Pinning down the linearly-polarised gluons inside unpolarised protons using quarkonium-pair production at the LHC

University of Groningen

Pinning down the linearly-polarised gluons inside unpolarised protons using quarkonium-pair

production at the LHC

Lansberg, Jean-Philippe; Pisano, Cristian; Scarpa, Florent; Schlegel, Marc

Published in:

Physics Letters B

DOI:

10.1016/j.physletb.2018.08.004

IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from

it. Please check the document version below.

Document Version

Publisher's PDF, also known as Version of record

Publication date:

2018

Link to publication in University of Groningen/UMCG research database

Citation for published version (APA):

Lansberg, J-P., Pisano, C., Scarpa, F., & Schlegel, M. (2018). Pinning down the linearly-polarised gluons

inside unpolarised protons using quarkonium-pair production at the LHC. Physics Letters B, 784, 217-222.

https://doi.org/10.1016/j.physletb.2018.08.004

Copyright

Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).

Take-down policy

If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim.

Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum.

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

Pinning

down

the

linearly-polarised

gluons

inside

unpolarised

protons

using

quarkonium-pair

production

at

the

LHC

Jean-Philippe Lansberg

a

,

∗

,

Cristian Pisano

b

,

c

,

Florent Scarpa

a

,

d

,

Marc Schlegel

e

,

f

a_{IPNO,CNRS-IN2P3,Univ.Paris-Sud,UniversitéParis-Saclay,91406OrsayCedex,France}

b_{DipartimentodiFisica,UniversitàdiPavia,andINFN,SezionediPaviaViaBassi6,I-27100Pavia,Italy}

c_{DipartimentodiFisica,UniversitàdiCagliari,andINFN,SezionediCagliariCittadellaUniversitaria,I-09042Monserrato(CA),Italy} d_{VanSwinderenInstituteforParticlePhysicsandGravity,UniversityofGroningen,Nijenborgh4,9747AGGroningen,theNetherlands} e_{InstituteforTheoreticalPhysics,UniversitätTübingen,AufderMorgenstelle14,D-72076Tübingen,Germany}

f_{DepartmentofPhysics,NewMexicoStateUniversity,LasCruces,NM88003,USA}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received14May2018

Receivedinrevisedform31July2018 Accepted3August2018

Availableonline6August2018 Editor: A.Ringwald

Weshowthattheproductionof J/ψorϒpairsinunpolarisedpp collisionsiscurrentlythebestprocess tomeasurethemomentumdistributionoflinearly-polarisedgluonsinsideunpolarisedprotonsthrough thestudyofazimuthalasymmetries. Notonlytheshort-distancecoefficientsforsuchreactionsinduce the largest possible cos 4φ modulations, but analysed data are alreadyavailable. Among the various finalstatespreviouslystudiedinunpolarisedpp collisionswithintheTMDapproach,di- J/ψproduction exhibits by far the largest asymmetries, in the region studied by the ATLAS and CMS experiments. In addition, weuse the veryrecent LHCb data at 13 TeV toperform the firstfit ofthe unpolarised transverse-momentum-dependentgluondistribution.

©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

1. Introduction

Probablyoneofthemoststrikingphenomenaarising fromthe extensionofthecollinearfactorisation–inspiredfromFeynman’s and Bjorken’sparton model – to Transverse Momentum Depen-dent (TMD) factorisation [1–4] is the appearance of azimuthal modulationsinduced bythe polarisationofpartonswithnonzero transversemomentum – even inside unpolarised hadrons. In the caseofgluonsin a proton,whichtrigger mostof thescatterings at highenergies, this new dynamics is encoded in the distribu-tionh⊥₁g

(

x

,

kT2

, μ)

oflinearly-polarisedgluons [5].Inpractice,they

generate cos 2

φ

(cos 4

φ

) modulations in gluon-fusion scatterings wheresingle(double)gluon-helicityflipsoccur.Theycanalsoalter transverse-momentum spectra,such as that ofa H0 _{boson [6,7],}

viadoublegluon-helicityflips.

InthisLetter, weshow that di- J

/ψ

production,whichamong thequarkonium-associated-productionprocesseshasbeenthe ob-jectofthelargestnumberofexperimental studiesattheLHCand the Tevatron [8–12], is in fact the ideal process to perform the firstmeasurement ofh⊥₁ g

(

x

,

k2T

, μ)

.It indeedexhibits thelargest

*

Correspondingauthor.

E-mailaddress:Jean-Philippe.Lansberg@in2p3.fr(J.-P. Lansberg).

possibleazimuthalasymmetriesinregionsalreadyaccessedbythe ATLASandCMSexperimentswheresuchmodulationscanbe mea-sured.Alongthewayofourstudy,weperformthefirstextraction of f₁g

(

x

,

k_T2

, μ)

–itsunpolarised counterpart–using recentLHCb data.

2. TMDfactorisationforgluon-inducedscatterings

TMDfactorisationextendscollinearfactorisationbyaccounting forthepartontransverse momentum,generallydenotedby kT.It

appliestoprocessesinwhichamomentumtransferismuchlarger than

|

kT

|

, for instance at the LHC when a pair of particles (e.g.

twoquarkoniumstates

Q

)isproducedwithalargeinvariantmass (M_QQ)ascomparedtoitstransversemomentum( P_QQT).

