Production of forward di-jet in p+Pb collisions in the small-$x$ improved TMD factorization framework

0.34 MB·

by  Krzysztof Kutak

Production of forward di-jet in p+Pb collisions in the
small-x improved TMD factorization framework
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F Krzysztof Kutak
4 InstituteofNuclearPhysics,PolishAcademyofSciences,
2 Radzikowskiego152,31-342Kraków,Poland

E-mail: [email protected]
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Wereportonrecentstudyoftheproductionofforwarddi-jetsinproton-protonandproton-lead
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e collisionsattheLargeHadronColliderwithImprovedTransversalMomentumFactorization[24].
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[ The results as compared to results obtained within High Energy Factorization show noticable

effectsrelatedtodetailedtreatmentofnonlineareffects.
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38thInternationalConferenceonHighEnergyPhysics
3-10August2016
Chicago,USA
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommons
Attribution-NonCommercial-NoDerivatives4.0InternationalLicense(CCBY-NC-ND4.0). http://pos.sissa.it/
Forwarddi-jet
1. Introduction
Measurementsofforwardjetorparticleproductioninhigh-energyhadroniccollisionsprovide
uniqueopportunitiestostudytheQCDdynamics[1,2,3,4,5,6]andinparticularofthenon-linear
partonsaturationregime[7,8]. Suchprocesses,inwhich,forkinematicalreasons,high-momentum
partons from one of the colliding hadrons mainly scatter with small-momentum partons from the
other,arecalleddilute-densecollisions. Indeed,thedensityofthelarge-xpartonsintheprojectile
hadronissmall,whilethedensityofthesmall-xgluonsinthetargethadronislarge,andtheformer,
wellunderstoodinperturbativeQCD,canbeusedtoprobethedynamicsofthelatter. Thisistrue
alreadyinproton-protoncollisions, althoughusingatargetnucleusdoesenhancethedilute-dense
asymmetryofsuchcollisions.
RHICmeasurementshaveprovidedsomeevidenceforthepresenceofsaturationeffectsinthe
data, the most compelling of which is the successful description of forward di-hadron production
[9, 10, 11], using the most up-to-date theoretical tools available at the time in the Color Glass
Condensate (CGC) framework [12, 13]. In particular, this approach predicted the suppression of
azimuthalcorrelationsind+Aucollisionscomparedtop+pcollisions[7],whichwasobservedlater
experimentally[14,15].
Inthiscontext,weshallconsiderforwarddi-jetproductioninproton-leadversusproton-proton
collisions. Inthatcase,itwasshownin[16]thatthefullcomplexityoftheCGCmachineryisnot
needed. Indeed,forthedi-hadronprocessatRHICenergies,noparticularorderingofthemomen-
tum scales involved is assumed in CGC calculations, while at the LHC one can take advantage of
the presence of final-state partons with transverse momenta much larger than the saturation scale
to obtain simplifications. On the flip side, different complications - left for future studies - are
expected to arise due to QCD dynamics relevant at large transverse momenta and not part of the
CGCframework, suchasSudakovlogarithms[17,18,19,20]orcoherenceintheQCDevolution
ofthegluondensity[21,22,23].
The goal of this article is to report on application [24] of that new formulation, dubbed im-
provedTMD(ITMD)factorizationwhichisageneralizationforconsideredprocessofHighEnergy
Factorization[1,25]andTMDfactorization[26]. Bycomparingtheforwarddi-jetproductioncross
sectionsinproton-leadandproton-protoncollisions,wecanclearlyseetheonsetofpartonsatura-
tioneffects, aswegofromakinematicalregimeinwhichk ∼P towardsonewherek ∼Q , and
t t t s
we obtain a good estimation of the size of those effects where they are the biggest, which is for
nearly back-to-back jets. We note that probing non-linear effects of similar strength with single-
inclusiveobservablesrequirestomaketheonlytransversemomentuminvolvedinthoseprocesses
oftheorderofthesaturationscale,whichmaynotbeeasyexperimentally. Withdi-jets,assuming
