First measurement of the Sivers asymmetry for gluons using SIDIS data

Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

First

measurement

of

the

Sivers

asymmetry

for

gluons

using

SIDIS

data

The

COMPASS

Collaboration

C. Adolph

h

,

M. Aghasyan

y

,

R. Akhunzyanov

g

,

M.G. Alexeev

aa

,

G.D. Alexeev

g

,

A. Amoroso

aa

,

ab

,

V. Andrieux

ac

,

u

,

N.V. Anfimov

g

,

V. Anosov

g

,

A. Antoshkin

g

,

K. Augsten

g

,

s

,

W. Augustyniak

ad

,

A. Austregesilo

p

,

C.D.R. Azevedo

a

,

B. Badełek

ae

,

F. Balestra

aa

,

ab

,

M. Ball

c

,

J. Barth

d

,

R. Beck

c

,

Y. Bedfer

u

,

J. Bernhard

m

,

j

,

K. Bicker

p

,

j

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E.R. Bielert

j

,

R. Birsa

y

,

M. Bodlak

r

,

P. Bordalo

l

,

1

,

F. Bradamante

x

,

y

,

C. Braun

h

,

A. Bressan

x

,

y

,

M. Büchele

i

,

W.-C. Chang

v

,

C. Chatterjee

f

,

M. Chiosso

aa

,

ab

,

I. Choi

ac

,

S.-U. Chung

p

,

2

,

A. Cicuttin

z

,

y

,

M.L. Crespo

z

,

y

,

Q. Curiel

u

,

S. Dalla Torre

y

,

S.S. Dasgupta

f

,

S. Dasgupta

x

,

y

_,

_{O.Yu. Denisov}

ab

,

∗

_,

_{L. Dhara}

f

_,

_{S.V. Donskov}

t

_,

_{N. Doshita}

ag

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_{Ch. Dreisbach}

p

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V. Duic

x

,

W. Dünnweber

3

,

M. Dziewiecki

af

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A. Efremov

g

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P.D. Eversheim

c

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W. Eyrich

h

,

M. Faessler

3

,

A. Ferrero

u

,

M. Finger

r

,

M. Finger jr.

r

,

H. Fischer

i

,

C. Franco

l

,

N. du Fresne von Hohenesche

m

,

j

,

J.M. Friedrich

p

,

V. Frolov

g

,

j

,

E. Fuchey

u

,

4

,

F. Gautheron

b

,

O.P. Gavrichtchouk

g

,

S. Gerassimov

o

,

p

,

J. Giarra

m

,

F. Giordano

ac

,

I. Gnesi

aa

,

ab

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M. Gorzellik

i

,

5

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S. Grabmüller

p

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A. Grasso

aa

,

ab

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M. Grosse Perdekamp

ac

,

B. Grube

p

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T. Grussenmeyer

i

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A. Guskov

g

,

F. Haas

p

,

D. Hahne

d

,

G. Hamar

x

,

y

,

D. von Harrach

m

,

F.H. Heinsius

i

,

R. Heitz

ac

,

F. Herrmann

i

,

N. Horikawa

q

,

6

_,

_{N. d’Hose}

u

_,

_{C.-Y. Hsieh}

v

,

7

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S. Huber

p

,

S. Ishimoto

ag

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8

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A. Ivanov

aa

,

ab

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Yu. Ivanshin

g

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T. Iwata

ag

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V. Jary

s

,

R. Joosten

c

,

P. Jörg

i

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E. Kabuß

m

,

A. Kerbizi

x

,

y

,

B. Ketzer

c

,

G.V. Khaustov

t

,

Yu.A. Khokhlov

t

,

9

,

10

,

Yu. Kisselev

g

,

F. Klein

d

,

K. Klimaszewski

ad

,

J.H. Koivuniemi

b

,

ac

,

V.N. Kolosov

t

,

K. Kondo

ag

,

K. Königsmann

i

,

I. Konorov

o

,

p

,

V.F. Konstantinov

t

,

A.M. Kotzinian

ab

,

11

,

O.M. Kouznetsov

g

,

M. Krämer

p

,

P. Kremser

i

,

F. Krinner

p

,

Z.V. Kroumchtein

g

,

Y. Kulinich

ac

,

F. Kunne

u

,

K. Kurek

ad

,

R.P. Kurjata

af

,

A.A. Lednev

t

,

27

,

A. Lehmann

h

,

M. Levillain

u

,

S. Levorato

y

,

Y.-S. Lian

v

,

12

_,

_{J. Lichtenstadt}

w

_,

_{R. Longo}

aa

,

ab

_,

_{A. Maggiora}

ab

_,

_{A. Magnon}

ac

_,

