Azimuthal correlations of forward di-hadrons in d+Au collisions suppressed by saturation

0.09 MB·English

by  Javier L. Albacete

Azimuthal correlations of forward di-hadrons in
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h JavierL.Albacete
p InstitutdePhysiqueThéorique,CEA/Saclay,91191Gif-sur-Yvettecedex,France
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p E-mail: [email protected]
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h Cyrille Marquet
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PhysicsDepartment,TheoryUnit,CERN,1211Genève23,Switzerland

1 E-mail: [email protected]
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RHIC experiments have recently measured the azimuthal correlation function of forward di-
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hadrons. Thedatashowadisappearanceoftheaway-sidepeakincentrald+Aucollisions,com-
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. paredtop+pcollisions,aswaspredictedbysaturationphysics. Indeed,wearguethatthiseffect,
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0 absent at mid-rapidity, is a consequence of the small-x evolution into the saturation regime of
1 the Gold nucleus wave function. We show that the data are well described in the Color Glass
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Condensateframework.
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35thInternationalConferenceofHighEnergyPhysics-ICHEP2010,
July22-28,2010
ParisFrance
Speaker.
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(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Azimuthalde-correlationsofforwarddi-hadrons CyrilleMarquet
Hard processes in hadronic collisions, which resolve the partonic structure of hadrons, are
welldescribed bytheleading-twist approximation ofQCD.Inthisweak-coupling regime, partons
in the hadronic wave function scatter independently, this is the essence of collinear factorization.
However, since the parton densities grow with decreasing energy fraction x, the hadronic wave
function also features a low-x part where the parton density has become large and partons scatter
coherently, invalidating the leading-twist approximation. This weak-coupling regime, where non-
linearities areimportant,iscalledsaturation, anditcanbeprobedathigh-energies sinceincreasing
theenergy ofacollision allowstoprobelower-energy partons.
Inhadron-hadron collisions, bycontrastwithdeepinelasticscattering, hardprocesses aresin-
gled out by requiring that one or more particles are produced with a large transverse momentum,
much bigger than L . When in addition the particles are produced at forward rapidities, such
QCD
processes aresensitiveonlytohigh-momentum partonsinsideoneofthecolliding (dilute)hadron,
whoseQCDdynamicsiswellunderstood, whilemainlysmall-momentum (small-x)partonsinside
theotherdensehadroncontributetothescattering. Replacingthathadronbyalargenucleusfurther
enhances thegluondensity, andthepossibility toreachthesaturation regime.
Inthecaseofsingle-inclusivehadronproduction,thesuppressionofparticleproductionatfor-
ward rapidities in d+Au collisions compared to p+p collisions, experimentally observed at RHIC
[1], constitutes one of the most compelling indications for the presence of non-linear QCD evo-
lution effects in presently available data. The Color Glass Condensate (CGC) provides a robust
theoretical framework to describe the small-x degrees of freedom of hadronic/nuclear wave func-
tions. The good description of, among other observables, forward hadron production in d+Au
collisions at RHIC [2] indeed lends support to the idea that saturation effects may be a relevant
dynamical ingredient atpresentenergies.
However, alternative explanations of the suppressed forward hadron yield in d+Au collisions
were proposed [3], suggesting that one is not yet sensitive to the saturation of the nuclear gluon
densityatRHICenergies, butthatthesuppression isratherduetopartonicenergylossthroughthe
nuclearmatter,neglectedinCGCcalculations. Inspiteofthefactthatsaturation-based approaches
were the only ones to correctly predict this phenomenon, the existence of alternative scenarios
calls for the study of more complex observables. In the light of recent preliminary data in d+Au
collisions atRHIC,showingtheproduction offorwardmono-jets [4],calculating double-inclusive
forward hadron production in both frameworks could help pin down which is the correct picture.
In this work, we show that the CGC calculation predicts correctly the azimuthal de-correlation
of forward di-hadrons in d+Au collisions compared to p+p collisions [5], thus providing further
support forthepresence ofsaturation effectsinpresentdata.
