Gluon TMD studies at EIC

0.09 MB·

by  Daniël Boer

EPJWebofConferenceswillbesetbythepublisher
DOI:willbesetbythepublisher
c Ownedbytheauthors,publishedbyEDPSciences,2016
(cid:13)
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Gluon TMD studies at EIC
n
a
J
8 DaniëlBoer1,a

1VanSwinderenInstituteforParticlePhysicsandGravity,UniversityofGroningen
]
h Nijenborgh4,NL-9747AGGroningen,TheNetherlands
p
-
p Abstract.Ahigh-energyElectron-IonCollider(EIC)wouldofferamostpromisingtool
e tostudyindetailthetransversemomentumdistributionsofgluonsinsidehadrons. This
h applies to unpolarized as well as linearly polarized gluons inside unpolarized protons,
[
andtoleft-rightasymmetricdistributionsofgluonsinsidetransverselypolarizedprotons,

1 theso-calledgluonSiverseffect. Theinherentprocessdependenceofthesedistributions
v canbestudiedbycomparingtosimilar,butoftencomplementaryobservablesatLHC.
3
1
8
1 Introduction
1
0
Transverse momentumdependentparton distributions (TMDs) are currently under active investiga-
.
1 tion,boththeoreticallyandexperimentally.TypicalTMDprocessesaresemi-inclusiveDeepInelastic
0
Scattering (SIDIS) or the Drell-Yan process. The SIDIS process (ep e hX) is sensitive to the
6 → ′
transversemomentumofquarks,whileforinstanceD-mesonpairproduction(ep e DDX)issen-
1 ′
→
: sitivetothetransversemomentumofgluonsintheback-to-backcorrelationlimit. Forstudiesofthe
v
gluonTMDs,higherenergy(√s)orsmallerxisrequired.Ahigh-energyElectron-IonCollider(EIC)
i
X canoffercleanprobesofthedistributionsofunpolarizedandlinearlypolarizedgluonsinsideunpolar-
r izedprotons,andofthegluonSiverseffectfortransverselypolarizedprotons.Thesedistributionsand
a
whatwecanlearnaboutthematanEICwillbereviewedhere,withemphasisonthemostpromising
observables,theprocessdependence,andtheexpectedsmall-xbehaviorofthedistributions.
Describingthetransversemomentumofpartonsinaprocessisnotjustamatterofaddingatrans-
versemomentumdependenceincollineardistributions,i.e. f (x) f (x,k2),thatappearincollinear
1 → 1 T
factorizationexpressions. RatheronehastodealwithTMDfactorization,inwhichnewfactorsand
newdistributionsappear,suchastheSiverseffectTMDthatdescribesacorrelationbetweenthetrans-
versemomentumandtheprotonspin.
ForgluonsthereareeightleadingtwistTMDs[1]thatparametrizethegluoncorrelator
d(ξ P)d2ξ
Γgµν[U,U′](x,kT)≡Z (xP ·n)2(2πT)3ei(xP+kT)·ξhP|Trc Fnν(0)U[0,ξ]Fnµ(ξ)U[′ξ,0] |Piξ·P′=0. (1)
· h i
Thedependenceonthegaugelinks and willbediscussedlateron.Forunpolarizedhadronsthe
′
correlatorΓ isparametrizedbytwoUgluonUTMDs[1](here k2 = k2):
g T − T
1 kµkν k2
Γgµν(x,kT)=−2x(cid:26)gµTν f1g−(cid:18) MT 2T +gµTν2MT2(cid:19)h⊥1g(cid:27). (2)
ae-mail:[email protected]
EPJWebofConferences
TheunpolarizedgluonTMD fg andlinearlypolarizedgluonTMDh⊥g arebothfunctionsof x and
1 1