_{N. Makins}

ac

_,

N. Makke

x

,

y

,

G.K. Mallot

j

,

∗

,

B. Marianski

ad

,

A. Martin

x

,

y

,

J. Marzec

af

,

J. Matoušek

x

,

y

,

r

,

H. Matsuda

ag

,

T. Matsuda

n

,

G.V. Meshcheryakov

g

,

M. Meyer

ac

,

u

,

W. Meyer

b

,

Yu.V. Mikhailov

t

,

M. Mikhasenko

c

,

E. Mitrofanov

g

,

N. Mitrofanov

g

,

Y. Miyachi

ag

,

A. Nagaytsev

g

,

F. Nerling

m

,

D. Neyret

u

,

J. Nový

s

,

j

,

W.-D. Nowak

m

,

G. Nukazuka

ag

,

A.S. Nunes

l

,

A.G. Olshevsky

g

,

I. Orlov

g

,

M. Ostrick

m

,

D. Panzieri

ab

,

13

,

B. Parsamyan

aa

,

ab

,

S. Paul

p

,

J.-C. Peng

ac

,

F. Pereira

a

,

M. Pešek

r

,

D.V. Peshekhonov

g

,

N. Pierre

m

,

u

,

S. Platchkov

u

,

J. Pochodzalla

m

,

V.A. Polyakov

t

,

J. Pretz

d

,

14

_,

_{M. Quaresma}

l

_,

_{C. Quintans}

l

_,

S. Ramos

l

,

1

,

C. Regali

i

,

G. Reicherz

b

,

C. Riedl

ac

,

M. Roskot

r

,

N.S. Rogacheva

g

,

D.I. Ryabchikov

t

,

10

,

A. Rybnikov

g

,

A. Rychter

af

,

R. Salac

s

,

V.D. Samoylenko

t

,

A. Sandacz

ad

,

C. Santos

y

,

S. Sarkar

f

,

I.A. Savin

g

,

T. Sawada

v

,

G. Sbrizzai

x

,

y

,

P. Schiavon

x

,

y

,

K. Schmidt

i

,

5

,

H. Schmieden

d

,

K. Schönning

j

,

15

,

E. Seder

u

,

A. Selyunin

g

,

L. Silva

l

,

L. Sinha

f

,

S. Sirtl

i

,

M. Slunecka

g

,

J. Smolik

g

,

A. Srnka

e

,

D. Steffen

j

,

p

,

M. Stolarski

l

,

O. Subrt

j

,

s

,

M. Sulc

k

,

H. Suzuki

ag

,

6

,

A. Szabelski

x

,

y

,

ad

,

∗

,

T. Szameitat

i

,

5

,

P. Sznajder

ad

,

S. Takekawa

aa

,

ab

,

M. Tasevsky

g

,

S. Tessaro

y

,

F. Tessarotto

y

,

F. Thibaud

u

,

A. Thiel

c

,

F. Tosello

ab

,

V. Tskhay

o

,

S. Uhl

p

,

A. Vauth

j

,

J. Veloso

a

,

M. Virius

s

,

J. Vondra

s

,

S. Wallner

p

,

T. Weisrock

m

,

http://dx.doi.org/10.1016/j.physletb.2017.07.018

M. Wilfert

m

,

J. ter Wolbeek

i

,

5

,

K. Zaremba

af

,

P. Zavada

g

,

M. Zavertyaev

o

,

E. Zemlyanichkina

g

,

N. Zhuravlev

g

,

M. Ziembicki

af

,

A. Zink

h

a_{University of Aveiro, Dept. of Physics, 3810-193 Aveiro, Portugal}

b_{Universität Bochum, Institut für Experimentalphysik, 44780 Bochum, Germany}16,17 c_{Universität Bonn, Helmholtz-Institut für Strahlen- und Kernphysik, 53115 Bonn, Germany} d_{Universität Bonn, Physikalisches Institut, 53115 Bonn, Germany}16

e_{Institute of Scientific Instruments, AS CR, 61264 Brno, Czech Republic}18

f_{Matrivani Institute of Experimental Research & Education, Calcutta-700 030, India}19 g_{Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia}20 h_{Universität Erlangen-Nürnberg, Physikalisches Institut, 91054 Erlangen, Germany}16 i_{Universität Freiburg, Physikalisches Institut, 79104 Freiburg, Germany}16,17 j_{CERN, 1211 Geneva 23, Switzerland}

k_{Technical University in Liberec, 46117 Liberec, Czech Republic}18 l_{LIP, 1000-149 Lisbon, Portugal}21

m_{Universität Mainz, Institut für Kernphysik, 55099 Mainz, Germany}16 n_{University of Miyazaki, Miyazaki 889-2192, Japan}22

o_{Lebedev Physical Institute, 119991 Moscow, Russia}

p_{Technische Universität München, Physik Dept., 85748 Garching, Germany}16,3 q_{Nagoya University, 464 Nagoya, Japan}22

r_{Charles University in Prague, Faculty of Mathematics and Physics, 18000 Prague, Czech Republic}18 s_{Czech Technical University in Prague, 16636 Prague, Czech Republic}18

t_{State Scientific Center Institute for High Energy Physics of National Research Center ‘Kurchatov Institute’, 142281 Protvino, Russia} u_{IRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette, France}17

v_{Academia Sinica, Institute of Physics, Taipei 11529, Taiwan}23

w_{Tel Aviv University, School of Physics and Astronomy, 69978 Tel Aviv, Israel}24 x_{University of Trieste, Dept. of Physics, 34127 Trieste, Italy}

