Transcript Saturation dans les collisionneurs de Hadrons

Forward particle production
in proton-nucleus collisions
Cyrille Marquet
Institut de Physique Théorique – CEA/Saclay
C. Marquet, Nucl. Phys. B705 (2005) 319
C. Marquet, Nucl. Phys. A796 (2007) 41
C. Marquet and J. Albacete, in preparation
+ work in progress
The hadron wavefunction in QCD
hadron  qqq  qqqg  ...  qqq......ggggg
Three types of states:
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,
effective coupling constant:S ln(1/ x)
dense system of partons
mainly gluons (small-x gluons):
the saturation regime of QCD
not relevant for 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 more dense
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
hadron = a dense system of partons,
responsible for collective phenomena
The saturation regime of QCD:
the weakly-coupled regime that describes the collective
behavior of quarks and gluons inside a high-energy hadron
Balitsky Fadin Kuraev Lipatov
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 flavours, dileptons
Geometric scaling in DIS
geometric scaling can be easily understood as a consequence of large parton densities
the hadron in the (Q², x) plane:
lines parallel to the saturation line are lines of
constant densities along which scattering is constant
Stasto, Golec-Biernat and Kwiecinski (2001)
x < 10-2
  0.3
Contents
• The Color Glass Condensate formalism
- effective description of the small-x gluons
- the JIMWLK evolution equation
- scattering off the CGC and n-point functions
• Single particle production at forward rapidities
- probes the two-point functions
- inclusive spectra and modification factors at RHIC
- from qualitative to quantitative CGC description
• Two-particle production at forward rapidities
- probes more information about the CGC
- comparisons with recent RHIC data
The CGC formalism
The Color Glass Condensate
the idea of the CGC is to describe the saturation regime with strong classical fields
McLerran and Venugopalan (1994)
lifetime of the fluctuations
in the wave function ~

high-x partons ≡ static sources
low-x partons ≡ dynamical fields
• an effective theory to describe the saturation regime
hadron  qqq  qqqg  ... qqq......ggggg
valence partons
as static random
color source
small x gluons
as radiation field
classical Yang-Mills equations
D , F 


c
    c ( z  , z)
in the A+=0 gauge

hadron   D  x [  ]   CGC
CGC wave function
separation between
the long-lived high-x partons
and the short-lived low-x gluons
2
from  x [ ] , one can obtain
the unintegrated gluon distribution,
as well as any n-parton distributions
The small-x evolution
2
 x [ ] is mainly non-perturbative, but its evolution is known
• the JIMWLK equation
Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner
2
the evolution of  x [ ] with x is a renormalization-group equation
2
d
2
 x [ ]  H JIMWLK   x [ ]
d ln(1 / x)
the solution gives
Qs ( x, A) ~ A1/ 3 x0.3
2
for a given value of k², the saturation regime in a nuclear wave function
extends to a higher value of x compared to a hadronic wave function
• Observables
in the CGC framework, any cross-section is determined by colorless combinations of
2
Wilson lines S [ ] , averaged over the CGC wave function
S
x
  D  x [ ] S[ ]
the energy evolution of cross-sections is encoded in the evolution of  x [ ]
2
Scattering off the CGC
• this is described by Wilson lines
scattering of a quark:
dependence kept implicit in the following
• the 2-point function or dipole amplitude
the qq dipole scattering amplitude: Txy
x
1
Txy [ ]  1 
Tr (WF (y )WF (x))
Nc
x : quark space transverse
coordinate
y : antiquark space transverse coordinate
this is the most common average
for instance it determines deep inelastic scattering
• more complicated correlators for less inclusive observables
when only the two-point function enters in the formulation of
a cross-section, the so-called kT-factorization is applicable
it is used in many CGC calculations without precaution
The Balitsky-Kovchegov equation
• the BK equation
Balitsky (1996), Kovchegov (1999)
the BK equation is a closed equation for Txy
Y
obtained by assuming TxzTzy Y  Txz Y Tzy Y
2
(xy)2
d Txy   d z
 2 (xz)2(zy)2
Y
dY
T
xz
Y
 Tzy Y  Txy Y  Txz Y Tzy Y

robust only for impact-parameter independent solutions
• the unintegrated gluon distribution
r = dipole size