In practice, the gluon TMDs in an unpolarised proton witha momentum P andmass Mp are definedthroughthehadron

cor-relator



μνg

(

x

,

kT

, μ)

[5,13,14], parametrised in terms of two

in-dependentTMDs,theunpolariseddistribution f₁g

(

x

,

k2T

, μ)

andthe

distributionoflinearly-polarised gluonsh⊥₁g

(

x

,

k2T

, μ)

(see Fig. 1),

where the gluon four-momentum k is decomposed ask

=

xP

+

kT

+

k−n [n is anylight-like vector (n2

=

0) such that n

·

P

=

0], k2T

= −

k

2

T and g

μν

T

=

gμν

− (

P μnν

+

P νnμ

)/

P

·

n and

μ

isthe

fac-torisationscale. https://doi.org/10.1016/j.physletb.2018.08.004

0370-2693/©2018TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3_.

218 J.-P. Lansberg et al. / Physics Letters B 784 (2018) 217–222

Fig. 1. Representative Feynmandiagramforp(P1)+p(P2)→ Q(PQ,1)+Q(PQ,2)+

X viagluonfusionatLOintheTMDframework.

In the TMD approach and up to corrections suppressed by powers of the observed system transverse momentum over its invariant mass, the cross section for any gluon-fusion process (here g

(

k1

)

+

g

(

k2

)

→

Q(

P_Q,1

)

+

Q(

P_Q,2

)

) can be expressed as

acontraction andaconvolution ofa partonicshort-distance con-tribution,

M

μρ , with two gluon TMD correlators evaluated at

(

x1

,

k1T

, μ)

and

(

x2

,

k2T

, μ)

.

M

μρ issimplycalculated in

pertur-bativeQCDthrough aseriesexpansion in

α

s [15] using Feynman

graphs(see Fig.1).

Owing to process-dependent Wilson lines in the definition of the correlatorswhich they parametrise, the TMDs are in general not universal. Physicswise, theseWilson linesdescribe the non-perturbative interactions ofthe active parton – the gluon inour case–withsoftspectatorquarksandgluonsinthenucleonbefore orafterthehard scattering.Fortheproductionofdi-leptons,

γ γ

, di-

Q

orboson-

Q

pairsviaaColor-Singlet(CS)transitions [16–18] –i.e. forpurelycolorlessfinalstates–inpp collisions,only initial-state interactions (ISI)betweentheactive gluonsandthe specta-torscanoccur. Mathematically,theseISIcanbeencapsulated [19] inTMDswithpast-pointingWilsonlines–theexchangecanonly occur beforethehard scattering. Suchgluon TMDscorrespondto the Weizsäcker–Williams distributions relevant for the low-x re-gion[20,21].

Besides,in lepton-induced productionofcolourful final states, like heavy-quark pair, dijet or J

/ψ

(via Colour Octet (CO) tran-sitions or states) production [22–24], to be studied at a future Electron–Ion Collider (EIC) [25], only final-state interactions (FSI) take place. Yet, since f₁g and h⊥₁g are time-reversal symmetric (T -even),1 TMD factorisationtells us that one in fact probes the same distributions in both the production of colourless systems

inhadroproductionwithISI andofcolourful systems in leptopro-duction with FSI. In particular, one expects(see [29] for further discussions)that,

f₁g[γp→QQ X¯ ]

(x,

k2_T

,

μ

)

=

f₁g[pp→QQX]

(x,

k2_T

,

μ

),

h₁⊥,g[γp→QQ X¯ ]

(x,

k2_T

,

μ

)

=

h₁⊥,g[pp→QQX]

(x,

k2_T

,

μ

).

(1)

In practice,this means that one should measure theseprocesses at similar scales,

μ

. The virtuality of the off-shell photon, Q ,

should be comparable to the invariant mass of the quarkonium pair,M_QQ.Ifitisnotthecase,theextractedfunctionsshouldbe evolvedtoacommonscalebeforecomparingthem.

1 _{Unlike otherTMDs [26,27] suchasthegluondistributioninatransversally} po-larisedproton,alsocalledtheSiversfunction [28].

Extracting thesefunctionsindifferent reactionsis essential to test this universality property of the TMDs – akin to the well-known sign changeof the quark Sivers effect [19,30] –,in order tovalidateTMDfactorisation.

3. Di-

Q

production&TMDfactorisation

ForTMDfactorisationtoapply,di-

Q

productionshouldatleast satisfy both following conditions. First, it should result from a Single-Parton Scattering (SPS). Second, FSI should be negligible, which is satisfied when quarkonia are produced via CS transi-tions [15].Forcompleteness,wenotethataformalproofof factori-sationforsuchprocessesisstilllacking.Wealsonotethat,insome recentworks [31–33],TMD factorisationhasbeenassumedinthe description ofprocessesinwhichboth ISIandFSIarepresent. In thatregard,aswediscussbelow,theprocesseswhichweconsider herearesafer.

The contributions of Double-parton-scatterings (DPSs) leading to di- J

/ψ

is below10% for



y

∼

0 in theCMSand ATLAS sam-ples [11,34], that isaway fromthethreshold witha P_QT cut. In

suchacase,DPSsonlybecomesignificantatlarge



y.IntheLHCb acceptance, they cannot be neglected butcan be subtracted [12] assuming the J

/ψ

fromDPSstobeuncorrelated;thisisthe stan-dardprocedureatLHCenergies [35–41].

The CS dominance tothe SPS yieldis expectedsince eachCO transition goesalong witha relative suppressionon the orderof

v4_{[42–44] (see [45–47] forreviews)–v beingtheheavy-quark}

ve-locityinthe

Q

restframe.Fordi- J

/ψ

productionwithv2 c



0

.

25,

the CO/CS yield ratio,scaling as v8_c, isexpected tobe below the per-centlevelsinceboththeCOandtheCSyieldsappearatsame orderin

α

s,i.e.