y_{Trieste Section of INFN, 34127 Trieste, Italy} z_{Abdus Salam ICTP, 34151 Trieste, Italy}

aa_{University of Turin, Dept. of Physics, 10125 Turin, Italy} ab_{Torino Section of INFN, 10125 Turin, Italy}

ac_{University of Illinois at Urbana-Champaign, Dept. of Physics, Urbana, IL 61801-3080, USA}25 ad_{National Centre for Nuclear Research, 00-681 Warsaw, Poland}26

ae_{University of Warsaw, Faculty of Physics, 02-093 Warsaw, Poland}26

af_{Warsaw University of Technology, Institute of Radioelectronics, 00-665 Warsaw, Poland}26 ag_{Yamagata University, Yamagata 992-8510, Japan}22

a

r

t

i

c

l

e

i

n

f

o

a

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s

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Article history:

Received10January2017

Receivedinrevisedform26May2017 Accepted10July2017

Availableonline19July2017 Editor:M.Doser

Keywords:

Deepinelasticscattering Gluon

Sivers TMD PDF

TheSiversfunctiondescribesthecorrelationbetweenthetransversespinofanucleonandthetransverse motionofitspartons.Forquarks,itwasstudiedinpreviousmeasurementsoftheazimuthalasymmetry of hadrons produced in semi-inclusive deep inelastic scattering of leptons off transversely polarised nucleontargets, anditwasfoundtobenon-zero.InthislettertheevaluationoftheSiversasymmetry forgluonsispresented.Thecontributionofthephoton–gluonfusionsubprocessisenhancedbyrequiring two high transverse-momentum hadrons.The analysis methodis basedonaMonte Carlosimulation thatincludesthreehardprocesses:photon–gluonfusion,QCDComptonscatteringandtheleading-order virtual-photon absorption process. The Sivers asymmetries of the three processes are simultaneously extracted usingthe LEPTOevent generatorand aneuralnetwork approach.The methodisapplied to samplesofeventscontainingatleasttwohadronswithlargetransversemomentumfromtheCOMPASS data takenwith a160 GeV/c muon beamscatteredoff transversely polariseddeuteronsand protons. Withasignificance ofabouttwostandarddeviations,anegativevalueisobtainedforthegluonSivers asymmetry. Theresult ofasimilaranalysis foraCollins-likeasymmetry forgluonsis consistentwith zero.

©2017TheAuthor(s).PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3_.

*

Correspondingauthors.

E-mail addresses:oleg.denisov@cern.ch(O.Yu. Denisov),gerhard.mallot@cern.ch(G.K. Mallot),adam.szabelski@cern.ch(A. Szabelski). 1 _{AlsoatInstitutoSuperiorTécnico,UniversidadedeLisboa,Lisbon,Portugal.}

2 _{AlsoatDept.ofPhysics,PusanNationalUniversity,Busan609-735,RepublicofKoreaandatPhysicsDept.,BrookhavenNationalLaboratory,Upton,NY11973,USA.} 3 _{SupportedbytheDFGclusterofexcellence‘OriginandStructureoftheUniverse’(http://www.universe-cluster.de)(Germany).}

4 _{SupportedbytheLaboratoired’excellenceP2IO(France).}

5 _{SupportedbytheDFGResearchTrainingGroupProgrammes1102and2044(Germany).} 6 _{AlsoatChubuUniversity,Kasugai,Aichi487-8501,Japan.}

7 _{AlsoatDept.ofPhysics,NationalCentralUniversity,300JhongdaRoad,Jhongli32001,Taiwan.} 8 _{AlsoatKEK,1-1Oho,Tsukuba,Ibaraki305-0801,Japan.}

9 _{AlsoatMoscowInstituteofPhysicsandTechnology,MoscowRegion,141700,Russia.} 10 _{SupportedbyPresidentialGrantNSh-999.2014.2(Russia).}

11 _{AlsoatYerevanPhysicsInstitute,AlikhanianBr.Street,Yerevan,Armenia,0036.}

1. Introduction

Aninterestingandrecentlyexaminedpropertyofthequark dis-tributioninanucleonthatispolarisedtransverselytoits momen-tumisnotleft-rightsymmetricwithrespecttotheplane defined by thedirectionsof nucleonspin andmomentum. This asymme-try of thedistribution function is called the Sivers effect. It was first suggested [1] as an explanation of the large left-right sin-gle transverse spin asymmetries observed for pions produced in the reaction p↑p

→

π

X [2–4]. Soon after, the existence of such anasymmetricdistribution,knownasSiversdistributionfunction, was excluded on the basis of T-invariance arguments [5]. Only ten years later it was recognised that such a function may in-deedexist [6].At that time it was also predictedthat the Sivers functioninsemi-inclusivemeasurements ofhadronproductionin DIS (SIDIS) and in the Drell–Yan process should have opposite sign[7],a propertyreferred to as“restricteduniversality”. Afew yearslater, theSivers effectforquarkswas observedinSIDIS ex-perimentsusingtransversely polarisedprotontargets, firstbythe

HERMES Collaboration [8] and then by the COMPASS

Collabora-tion[9].FromcombinedanalysesofthefirstHERMESdataandthe earlyCOMPASS datatakenwithatransverselypolariseddeuteron target [10], first extractions of the Sivers functions foru and d-quarksemerged [11–13]. Sincethen, moreprecise measurements oftheSiverseffectwereperformedbytheHERMES[14]and COM-PASS[15–17]Collaborations,andnewmeasurementswitha trans-verselypolarised3He targetwere alsocarriedout atJLab[18,19]. Moreinformationcanbefoundinrecentreviews[20–22].