• modeling the unintegrated gluon distribution
the numerical solution of the BK equation is not useful for
phenomenology, because this is a leading-order calculation
instead, CGC-inspired parameterizations are used for
with a few parameters adjusted to reproduce the data
,
BK evolution at NLO
• running coupling (RC) corrections to the BK equation
taken into account by the substitution
Kovchegov
Weigert
Balitsky
(2007)
RC corrections represent most of the NLO contribution
• the begining of the NLO-CGC era
first numerical solution
Albacete and Kovchegov (2007)
first phenomenological implementation
Albacete, Armesto, Milhano and Salgado (2009)
to successfully describe the proton structure function F2 at small x
Single particle production
Forward particle production
kT , y
transverse momentum kT, rapidity y > 0
values of x probed in the process:
x1 s  kT e y
x 2 s  kT e  y
the large-x hadron should be described by
standard leading-twist parton distributions
the small-x hadron/nucleus should be
described by all-twist parton distributions
the cross-section:
d
k 2
 g ( x1 , kT2 ) f ( x2 , kT2 )
d kT dy
2
T
if the emitted particle is a (valence) quark,
if the emitted particle is a gluon,
single gluon production
probes only the unintegrated
gluon distribution (2-point function)
involves
involves
Kovner and Wiedemann (2001), Kovchegov and Tuchin (2002), Dumitru and McLerran (2002)
Blaizot, Gélis and Venugopalan (2004), Marquet (2005), Gélis and Mehtar-Tani (2006)
The suppression of RdA
• the suppression of RdA was predicted
RdA 
xA decreases
(y increases)
1
N coll
dN dAhX
d 2 kdy
dN pp hX
d 2 kdy
in the absence of nuclear
effects, meaning if the gluons in the
nucleus interact incoherently like in A protons
• what we learned
forward rapidities are needed to see the
suppression Q 2 (0.01, Au) ~ 2 GeV2
s
if forward rapidity data are included in npdfs fit,
the resulting gluon distribution is over suppressed
RdA and forward pion spectrum
RdA
first comparisons to data:
Kharzeev, Kovchegov and Tuchin (2004)
Kharzeev, Levin and Nardi (2005)
more recent work:
from qualitative to quantitative agreement
Dumitru, Hayashigaki and
Jalilian-Marian (2006)
pT - spectrum
shows the importance of both evolutions:
xA (CGC) and xd (DGLAP)
shows the dominance
of the valence quarks
New NLO-BK description
Albacete and C.M, in preparation
the shapes and normalizations are well
reproduced, except the 0 normalization
the speed of the x evolution and of
the pT decrease are now predicted
this fixes the two parameters of the theory:
- the value of x at which one starts to trust (and therefore use) the CGC description
- and the saturation scale at that value of x
in very forward particle production in p+p collisions at RHIC, (where NLO DGLAP fails) using
the CGC to describe the (small-x) proton also works
Betemps, Goncalves, de Santana Amaral (2009)
Two particles at forward rapidities
Motivation
- after the first d-Au run at RHIC, there was a lot of new results on
d Au → h X
single inclusive particle production at forward rapidities
the spectrum
and
1 dN dAhX
the modification factor RdA 
N coll d 2 kdy
dN pp hX
d 2 kdy
were studied
the suppressed production (RdA < 1) was predicted in the
Color Glass Condensate picture of the high-energy nucleus
- but single particle production probes limited information about the CGC
(only the 2-point function)
to strengthen the evidence, we need to study
more complex observables to be measured with the next d-Au run
- my calculation: two-particle production at forward rapidities
d Au → h1 h2 X
C. Marquet, NPA 796 (2007) 41
I computed
(probes up to a 6-point function)
Central/forward correlations
• first measurements of azinuthal correlations
coincidence
probability
STAR, PRL 97 (2006) 152302
PHENIX, PRL 96 (2006) 222301
a measurement sensitive to possible modifications
of the back-to-back emission pattern in a hard process
signal
• difficult to make robust predictions
- the values of xA are at the limit of the CGC applicability
(trigger at central rapidity  high x)
- the fragmentation of low energy particles is not well known
(fragmentation functions are not constrained at low z)
Two particles at forward rapidities