α

s4.Thishasbeencorroboratedbyexplicit

compu-tations [34,48,49] withcorrectionsfrom theCO statesbelow the per-cent level in the region relevant for our study. Only in re-gionswhereDPSsareanyhowdominant(large



y) [34,50,51] such CO contributions might become non-negligible because of spe-cific kinematicalenhancements [34] whicharehoweverirrelevant whereweproposetomeasuredi- J

/ψ

productionasaTMDprobe. We furthernotethat thedi- J

/ψ

CSyieldhasbeenstudiedup to next-to-leading(NLO)accuracyin

α

s[52–54] incollinear

factorisa-tion.Thefeeddownfromexcitedstatesisalsonotproblematicfor TMDfactorisationtoapply: J

/ψ

+

χ

cproductionissuppressed [34]

and J

/ψ

+ ψ

 can be treatedexactly like J

/ψ

+

J

/ψ

. Fordi-

ϒ

, theCSyieldshouldbeevenmoredominantandtheDPS/SPSratio shouldbesmall.

Following [55], thestructureoftheTMDcrosssection for

QQ

productionreads d

σ

dM_QQdY_QQd2_P QQTd

=



M2_QQ

−

4M_Q2

(2

π

)

2_{8s M}2 QQ



F1

C



f₁gf₁g



+

F2

C



w2h⊥1gh⊥ g 1



+

cos 2φCS



F3

C



w3f1gh⊥ g 1



+

F₃

C



w₃h₁⊥gf₁g



+

cos 4φCSF4

C



w4h⊥1gh⊥ g 1

 

,

(2)

whered



=

dcos

θ

CSd

φ

CS,

{θ

CS

,

φ

CS

}

aretheCollins–Soper(CS)

an-gles [56] and Y_QQ is the pair rapidity – P_QQT and YQQ are

defined in thehadron c.m.s.In the CS frame, the

Q

directionis along e

= (

sin

θ

CScos

φ

CS

,

sin

θ

CSsin

φ

CS

,

cos

θ

CS

)

. The overall factor

isspecifictothemassofthefinal-stateparticlesandtheanalysed differentialcrosssections,andthehardfactors Fi dependneither

on Y_QQnoron P_QQT.Inaddition,letusnotethat –awayfrom

thatis ourpreferredregionto avoidDPScontributions.The TMD convolutionsin Eq. (2) aredefinedas

C[

w f g

] ≡

d2k1T

d2k2T

δ

2

(

k1T

+

k2T

−

PQQT

)

×

w(k1T

,

k2T

)

f

(x

1

,

k21T

,

μ

)

g(x2

,

k 2 2T

,

μ

) ,

(3)

where w

(

k1T

,

k2T

)

are generic transverse weights and x1,2

=

exp

[±

Y_QQ

]

M_QQ

/

√

s,withs

= (

P1

+

P2

)

2.TheweightsinEq. (2)

areidenticalforallthegluon-inducedprocessesandcanbefound in [55].

4. Theshort-distancecoefficientsFi

The factors Fi are calculable process by process and we

re-fer to [55] for details on how to obtain them from the helicity amplitudes.As such, they can be derived from the uncontracted amplitudegivenin [57].Foranyprocess, F₂(_,)_3,4

≤

F1.For

QQ

pro-duction,theyread F1

=

N

D

M_Q2 6

n=0 f1,n

(cosθ

CS

)

2n

,

F2

=

24_3M2 Q

N

D

M4_QQ 4

n=0 f2,n

(cos

θ

CS

)

2n

,

F₃

=

F3

=

−

23

₍₁

₋

_α

2

₎

_N

D

M2_QQ 5

n=0 f3,n

(cos

θ

CS

)

2n

,

F4

=

(1

−

α

2

₎

2

_N

D

M2_Q 6

n=0 f4,n

(cos

θ

CS

)

2n

,

(4) with

α

=

2M_Q

/

M_QQ,

N =

2113−4

(

N2c

−

1

)

−2

π

2

α

s4

|

RQ

(

0

)

|

4,

D =

M4

QQ

(

1

−(

1

−

α

2

)

cos

θ

CS2

)

4andwhereRQ

(

0

)

isthe

Q

radialwave

functionattheoriginand Nc

=

3 . Notethat theexpressions are

symmetricabout

θ

CS

=

π

/

2 sincetheprocessisforward-backward

symmetric. The coefficient fi,n which are simple polynomials in

α

aregivenintheAppendix A.Likeincollinearfactorisation,the Born-ordercrosssectionscalesas

α

4

s.

Bothlargeandsmall

QQ

mass,M_QQ,limitsarevery interest-ing. Indeed,when M_QQ becomes much larger than the quarko-niummass,M_Q,onefindsthat,forcos

θ

CS

→

0,

F4

→

F1

→

256

N

M4_QQM2_Q

,

(5) F2

→

81M_Q4 cos

θ

_CS2 2M4_QQ

×

F1

,

(6) F3

→

−

24M_Q2 cos

θ

_CS2 M2_QQ

×

F1

.

(7)

Onefirstobservesthat F4

→

F1,forcos

θ

CS

→

0 awayfromthe

threshold–where theCMS andATLAS datalie. Thisis the most importantresultofthisstudyandis,tothebestofourknowledge, auniquefeatureofdi- J

/ψ

anddi-

ϒ

production.Fromthis,it read-ilyfollowsthat,foragivenmagnitudeofh⊥₁g,theseprocesseswill exhibitthe largestpossible cos 4

φ

CS modulation, thusthe highest

possiblesensitivityonh⊥₁g.