At this point, the question arises whether or not the gluon distribution in a transversely polarised nucleon is also left-right asymmetric, i.e.exhibits a Sivers effect similar to that found for thequarkdistributions.Recently,theissuehasbeendiscussed re-peatedly in the literature and the properties ofthe gluon Sivers distributions have been studied in great detail [23,24]. While it wasfound thatanon-zeroSivers functionimpliestransverse mo-tionofpartons inthenucleon,presently theconnection between theSivers functionandthe partonorbital angular momentumin thenucleoncanonlybedescribedinamodel-dependentway[25]. Originally, the correspondence between the left-right asymmetry now known asSivers effect and parton orbital angular momen-tumwasproposedbySivershimself

[1,26]

,andthenitwasfurther elaboratedbyseveralauthors

[27–29]

.

Presently, the information on the gluon Sivers function is scarce. An important theoretical constraint comes from the so-called Burkardt sum rule [30]. It states, based on the presence of QCD colour-gauge links, that the total transverse momentum of all partons in a transversely polarised proton should vanish.

14 _{Presentaddress: RWTH Aachen University, III.Physikalisches Institut, 52056} Aachen,Germany.

15 _{Presentaddress:UppsalaUniversity,Box516,75120Uppsala,Sweden.} 16 _{Supportedby BMBF – Bundesministerium für Bildung und Forschung} (Ger-many).

17 _{SupportedbyFP7,HadronPhysics3,Grant283286(EuropeanUnion).} 18 _{SupportedbyMEYS,GrantLG13031(CzechRepublic).}

19 _{SupportedbySAIL(CSR)andB.Senfund(India).} 20 _{SupportedbyCERN–RFBRGrant12-02-91500.}

21 _{SupportedbyFCT–FundoRegionalparaaCiênciaeTecnologia,COMPETEand} QREN, GrantsCERN/FP116376/2010,123600/2011 and CERN/FIS-NUC/0017/2015 (Portugal).

22 _{SupportedbyMEXTandJSPS,Grants18002006,20540299and18540281,the} DaikoandYamadaFoundations(Japan).

23 _{SupportedbytheMinistryofScienceandTechnology,Taiwan (Taiwan).} 24 _{SupportedbytheIsraelAcademyofSciencesandHumanities(Israel).} 25 _{SupportedbyNSF–NationalScienceFoundation(USA).}

26 _{SupportedbyNCN,Grant2015/18/M/ST2/00550(Poland).} 27 _Deceased.

Fits to the Sivers asymmetry using SIDIS data [31] almost fulfil, within uncertainties,theBurkardtsumruleandhenceleave little space foragluon contribution. Fromthe nullresultof the COM-PASSexperimentfortheSiversasymmetryofpositiveandnegative hadronsproducedonatransverselypolariseddeuterontarget[10], together with additional theoretical considerations, it was stated

[32]thatthegluoncontributiontopartonorbitalangular momen-tumshould be negligible,andconsequentlythat the gluonSivers effect should be small. Also, using the so-called transverse mo-mentumdependent(TMD)generalisedpartonmodelandthemost recent phenomenological information on the quark Sivers distri-butions coming from SIDIS data,constraints on the gluon Sivers function were derived [33] from the recent precise data on the transversesinglespinasymmetry AN

(

p↑p

→

π

0X

)

thatwas

mea-suredatcentralrapiditybythePHENIXCollaborationatRHIC[34]. Altogether, it is ofgreat interest to know whetherthere exists a Siverseffectforgluonsornot.

In DIS,the leading-ordervirtual-photon absorptionsubprocess (LP)doesnotprovidedirectaccesstothegluon distributionsince the virtual photon does not couple to the gluon. Hence higher-ordersubprocesseshavetobestudied,i.e. QCDComptonscattering (QCDC) andPhoton–Gluon Fusion (PGF).It iswell known that in lepton–proton scatteringone ofthe mostpromising processes to directly probethegluon isopen charmproduction,



p↑

→ 

cc X .

