong
(2005) C.M., Mueller, Shoshi and Wong
(2006) Hatta, Iancu, McLerran and Stasto (2006)
12
Scattering of projectile dipoles
high-energy scattering of dipoles off this dilute
hadron
obtained from T a after inverting
?
13
A new hierarchy of equations
for dipoles scattering off a dilute target
I denote
The equation for T(n) reads
k 1 ? the BFKL equation
k gt 1 ? fluctuation terms
14
Structure of the fluctuation term
general structure
BFKL
fluctuation, important when
except for n 1, there is more than BFKL
analogous to recent toy models
Kovner and Lublinsky (2006) Blaizot, Iancu and
Triantafyllopoulos (2006) Iancu, de Santana
Amaral, Soyez and Triantafyllopoulos (2006)
differences to understand
work in progress
previous hierarchy of Iancu and
Triantafyllopoulos
Iancu and Triantafyllopoulos (2005)
obtained requiring that the target dipoles
scatter only once
15
The Pomeron-loop equationscombining dense and
dilute evolution
16
A stochastic evolution
by combining the evolution equations of the dense
and dilute regimes, (counting the BFKL term only
once), one gets
the QCD evolution is equivalent to a stochastic
process
17
The sF-KPP equation
high-energy QCD evolution stochastic process in
the universality class of reaction-diffusion
processes, of the sF-KPP equation
Iancu, Mueller and Munier (2005)
noise
r dipole size
the reduction to one dimension introduces the
noise strength parameter ?
18
A stochastic saturation scale
The noise term introduce a stochastic saturation
scale
the saturation scale is a stochastic variable
distributed according to a Gaussian probability
law
average saturation scale
v average speed of the waves
D dispersion coefficient
19
A new scaling law
the average dipole scattering amplitude
the diffusion is negligible and with
we obtain geometric scaling
the diffusion is important and
new regime diffusive scaling
20
Phenomenological consequencesdiffusive scaling
21
Geometric scaling and DIS data
Stasto, Golec-Biernat and Kwiecinski (2001)
2

• photon virtuality Q2 - (k-k)2 gtgt ?QCD
• ?p collision energy W2 (k-kp)2

this is seen in the data with
22
High-energy DIS
Y. Hatta, E. Iancu, C.M., G. Soyez and D.
Triantafyllopoulos (2006)
an intermediate energy regime geometric scaling
it seems that HERA is probing the geometric
scaling regime
23
Consequences for the observables
Y. Hatta, E. Iancu, C.M., G. Soyez and D.
Triantafyllopoulos (2006)
dipole size r
24
Inclusive gluon production

• gluon production is effectively described by a
  gluonic dipole (gg)

scattering amplitude
with
adjoint Wilson line
the other Wilson lines (coming from
the interaction of non-mesured partons) cancel
h
h
q gluon transverse momentum yq gluon rapidity
25
Forward particle production
important in view of the LHC large kT , small
values of x
kT , y
particle production at forward rapidities y (in
hadron-hadron and heavy-ion collisions)
in forward particle production, the transverse
momentum spectrum is obtained from the
unintegrated gluon distribution of the small-x
hadron
26
Consequences in particle production
E. Iancu, C.M. and G. Soyez (2006)
In the diffusive scaling regime,
flattens with increasing Y
Is diffusive scaling within the LHC energy range?
hard to tell theoretically, we have a poor
knowledge of the coefficient D
Consequences for RpA ( ratio of gluon
distribution)
Kozlov, Shoshi and Xiao (2006)
27
Conclusions

• Scattering off a dense targetB-JIMWLK
  equations
• Scattering off a dilute targetdipole model
  equations
• Pomeron loop equationscombining the dense and
  dilute regimes high-energy QCD evolution ?
  stochastic processthis implies geometric
  scaling at intermediate energies diffusive
  scaling at higher energies
• Phenomenological consequencesnew scaling laws in
  DIS and particle production for large momenta and
  small xof strong interest in view of the LHC