Onealso observesthat F2 (F3) scales like M−_QQ4 (M−_QQ2 )

rela-tiveto F1 and F4.In other words,themodificationof the P_QQT

dependence due to the linearly-polarised gluons encoded in F2

vanishesatlargeinvariantmasses.Infact,itisalsosmallat thresh-old,M_QQ

→

2M_Q,whereonegets:

F1

→

787

N

16M6 Q

,

F2

→

3F1 787

,

F3,4

→

0. (8)

F2 canthusbeneglectedforallpurposesinwhatfollows.

GoingbacktothecasewhereM_QQ2

4M2_Q,the massscaling inEq. (5) also indicatesthat thecos 4

φ

CS modulation(double

he-licity flip)quicklytakesoverthe cos 2

φ

CS one(singlehelicityflip)

andthecos

θ

CSdependenceindicatesthatF2,3 aresuppressednear



y

∼

0.

As such, and thanks to the collected di- J

/ψ

data, we con-clude that this process is indeed the ideal one to extract the linearly-polarised gluon distributions. The previously studied

γ γ

[58], H0

₊

_{jet [31],}

_Q

₊

_γ

_[59],

_Q

₊

_γ

_or

_Q

₊

_{Z [55] processes}

showsignificantlysmallervaluesofF4

/

F1,thusastronglyreduced

sensitivityonh⊥₁g.

KnowingtheFiandanobserveddifferentialyield,onecanthus

extract the various TMD convolutions of Eq. (3) from their az-imuthal(in)dependentparts.Whenthecrosssectionisintegrated over

φ

CS, the contribution from F3,4 drops out from Eq. (2) and

only dependson

C



f₁gf₁g



and

C



w2h⊥1gh⊥ g 1



. Togo further, we define cos n

φ

CS [for n

=

2

,

4] weighted differential cross sections

normalisedtotheazimuthallyindependenttermas:



cos nφCS

 =



dφCScos nφCS_dM dσ QQdY_QQd2_P QQTd



dφCS_dM dσ QQdY_QQd2_P QQTd

.

(9)

It is understood that



cos n

φ

CS



computed in a range of MQQ,

Y_QQ, P_QQT orcos

θ

CS istheratioofcorrespondingintegrals.

Us-ingEq. (2),onegetsinasinglephase-spacepoint:



cos 2φCS

 ∝

F3

C



w3f₁gh⊥₁g

+

1

↔

2



(10)



cos 4φCS

 ∝

F4

C



w4h⊥1gh⊥ g 1



(11)

5. Thetransverse-momentumspectrum

Before discussingtheexpectedsizeofthe azimuthal asymme-tries, let ushave a closer lookat the transverse-momentum de-pendence of Eq. (2), entirelyencoded inthe

C[

wf g

]

, whichare process-independent,unlikethe Fi.SincethegluonTMDsarestill

unknown,weneedtoresorttomodels.

Following[60],onecanassume asimpleGaussiandependence onk2T for f g 1,namely f₁g

(x,

k2_T

,

μ

)

=

g(x,

μ

)

π



k2T



exp



−

k2T



k2T



,

(12)

where g

(

x

)

isthecollineargluonPDFand



k2T



implicitlydepends

onthescale

μ

.

Since F2 is always small compared to F1, the P_QQT

spec-truminpracticefollowsfromtheTMDconvolution

C[

f1f1

]

which

only dependson



k2

T



.Conversely, one can thus fit



k

2

T



from the P_QQT spectrum recentlymeasured by the LHCbCollaboration at 13 TeV [12] (seeFig.2)fromwhichwehavethesubtractedtheDPS contributionsevaluatedbyLHCb.SuchDPSsareindeedexpectedto yieldadifferent



P_QQ2

T



sincetheyresultfromtheconvolutionof

twoindependent2

→

2 scatterings.

Wefurther notethat, forTMDAnsätze withfactorised depen-dences on x and k2T, the normalised PQQT spectrum depends

neitheron x noronothervariables.The dataonthe P_QQT

spec-trumarefitteduptoM_QQ

/

2,employinganon-linearleast-square minimisationprocedurewiththeLHCbexperimentaluncertainties usedtoweightthedata.Weobtain



kT2



=

3

.

3

±

0

.

8 GeV

2_.The

220 J.-P. Lansberg et al. / Physics Letters B 784 (2018) 217–222

Fig. 2. The normalised PQQT dependenceofthe di- J/ψ yieldobtainedwith a Gaussian f1g withk2TfittothenormalisedLHCbdataat13 TeV [12] [Thedata inthegreyzonewerenotusedforthefitsincetheTMDframeworkdoesnotapply there].

This is the first time that experimental information on gluon TMDsisextractedfromagluon-inducedprocesswithacolourless finalstate,forwhichTMDfactorisationshouldapply.The discrep-ancybetween the TMD curve and thedata for P_QQT



MQQ

/

2

is expected, asit leaves room for hard final-state radiations not accountedforintheTMDapproachoutsideofitsrangeof applica-bility.

The data usedfor our



k2T



fitcorrespond to a scale,

μ

, close

to M_QQ

∼

8 GeV. As such, it should be interpreted as an effec-tive value, including both nonperturbative and perturbative con-tributions. Thelatter, through TMD QCD evolution,increases



k2T



with

μ

[6,61,62]. Extracting a genuine nonpertubative



k2T



[at

μ



1 GeV]thusrequirestoaccountforTMDevolutionalongwith afittodataatdifferentscales. Di- J

/ψ

datafromLHCb,CMSand ATLASshould inprinciplebe enoughto disentanglethese pertur-bativeandnonperturbativeevolutioneffects,yetrequiringacareful accountforacceptanceeffectsaswellasperturbativecontributions beyondTMDfactorisation;thesedataareindeednotdouble differ-entialinP_QQT andMψ ψ.Thisisleftforafuturestudy.