moderate values of xd, typically 0.5
dominant partonic process :
|k1|, |k2| >> QCD  collinear
factorization of the quark density
h+T  h1+h2+X
the CGC cannot be described
by a single gluon distribution
y1 ~ y2 ~ 3 : both h1 and h2
in forward hemisphere
feasible in d-Au collisions at RHIC
(or p-Pb at LHC, but then xp ~ 0.1,
and
or
important)
very low values of xA, typically < 10-4
need CGC resummation of large logarithms αS ln(1/xA) ~ 1 and large gS A ~ 1
The two-particle spectrum
b: quark in the amplitude
x: gluon in the amplitude
b’: quark in the comp. conj. amplitude
x’: gluon in the comp. conj. amplitude
collinear factorization of quark density in deuteron
Fourier transform k┴ and q┴
into transverse coordinates
pQCD q → qg
wavefunction
interaction with hadron 2 / CGC
n-point functions that resums the powers of gS A and the powers of αS ln(1/xA)
I obtain a formula similar to that of
Nikolaev, Schäfer, Zakharov and Zoller (2005)
2- 4- and 6-point functions
the scattering off the CGC is expressed through the following correlators of Wilson lines:
if the gluon is emitted before the interaction, four partons scatter off the CGC
if the gluon is emitted after the interaction, only the quarks interact with the CGC
interference terms, the gluon interacts in the amplitude only (or c.c. amplitude only)
need more than the 2-point function: no kT factorization
same conclusions in sea quark production
and two-gluon production
Blaizot, Gélis and Venugopalan (2004)
Jalilian-Marian and Kovchegov (2004)
using Fierz identities that relate WA and WF, we recover the z → 0 (soft gluon) limit
Baier, Kovner, Nardi and Wiedemann (2005)
we will now include the xA evolution
Performing the CGC average
• a Gaussian distribution of color sources
characterizes the density of color charges along the projectile’s path
with this model for the CGC wavefunction squared, it is possible to compute n-point functions
• applying Wick’s theorem
Fujii, Gelis and Venugopalan (2006)
when expanding
in powers of α and averaging,
all the field correlators can be expressed in terms of  c ( z  , x) d ( z ' , y )
is the two-dimensional massless propagator
the difficulty is to deal with the color structure
MV model and BK evolution
With this model for the CGC wavefunction squared, it is possible to compute the
n-point functions:
Blaizot, Gélis and Venugopalan (2004)
is related to
in the following way
in the large-Nc limit
and obeys the BK equation:
we will use the MV initial condition:
McLerran and Venugopalan (1994)
→
with
the initial saturation scale
Final expression
the final expression for the cross-section can be decomposed into three pieces:
quark density in dilute hadron
modified q → qg vertex
due to multiple scattering
unintegrated gluon density of CGC
(Fourier transform of 2-point function)
: pQCD q → qg wavefunction in momentum space
with zero quark masses, I reduces to
with
goal: study the CGC evolution  try to avoid the competition between the
xd (DGLAP) evolution of
and the small xA evolution of
and
Forward/forward correlations
• the focus is on the away-side peak
typical coincidence probability
where non-linearities have the biggest effect
to calculate the near-side peak,
one needs di-pion fragmentation functions
•
pT dependence
the away-side peak is restored at higher pT
suppressed away-side peak
Centrality dependence
•
comparison with data for central collisions
there is a very good agreement with STAR data
(an offset is needed to account for the background)
•
the centrality dependence
for a given impact parameter,
the initial saturation scale used is
this shows the qualitative
behavior of the correlation
Conclusions
• Forward particle production in d+Au collisions
- the suppressed production at forward rapidities was predicted
- there is a good agreement with CGC calculations
- now that NLO-BK is known, one should stop using models
• Two-particle correlations at forward rapidities
- probe the theory deeper than single particle measurements
- forward/forward correlations probe x as small as in the RdA measurement
- jet quenching seen in central d+Au collisions
- first theory(CGC)/data comparison successful, more coming