Transcript Heavy-quark energy loss in finite extend SYM plasma

Small-x physics
3- Saturation phenomenology
at hadron colliders
Cyrille Marquet
Columbia University
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
The hadronic/nuclear
wave function
The hadron wave function in QCD
hadron  qqq  qqqg  ...  qqq......ggggg
• one can distinguish three regimes
hadron  kT ~ QCD  kT  QCD , x  1  kT  QCD , x  1
S (kT ) << 1
non-perturbative
regime: soft QCD
relevant for instance for
the total cross-section in
hadron-hadron collisions
perturbative regime,
dilute system of partons:
hard QCD (leading-twist
approximation)
relevant for instance for
top quark production
weakly-coupled regime,
dense system of partons (gluons)
non linear QCD
the saturation regime
not relevant to experiments
until the mid 90’s
with HERA and RHIC: recent gain of interest for saturation physics
The dilute regime
hadron  kT ~ QCD  kT  QCD , x  1  kT  QCD , x  1
as kT increases, the hadron gets more dilute
the dilute (leading-twist) regime:
QCD
 1
kT
hadron =a dilute system of partons
which interact incoherently
for instance, the total cross-section in DIS
 DIS (xBj, Q ) 
2
1
 dx 
(x, Q2)ˆa(xBj/ x, Q2)
a/ p
partons a x
Bj
parton density
partonic cross-section
transverse
view
1/kT ~
parton transverse
size
of the hadron
leading-twist
regime
Dokshitzer Gribov
Lipatov Altarelli Parisi
The saturation regime
hadron  kT ~ QCD  kT  QCD , x  1  kT  QCD , x  1
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:
Q (x)
QCD
~1
 1 , s
kT
kT
Balitsky Fadin Kuraev Lipatov
hadron = a dense system of partons
which interact coherently
the saturation regime of QCD:
the weakly-coupled regime that describes the collective
behavior of quarks and gluons inside a high-energy hadron
Geometric scaling from BK
• what we learned about the transition to saturation:
the dipole scattering amplitude
N=1
N << 1
the saturation scale:
traveling wave solutions  geometric scaling
the amplitude is invariant along any
line parallel to the saturation line
When is saturation relevant ?
in processes that are sensitive to the small-x part of the hadron wavefunction
• deep inelastic scattering at small xBj :
xBj 
Q2
W  Q2
2
at HERA, xBj ~10-4 for Q² = 10 GeV²
in DIS small x corresponds to high energy
saturation relevant for inclusive,
diffractive, exclusive events
• particle production at forward rapidities y :
pT , y
x1 s  pT e y
x2 s  pT e y
at RHIC, x2 ~10-4 for pT ² = 10 GeV²
in particle production, small x corresponds
to high energy and forward rapidities
saturation relevant for the production of
jets, pions, heavy flavors, photons
The dipole models
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
this is also what is obtained in the MV model for the
CGC wave function, the
behavior is recovered
standard leading-twist
gluon distribution
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
• improvement with the inclusion of heavy quarks
Soyez (2007)
fixed numbers:
matching point
size of scaling violations
quark masses
the parameters:
originally, this was fixed at the leading-log value
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
• the b-CGC model
impact parameter profile
Kowalski, Motyka and Watt (2006)
IIM model with the saturation scale is replaced by
• the t-CGC model
C.M., Peschanski and Soyez (2007)
the idea is to Fourier transform
where
is directly related to the measured momentum transfer
the hadron-size parameter is always of order
The KKT parametrization
• build to be used as an unintegrated gluon distribution
the idea is to modify the saturation exponent
• the DHJ version
Kovchegov, Kharzeev and Tuchin (2004)
Dumitru, Hayashigaki and Jalilian-Marian (2006)
KKT modified to better account for geometric scaling violations
• the BUW version
Boer, Utermann and Wessels (2008)
KKT modified to feature exact geometric scaling
in practice
is always replaced by
before the Fourier transformation
Deep inelastic scattering (DIS)
Kinematics of DIS
k’
lh center-of-mass energy
S = (k+p)2
k
*h center-of-mass energy
W2 = (k-k’+p)2
photon virtuality
Q2 = - (k-k’)2 > 0
p
size resolution 1/Q
x
Q2
2 p.(k  k ' )

Q2
W 2  M h2  Q 2
2
p.(k  k ' ) Q / x
y

p.k
S  M h2
x ~ momentum fraction of the struck parton
• the measured cross-section
experimental data are often shown in terms of
y ~ W²/S
The virtual photon wave functions
• computable from perturbation theory
*

  d 3k   (k , Q 2 ) q(k ); q ( P  k )
wave function computed from QED at lowest order in em
• as usual we go to the mixed space
where the interaction with the CGC is diagonal
*

  dk  d 2 xd 2 y  (k  , x  y, Q 2 ) q(x); q (y )
x : quark transverse coordinate
y : antiquark transverse coordinate
in DIS we need the overlap function
The dipole factorization
• the virtual photon overlap functions
• scattering off the CGC
we already computed the dipole-CGC scattering amplitude
1
Tr (WF (b  r / 2)WF (b  r / 2))
Nc
average over the CGC wave function
N ( r , x, b )  1 
then
up to deviations due to quark masses
the geometric scaling
implies
x
at small x, the dipole cross
section is comparable to that
of a pion, even though
r ~ 1/Q << 1/QCD
HERA data and geometric scaling
Soyez (2007)
IIM fit (~250 points)
Stasto, Golec-Biernat and Kwiecinski (2001)
geometric scaling seen in the data, but
scaling violations are essential for a good fit
Diffractive DIS
Inclusive diffraction in DIS
k’
k’
some events
k
k
are diffractive
when the hadron
remains intact
p
diffractive mass
MX2 = (p-p’+k-k’)2