In the above extraction of



k2

T



, we have neglected the

influ-enceofh⊥₁g onthe P_QQT spectrum.TheLHCbmeasurementwas

madewithoutanytransverse-momentumcuts,thusnearthreshold whereM_QQ

∼

2M_Qandwhere F2

/

F1iscloseto0.4%(cf.Eq. (8)).

Thesituation isanalogousto

Q

+

γ

[59],

Q

+

γ

_or

_Q

₊

_{Z [55]}

withanegligibleimpactofh⊥₁g ontheTMspectrabutsignificantly differentfromthatfordi-photon [58],single

η

c [65],di-

η

c[66] and

H0

₊

_{jet [31] production.Datanonethelessdonotexistyetforany}

ofthesechannels.Unfortunately,theCMSdi-

ϒ

sample [67] isnot largeenough(40events) toperforma



k2T



fitatMQQ

∼

20 GeV.

With100fb−1 of13TeVdata,thisshouldbepossible. 6. Azimuthaldependences

Intheperturbativeregime, particularlyatlargekT,h⊥1g canbe

connected [61,62] tog

(

x

)

witha

α

spre-factor.Inthe

nonperturba-tiveregime,thisconnectionislostandwecurrentlydonotknow whetheritisalso

α

s-suppressed.Assuch,itremainsusefulto

con-siderthemodel-independentpositivitybound [5,63]:

|

h⊥₁g

(x,

k2_T

,

μ

)

| ≤

2M 2 p k2_T f g 1

(x,

k 2 T

,

μ

)

(13)

holdingforanyvalueofx andk2_T. Thisboundissatisfied [6] by h⊥₁g

(x,

k2_T

,

μ

)

=

2M 2 p



k2T



(1

−

r) r g(x,

μ

)

π



k2T



exp



1

−

k 2 T r



k2T



(14)

Fig. 3. Various ratiosoftheTMDconvolutionsusingbothourmodelsofh⊥1gfor k2

T=3.3 GeV2(centralcurves)variedby0.8 GeV2(bands).

with r

<

1. We take r

=

2

/

3 maximising thesecond kT moment

ofh⊥₁g.Wenotethat suchachoiceismotivatedbyprevious TMD studies [6,65] wheretheeffectsofh⊥₁gwerealsopredicted.In gen-eral,valuesofr smallerthan2

/

3 will leadtoasymmetrieswhich are narrowerin P_QQT,butwithalargermaximum.Ontheother

hand, for r

>

2

/

3, the asymmetries will be broader and with a smaller peak. Withthis choice, all 4 TMD convolutions are sim-ple analytical functions whose P_QQT dependence is shown on

Fig. 3.Beside,computationsinthe high-energy(low-x) limit(see

e.g. [20,64])suggesttotake h⊥₁g

(x,

k2_T

,

μ

)

=

2M 2 p k2_T f g 1

(x,

k2T

,

μ

).

(15)

The corresponding convolutions can easily be calculated nu-merically.Their P_QQT dependenceisshownon Fig. 3for



k

2

T



=

3

.

3 GeV2 (whichfollows fromour fitof f₁g).As we discusslater, havingboththesemodelsathandisveryconvenient,asitallows us to assess the influence of the variation of h⊥₁g – e.g. due to the scale evolution –on the observables.“Model1” will refer to the Gaussian form withr

=

2

/

3 and “Model 2” to theform sat-urating thepositivitybound. Thebands inFig.3 correspondto a variation of



k2T



about3.3GeV

2 _{by 0.8GeV}2 _{(which alsoresults}

fromourfit).Wenotethatthesebandsareingeneralsignificantly smallerthanthedifferencebetweenthecurvesforModel1and2. Assuch,wewillusetheresultsfromModel1and2toderive un-certainty bandswhichhowevershouldremainindicativesince,as statedabove,nearlynothingisknownaboutthesedistributions.

Having fixed the functional form of the TMDs and



k2

T



and

having computed the factors Fi, we are now ready to provide

predictionsfortheazimuthalmodulationsthrough



cos n

φ

CS



asa

functionof P_QQT,cos

θ

CS orMQQ.Figs.4a&4b show



cos n

φ

CS



(n

=

2

,

4)asa functionof P_QQT forboth ourmodelsofh⊥₁ g for 3 values of M_QQ, 8, 12 and21 GeVfor

|

cos

θ

CS

|

<

0

.

25. These

values are relevant respectivelyfor the LHCb [12], CMS [10] and ATLAS [11] kinematics. Stillto keeptheTMD description applica-ble, we haveplottedthe spectra up to M_QQ

/

2.Let usalsonote that withour factorised TMD Ansätze,



cos n

φ

CS



do not depend

onY_QQ.Indeed,thepairrapidityonlyenterstheevaluationofd

σ

viathemomentumfractionsx1,2 intheTMDs.Itthussimplifiesin

theratios.

Thesize oftheexpectedazimuthalasymmetriesisparticularly large, e.g. for P2

QQT

 

k

2

T



.



cos 4

φ

CS



even gets closeto 50% in

the P_QQT region probed byCMSandATLAS for

|

cos

θ

CS

|

<

0

.

