1. Small-x physics 3- Saturation phenomenology at hadron colliders
Cyrille Marquet
Columbia University

2. Outline of the third lecture
The hadronic wave function summary of what we have learned
The saturation models from GBW to the latest ones
Deep inelastic scattering (DIS) the cleanest way to probe the CGC/saturation allows to fix the model parameters
Diffractive DIS and other DIS processes these observables are predicted
Forward particle production in pA collisions and the success of the CGC picture at RHIC

3. The hadronic/nuclear wave function

4. The hadron wave function in QCD
one can distinguish three regimes
S (kT ) << 1
perturbative regime,
dilute system of partons:
hard QCD (leading-twist
approximation)
weakly-coupled regime,
dense system of partons (gluons)
non linear QCD
the saturation regime
non-perturbative
regime: soft QCD
relevant for instance for
the total cross-section in
hadron-hadron collisions
relevant for instance for
top quark production
not relevant to experiments
until the mid 90’s
with HERA and RHIC: recent gain of interest for saturation physics

5. The dilute regime as kT increases, the hadron gets more dilute
the dilute (leading-twist) regime:
1/kT ~ parton transverse size
transverse view of the hadron
leading-twist
regime
hadron = a dilute system of partons which interact incoherently
Dokshitzer Gribov
Lipatov Altarelli Parisi
for instance, the total cross-section in DIS
partonic cross-section
parton density

6. Balitsky Fadin Kuraev Lipatov
The saturation regime
as x decreases, the hadron gets denser
the separation between the dilute and dense
regimes is caracterized by a momentum scale:
the saturation scale Qs(x)
the saturation regime of QCD: the weakly-coupled regime that describes the collective behavior of quarks and gluons inside a high-energy hadron
the saturation regime:
hadron = a dense system of partons
which interact coherently
Balitsky Fadin Kuraev Lipatov

7. Geometric scaling from BK
what we learned about the transition to saturation:
the dipole scattering amplitude
N = 1
N << 1
the amplitude is invariant along any
line parallel to the saturation line
the saturation scale:
traveling wave solutions  geometric scaling

8. When is saturation relevant ?
in processes that are sensitive to the small-x part of the hadron wavefunction
deep inelastic scattering at small xBj :
particle production at forward rapidities y :
at HERA, xBj ~10-4 for Q² = 10 GeV²
in DIS small x corresponds to high energy
saturation relevant for inclusive,
diffractive, exclusive events
at RHIC, x2 ~10-4 for pT ² = 10 GeV²
pT , y
in particle production, small x corresponds
to high energy and forward rapidities
saturation relevant for the production of
jets, pions, heavy flavors, photons

9. The dipole models

10. The GBW parametrization
the original model for the dipole scattering amplitude
Golec-Biernat and Wusthoff (1998)
it features geometric scaling:
the saturation scale:
the parameters:
fitted on F2 data
λ consistent with BK + running coupling
main problem: the Fourier transform behaves badly at large momenta:
improvement for small dipole sizes
Bartels, Golec-Biernat and Kowalski (2002)
obtained by including DGLAP-like geometric scaling violations
standard leading-twist
gluon distribution
this is also what is obtained in the MV model for the
CGC wave function, the behavior is recovered

11. The IIM parametrization
a BK-inspired model with geometric scaling violations
Iancu, Itakura and Munier (2004)
α and β such that N and its derivative are continuous at
the saturation scale:
main problem: the Fourier transform features oscillations
matching point
size of scaling violations
quark masses
Soyez (2007)
improvement with the inclusion of heavy quarks
the parameters:
fixed numbers:
originally, this was fixed at the leading-log value

12. Impact parameter dependence
the impact parameter dependence is not crucial for F2, it only affects the normalization
however for exclusive processes it must be included
the IPsat model
Kowalski and Teaney (2003)
same as before
impact parameter profile
the b-CGC model
Kowalski, Motyka and Watt (2006)
IIM model with the saturation scale is replaced by
the t-CGC model
the hadron-size parameter is always of order
C.M., Peschanski and Soyez (2007)
the idea is to Fourier transform where
is directly related to the measured momentum transfer

13. The KKT parametrization
build to be used as an unintegrated gluon distribution
Kovchegov, Kharzeev and Tuchin (2004)
the idea is to modify the saturation exponent
the DHJ version
the BUW version
KKT modified to feature exact geometric scaling
Dumitru, Hayashigaki and Jalilian-Marian (2006)
Boer, Utermann and Wessels (2008)
in practice is always replaced by before the Fourier transformation
KKT modified to better account for geometric scaling violations

14. Deep inelastic scattering (DIS)

15. Kinematics of DIS
size resolution 1/Q k k’ p
lh center-of-mass energy S = (k+p)2 *h center-of-mass energy W2 = (k-k’+p)2 photon virtuality Q2 = - (k-k’)2 > 0
x ~ momentum fraction of the struck parton y ~ W²/S
the measured cross-section
experimental data are often shown in terms of

16. The virtual photon wave functions
computable from perturbation theory
wave function computed from QED at lowest order in em
x : quark transverse coordinate y : antiquark transverse coordinate
as usual we go to the mixed space
where the interaction with the CGC is diagonal
in DIS we need the overlap function

17. The dipole factorization
the virtual photon overlap functions
scattering off the CGC
we already computed the dipole-CGC scattering amplitude
average over the CGC wave function
then
up to deviations due to quark masses
the geometric scaling implies
at small x, the dipole cross
section is comparable to that
of a pion, even though
r ~ 1/Q << 1/QCD