1. Formulation
Inthecaseofdouble-inclusive hadronproduction, denoting p , p andy ,y thetransverse
1 2 1 2
⊥ ⊥
momenta and rapidities of the final state particles, the partons that can contribute to the cross
sectionhaveafractionoflongitudinalmomentumboundedfrombelow,byx (forpartonsfromthe
p
deuteron wavefunction) andx (forpartonsfromthenucleus wavefunction), whicharegivenby
A
p
xp=x1+x2 , xA=x1 e−2y1+x2 e−2y2 , with xi= | i⊥| eyi . (1.1)
√s
2
Azimuthalde-correlationsofforwarddi-hadrons CyrilleMarquet
The kinematic range for forward particle detection at RHIC is such that, with √s = 200 GeV,
x 0.4andx 10 3.Therefore the dominant partonic subprocess isinitiated byvalence quarks
p A −
∼ ∼
in the deuteron and, at lowest order in a , the dAu h h X cross-section is obtained from the
s 1 2
→
qA qgX cross-section,thevalencequarkdensityinthedeuteron f ,andtheappropriatehadron
q/d
→
fragmentation functions D andD :
h/q h/g
1 1 1 p p
dNdAu h1h2X = dz dz dx dNqA qgX xP, 1, 2 D (z ,m )D (z ,m )+
→ Zx1 1Zx2 2Zxz11+xz22 (cid:20) → (cid:18) z1 z2(cid:19) h1/q 1 h2/g 2
p p
dNqA qgX xP, 2, 1 D (z ,m )D (z ,m ) f (x,m )(.1.2)
→ (cid:18) z z (cid:19) h1/g 1 h2/q 2 (cid:21) q/d
2 1
HerewewillusetheCTEQ6NLOquarkdistributions andtheKKPNLOfragmentation functions.
The factorization and fragmentation scales are both chosen equal to the transverse momentum of
the leading hadron, which we choose to denote hadron 1, m = p . Note that we have assumed
1
| ⊥|
thatthetwofinal-statehadronscomefrompartonswhichhavefragmentedindependently, therefore
formula (1.2) cannot be used when R2 =(y y )2+(D f )2 is too small, where D f is the differ-
2 1
−
ence between the hadrons azimuthal angles. Computing the cross section for small R requires the
introduction ofpoorly-known di-hadronfragmentation functions, weshallnotdoit,becauseaswe
shallseethenon-linear QCDeffectsweareinterested inmanifestthemselves aroundDF =p .
As usual, due to parton fragmentation, the values of x’s probed are generically higher than
x and x defined in (1.1). For the proton, one has x <x<1, and if x would be smaller (this
p A p p
will be the case at the LHC), the gluon initiated processes gA qq¯X and gA ggX should also
→ →
be included in (1.2), they have been computed recently [6] . For the nucleus, we shall see that
the parton momentum fraction varies between xA and e−2y1+e−2y2. Therefore with large enough
rapidities, onlythesmall-xpartofthenuclearwavefunctionisrelevantwhencalculatingtheqA
→
qgX crosssection,andthatcrosssectioncannotbefactorizedfurther: dNqA qgX = f dNqg qg.
→ g/A →
6 ⊗
Indeed, when probing the saturation regime, dNqA qgX is expected to be a non-linear function of
→
thenucleargluondistribution, whichisitself,throughevolution,anon-linear functionofthegluon
distribution athigherx.
UsingtheCGCapproach todescribe thesmall-x partofthenucleus wavefunction, theqA
→
qgX cross section was calculated in [7, 8]. It was found that the nucleus cannot be described
by only the single-gluon distribution, a direct consequence of the fact that small-x gluons in the
nuclear wave function behave coherently, and not individually. The qA qgX cross section is
→
instead expressed in terms ofcorrelators of Wilson lines (which account for multiple scatterings),
withuptoasix-pointcorrelatoraveragedovertheCGCwavefunction,whilethegluondistribution
istheFouriertransform ofonlyatwo-point(dipole) correlator N :
d2r
F(x,k )= e ik r [1 N (x,r)], (1.3)
⊥ Z (2p )2 − ⊥· −
whererdenotesthedipoletransversesize. F(x,k )isactuallycalledtheunintegrated gluondistri-
⊥
bution, duetothe fact that itisk dependent, afeature known tobenecessary todescribe small-x
⊥
partons, eveninthelinearregime.