25;

thisisprobablythehighestvalueeverpredictedforagluon-fusion processwhichdirectlyfollowsfromtheextremelyfavourablehard coefficient F4 – as large as F1. Such values are truly promising

Fig. 4.cos nφCSforn=2,4 computedfor|cosθCS|<0.25 andfor0.25<cosθCS<0.5 fork2T=3.3 GeV

2_{for3valuesofM}

QQ(8,12and21GeV)relevantrespectively fortheLHCb [12],CMS [10] andATLAS [11] kinematics.ThespectraareplotteduptoMQQ/2.OurresultsdonotdependonYQQ.Theuncertaintybandsresultfromthe useofbothourmodelsofh1⊥g.Thesolidline,whichshowsthelargestasymmetriescorrespondstotheModel2(saturationofthepositivitybound)andthedashedlineto Model1.

toextract thedistribution h⊥₁g oflinearly-polarised gluonsinthe protonwhichappearsquadraticallyin



cos 4

φ

CS



.Inview ofthese

results,itbecomesclearthatthekinematicsofCMSandATLASare bettersuitedwithmuchlargerexpectedasymmetriesthanthatof LHCb,notfarfromthreshold,unlessLHCbimposes PψT cuts.



cos 2

φ

CS



allows one to lift the sign degeneracy of h⊥₁g in



cos 4

φ

CS



butis below 10% for

|

cos

θ

CS

|

<

0

.

25 (Fig. 4a). This is

expected since F3 vanishes for small cos

θ

CS (Eq. (5)). It would

thusbeexpedienttoextendtherangeof

|

cos

θ

CS

|

pendingtheDPS

contamination.Indeed,in viewof recentdi- J

/ψ

phenomenologi-calstudies [34,68,69], one expectstheDPSsto becomedominant atlarge



y while thesecannot be treatedalongthe linesofour analysis.ToensuretheSPSdominance,itisthusjudicioustoavoid theregion



y

>

2,and probably



y

>

1 tobe on the safeside. Eventhoughtherelationbetween



y –measuredinthehadronic c.m.s. – and cos

θ

CS is in general not trivial, it strongly

simpli-fies when P2_QT

(

M_Q2

,

P2_QQ

T

)

, such that cos

θ

CS

=

tanh



y

/

2. 2 Up to

|

cos

θ

CS

|

∼

0

.

5, the sample should thus remain SPS

domi-natedinparticularwiththe CMSandATLAS P_QT cuts.Infact,in

abin0

.

25

<

|

cos

θ

CS

|

<

0

.

5,



cos 2

φ

CS



nearlyreaches30%(Fig.4c).

Onthe contrary,



cos 4

φ

CS



exhibits a node closeto cos

θ

CS

∼

0

.

3

(Fig. 4d). As such, measuring



cos 4

φ

CS



for

|

cos

θ

CS

|

<

0

.

25 and

0

.

25

<

|

cos

θ

CS

|

<

0

.

5 wouldcertainlybeinstructive.Ifourmodels

forh⊥₁g arerealistic,thisisdefinitelywithinthereachofCMSand ATLAS,probablyevenwithdataalreadyontape.

TMD evolution will affect the size of these asymmetries, al-though in a hardly quantifiable way. In fact, TMD evolution has never been applied to any 2

→

2 gluon-induced process and is beyond the scope of our analysis. One can however rely on an analogywitha

η

b-productionstudy [62] (a 2

→

1 gluon-induced

process at

μ

∼

9 GeV) where the ratio

C[

w2h₁⊥gh⊥₁g

]/

C[

f₁gf₁g

]

2 _{Infact,y/2 thencoincideswiththeusualdefinitionofthepseudorapidityof} onequarkoniumsincey isnotsensitivetothelongitudinalboostbetweentheCS frameandthec.m.s.

was found torangebetween0.2and0.8. This arisesfroma sub-tle interplay between the evolution and the nonperturbative be-haviourof f₁g andh₁⊥g.Weconsiderthattheuncertaintyspanned by our Model 1 and 2gives a fair account of the typical uncer-tainty ofan analysiswithTMD evolution,hencethebandsinour plots.

7. Conclusions

We have found out that the short-distance coefficients to the azimuthal modulations of J

/ψ

(

ϒ

) pair yields equate the az-imuthallyindependentterms,whichrenderstheseprocessesideal probes of the linearly-polarised gluon distributions in an unpo-larisedproton,h⊥₁g.Experimentaldataalreadyexist–morewillbe recordedinthenearfuture–anditonlyremainstoanalysethem alongthelinesdiscussedabove,byevaluatingtheratios



cos 2

φ

CS



and



cos 4

φ

CS



.Infact, wehavealreadyhighlighted therelevance

oftheLHCdatafordi- J

/ψ

productionbyconstraining,forthefirst time,thetransverse-momentumdependenceof f₁gatascaleclose to2Mψ.

Letusalsonote thatsimilar measurementscan becarriedout atfixed-targetset-upswhereluminositiesarelargeenoughto de-tect J

/ψ

pairs. The COMPASS experiment withpion beams may alsorecorddi- J

/ψ

eventsasdidNA3inthe80’s [70,71].Whereas single- J

/ψ

productionmay partlybe fromquark-antiquark anni-hilation, di- J

/ψ

production should mostly be from gluon fusion andthusanalysablealongthe abovediscussions. Usingthe7TeV LHCbeams [72] in thefixed-targetmodewithaLHCb-like detec-tor [73–75], one can expect 1000eventsper 10fb−1_{, enough to}

measureapossiblex dependenceof



k2

T



aswellastolookfor

az-imuthal asymmetriesgenerated by h⊥₁g.Such analyses could also becomplementedwithtarget-spinasymmetrystudies [76–78], to extractthegluonSiversfunction f_1T⊥gaswellasthegluon transver-sitydistributionh_1Tg orthedistributionoflinearly-polarisedgluons inatransverselypolarisedproton,h⊥_1Tg,pavingthewayforan in-depthgluontomographyoftheproton.

