Forward particle production in proton-nucleus collisions
Cyrille MarquetInstitut de Physique Théorique – CEA/Saclay
C. Marquet, Nucl. Phys. B705 (2005) 319C. Marquet, Nucl. Phys. A796 (2007) 41
C. Marquet and J. Albacete, in preparation+ work in progress
The hadron wavefunction 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
Three types of states:
S (kT ) << 1
weakly-coupled regime,
effective coupling constant:
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
)/1ln( xS
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,responsible for collective phenomena
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 more dense
• 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 flavours, dileptons
at HERA, xBj ~10-4 for Q² = 10 GeV²
at RHIC, x2 ~10-4 for pT ² = 10 GeV²
Geometric scaling in DISgeometric scaling can be easily understood as a consequence of large parton densities
the hadron in the (Q², x) plane:
0.3
Stasto, Golec-Biernat and Kwiecinski (2001)
x < 10-2
lines parallel to the saturation line are lines ofconstant densities along which scattering is constant
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 Condensatethe 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
small x gluons as radiation field
),(,
z zFD cc
valence partonsas static random
color source separation between
the long-lived high-x partons
and the short-lived low-x gluons
CGC wave function
classical Yang-Mills equations
• an effective theory to describe the saturation regime
gggggqqqqqqgqqq .........hadron CGC][hadron xD
from , one can obtainthe unintegrated gluon distribution,
as well as any n-parton distributions
2][x
in the A+=0 gauge
The small-x evolution
the solution gives 3.03/12 ~),(Q xAAxs
the evolution of with x is a renormalization-group equation2
][x
for a given value of k², the saturation regime in a nuclear wave functionextends to a higher value of x compared to a hadronic wave function
22][][
)/1ln( x
JIMWLKx H
xd
d
• the JIMWLK equation
is mainly non-perturbative, but its evolution is known
Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner
2][x
the energy evolution of cross-sections is encoded in the evolution of2
][x
in the CGC framework, any cross-section is determined by colorless combinations ofWilson lines , averaged over the CGC wave function
][][2
SDS xx ][S
• Observables
Scattering off the CGC
scattering of a quark:
• this is described by Wilson lines
dependence kept implicit in the following
))()((1
1][ xyxy FFc
WWTrN
T x : quark space transverse coordinate
y : antiquark space transverse coordinate
the dipole scattering amplitude:qq
this is the most common averagefor instance it determines deep inelastic scattering
• the 2-point function or dipole amplitude
xTxy
it is used in many CGC calculations without precaution
when only the two-point function enters in the formulation of
a cross-section, the so-called kT-factorization is applicable
• more complicated correlators for less inclusive observables
The Balitsky-Kovchegov equation
YYYTTTT zyxzzyxz the BK equation is a closed equation for obtained by assuming
YTxy
YYYYYY
TTTTTzd
TdYd
zyxzxyzyxzxy yzzxyx
22
22
)()()(
2
robust only for impact-parameter independent solutions
• the BK equation
r = dipole size• the unintegrated gluon distribution
• 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
Balitsky (1996), Kovchegov (1999)
BK evolution at NLO• running coupling (RC) corrections to the BK equation
taken into account by the substitution
Kovchegov
Weigert
Balitsky
RC corrections represent most of the NLO contribution
(2007)
• the begining of the NLO-CGC era
first numerical solution
first phenomenological implementation
Albacete and Kovchegov (2007)
to successfully describe the proton structure function F2 at small x
Albacete, Armesto, Milhano and Salgado (2009)
Single particle production
Forward particle production
),(),( 22
212
2TT
TT kxfkxg
dykd
dk
kT , y
yT eksx 1
transverse momentum kT, rapidity y > 0
yT eksx 2
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
values of x probed in the process:
the cross-section:single gluon production
probes only the unintegrated
gluon distribution (2-point function)
Kovner and Wiedemann (2001), Kovchegov and Tuchin (2002), Dumitru and McLerran (2002)Blaizot, Gélis and Venugopalan (2004), Marquet (2005), Gélis and Mehtar-Tani (2006)
if the emitted particle is a (valence) quark, involves
if the emitted particle is a gluon, involves
The suppression of RdA
kdyddN
kdyddN
NR hXpp
hXdA
colldA
2
21
xA decreases(y increases)
• the suppression of RdA was predicted
in the absence of nucleareffects, meaning if the gluons in the
nucleus interact incoherently like in A protons
• what we learned
if forward rapidity data are included in npdfs fit, the resulting gluon distribution is over suppressed
forward rapidities are needed to see the suppression 22 GeV 2~),01.0(Q Aus