18. HERA data and geometric scaling
Soyez (2007)
Stasto, Golec-Biernat and Kwiecinski (2001)
geometric scaling seen in the data, but
scaling violations are essential for a good fit
IIM fit (~250 points)

19. Diffractive DIS

20. Inclusive diffraction in DISk k’ p k k’ p p’
when the hadron
remains intact
rapidity gap
some events
are diffractive
momentum fraction of the exchanged object
(Pomeron) with respect to the hadron
diffractive mass
MX2 = (p-p’+k-k’)2
the measured cross-section
momentum transfer t = (p-p’)2 < 0

21. The dipole picture the contribution
the diffractive final state is decomposed into contributions
the contribution
double differential cross-section
(proportional to the structure function)
for a given photon polarization:
comes from
Fourier transform to MX2
overlap of
wavefunctions
Fourier transform to t
dipole amplitudes
geometric scaling implies

22. Hard diffraction and saturation
the total cross sections
recall the dipole scattering amplitude
in DIS
in DDIS
contribution of the different r
regions in the hard regime
DIS dominated by relatively hard sizes
DDIS dominated by semi-hard sizes
diffraction directly sensitive to saturation
dipole size r

23. Comparison with HERA data
with proton tagging e p  e X p
H1 FPS data (2006)
ZEUS LPS data (2004)
without proton tagging e p  e X Y
H1 LRG data (2006) MY < 1.6 GeV
ZEUS FPC data (2005) MY < 2.3 GeV
parameter-free predictions
with IIM model
(~450 points)
C.M. (2007)

24. Important features the β dependence geometric scaling
C.M. and Schoeffel (2006)
geometric scaling
contributions of the different final states to the diffractive structure function:
tot = F2D
at small  : quark-antiquark-gluon
at intermediate  : quark-antiquark (T)
at large  : quark-antiquark (L)

25. Hard diffraction off nuclei
the dipole-nucleus cross-section
Kowalski and Teaney (2003) 
averaged with the Woods-Saxon distribution
position of the nucleons
the Woods-Saxon averaging
in diffraction, averaging at the level of the amplitude
corresponds to a final state where the nucleus is intact
Kowalski, Lappi, C.M. and Venugopalan (2008)
nuclear effects
enhancement at large 
suppression at small 
averaging at the cross-section level
allows the breakup of the nucleus into nucleons

26. Exclusive vector meson production
sensitive to impact parameter
the overlap function:
instead of
lots of data from HERA
rho
J/Psi
success of the dipole models
t-CGC
b-CGC appears to work well also but no given
predictions for DVCS are available
measurements:

27. Forward particle production in pA collisions

28. Forward particle production
forward rapidities probe small values of x
kT , y
transverse momentum kT, rapidity y > 0
values of x probed in the process:
the large-x hadron should be described by
standard leading-twist parton distributions
the small-x hadron/nucleus should be
described by CGC-averaged correlators
the cross-section:
single gluon production
probes only the unintegrated
gluon distribution (2-point function)

29. RHIC vs LHC
typical values of x being probed at forward rapidities (y~3) xA xp xd
RHIC
deuteron dominated by valence quarks
nucleus dominated by early CGC evolution
LHC
the proton description should include both quarks and gluons
on the nucleus side, the CGC
picture would be better tested
RHIC
LHC
if the emitted particle is a quark, involves
if the emitted particle is a gluon, involves
how the CGC is being probed

30. Inclusive gluon production
effectively described by a gluonic dipole h
gg dipole scattering amplitude:
adjoint
Wilson line
with
the other Wilson lines and (coming
from the interaction of non-mesured partons)
cancel when summing all the diagrams
this derivation is for dipole-CGC scattering
but the result valid for any dilute projectile
q : gluon transverse momentum
yq : gluon rapidity
the transverse momentum spectrum
is obtained from a Fourier
transformation of the dipole size r
very close to the unintegrated
gluon distribution introduced earlier
the gluon production cross-section

31. A CGC prediction the unintegrated gluon distribution y
in the geometric scaling regime
is peaked around QS(Y)
the infrared diffusion problem of the BFKL
solutions has been cured by saturation
the suppression of RdA was predicted
xA decreases
(y increases)
the suppression of RdA
in the absence of nuclear effects,
meaning if the gluons in the nucleus interact incoherently like in A protons

32. RdA and forward pion spectrum
first comparison to data
RdA
Kharzeev, Kovchegov and Tuchin (2004)
qualitative agreement
with KKT parametrization
Dumitru, Hayashigaki and Jalilian-Marian (2006)
shows the importance of both
evolutions: xA (CGC) and xd (DGLAP)
shows the dominance
of the valence quarks
for the pT – spectrum
with the DHJ model
quantitative agreement

33. 2-particle correlations in pA
inclusive two-particle production
at forward rapidities in order to probe small x
final state :
probes 2-, 4- and 6- point functions
one can test more information about the CGC compared to single particle production
as k2 decreases, it gets closer to QS and the
correlation in azimuthal angle is suppressed
some results for azimuthal correlations
obtained by solving BK, not from model
k2 is varied from 1.5 to 3 GeV
C.M. (2007)

34. What is going on now in this field
Link with the MLLA ? we would like to understand the differences between the pictures similar objects have already been identified (triple Pomeron vertex)
Higher order corrections running coupling corrections to BK are known, but not the full non linear equation at next-to-leading log
Heavy ion collisions what is the system at the time ~1/Qs after the collision crucial for the rest of the space-time evolution
Calculations for RHIC/LHC total multiplicities, jets, pions, heavy flavors, photons, dileptons