Small-x physics 3- Saturation phenomenology at hadron colliders Cyrille Marquet Columbia University.
-
Upload
lawrence-lee -
Category
Documents
-
view
214 -
download
1
Transcript of Small-x physics 3- Saturation phenomenology at hadron colliders Cyrille Marquet Columbia University.
Small-x physics
3- Saturation phenomenologyat hadron colliders
Cyrille Marquet
Columbia University
Outline of the third lecture
• The hadronic wave functionsummary of what we have learned
• The saturation modelsfrom GBW to the latest ones
• Deep inelastic scattering (DIS)the cleanest way to probe the CGC/saturationallows to fix the model parameters
• Diffractive DIS and other DIS processesthese observables are predicted
• Forward particle production in pA collisionsand the success of the CGC picture at RHIC
The hadronic/nuclearwave function
The hadron wave function in QCDgggggqqqqqqgqqq .........hadron
non-perturbative
regime: soft QCD
1, 1, ~hadron xkxkk QCDTQCDTQCDT
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
S (kT ) << 1
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
• one can distinguish three regimes
The dilute regime1, 1, ~hadron xkxkk QCDTQCDTQCDT
1T
QCD
kthe dilute (leading-twist) regime:
hadron =a dilute system of partons which interact incoherently
)Q,/(ˆ)Q,()Q,( 22/
12 xxxdxx Bjapa
apartons x
BjDIS
Bj
for instance, the total cross-section in DIS
partonic cross-sectionparton density
leading-twistregime
1/kT ~ parton transverse size
as kT increases, the hadron gets more dilute
Dokshitzer GribovLipatov Altarelli Parisi
transverse view of the hadron
The saturation regime1, 1, ~hadron xkxkk QCDTQCDTQCDT
the saturation regime of QCD:the weakly-coupled regime that describes the collective behavior of quarks and gluons inside a high-energy hadron
1~)(Q
, 1T
s
T
QCD
kx
kthe saturation regime:
hadron = a dense system of partons which interact coherently
the separation between the dilute and dense
regimes is caracterized by a momentum scale:
the saturation scale Qs(x)
Balitsky Fadin Kuraev Lipatov
as x decreases, the hadron gets denser
N = 1
N << 1
Geometric scaling from BK• what we learned about the transition to saturation:
the amplitude is invariant along anyline parallel to the saturation line
the saturation scale:
traveling wave solutions geometric scaling
the dipole scattering amplitude
• deep inelastic scattering at small xBj :
• particle production at forward rapidities y :
When is saturation relevant ?in processes that are sensitive to the small-x part of the hadron wavefunction
22
2
Q
Q
WxBj
in DIS small x corresponds to high energy
saturation relevant for inclusive,diffractive, exclusive events
pT , y
yT epsx 2
yT epsx 1 in particle production, small x corresponds
to high energy and forward rapidities
saturation relevant for the production ofjets, pions, heavy flavors, photons
at HERA, xBj ~10-4 for Q² = 10 GeV²
at RHIC, x2 ~10-4 for pT ² = 10 GeV²
The dipole models
The GBW parametrization• the original model for the dipole scattering amplitude
Golec-Biernat and Wusthoff (1998)
main problem: the Fourier transform behaves badly at large momenta:
it features geometric scaling:
fitted on F2 data
the saturation scale:
the parameters:
λ consistent with BK + running coupling
• improvement for small dipole sizesBartels, Golec-Biernat and Kowalski (2002)obtained by including DGLAP-like geometric scaling violations
standard leading-twistgluon distribution
this is also what is obtained in the MV model for theCGC wave function, the behavior is recovered
The IIM parametrization
α and β such that N and its derivative are continuous at
• a BK-inspired model with geometric scaling violations
main problem: the Fourier transform features oscillations
Iancu, Itakura and Munier (2004)
the saturation scale:
matching pointsize of scaling violationsquark masses
Soyez (2007)• improvement with the inclusion of heavy quarks
the parameters:
fixed numbers:
originally, this was fixed at the leading-log value
Impact parameter dependencethe 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)
• the b-CGC model
same as beforeimpact parameter profile
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
The KKT parametrization• build to be used as an unintegrated gluon distribution
the idea is to modify the saturation exponentKovchegov, Kharzeev and Tuchin (2004)
• 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
Deep inelastic scattering (DIS)
Kinematics of DIS
size resolution 1/Q
k
k’
p
lh center-of-mass energyS = (k+p)2
*h center-of-mass energyW2 = (k-k’+p)2
photon virtualityQ2 = - (k-k’)2 > 0
222
22
Q
Q
)'.(2
Q
hMWkkpx
x ~ momentum fraction of the struck parton y ~ W²/S
2
2 /Q
.