¯

Thischannel was studiedindetailby theCOMPASS Collaboration inordertomeasure



g

/

g,i.e. thegluonpolarisationina longitu-dinallypolarisednucleon[35].Taggingthecharmquarkby identi-fyingD-mesonsinthefinalstatehastheadvantagethatinlowest orderofthestrongcouplingconstanttherearenoother contribu-tionstothecrosssection,sothatonebecomesessentiallysensitive to the gluon distribution function. An alternative method to tag

the gluon in DIS, which was also developed and used by

COM-PASS is the production of high-pT hadrons [36,37] that has the

advantage of higherstatistics. In theleading process, thehadron transverse momentum pT withrespect to the virtual photon

di-rection (in the frame where the nucleon momentum is parallel to thisdirection) originatesfromtheintrinsictransverse momen-tumkT ofthestruck quarkinthe nucleonanditsfragmentation,

whichbothleadtoasmalltransversecomponent.Onthecontrary,

both the QCDC and PGF hard subprocesses can provide hadrons

with high transverse momentum. Therefore, tagging events with hadronsofhightransversemomentum pT enhancesthe

contribu-tion ofhigher-order subprocesses. Nevertheless,although insuch a high-pT sample the PGF fractionis enriched, the contributions

from LP and QCDChave to be subtracted in order to single out thecontributionofthePGFsubprocesstothemeasured asymme-try[38].

Inthisletter,thegluonSiverseffectisinvestigatedusing

COM-PASS data collected by scattering a 160 GeV/c muon beam off

transversely polarised deuterons and protons. The experimental set-up and thedata selection are described in Section 2. In Sec-tion3themeasurementisdescribed.Thedetailsoftheanalysisare giveninSection 4.Theprocedureofneuralnetwork(NN)training witha MonteCarlodatasampleis showninSection5.Section 6

contains the overviewofthe systematicstudies. InSection 7 the resultsarepresented.Summary andconclusions aregivenin Sec-tion8.

2. Experimentalset-upanddataselection

The COMPASS experiment uses a fixed target set-up and the

naturally polarised muon beam delivered by the M2 beam line

witha10 cmgapinbetween.In2010, thetransverselypolarised proton target consisted of three cells, 30 cm, 60 cm and30 cm long with 5 cm gaps between them. The polarisation direction in the central cell was opposite to that in the downstream and upstream cells. During all data taking periods, the polarisation directionwas reversed once per week in order tominimise sys-tematiceffectsdue toacceptance.For themeasurements using a deuterontargetthecellswerefilledwith6LiD.The6Linucleuscan be regardedto be composed of a quasi-free deuteronand a 4He core. The average dilutionfactor fd,which is definedas ratioof

theDIScrosssection foronlythepolarisable nucleonsinthe tar-getoverthe DIScrosssection forall target nucleons,amounts to 0.36 including also electromagneticradiative corrections. The av-eragedeuteronpolarisationwas0.50.Forthemeasurementsusing aprotontarget, thecells were filledwithNH3.Theaverage dilu-tionfactor fp amountsto0.15 andtheaverageprotonpolarisation

was 0.80.A muon beamenergyof160GeV/c was chosen forall measurements.ThebasicfeaturesoftheCOMPASSspectrometer,as describedinRef. [40],arethesamefor2003–2004and2010data taking.Severalupgradeswereperformedin2005,wherethemain onewastheinstallationofanewtargetmagnetthatallowedusto increasethepolarangleacceptancefrom70 mradto180 mrad.

A crucial point of this analysis is the search for an observ-able that is strongly correlated with the azimuthal angle

φ

g of

thegluon.IntheMC simulationsusing theLEPTOgenerator[41], gluonsareprobed throughthe PGFsubprocessthat hasa quark– antiquarkpairinthefinal state,andthefragmentationprocessis describedby theLund model[42].As aresultofMC studies,the bestcorrelationisfoundbetween

φ

g and

φ

P,wherethelatter

de-notestheazimuthalangleofthevectorsum P ofthetwohadron momenta.For the presentanalysis, two charged hadronsare se-lectedineachevent.Ifmorethantwochargedhadronsare recon-structedin an event,only thehadron withthe largesttransverse momentum, pT 1, andtheone with thesecond-largest transverse

momentum, pT 2,aretakeninto account.Inorderto enhancethe

PGF fractioninthe sample andat thesame time the correlation between

φ

gand

φ

P,furtherrequirementsareappliedtothe

trans-versemomentaofthetwohadrons,i.e. pT 1

>

0.7 GeV/c andpT 2

>

0.4 GeV/c. Moreover, the fractional energies of the two hadrons must fulfil the conditions zi

>

0.1

(

i

=

1,2) and z1

+

z2

<

0.9, wherethelast requirementrejectsevents fromdiffractive vector-meson production. Hadron pairs are selected without constrain-ingtheircharge. Withtheabovechoice thecorrelationcoefficient amounts to 0.54. The Sivers asymmetry is obtained as the sine modulationintheSiversangle,whichisdefinedas

φ

Siv

= φ

P

− φ

S

inthisanalysis.Here,

φ

S istheazimuthalangleofthenucleonspin

vector.