At the moment, it is not known how to practically evaluate the six-point function. In [8],
an approximation was made which allows to express the higher-point correlators in terms of the
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Azimuthalde-correlationsofforwarddi-hadrons CyrilleMarquet
two-point function (1.3). This isdone assuming aGaussian distribution of the color sources, with
a non-local variance. The resulting cross section for the inclusive production of the quark-gluon
system inthescattering ofaquarkwithmomentumxP+ offthenucleusAreads[8]:
dNqA→qgX = a SCF d (xP+ k+ q+)F(x˜ ,D ) (cid:229) Il (z,k D ;x˜ ) y l (z,k zD ) 2 , (1.4)
d3kd3q 4p 2 − − A lab (cid:12) ab ⊥− A − ab ⊥− (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where q and k are the momenta the quark and gluon respectively, and with D = k +q and
⊥ ⊥
z = k+/xP+. In this formula, x˜ denotes the longitudinal momentum fraction of the gluon in
A
the nucleus, and x˜A =x1 e−2y1/z1+x2 e−2y2/z2 >xA when the cross section (1.4) is plugged into
formula(1.2).
Thesecondlineofformula(1.4)features theso-called k -factorization breaking term,with
T
Il (z,k ;x)= d2q y l (z,q )F(x,k q ), (1.5)
ab ab
⊥ Z ⊥ ⊥ ⊥− ⊥
and where y l is the well-known amplitude for q qg splitting (l , a and b are polarization
ab
→
and helicity indices). While no additional information than the two-point function is needed to
compute (1.4), since higher-point correlators needed in principle have been expressed in terms
of F(x,k ), the cross section is still a non-linear function of that gluon distribution, invalidating
⊥
k -factorization. The rather simple form of the k -factorization breaking term is due to the use
T T
of a Gaussian CGC color source distribution, and to the large-N limit. However, the validity of
c
this approximation has been critically examined [9], as it does not allow to correctly implement
the non-linear QCD evolution, even in the large N limit. Finally, let us comment on the factor
c
d (xP+ k+ q+)informula (1.4). Thisdeltafunction isamanifestation ofthefactthatinahigh-
− −
energy hadronic collision, the momentum transfer is mainly transverse, and it appears because
the eikonal approximation was used to compute the qA qgX cross section. This is valid in the
→
high-energy limit,asforinstance theenergylossoftheincomingquarkisneglected.
TheCGCis endowed withaset ofnon-linear evolution equations whichin thelarge-N limit
c
reduce to the Balitsky-Kovchegov (BK) equation [10]. These equations can be interpreted as a
renormalization groupequationforthexevolutionoftheunintegratedgluondistribution, andmore
generally of n-point correlators, in which both linear radiative processes and non-linear recombi-
nationeffectsareincluded. Inthiswork,wecomputethesmall-xdynamicsofthedipolecorrelator
N by solving the running-coupling (rc) BK equation. The evolution kernel is evaluated accord-
ing to the prescription of Balitsky. Explicit expressions, together with a detailed discussion on
the numerical method used to solve the rcBK equation can be found in [11], along with detailed
discussions aboutotherprescriptions proposed todefinetherunning-coupling kernel.
The only piece of information left to fully complete all the ingredients in (1.2) are the initial
conditions for the rcBK evolution of N (x,r). This non-perturbative input has been constrained
by single-inclusive forward hadron production data in [2]. The two parameters are x =0.02, the
0
value of x below which one starts to trust, and therefore use, the CGC framework, and the value
A
of the saturation scale at the starting point of the evolution Q¯2(x ) Q¯2 =0.4 GeV2. Then, this
s 0 ≡ s0
information wassimplytaken overin[5]. Inthis respect, theseforward di-hadron calculations are
predictions, therearenofreeparametertoplaywith.
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Azimuthalde-correlationsofforwarddi-hadrons CyrilleMarquet
DfCP() 0.02 1 GpTe,LV>/2c G< epVT,/Sc<pT,L Spd++TpAA uR(− c 0Pe.Rn0t0Er4aL5lI M()−I0N.A01R4Y5) DfCP() 000...000000678 pdTA,L>u2fi G peV0p/c0,X 1 GeV/c < pT,S<pT,L bbb===034 .ff4mm fm
0.015 b=5.1 fm
0.005
0.004
0.01
0.003
0.005 0.002
0.001
0
0
0 1 2 3 4 5 6 0 1 2 3 4 5 6
D f D f
Figure1: ThecoincidenceprobabilityatafunctionofD f . Left: CGCcalculations[5]forp+pandcentral
d+Au collisions, the disappearanceofthe away-sidepeakisquantitativelyconsistentwiththe STAR data.
Right: CGC predictionsfordifferentcentralitiesofthed+Aucollisions, thenear-sidepeakisindependent
ofthecentrality,whiletheaway-sidepeakreappearsascollisionsaremoreandmoreperipheral.