222 J.-P. Lansberg et al. / Physics Letters B 784 (2018) 217–222

Acknowledgements

WethankA.Bacchetta,D.Boer,M.EchevarriaandH.S.Shaofor usefulcommentsandL.P.Sunfordiscussionsabout [57].Thework of J.P.L. andF.S. is supported in part by the French IN2P3–CNRS viatheLIAFCPPL(Quarkonium4AFTER)andtheprojectTMD@NLO. The work of C.P. is supported by the European Research Council (ERC)undertheEuropean Union’s Horizon2020research and in-novationprogram(grantagreementNo.647981,3DSPIN).Thework ofM.S.issupportedinpartbytheBundesministeriumfürBildung undForschung(BMBF)grant05P15VTCA1.

Appendix A. Thefullexpressionsofthe fi,n

Thefactors fi,n aresimplepolynomialsin

α

,i.e. f1,0

=

6

α

8

−

38

α

6

+

83

α

4

+

480

α

2

+

256, f1,1

=

2(1

−

α

2

)(6

α

8

+

159

α

6

−

2532

α

4

+

884

α

2

+

208), f1,2

=

2(1

−

α

2

)

2

(3

α

8

+

19

α

6

+

7283

α

4

−

8448

α

2

−

168), f1,3

= −

2(1

−

α

2

)

3

(159

α

6

+

6944

α

4

−

17064

α

2

+

3968), f1,4

= (

1

−

α

2

)

4

(4431

α

4

−

27040

α

2

+

17824), f1,5

=

504(1

−

α

2

)

5

(15

α

2

−

28), f1,6

=

3888(1

−

α

2

)

6

,

(A.1) f2,0

=

α

4

,

f2,1

= −

2(

α

6

+

17

α

4

−

126

α

2

+

108), f2,2

= (

1

−

α

2

)

2

(

α

4

+

756), f2,3

= −

36(1

−

α

2

)

3

(

α

2

+

24), f2,4

=

324(1

−

α

2

)

4

,

(A.2) f3,0

=

α

2

(16

−

3

α

2

),

f3,1

=

6

α

6

+

159

α

4

−

1762

α

2

+

1584, f3,2

= (

1

−

α

2

)(3

α

6

+

19

α

4

+

5258

α

2

−

6696), f3,3

= −(

1

−

α

2

)

2

(159

α

4

+

5294

α

2

−

10584), f3,4

=

18(1

−

α

2

)

3

(99

α

2

−

412), f3,5

=

1944(1

−

α

2

)

4

,

(A.3) f4,0

=

3

α

4

−

32

α

2

+

256, f4,1

= −(

6(

α

4

+

36

α

2

−

756)

α

2

+

4768), f4,2

=

3

α

8

+

38

α

6

+

11994

α

4

−

32208

α

2

+

20400, f4,3

= −

2(1

−

α

2

)(105

α

6

+

5512

α

4

−

23120

α

2

+

19520), f4,4

= (

1

−

α

2

)

2

(3459

α

4

−

30352

α

2

+

38560), f4,5

=

72(1

−

α

2

)

3

(105

α

2

−

268), f4,6

=

3888(1

−

α

2

)

4

.

(A.4) References

[1]J.Collins,FoundationsofPerturbativeQCD,CambridgeUniversityPress,2013.

[2]S.M.Aybat,T.C.Rogers,Phys.Rev.D83(2011)114042.

[3]M.G.Echevarria,A.Idilbi,I.Scimemi,J.HighEnergyPhys.07(2012)002.

[4]R.Angeles-Martinez,etal.,ActaPhys.Pol.B46(2015)2501.

[5]P.J.Mulders,J.Rodrigues,Phys.Rev.D63(2001)094021.

[6]D.Boer,W.J.denDunnen,C.Pisano,M.Schlegel,W.Vogelsang,Phys.Rev.Lett. 108(2012)032002.

[7]D.Boer,W.J.denDunnen,C.Pisano,M.Schlegel,Phys.Rev.Lett.111(2013) 032002.

[8]R.Aaij,etal.,Phys.Lett.B707(2012)52.

[9]V.M.Abazov,etal.,Phys.Rev.D90(2014)111101.

[10]V.Khachatryan,etal.,J.HighEnergyPhys.09(2014)094.

[11]M.Aaboud,etal.,Eur.Phys.J.C77(2017)76.

[12]R.Aaij,etal.,J.HighEnergyPhys.06(2017)047.

[13]S.Meissner,A.Metz,K.Goeke,Phys.Rev.D76(2007)034002.

[14]D.Boer,etal.,J.HighEnergyPhys.10(2016)013.

[15]J.P.Ma,J.X.Wang,S.Zhao,Phys.Rev.D88(2013)014027.

[16]C.-H.Chang,Nucl.Phys.B172(1980)425.

[17]R.Baier,R.Ruckl,Phys.Lett.B102(1981)364.

[18]R.Baier,R.Ruckl,Z.Phys.C19(1983)251.

[19]J.C.Collins,Phys.Lett.B536(2002)43.

[20]A.Dumitru,T.Lappi,V.Skokov,Phys.Rev.Lett.115(2015)252301.

[21]F.Dominguez,C.Marquet,B.-W.Xiao,F.Yuan,Phys.Rev.D83(2011)105005.

[22]D.Boer,S.J.Brodsky,P.J.Mulders,C.Pisano,Phys.Rev.Lett.106(2011)132001.

[23]D.Boer,P.J.Mulders,C.Pisano,J.Zhou,J.HighEnergyPhys.08(2016)001.