)'.(
hMS
x
kp
kkpy
experimental data are often shown in terms of
• the measured cross-section
The virtual photon wave functions
wave function computed from QED at lowest order in em
)();()Q,( 23* kPqkqkkd
• computable from perturbation theory
)();()Q,,( 222* yxyx qqkyxdddk
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
The dipole factorization
we already computed the dipole-CGC scattering amplitude
• the virtual photon overlap functions
• scattering off the CGC
x
FFc
rbWrbWTrN
bxrN ))2/()2/((1
1),,(
average over the CGC wave functionthen
up to deviations due to quark massesthe geometric scaling implies
at small x, the dipole crosssection is comparable to that
of a pion, even though
r ~ 1/Q << 1/QCD
HERA data and geometric scaling
geometric scaling seen in the data, butscaling violations are essential for a good fit
Stasto, Golec-Biernat and Kwiecinski (2001)
IIM fit (~250 points)
Soyez (2007)
Diffractive DIS
Inclusive diffraction in DIS
k
k’
p
k
k’
p
p’
when the hadronremains intact rapidity gap
some events
are diffractive
22
22
Q
Q
)').('(2
Q
tMkkpp X
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 transfert = (p-p’)2 < 0
• the contribution
The dipole picturethe diffractive final state is decomposed into contributions
comes from Fourier transform to MX2
overlap ofwavefunctions Fourier transform to t dipole amplitudes
double differential cross-section(proportional to the structure function)
for a given photon polarization:
geometric scaling implies
Hard diffraction and saturation
dipole size r
recall the dipole scattering amplitude• the total cross sections
in DIS
in DDIS
contribution of the different rregions in the hard regime
DIS dominated by relatively hard sizes
DDIS dominated by semi-hard sizes Sr Q1~Sr Q1Q1
22 QQ S
1 )/Qln(Q 1 Q 2S
22 DIS
1 1 Q
1 Q
22 DDIS
• diffraction directly sensitive to saturation
Comparison with HERA data
(~450 points)parameter-free predictionswith IIM model
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
C.M. (2007)
Important features
tot = F2D
contributions of the different final statesto the diffractive structure function:
at small : quark-antiquark-gluon
at intermediate : quark-antiquark (T)
at large : quark-antiquark (L)
• the β dependenceC.M. and Schoeffel (2006)
• geometric scaling
Hard diffraction off nuclei
in diffraction, averaging at the level of the amplitudecorresponds to a final state where the nucleus is intact
averaging at the cross-section levelallows the breakup of the nucleus into nucleons
averaged with the Woods-Saxon distribution
position of the nucleons
• the dipole-nucleus cross-section Kowalski and Teaney (2003)
• the Woods-Saxon averaging
Kowalski, Lappi, C.M. and Venugopalan (2008)
• nuclear effects
enhancement at large
suppression at small
Exclusive vector meson production• sensitive to impact parameter
)M,,()Q,,()M,Q,( 2V
22V
2 zrzrdzr V
22V
2.22*
)M,Q,();,(16
1 rexbrTbdrddt
d biqqq
VppVM
the overlap function:instead of
)Q,,( 2zr
)M,,( 2VzrV
lots of data from HERA
)²,Q,(*
txdt
d VppVM
²)Q,(* xVppVM
measurements: rho J/Psi
• success of the dipole models
t-CGC
b-CGC appears to work wellalso but no given
predictions for DVCS are available
Forward particle productionin pA collisions
Forward particle production
),(),( 22
212
2TT
TT kxfkxg
dykd
dk
kT , y
yT eksx 1
transverse momentum kT, rapidity y > 0
yT eksx 2
• forward rapidities probe small values of x
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
values of x probed in the process:
the cross-section:single gluon production
probes only the unintegrated
gluon distribution (2-point function)
RHIC vs LHC
xA xA xp xdLHCRHIC
deuteron dominated by valence quarks
• typical values of x being probed at forward rapidities (y~3)
RHIC
LHC
nucleus dominated by early CGC evolution
on the nucleus side, the CGC
picture would be better tested
the proton description shouldinclude both quarks and gluons
if the emitted particle is a quark, involves
if the emitted particle is a gluon, involves
• how the CGC is being probed
Inclusive gluon production
q : gluon transverse momentum
yq : gluon rapidity
))()((1
11][
~2
zzzz AAc
' W'WTrN
AT
gg dipole scattering amplitude:
with
adjointWilson line
Y'Tzz
~
this derivation is for dipole-CGC scattering
but the result valid for any dilute projectile
hh
• effectively described by a gluonic dipole
the other Wilson lines and (coming
from the interaction of non-mesured partons)
cancel when summing all the diagrams
)(xFW )(yFW
),()(~2.2
222 q
y
i yqrTerdbqdd
dqq
r
rq the transverse momentum spectrum
is obtained from a Fourier
transformation of the dipole size rvery close to the unintegrated
gluon distribution introduced earlier
• the gluon production cross-section
A CGC prediction
in the geometric scaling regime
is peaked around QS(Y)
y
),( yk
• the unintegrated gluon distribution
kdyddN
kdyddN
NR hXpp
hXdA
colldA
2
21
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
the infrared diffusion problem of the BFKL
solutions has been cured by saturation
RdA and forward pion spectrum
Kharzeev, Kovchegov and Tuchin (2004)
RdA• first comparison to data
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
2-particle correlations in pA• inclusive two-particle production
11 , yk 22 , yks
ekekx
yy
p
21 21
s
ekekx
yy
A
21 21
probes 2-, 4- and 6- point functions
final state :
one can test more information about the CGC compared to single particle production
as k2 decreases, it gets closer to QS and thecorrelation 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)
at forward rapidities in order to probe small x
What is going on now in this field
• Link with the MLLA ?we would like to understand the differences between the picturessimilar objects have already been identified (triple Pomeron vertex)
• Higher order correctionsrunning coupling corrections to BK are known,but not the full non linear equation at next-to-leading log
• Heavy ion collisionswhat is the system at the time ~1/Qs after the collisioncrucial for the rest of the space-time evolution
• Calculations for RHIC/LHCtotal multiplicities, jets, pions, heavy flavors, photons, dileptons