k2T. Nonzeroh⊥1grequiresnonzerotransversemomentumandstemsfromaninterferencebetween±1
gluonhelicities. Forpositiveh⊥1gthegluonpolarizationǫT isdistributedaroundkT withacos2φdis-
tribution(φ=∠(k ,ǫ )). Lineargluonpolarizationmodifiesamongothersthetransversemomentum
T T
distributionofHiggsproduction(perturbativelyatNNLO[2,3]andnonperturbativelyatLOinpQCD
[4,5]),whichcanbestudiedatLHC.InTMDfactorizationthecrosssectiontakestheform[5]
Edσpp HX q
d3~q→ (cid:12)(cid:12)qT≪mH ∝(cid:16)Chf1g f1gi+ChwHh⊥1gh⊥1gi(cid:17) +O mTH!, (3)
(cid:12)
where denotesaconvolution(cid:12)ofTMDsandw = (k k )2 1k2 k2 /2M4. Includingresum-
C H 1T · 2T − 2 1T 2T
mationoflargelogarithmsthecontributionoflinear(cid:16)lypolarizedgluonsrelat(cid:17)ivetounpolarizedgluons
isgivenby(forexplanationscf.[5]):
R(QT)≡ C[wCH[fh1g⊥1gf1gh]⊥1g] = R d2db2beiebi·bq·TqeT−eS−AS(bA∗(,bQ∗),Q−S)−NSPN(bP,(Qb),Qeh)⊥1f1gg((xxAA,,bb2∗2;;µµbb∗))ehf1⊥1g(gx(Bx,Bb,2b;2∗µ;bµb)∗), (4)
R ∗ ∗ ∗ ∗
where fgdenotestheFouriertransformof fgand e e
1 1
e (b k )2 1b2k2 k2
h⊥1g(x,b2)≡Z d2kT · Tb2M−22 T e−ib·kT h⊥1g(x,kT2)=−πZ dkT22MT2J2(bkT)h⊥1g(x,kT2). (5)
e
Theintegrandinbspacehasbeensplitintoacalculableperturbativepartandanonperturbative(NP)
partthatshouldbeobtainedfromfitstodata. AlthoughthenonperturbativeSudakovfactorS for
NP
gg Hisunknown,attheHiggsscaleitdoesnotmattertoomuch.Whatmattersmostisthesmall-b
par→t of the TMDs, which is perturbatively calculable [2, 4, 6]: fg(x,b2;µ ) = f (x;µ )+ (α ),
1 b g/P b O s
while
α (µ )C 1 dxˆ xˆ e
h⊥1g(x,b2;µb)= s 2bπ A Z xˆ x −1! fg/P(xˆ;µb)+O(α2s). (6)
x
Notethattheperturbeativetailofh⊥1gisdrivenbytheunpolarizedcollineargluondistribution fg/P(x;µ).
In[7]and[8]theaboveexpressionswerestudiednumerically(cf.[9]fortherangesofthepredic-
tions).Theconclusionfromthosestudiesisthat (Q )isontheorderof2-5%inHiggsproductionat
T
lowQT. ThisprobablymeansthattheextractionRofh1⊥g fromHiggsproductionwillbetoochalleng-
ing.In[7]and[8]alsoheavy(pseudo-)scalarC =+quarkoniumproduction,pp [QQ]X,hasbeen
→
studied.Muchlargereffectsfromlineargluonpolarizationarepossibleinthiscase,buttherearevery
largeuncertainties(cf.[9]). ItismuchmoresensitivetotheunknownNPpartthanHiggsproduction.
Fromthisperspectivethe heavierbottomoniumstates areprobablybestto consider. Employingthe
colorsingletmodel[10] and LO NRQCD results [11, 12], the differentialcrosssectionsfor η , χ
b b0
and χ production have been obtained in [13]. By forming ratios of ratios, in which the hadronic
b2
uncertaintiescancel,itbecomesinprinciplepossibletoprobe (Q )directly:
T
R
σ(χ )dσ(χ )/d2q σ(χ ) dσ(η )/d2q 1 (Q )
b2 b0 T 1+ (Q ), b0 b T −R T . (7)
σ(χ )dσ(χ )/d2q ≈ R T σ(η ) dσ(χ )/d2q ≈ 1+ (Q )
b0 b2 T b b0 T R T
Thesearecolorsingletmodelexpressions,whichmaybejustifiedforC =+bottomoniumstatesfrom
NRQCDconsiderations[12,14]andbyseveralnumericalstudiesofcoloroctetcontributions[14–16].