[24] S.Rajesh,R.Kishore,A.Mukherjee,2018. [25]A.Accardi,etal.,Eur.Phys.J.A52(2016)268.

[26]D.Boer,P.J.Mulders,Phys.Rev.D57(1998)5780.

[27]D.Boer,P.J.Mulders,F.Pijlman,Nucl.Phys.B667(2003)201.

[28]D.W.Sivers,Phys.Rev.D41(1990)83.

[29]D.Boer,Few-BodySyst.58(2017)32.

[30]S.J.Brodsky,D.S.Hwang,I.Schmidt,Nucl.Phys.B642(2002)344.

[31]D.Boer,C.Pisano,Phys.Rev.D91(2015)074024.

[32]A.Mukherjee,S.Rajesh,Phys.Rev.D95(2017)034039.

[33]U.D’Alesio,F.Murgia,C.Pisano,P.Taels,Phys.Rev.D96(2017)036011.

[34]J.-P.Lansberg,H.-S.Shao,Phys.Lett.B751(2015)479.

[35]T.Akesson,etal.,Z.Phys.C34(1987)163.

[36]J.Alitti,etal.,Phys.Lett.B268(1991)145.

[37]F.Abe,etal.,Phys.Rev.D47(1993)4857.

[38]F.Abe,etal.,Phys.Rev.D56(1997)3811.

[39]V.M.Abazov,etal.,Phys.Rev.D81(2010)052012.

[40]G.Aad,etal.,NewJ.Phys.15(2013)033038.

[41]S.Chatrchyan,etal.,J.HighEnergyPhys.03(2014)032.

[42]G.T.Bodwin,E.Braaten,G.P.Lepage,Phys.Rev.D51(1995)1125; G.T.Bodwin,E.Braaten,G.P.Lepage,Phys.Rev.D55(1997)5853,Erratum.

[43]P.L.Cho,A.K.Leibovich,Phys.Rev.D53(1996)6203.

[44]P.L.Cho,A.K.Leibovich,Phys.Rev.D53(1996)150.

[45]A.Andronic,etal.,Eur.Phys.J.C76(2016)107.

[46]N.Brambilla,etal.,Eur.Phys.J.C71(2011)1534.

[47]J.P.Lansberg,Int.J.Mod.Phys.A21(2006)3857.

[48]P.Ko,C.Yu,J.Lee,J.HighEnergyPhys.01(2011)070.

[49]Y.-J.Li,G.-Z.Xu,K.-Y.Liu,Y.-J.Zhang,J.HighEnergyPhys.07(2013)051.

[50]Z.-G.He,B.A.Kniehl,Phys.Rev.Lett.115(2015)022002.

[51]S.P.Baranov,A.H.Rezaeian,Phys.Rev.D93(2016)114011.

[52]J.-P.Lansberg,H.-S.Shao,Phys.Rev.Lett.111(2013)122001.

[53]L.-P.Sun,H.Han,K.-T.Chao,Phys.Rev.D94(2016)074033.

[54]A.K.Likhoded,A.V.Luchinsky,S.V.Poslavsky,Phys.Rev.D94(2016)054017.

[55]J.-P.Lansberg,C.Pisano,M.Schlegel,Nucl.Phys.B920(2017)192.

[56]J.C.Collins,D.E.Soper,Phys.Rev.D16(1977)2219.

[57]C.-F.Qiao,L.-P.Sun,P.Sun,J.Phys.G37(2010)075019.

[58]J.-W.Qiu,M.Schlegel,W.Vogelsang,Phys.Rev.Lett.107(2011)062001.

[59]W.J.den Dunnen, J.P.Lansberg,C. Pisano,M. Schlegel, Phys. Rev.Lett.112 (2014)212001.

[60]P.Schweitzer,T.Teckentrup,A.Metz,Phys.Rev.D81(2010)094019.

[61]D.Boer,W.J.denDunnen,Nucl.Phys.B886(2014)421.

[62]M.G.Echevarria,T.Kasemets,P.J.Mulders,C.Pisano,J.HighEnergyPhys.07 (2015)158;

M.G.Echevarria,T.Kasemets,P.J.Mulders,C.Pisano,J.HighEnergyPhys.05 (2017)073,Erratum.

[63]S.Cotogno,T.vanDaal,P.J.Mulders,J.HighEnergyPhys.11(2017)185.

[64]A.Metz,J.Zhou,Phys.Rev.D84(2011)051503.

[65]D.Boer,C.Pisano,Phys.Rev.D86(2012)094007.

[66]G.-P.Zhang,Phys.Rev.D90(2014)094011.

[67]V.Khachatryan,etal.,J.HighEnergyPhys.05(2017)013.

[68]C.H.Kom,A.Kulesza,W.J.Stirling,Phys.Rev.Lett.107(2011)082002.

[69]S.P.Baranov,A.M.Snigirev,N.P.Zotov,Phys.Lett.B705(2011)116.

[70]J.Badier,etal.,Phys.Lett.B114(1982)457.

[71]J.Badier,etal.,Phys.Lett.B158(1985)85[401(1985)].

[72]J.-P.Lansberg,H.-S.Shao,Nucl.Phys.B900(2015)273.

[73]L.Massacrier,etal.,Adv.HighEnergyPhys.2015(2015)986348.

[74]L.Massacrier,etal.,Int.J.Mod.Phys.Conf.Ser.40(2016)1660107.

[75]J.P.Lansberg,etal.,EPJWebConf.85(2015)02038.

[76]D.Kikola,etal.,Few-BodySyst.58(2017)139.

[77]J.P.Lansberg,etal.,PoSDIS2016(2016)241.