P ∼20GeVandk ∼Q ∼2GeV,wecanreachR ∼0.5.
t t s pPb
2. TheITMDfactorizationformulaforforwarddi-jetsindilute-densecollisions
Weconsidertheprocessofinclusiveforwarddi-jetproductioninhadroniccollisions
p(p )+A(p )→ j (p )+ j (p )+X , (2.1)
p A 1 1 2 2
wherethefour-momentaoftheprojectileandthetargetaremasslessandpurelylongitudinal. The
longitudinalmomentumfractionsoftheincomingpartonfromtheprojectile,x ,andthegluonfrom
1
1
Forwarddi-jet
thetarget,x ,canbeexpressedintermsoftherapidities(y ,y )andtransversemomenta(p ,p )
2 1 2 t1 t2
oftheproducedjetsas
x = p+1 +p+2 = √1 (|p |ey1+|p |ey2) , x = p−1 +p−2 = √1 (cid:0)|p |e−y1+|p |e−y2(cid:1) . (2.2)
1 p+ s 1t 2t 2 p− s 1t 2t
p A
By looking at jets produced in the forward direction, we effectively select those fractions to be
x ∼1 and x (cid:28)1. Since the target A is probed at low x , the dominant contributions come from
1 2 2
thesubprocessesinwhichtheincomingpartononthetargetsideisagluon
qg→qg, gg→qq¯, gg→gg. (2.3)
Moreover,thelarge-xpartonsofthediluteprojectilearedescribedintermsoftheusualparton
distributionfunctionsofcollinearfactorization f (x )whilethesmall-xgluonsofthedensetarget
a/p 1
are described by TMD distributions Φ (x ,k ). Indeed, the momentum of the incoming gluon
g/A 2 t
fromthetargetisnotonlylongitudinalbutalsohasanon-zerotransversecomponentofmagnitude
k =|p +p | (2.4)
t 1t 2t
which leads to imbalance of transverse momentum of the produced jets: k2 = |p |2+|p |2+
t 1t 2t
2|p ||p |cos∆φ. ThevaliditydomainofITMDfactorizationis
1t 2t
Q (x )(cid:28)P (2.5)
s 2 t
whereP isthehardscaleoftheprocess,relatedtotheindividualjetmomentaP ∼|p |,|p |. By
t t 1t 2t
contrast,thevalueofk canbearbitrary.
t
TheITMDfactorizationformulareads[16]
dσpA→dijets+X = αs2 ∑ x1fa/p(x1)∑2 K(i) (P,k )Φ(i) (x ,k ). (2.6)
d2Pd2k dy dy (x x s)2 1+δ ag∗→cd t t ag→cd 2 t
t t 1 2 1 2 a,c,d cd i=1
(i)
ItinvolvesseveralgluonTMDsΦ (2perchannel),withdifferentoperatordefinitions,thatare
ag→cd
(i)
accompaniedbydifferenthardfactorsK . Thosewherecomputedin[16]usingeitherFeyn-
ag∗→cd
man diagram techniques, or color-ordered amplitude methods. They encompass the improvement
overtheTMDfactorizationformuladerivedinRef.[27]wherethematrixelementswereon-shell
andafunctionofP only.
t
WewouldliketopointoutthattheITMDfactorizationformula2.6wasbuildinordertocon-
tainboththeHEFandtheTMDexpressionsasitslimitingcases,andassuchshouldbeconsidered
no more than an interpolating formula. We note however, that if one would be able to directly
derive a factorization formula valid for Q (cid:28)P regardless of the value of k , any additional term
s t t
comparedto2.6shouldvanishinbothlimitsQ ∼k (cid:28)P andQ (cid:28)k ∼P.
s t t s t t
3. Numericalstudiesoftheforwarddi-jetcrosssection
We move now to the numerical results1 for forward di-jet production in p+p and p+Pb colli-
sionsattheLHC.Weconsideracenter-of-massenergyof8.16TeV,andgenerateallourpredictions
1thecalculationswereperformedusingMonteCarloprograms[28,29]
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Forwarddi-jet
1.8
ITMD (KS) with S(x), p+p ITMD, d=0.5
ITMD (KS) with S(x), d=0.75, p+Pb ITMD, d=0.75
1.6
100.0 √S = 8.16 TeV
√S = 8.16 TeV 1.4 pT1>pT2 > 20 GeV
pT1>pT2 > 20 GeV 3.5<y1,y2<4.5
] 3.5<y1,y2<4.5 1.2
b
n
[Δφ 10.0 RpA1.0
d
/
σ
d 0.8
1.0
0.6
0.4
0.1 0.2
0.0 0.5 1.0 1.5 2.0 2.5 3.0 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0
Δφ Δφ
Figure1: Leftplot: differentialcrosssectionasafunctionoftheazimuthalanglebetweenthejetsforp+p
andp+Pbcollisions(rescaledbythenumberofnucleons). Thedistributionsareidenticaleverywhereexpect
near∆φ (cid:39)π,wheresaturationarethestrongest. Rightplot: nuclearmodificationfactorsfortwovaluesof
thenuclearsaturationscale,providinganuncertaintyband.
withtheforwardregiondefinedastherapidityrange3.5<y<4.5ononesideofthedetector. The
two hardest jets are required to lie within this region and we also impose a cut on the minimal