TMDfactorizationforthe p-wavestatesχ hasbeencalledintoquestionthough[17]. Consistency
bJ
betweentheexperimentalresultsfor (Q )from(7),e.g.atLHCb,canbeusedtoassessthepossible
T
R
factorizationbreakingcontributions. Becauseofthesmallenergyscaledifferences(m = 9.4GeV,
ηb
m =9.9GeV,m =10.3GeV),evolutioneffectsshouldbenegligibleinthiscomparison.
χb0 χb2
6thInternationalconferenceonPhysicsOpportunitiesatanElecTron-IonCollider(POETICVI)
2 EIC probes
AtEICh⊥1g canbeprobedinopencharmandbottomquarkpairelectro-production,ep → e′QQX,
whereQandQarealmostback-to-backinthetransverseplane. UnlikeHiggsproductiononeneeds
tostudyangulardistributionsnow,e.g.acos2φasymmetrywhereφ=φ φ andφ aretheangles
T T/
− ⊥ ⊥
of KQ KQ [18], underthe restrictionq (KQ +KQ) K (KQ KQ)/2. Inthe asymmetry
T
expre⊥ss±ion,⊥h⊥g appears by itself, as oppos≡ed to⊥in a⊥pro≪duct⊥of≡two.⊥ T−her⊥efore, larger effects are
1
expectedandthesignofh⊥g canbedetermined.TheasymmetrydependsonQ2,K2 andM2,butthe
1 Q
maximumof the asymmetry is to a large extentindependentof these scales, and⊥around15% [19].
Therearealsoangularasymmetriesw.r.t.theleptonscatteringplanethatprobeh⊥g. Thesearemostly
1
relevant at smaller K [19]. Dijet DIS, ep e jetjetX, is similar except that also quark TMDs
′
| ⊥| →
enter.Theanalogousprocessespp QQXand pp jetjetXatRHICorLHCarenotexpectedto
→ →
beTMDfactorizing[20].
At an EIC one also can consider transversely polarized protons, where ep e QQX is a
↑ ′
→
verypromisingprocessforprobingthegluonSiverseffect. Forareviewofthestatusandprospects
of the gluon Sivers distribution, cf. [21], and for specific model studies, cf. [22]. There are also
suggestionstomeasurethegluonSiverseffectinproton-proton/ioncollisions(RHIC,AFTER@LHC),
inprocessesforwhichTMDfactorizationmayhold: p p γjetX [23,24], p p J/ψγX [25],
↑ ↑
→ →
p p J/ψJ/ψX [26, 27]. According to [25, 26], the color singlet contribution to a large extent
↑
→
dominatesoverthecoloroctetoneintheJ/ψproductionprocesses.
Such gluonSivers effect measurementsin pp collisionsare complementaryto EIC studies, be-
causeTMDsareactuallyprocessdependent,aswillbediscussednext. Althoughthisprocessdepen-
dencecanbecalculated,notallSiversfunctionsfromallprocessescanberelatedtoeachother!
3 Process dependence
IthasbeenrealizedthatTMDsin generalarenotuniversal[28–30]. Gluonrescatteringcorrections
can be summed into path-orderedexponentials in TMD correlators [31], where the gauge link
UC
orWilsonlineisalongapath . ThepathinthisWilsonlinedependsonwhethercolorchargesare
C
comingfromtheinitialstateorgoingintothefinalstate[29,30,32–35]. Surprisingly,ithasturned
out that in certain cases the shape of the Wilson lines affects observables, such as the Sivers effect
asymmetries.InSIDISthequarkTMDcorrelatorhasafuturepointingstaple-likeWilsonlinearising
from final state interactions (FSI), referred to as a + link. In the Drell-Yan (DY) process it is past
pointingfrominitialstateinteractions(ISI),a link. ThequarkSiversfunctionswith+and links
arerelatedbyparityandtimereversalinvarian−cebyanoverallminussign: f1⊥T[SIDIS] = −f1⊥T[DY−] [29].
Ingeneral,themorehadronsobservedinaprocess,themorecomplicatedtheresultingWilsonlines
andthepossiblerelationsamongTMDsofvariousprocesses[36–38].Wilsonlinesmayevenbecome
entangledortrapped,leadingtofactorizationbreaking[20,39].