RdA and forward pion spectrum
first comparisons to data:
Kharzeev, Kovchegov and Tuchin (2004)Kharzeev, Levin and Nardi (2005)
Dumitru, Hayashigaki andJalilian-Marian (2006)
more recent work:
from qualitative to quantitative agreement
shows the importance of both evolutions:
xA (CGC) and xd (DGLAP)
shows the dominance
of the valence quarks
RdA
pT - spectrum
New NLO-BK description
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
Albacete and C.M, in preparation
Betemps, Goncalves, de Santana Amaral (2009)
the shapes and normalizations are wellreproduced, except the 0 normalization
the speed of the x evolution and of
the pT decrease are now predicted
Two particles at forward rapidities
the spectrum and
Motivation- after the first d-Au run at RHIC, there was a lot of new results on
single inclusive particle production at forward rapidities
kdyddN
kdyddN
NR
hXpphXdA
colldA 22
1
the suppressed production (RdA < 1) was predicted in the Color Glass Condensate picture of the high-energy nucleus
d Au → h X
the modification factor were studied
- my calculation: two-particle production at forward rapidities
- but single particle production probes limited information about the CGC(only the 2-point function)
to strengthen the evidence, we need to studymore complex observables to be measured with the next d-Au run
d Au → h1 h2 XI computed C. Marquet, NPA 796 (2007) 41
(probes up to a 6-point function)
Central/forward correlations• first measurements of azinuthal correlations
signal
STAR, PRL 97 (2006) 152302
PHENIX, PRL 96 (2006) 222301
coincidenceprobability
• difficult to make robust predictions
- the fragmentation of low energy particles is not well known(fragmentation functions are not constrained at low z)
- the values of xA are at the limit of the CGC applicability(trigger at central rapidity high x)
a measurement sensitive to possible modifications
of the back-to-back emission pattern in a hard process
moderate values of xd, typically 0.5
dominant partonic process :
Two particles at forward rapidities
feasible in d-Au collisions at RHIC
(or p-Pb at LHC, but then xp ~ 0.1,and or important)
|k1|, |k2| >> QCD collinearfactorization of the quark density
h+T h1+h2+X
y1 ~ y2 ~ 3 : both h1 and h2
in forward hemisphere
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 CGC cannot be describedby a single gluon distribution
The two-particle spectrum
collinear factorization of quark density in deuteron Fourier transform k┴ and q┴
into transverse coordinates
pQCD q → qg wavefunction
b: quark in the amplitudex: gluon in the amplitudeb’: quark in the comp. conj. amplitudex’: gluon in the comp. conj. amplitude
interaction with hadron 2 / CGCn-point functions that resums the powers of gS A and the powers of αS ln(1/xA)
Nikolaev, Schäfer, Zakharov and Zoller (2005)I obtain a formula similar to that of
2- 4- and 6-point functionsthe 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)
Blaizot, Gélis and Venugopalan (2004)
need more than the 2-point function: no kT factorization same conclusions in sea quark
production
and two-gluon productionusing Fierz identities that relate WA and WF, we recover the z → 0 (soft gluon) limit
Jalilian-Marian and Kovchegov (2004)
Baier, Kovner, Nardi and Wiedemann (2005)
we will now include the xA evolution
Performing the CGC average
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
• a Gaussian distribution of color sources
is the two-dimensional massless propagator
• applying Wick’s theorem
when expanding in powers of α and averaging,
all the field correlators can be expressed in terms of ),'(),( yx zz dc
the difficulty is to deal with the color structure
Fujii, Gelis and Venugopalan (2006)
MV model and BK evolution
in the large-Nc limitis related to in the following way
With this model for the CGC wavefunction squared, it is possible to compute then-point functions:
Blaizot, Gélis and Venugopalan (2004)
and obeys the BK equation:
we will use the MV initial condition: McLerran and Venugopalan (1994)
with the initial saturation scale
→
Final expression
quark density in dilute hadron
unintegrated gluon density of CGC(Fourier transform of 2-point function)
the final expression for the cross-section can be decomposed into three pieces:
modified q → qg vertexdue to multiple scattering
: pQCD q → qg wavefunction in momentum space
with zero quark masses, I reduces towith
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
where non-linearities have the biggest effect
• pT dependence
the away-side peak is restored at higher pT
typical coincidence probability
to calculate the near-side peak,one needs di-pion fragmentation functions
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
this shows the qualitativebehavior of the correlation
for a given impact parameter,the initial saturation scale used is
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