transverse momentum of each two jets: p =20GeV. In such a setup, the cross section still may
t0
be divergent due to collinear singularities. These are cut-off by applying a jet algorithm on the
final state momenta with a delta-phi-rapidity cut R = 0.5. Finally, we require the jets to be or-
dered according to increasing transverse momentum, that is we have |p |>|p |> p . For the
t1 t2 t0
collinearpartondistributionsthatentertheITMDformula,wechosethegeneral-purposeCT10set.
For the central value of the factorization and renormalization scale, we choose the average trans-
versemomentumofthetwoleadingjets, µ =µ = 1(|p |+|p |). Wewillproduceerrorbands
F R 2 t1 t2
corresponding to the renormalization and factorization scale uncertainties by varying the central
numbersfromhalftotwicetheirvalue.
ForthevariousobservablesO shownbelow,wealsoconsiderthenuclearmodificationfactors
definedas
dσp+Pb
dO
R = . (3.1)
pPb dσp+p
A
dO
withA=208forPb. Inourapproach, intheabsenceofsaturationeffects, orinthecaseinwhich
theyareequallystronginthenucleusandintheproton,thisratioisequaltounity. If,however,the
non-linear evolution plays a more important role in the case of the nucleus, the R ratio will be
pPb
suppressedbelow1.
Westartbyinvestigatingtheazimuthalcorrelations,withtheazimuthalanglebetweenthejets
∆φ definedtoliewithin0<∆φ <π.
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Forwarddi-jet
1.8 1.8
ITMD, d=0.5 ITMD, d=0.5
HEF, d=0.5 HEF, d=0.5
1.6 1.6
√S = 8.16 TeV √S = 8.16 TeV
1.4 pT1>pT2 > 20 GeV 1.4 pT1>pT2 > 20 GeV
3.5<y1,y2<4.5 3.5<y1,y2<4.5
1.2 1.2
A A
p1.0 p1.0
R R
0.8 0.8
0.6 0.6
0.4 0.4
0.2 0.2
20.0 25.0 30.0 35.0 40.0 45.0 50.0 20.0 25.0 30.0 35.0 40.0 45.0 50.0
pT1 pT2
Figure2: Nuclearmodificationfactorsasafunctionofthetransversemomentumoftheleading(left)and
subleading(right)jet,comparingthenewITMDapproachwithpreviouslyobtainedHEFresults.
InFig.1wecomparethe∆Φdistributioninp+pandp+Pbcollisions. Afterrescalingthep+Pb
crosssectionbythenumberofnucleons,weobtainidenticaldistributionsalmosteverywhere. Itis
onlyfornearlyback-to-backjets,around∆φ (cid:39)π,thatsaturationeffectsinduceadifference. This
difference is better appreciated on the nuclear modification factor, which goes from unity to 0.6,
as ∆φ varies from ∼2.7 to π. Two values of the parameter c have been considered, which makes
upanuncertaintybandthatturnsouttoberathersmall. Thismeansthattheuncertaintyrelatedto
thevalueofthesaturationscaleoftheleadnucleusdoesnotstronglyinfluencethepredictedR
pPb
suppression.
Finally, in Fig. 2 we display the nuclear modification factors as a function of the transverse
momentum of the leading and sub-leading jet. Our conclusions are similar for these observables:
thenewITMDpredictionsaresimilartothepreviouslyobtainedHEFresults,duetothefactthatthe
ITMD/HEFratioissimilarinp+pandp+Pbcollisions. ThismeansthattheHEFframework,which
isincorrectfornearlyback-to-backjets-sinceinthisformalismallthegluonTMDsareconsidered
equalregardlessofthekinematics-canneverthelessbesafelyusedforR calculations.
pPb
4. Conclusions
In this paper, we have studied forward di-jet production in proton-proton and proton-lead
collisions, using the small-x improved TMD factorization framework Eq. (2.6. We have obtained
thefirstnumericalimplementationofthisformalism,andthefirstpredictionsforforwarddi-jetsat
theLHC,aprocesswhichisparticularlyinterestingfromsmall-xpointofview. Ourresultsforthe
nuclearmodificationfactorsinp+Pbvsp+pcollisionsconfirmtheconclusionsobtainedin[30]in
theHEFframework,thatfornearlyback-to-backjets,nonnegligibleeffectsofgluonsaturationare
tobeexpectedasonegoesfromp+ptop+Pbcollisions.
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Forwarddi-jet
Acknowledgements
K.KacknowledgessupportbyNarodoweCentrumNaukiwithSonataBisgrantDEC-2013/10/E/ST2/00656.
The article is based on paper written with P. Kotko, C. Marquet, E. Petreska, S. Sapeta, A. van
Hameren
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