Theprocessesthatallowaccessto thelinearlypolarizedgluondistributionandthe gluonSivers
distributiondependontwogaugelinksasinEq.(1). Thesubprocessγ g QQforep e QQX
∗ ′
→ →
probesagluoncorrelatorwithtwo+links,i.e.botharefuturepointing.Inthekinematicregimewhere
gluonsinoneprotonin pp γjetXdominate,oneeffectivelyselectsthesubprocessqg γq. The
→ →
lattersubprocessprobesagluoncorrelatorwitha+and link(futureandpastpointing),enclosingan
−
area. Asaconsequence,thesetwoprocessesprobetwodistinct,independentgluonSiversfunctions.
Theycorrespondtoantisymmetric(f )andsymmetric(d )colorstructuresasdiscussedin[40].
abc abc
AtLHCgg H andgg [QQ]bothprobeagluoncorrelatorwithtwo links. Ash⊥g[+,+] =
→ → − 1
h⊥g[−,−](andh⊥g[+,−] =h⊥g[−,+]),oneconcludesthatEICandLHCcanprobethesameh⊥gfunction.
1 1 1 1
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Table1.ListofprocessesthatprobetheWWand/orDPunpolarizedgluonTMDatsmallx[41].
DIS&DY SIDIS pA hX pA γjetX DijetinDIS Dijetin pA
→ →
fg[+,+](WW) √ √
1 × × × ×
fg[+,−](DP) √ √ √ √ √
1 ×
Bute.g.gg Hgprobesamorecomplicatedlinkstructure. Ontheotherhand,forthegluonSivers
function it h→olds that f⊥g[+,+] = f⊥g[−,−] and f⊥g[+,−] = f⊥g[−,+]. One thus concludesthat the
1T − 1T 1T − 1T
proposedgluonSiversTMDstudiesatEICandatRHICorAFTER@LHCarecomplementary[21].
ThisTMDnonuniversalityisnotjustapolarizationissue. Itwasfirstrealizedinasmall-xcontext
thatthisprocessdependencealsoappliestotheunpolarizedgluonTMD fg[41].
1
4 Small-x: a tale of two gluon distributions
At small x (and large N ) there are two unpolarized gluon distributions that matter [41], the gluon
c
correlator with two + links (for which fg[+,+] = fg[−,−]) and the one with a + and a link (for
1 1 −
which fg[+,−] = fg[−,+]). In[41]theseweredenotedbyG(1) andG(2), respectively. Atsmall xthey
1 1
correspond to the Weizsäcker-Williams (WW) and dipole (DP) distributions, which are in general
different. The fact that there are two distinct butequally valid definitionsfor the gluondistribution
was noted first in “A tale of two gluondistributions” by Kharzeev, Kovchegov& Tuchin (KKT) in
[42],wheretheauthorssaythatthey“cannotofferanysimplephysicalexplanationofthisparadox”.
The explanation turns out to be the process dependence of the gluon distribution, in other words,
its sensitivity to the ISI/FSI in a process. Here it is not so much the direction, but rather whether
a process is only sensitive to either ISI or FSI or to both ISI and FSI. The difference between the
WW and DP distributionswould disappearwithout ISI/FSI. In the MV model consideredby KKT,
onemaynotnoticetheoriginforthedifference,becausethetwogluondistributionsbecomerelated:
xG(2)(x,q ) MV q2 2 xG(1)(x,q )[41,42]. Forinstance,theprocessγA QQX hasbeenstudied
in [43] in⊥the∝MV⊥∇mq⊥odel, whe⊥re the cross section dσ /dydk is expre→ssed in terms of C(k ) =
T
d2x eik x U(0)U (x ) G(2)whichistheDPdistribution,w⊥hereastheprocessratherprobe⊥sthe
WR Wd⊥istr⊥i·bu⊥htionG(1†) ([⊥+,i+∼]).