The same kinematic data selection is used for both deuteron and proton data. The requirement on photon virtuality, Q2

_>

1

(GeV/

c

)

2,selectseventsintheperturbative regionofQCD. The requirementonthemassofthehadronicfinalstate,W

>

5 GeV/c2, removestheregionofexclusivenucleonresonanceproduction.The Bjorken-x variable covers the range 0.003

<

xB j

<

0.7. The

frac-tional energy of the virtual photon, y, is limited by y

>

0.1 to remove aregion sensitive to experimental biasesandby y

<

0.9 toeventswithlargeelectromagneticradiativecorrections. 3. Siversasymmetryintwo-hadronproduction

In order to extract the gluon Sivers asymmetry, two-hadron eventsproducedinmuon-nucleonscattering,

μ

+

N

→

μ



+

2h

+

X ,

areselectedasdescribedinSection2.Bylabellingwiththesymbol

d7

σ

↑ (

d7

σ

↓)

thecrosssectionforeventsthatwereproduced us-ingatargetcell withpolarisationdirectionupwards(downwards) inthelaboratory,theSiversasymmetrycanbewrittenas

A2h_T

(



x, φSiv

)

=



σ

(



x, φSiv

)

σ

(



x)

,

(1)

where



x

= (

xB j

,

Q2

,

pT 1

,

pT 2

,

z1

,

z2

),



σ

≡

d7

σ

↑ −

d7

σ

↓

and

σ

≡

d7

_σ

_{↑ +}

_d7

_σ

_↓

_{. All cross sections are integratedover the two} az-imuthalangles

φ

S and

φ

R,where

φ

R istheazimuthalangleofthe

vector difference R

=

P1

−

P2,which describesthe relative mo-mentum ofthetwohadrons. Thenumberofeventsina

φ

Siv bin

isgivenby

N(



x, φSiv

)

=

α

(



x, φSiv

)



1

+

f

(



x)PTASiv

(



x)sin

φ

Siv



.

(2)

Here f

(



x

)

is the dilution factor, PT the target polarisation and

α

=

an



σ

0 an acceptance-dependent factor, where a is the to-tal spectrometer acceptance, n the density of scattering centres,



thebeamflux and

σ

0 the spin-averaged partofthecross sec-tion.Fromhereon,theSiversasymmetry A_T2h

(



x

,

φ

Siv

)

isfactorised

intotheazimuth-independentamplitude ASiv

₍

_

_x

₎

_andthe

modula-tionsin

φ

Siv.

In order to evaluate the Sivers asymmetry of the gluon, the amplitude ofthe sin

φ

Siv modulation is extractedfrom the data.

The general expression forthe cross section of SIDIS production with at least one hadron in the final state is well known [43]. It contains eight azimuthal modulations, which are functions of thesingle-hadronazimuthal angleand

φ

S.Inthe absenceof

cor-relationspossiblyintroducedbyexperimental effects,alleightare orthogonal.TheSiversasymmetrycaneitherbeextractedbyfitting onlytheamplitudeofthesin

φ

Siv modulationoronecanperform

a simultaneous fit ofall eight amplitudes. In the caseof heavy-quarkpairanddijetproductioninlepton–nucleoncollisions,all az-imuthalasymmetriesassociatedtothegluondistributionfunction havebeenrecentlyworkedoutinRef.[44].There,theSivers asym-metryisdefinedastheamplitudeofthesin

(φ

T

− φ

S

)

modulation,

where

φ

T istheazimuthalangleofthetransverse-momentum

vec-torofthequark–antiquarkpair,qT.Inouranalysis,

φ

T isreplaced

by

φ

P exploitingitscorrelationwiththegluonazimuthalangle

φ

g,

andtheSiversasymmetryisextractedtakingintoaccountonlythe sin

(φ

P

− φ

S

)

modulationinthe crosssection.Itwas verifiedthat

includinginthecrosssectionthesameeighttransverse-spin mod-ulations as in SIDISsingle-hadron production [43] and fittingall eight amplitudes simultaneously leads to the sameresult onthe gluonSiversasymmetry.

In order to determinethe gluon Sivers asymmetry from two-hadron production in SIDIS, it is necessary to assume that the maincontributorstomuon-nucleonDISarethethreesubprocesses (Fig. 1) aspresented inRef. [41].Thismodel issuccessful in de-scribingtheunpolariseddata.AtCOMPASSkinematics,theleading process appears at zero-orderQCD inthe total DIScross section and it is the dominant subprocess,while the other two subpro-cesses,photon–gluonfusionandQCDCompton,arefirst-orderQCD processes andhencesuppressed. However, their contributioncan be enhancedby requiringhightransverse momentumofthe pro-ducedhadrons,asmentionedabove.

Introducing the subprocess fractions Rj

=

σ

j

/

σ

( j

∈

{PGF,

QCDC, LP}), the amplitude of the Sivers asymmetry can be ex-pressedintermsoftheamplitudesofthethreecontributing sub-processes:

f PTASivsin

φ

Siv

=



σ

σ

=

σ

PGF

σ



σ

PGF

σ

PGF

+

σ

QCDC

σ



σ

QCDC

σ

QCDC

+

σ

LP

σ



σ

LP

σ

LP

=

f PT

(R

PGFASivPGF

+

RQCDCAQCDCSiv

+

RLPASivLP

)

sin

φ

Siv

,

(3)

with

σ

=



j

σ

j,



σ

=



j



σ

j and f PTASiv_j sin

φ

Siv

= 

σ

j

/

σ

j.