Q2
2( p  p' ).(k  k ' )
momentum transfer
t = (p-p’)2 < 0

Q2
M t Q
2
X
2
• the measured cross-section
rapidity gap
p
p’
momentum fraction of the exchanged object
(Pomeron) with respect to the hadron
The dipole picture
the diffractive final state is decomposed into
• the
contributions
contribution
double differential cross-section
(proportional to the structure function)
for a given photon polarization:
overlap of
wavefunctions
geometric scaling implies
comes from
Fourier transform to t
Fourier transform to MX2
dipole amplitudes
Hard diffraction and saturation
• the total cross sections
recall the dipole scattering amplitude
in DIS
in DDIS
• diffraction directly sensitive
to saturation
contribution of the different r
regions in the hard regime
Q Q
2
2
S
dipole size r
Q2 DIS 
Q  DDIS
2
DIS dominated by relatively hard sizes
DDIS dominated by semi-hard sizes
1
1

Q2
 ln(Q2 /QS2 ) 
1

1
1 Q  r  1 QS
r ~ 1 QS
1

Comparison with HERA data
with proton tagging e p  e X p
H1
ZEUS
FPS data (2006)
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 (~450 points)
with IIM model
C.M. (2007)
Important features
• the β dependence
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)
• geometric scaling
C.M. and Schoeffel (2006)
Hard diffraction off nuclei
• the dipole-nucleus cross-section
Kowalski and Teaney (2003)

averaged with the Woods-Saxon distribution
• the Woods-Saxon averaging
in diffraction, averaging at the level of the amplitude
corresponds to a final state where the nucleus is intact
averaging at the cross-section level
allows the breakup of the nucleus into nucleons
• nuclear effects
Kowalski, Lappi, C.M. and Venugopalan (2008)
enhancement at large 
suppression at small 
position of the nucleons
Exclusive vector meson production
• sensitive to impact parameter
2
 * pVp
dVM
iq.b
2
1
2
2
2

d r d b Tqq(r, b; x ) e  (r, Q , MV )
dt
16 
instead of
V (r, z, M2V )
 (r, z, Q2)
the overlap function:
 (r, Q2, M2V )  dz  (r, z, Q2) V (r, z, M2V )
measurements:
 *pVp
dVM
(x, Q², t)
dt
 *pVp
VM
(x, Q²)
• success of the dipole models
t-CGC
b-CGC
appears to work well
also but no
given
predictions for DVCS are available
rho
lots of data from HERA
J/Psi
Forward particle production
in pA collisions
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:
x1 s  kT e y
the large-x hadron should be described by
standard leading-twist parton distributions
the cross-section:
kT2
d
2
2

g
(
x
,
k
)
f
(
x
,
k
)
1
T
2
T
2
d kT dy
x 2 s  kT e  y
the small-x hadron/nucleus should be
described by CGC-averaged correlators
single gluon production
probes only the unintegrated
gluon distribution (2-point function)
RHIC vs LHC
• typical values of x being probed at forward rapidities (y~3)
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 xA
• how the CGC is being probed
if the emitted particle is a quark,
involves
if the emitted particle is a gluon,
involves
xA
xp
xd
Inclusive gluon production
• effectively described by a gluonic dipole
gg dipole scattering amplitude: T~zz'
~
with Tzz' [ A]  1 
Y

1
Tr
(
W
A (z' )WA ( z ))
N c2  1
adjoint
Wilson line

the other Wilson lines WF (x) and WF (y) (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
h
h
q : gluon transverse momentum
yq : gluon rapidity
• the gluon production cross-section
iq.r
2 ~
d

2
q 2 2  d r e r T(r)  (q, yq)
yq
d qd b
2
very close to the unintegrated
gluon distribution introduced earlier
the transverse momentum spectrum
is obtained from a Fourier
transformation of the dipole size r
A CGC prediction
• the unintegrated gluon distribution
in the geometric scaling regime
(k , y) is peaked around QS(Y)
the infrared diffusion problem of the BFKL
solutions has been cured by saturation
y
• the suppression of RdA
the suppression of RdA was predicted
RdA 
1
N coll
dN dAhX
d 2 kdy
dN pp hX
d 2 kdy
xA decreases
(y increases)
in the absence of nuclear effects,
meaning if the gluons in the nucleus interact incoherently like in A protons
RdA and forward pion spectrum
• first comparison to data
RdA
Kharzeev, Kovchegov and Tuchin (2004)
qualitative agreement
with KKT parametrization
• quantitative agreement
Dumitru, Hayashigaki and Jalilian-Marian (2006)
for the pT – spectrum
with the DHJ model
shows the importance of both
evolutions: xA (CGC) and xd (DGLAP)
shows the dominance
of the valence quarks
2-particle correlations in pA
• inclusive two-particle production
at forward rapidities in order to probe small x
final state :
k1 , y1
k2 , y2
k1 e y1  k2 e y2
xp 
s
k1 e  y1  k2 e  y2
xA 
s
probes 2-, 4- and 6- point functions
one can test more information about the CGC compared to single particle production
• some results for azimuthal correlations
k2 is varied from 1.5 to 3 GeV
as k2 decreases, it gets closer to QS and the
correlation in azimuthal angle is suppressed
obtained by solving BK, not from model
C.M. (2007)
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