DifferentprocessesprobeG(1)orG(2)oramixture,aslistedinTable1. Fordijetproductionin pA
collisions,theresultrequireslargeN ,otherwise(four)additionalfunctionsappear(cf.[44]).
c
ThisprocessdependenceofTMDsimpliesthatalso their p -widthsare processdependent,and
T
asaconsequence,itgivesanadditionalprocessdependenceto p -broadening[45].
T
The WW and DP h⊥g distributions will be different too. Within the MV model [46] the DP
1
h⊥g distributionisfoundtobemaximalforalltransversemomenta,whiletheWWh⊥g distribution
1 1
is maximal only at large k ( Q ) and suppressed w.r.t. fg in the saturation region (k Q ).
T ≫ s 1 T ≪ s
Maximal(positive)linearpolarizationalsoarisesinthesmall-x“k -factorization”approach[47]:
T
1 kµkν
Γµν(x,k ) = T T fg(x,k2). (8)
g T maxpol x k2 1 T
T
Finally, the perturbative tail of h⊥g in Eq. (6) has a 1/x growth, which keeps up with fg towards
1 1
small x. Clearly there is no theoretical reason why h⊥g should be small, especially at small x. In
1
analogyto fg, in Table 2 we list processeswherethe WW and DP h⊥g distributionsenter (or not).
1 1
It turns out that the processes DIS, DY, SIDIS, hadron and γ + jet production in pA collisions do
6thInternationalconferenceonPhysicsOpportunitiesatanElecTron-IonCollider(POETICVI)
Table2.ListofprocessesthatprobetheWWand/orDPlinearlypolarizedgluonTMDatsmallx.
DIS&DY SIDIS pA hX pA γjetX DijetinDIS Dijetin pA
→ →
h⊥g[+,+](WW) √ √
1 × × × ×
h⊥g[+,−](DP) √
1 × × × × ×
notprobeh⊥g inleadingpower[48]. DijetproductioninepandeAcollisionsatsmall x probesthe
1
WW distribution. Since there are different expectations inside and outside the saturation region, it
would thus be very interesting to study h⊥g (via the cos2φ asymmetries discussed earlier) in dijet
1
DIS at a high-energyEIC. The relevant expressions for general x can be found in [19] and small-
x expressions in [46, 49]. As said, these expressions involve only the WW-type distributions (at
any N ). In contrast, dijet and open heavy quark pair production in pp and pA collisions suffer
c
from factorization breaking[20]. Although at small x the factorizationbreaking contributionsmay
becomesuppressed,effectivelyrestoringTMDfactorization[44,50],stillacombinationofsixdistinct
distributionsisprobed,complicatingtheanalysisconsiderably,probablytoomuch.
5 Summary
Productionof(pseudo-)scalarparticlesatLHCisagoodwaytoprobegluondistributions,butunfor-
tunatelytheeffectoflineargluonpolarizationonHiggsproductionissmall(2-5%level),smallerthan
thecurrenttheoreticaluncertaintyintheperturbativetreatment(NNLL+NNLO).C = +quarkonium
statesmayofferalternativeprobes,butinthiscasethepredictionshavelargetheoreticaluncertainties.
FutureLHCdataonbottomoniumstatesχ andη aremostpromising. Lineargluonpolarization
b0/2 b
isexpectedtoleadtolargedifferencesbetweenthesethreestates.
HeavyquarkpairanddijetproductioninDISatahigh-energyEICoffercleanchannelsforprobing
linearlypolarizedgluonsandthegluonSiverseffect. Specificcos2φasymmetriesmayexhibitlarge
h⊥g effects, allowing to study its sign, its small-x behavior. It may even show saturation effects,
1
as itprobesthe WW or f-type([+,+])distribution, whichis expectedto show a significantchange
in behavior around k Q . This same distribution happens to appear in Higgs or JPC = 0 +
s ±
⊥ ∼
quarkoniumproductionatLHC.Incontrast,forthegluonSiversTMDthecleanestprobesatEICand
atRHICand/orAFTER@LHCareactuallyentirelycomplementary.
Acknowledgements
IwouldliketothankMaartenBuffing,WilcodenDunnen,Jean-PhilippeLansberg,PietMulders,ElenaPetreska,
CristianPisano,andJianZhou,forfruitfuldiscussionsand/orcollaborations.
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