Fig. 1. Feynman diagrams considered forγ∗N scattering: a) photon–gluon fusion (PGF), b) gluon radiation (QCD Compton scattering), c) Leading order process (LP).

basisusingneuralnetworks(NNs)trainedonMonteCarlodata,as willbedescribedinSection5.

4. Asymmetryextractionusingthemethodsofweights

The method adopted in the present analysiswas already ap-plied to extract the gluon polarisation from the longitudinal double-spinasymmetryintheSIDISmeasurementofsingle-hadron

production [38]. While the methodmay appear somewhat

com-plex,thebasicideabehindisrathersimple.Equation(3)contains threeunknowns, i.e. the asymmetries ASiv

j , sothat fora solution

atleast threeequations ofthe type of Eq.(3) are needed. These threeequationsareconstructedusingtheweightingprocedure de-scribed in thissection, which in addition allows us to achieve a nearlyoptimalstatisticalaccuracy(inthesenseoftheCramer–Rao bound

[39]

)inamultidimensionalanalysis.

Both for the deuteron data (two target cells) and the proton data (three target cells), four target configurations can be intro-duced. In the case of the two-cell target: 1 – upstream, 2 – downstream,3 – upstream, 4 – downstream.In thecaseofthe three-cell target: 1 – (upstream

+

downstream), 2 –centre, 3 – (upstream

+

downstream), 4 –centre.Here upstream, centre and downstream denote the cells after the polarisation rever-sal andconfiguration 1has the polarisationpointing upwardsin thelaboratory.Whendecomposing theSiversasymmetry intothe asymmetriesofthecontributingsubprocesses (Eq.(3)) and intro-ducingtheSiversmodulation

β

t_j

(



x, φSiv

)

=

Rj

(

x)



f

(



x)PtTsin

φ

Siv

,

(4) whichisspecificforsubprocess j,onecanrewriteEq.(2):

Nt

(



x, φSiv

)

=

α

t

(



x, φSiv

)



1

+ β

_PGFt

(



x, φSiv

)A

SivPGF

(



x)

+ β

t

QCDC

(



x, φSiv

)A

SivQCDC

(



x)

+ β

LPt

(



x, φSiv

)

ALPSiv

(



x)



.

(5)

Heret

=

1,2,3,4 denotesthetargetconfiguration.

Inordertominimisethestatisticaluncertainties foreach sub-process,aweightingfactorisintroduced.Itisknown[45]thatthe choice

ω

j

= β

j fortheweightoptimises thestatisticaluncertainty

butvariations ofthe target polarisation PT over time may

intro-duceabiastothefinalresult.Therefore,theweightingfactor

ω

j

(



x)

=

Rj

(



x)f

(



x)sin

φ

Siv (6) isusedinstead.Byweightingeach ofthefourequations

(5)

three timeswith

ω

j depending on thesubprocess j

∈ {

PGF, QCDC, LP

}

,

andbyintegratingover

φ

Siv and



x,twelve observedquantitiesqt_j areobtained:

qt_j

=



d



xdφSiv

ω

j

(

x, φ



Siv

)N

t

(



x, φSiv

)

= ˜

α

t_j



1

+ {β

_PGFt

}

ωj



ASiv_PGF



_β PGFωj

+ {β

t QCDC

}

ωj



A_QCDCSiv



_β QCDCωj

+ {β

t LP

}

ωj



A_LPSiv



βLPωj



,

(7) where

α

˜

t

j is the

ωj

-weighted acceptance-dependent factor. The

quantities

{β

_it

}

ωj and

{

A

Siv

i

}

βt

iωj areweightedaverages:

{β}

ω

=



α

β

ω

d



xdφSiv



αω

d



xdφSiv

,

{

A

}

βω

=



A

α

β

ω

d



xdφSiv



α

β

ω

d



xdφSiv

.

(8)

The acceptancefactors

α

˜

t_j cancel iffortheasymmetry extrac-tionthedoubleratio

rj

:=

q1_jq4_j

q2_jq3_j (9)

is used, as the data taking was performed in such a way that

˜

α

1 j

α

˜

4 j

/

α

˜

2 j

α

˜

3

j

=

1.Ifthisconditionisnotfulfilled,falseasymmetries

mayoccur.Itischeckedthatthisisnotthecase(seeSection6). Inthisanalysis,anunbiasedestimatorofqt_jisselected:

qt_j

=

Nt

k=1

ω

k_j

,

(10) and

β

_it isapproximatedby

{β

t i

}

ωj

≈

Nt



k=1

β

t_i,k

ω

k_j Nt



k=1

ω

k_j

.

(11)

Thisapproximationholdsforsmallobservedrawasymmetries,i.e.

β

A



1. It can lead to a bias of the order of d A

/

A

≈

0.2

β

2

,

β

2

P G F

≈

4

×

10−6,whichisnegligiblecomparedtoothersources

ofsystematicuncertainties,see

Table 1

.Inordertoavoid numeri-calinconsistenciesinEq.(11)duetoazero-polewhenintegrating over the full range of

φ

Siv,28 two bins in

φ

Siv

(

[

0

;

π

],

[

π

;

2

π

])

areintroduced.Intheaforementionedthreedoubleratiosgivenin Eq.(9)onlyasymmetriesareunknown.However,inordertosolve the system ofequations one needs to assume that the weighted asymmetry foragivensubprocessi isthesameforthethree dif-ferent weights

ω

j

β

i, i.e.