2. Comparisonwithdata
Wewillnowinvestigate theprocessdAu h h X,with√s=200GeV.Inparticular weshall
1 2
→
studytheD f dependenceofthespectrum,whereD f isthedifferencebetweentheazimuthalangles
ofthemeasuredforwardparticlesh andh .Tobemorespecific,weshallcomputethecoincidence
1 2
probability to,givenatriggerparticleinacertainmomentumrange,produceanassociated particle
inanothermomentumrange. Itisgivenby
CP(D f )= Npair(D f ) , with N (D f )= dNpA→h1h2X , N = dNpA→hX . (2.1)
N pair Z d3p d3p trig Z d3p
trig 1 2
yi,|pi⊥| y, p⊥
In order to compare with the STARmeasurement, the integration bounds for the rapidities are set
to 2.4<y<4, which also ensures that only small-momentum partons are relevant in the nucleus
wave function. In addition, for thetrigger (leading) particle p >2GeV and for the associated
1
| ⊥|
(sub-leading) hadron 1 GeV< p < p . The single-inclusive hadron production spectrum,
2 1
| ⊥| | ⊥|
usedtonormalize thecoincidence probability, iscalculated asexplained in[2].
To deal with the centrality dependence, we identify the centrality averaged initial saturation
scaleQ¯2 ,extractedfromminimum-biassingle-inclusive hadronproduction data,withthevalueof
s0
Q2 atb=5.47fm,andusetheWoods-Saxondistribution T (b)tocalculate thesaturation scaleat
s0 A
othercentralities:
Q¯2 T (b)
Q2 (b)= s0 A , Q¯2 =0.4GeV2 . (2.2)
s0 T (5.47fm) s0
A
The result for central d+Au collisions is displayed in Fig.1, left plot, along with preliminary data
from the STARcollaboration. As mentioned before, we do not calculated the complete near-side
peak,asourformuladoesnotapplyaroundD f =0. Weseethatthedisappearanceoftheaway-side
peak in central d+Au collisions, compared to p+p collisions, is quantitatively consistent with the
CGC calculations. The latter are only robust for central d+Au collisions, but the extrapolation to
p+p collisions is displayed in order to show that it is qualitatively consistent with the presence of
theaway-sidepeak,andalsowiththefactthatthenear-sidepeakisidenticalinthetwocasesandis
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Azimuthalde-correlationsofforwarddi-hadrons CyrilleMarquet
notsensitivetosaturation physics. Sinceuncorrelated background hasnotbeenextractedfromthe
data,theoverallnormalization ofthedatapointshasbeenadjusted bysubtracting aconstant shift.
In Fig.1, right plot, we show the centrality dependence of the coincidence probability. Al-
though it isdifficult to trust our formalism all the way to peripheral collisions, wepredict that the
near-side peakdoesnotchangewithcentrality, andthattheaway-sidepeakreappears forlesscen-
tral collisions. This is consistent with the fact that peripheral d+Au collisions are p+p collisions.
Thefactthattheaway-sidepeakdisappearsfromperipheral tocentralcollisions showsthatindeed
this effect is correlated with the nuclear density. Moreover di-hadron correlations at mid-rapidity,
which are sensitive to larger values of x , feature an away-side peak whatever the centrality. The
A
factthatforcentralcollisionstheaway-sidepeakdisappearsfromcentraltoforwardrapiditiesalso
shows that the effect is correlated with the nuclear gluon density. In a similar way, we predict
that for higher transverse momenta, the away-side peak will reappear, as larger values of x will
A
beprobed. Thesemodifications oftheaway-side peak withtransverse momentaandrapidity were
already predicted in[8]asatimewheretherewasnodata.
Wearenotawareofanydescriptionsofthisphenomenathatdoesnotinvokesaturationeffects.
WenotethatapartfromourCGCcalculation,asuccessfuldescriptionbasedontheKLNsaturation
model was also recently proposed [12]. While more differential measurements ofthe coincidence
probability,asafunctionoftransversemomentumorrapidity,willprovidefurthertestsofourCGC
predictions and help understand better this theory of saturation, the analysis of forward di-hadron
correlations presented in this work adds further support to the idea that the saturation regime of
QCDhasbeenprobed atRHIC.Futurep+Pbcollisions attheLHCwillallowdefinitivetests.
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