{

Ai

}

βiωPGF

= {

Ai

}

βiωQCDC

= {

Ai

}

βiωLP. This

means that the values of

ω

j and Ai must be uncorrelated. For

example, since

ω

j is proportional to Rj, which in turn strongly

depends on thehadron transversemomentum, one has to use a

kinematic region where the asymmetries Ai are expected to be

independent of pT.Under theseassumptions,the numberof

un-knownweighted asymmetriesisthree,whichexactlycorresponds tothenumberofequationsoftype

(9)

.Theseequationsaresolved bya

χ

2 _{fitthatincludessimultaneouslybothbinsin}

_φ

Siv. Assumingthat Ai can beapproximatedby alinearfunctionof xi andthatxiisnotcorrelatedwith

ωj

,resultsin

28 _Notethatωk

Table 1

SummaryonsystematicuncertaintiesofthefinalvaluesofthegluonSivers asym-metryfordeuteronandprotondata.

Source Deuteron data Proton data Uncertainty Fraction

ofσstat

Uncertainty Fraction ofσstat

MonteCarlosettings 0.060 40% 0.054 64% Radiativecorrections 0.018 12% 0.018 21% Oneortwo

x

B jbins 0.07 47% 0.011 13%

Include7other asymmetries 0.003 2% 0.005 6% Targetpolarisation 0.0075 5% 0.0043 5% Dilutionfactor 0.003 2% 0.0018 2% Totalσ2 i 0.10 63% 0.06 69%

{

Ai}βiωi

=

Ai

(

{

xi}βiωi

).

(12)

Thisapproximationallows to interprettheobtainedresults asan asymmetryvalue measured attheweighted value of xi.Foreach

subprocess,the weighted value of xi is obtainedfrom MC using

therelation

{

xi}βiωi

=

Ni



k=1 xk_i

β

_ik

ω

k i Ni



k=1

β

_ik

ω

k i

.

(13)

Here, Ni is the number ofevents of type i in the MC data. The

assumptionthatthevaluesofxi arenotcorrelatedwith

ωj

,which

allowsustoconsideronly

{

xi

}

ωiβi,wasverifiedusingMCdata.The

detailsoftheanalysisaregivenin[46].

5. MonteCarlooptimisationandneuralnetworktraining The present analysis is very similar to the one used for the



g

/

g extraction using high-pT hadron pairs [37] and single

hadrons[38].ThepackageNetMaker[47]isusedfortheNN train-ing withinput,output andtarget vector. The NNistrainedusing aMonteCarlosamplewithsubprocessidentification.The subpro-cess type defines the target vector. The following six kinematic variables are chosen as input vector: xB j

,

Q2

,

pT 1

,

pT 2

,

pL1

,

pL2.

Thelattertwoarethelongitudinalcomponentsofthehadron mo-menta.Thetrainedneuralnetworkisappliedtothedatabytaking as input vector of the aforementioned sixvariables, and its out-put vector is interpreted as probabilities that the given event is a resultofoneofthe threecontributingsubprocesses. Hence the simulateddistributionsofthesevariablesneedtobeinagreement withthecorrespondingdistributionsinthedata.

UsingtheLEPTOgenerator(version6.5)[41],twoseparateMC data samples were produced to simulate the deuteron and

pro-ton data. The generator is tuned to the COMPASS data sample

obtained withthe high-pT hadron-pair selection asdescribed in

Ref. [37]. The MSTW08 parametrisation of input PDFs [48] was chosen as it gives a good description of the structure function

F2 in the COMPASS kinematic range, and it is valid down to

Q2

=

1

(GeV/

c

)

2.Electromagnetic radiativecorrections[49] were applied as a weighting factor to the MC distributions shown in

Figs. 2 and 3butnotintheMCsamplesusedinNNtraining.This difference wasstudied andit wasestimatedtobe negligible.The

Fig. 3. Comparison of distributions of kinematic variables between experimental and MC high-pT proton data.

generatedeventswereprocessedbyCOMGEANT,theCOMPASS de-tectorsimulationprogrambasedonGEANT3.The MCsamplesfor theprotonanddeuterondata differinthetarget material andin thespectrometerset-up.TheFLUKApackage[50]isusedinorder tosimulatesecondaryinteractions.Asthenextstep,theCOMPASS reconstructionprogramCORALwasapplied.ForMCdata,thesame data selection as for real events was used. The comparison be-tweenexperimentalandMCdistributions ofthevariablesusedin theNNtraining,i.e. xB j

,

Q2

,

pT 1

,

pT 2

,

pL1

,

pL2,isshownin

Figs. 2

and 3fordeuteronandprotondata,respectively.Discrepancies be-tweentheMCandexperimentaldatakinematicaldistributionsare seenatthelevel of10%,mostly intheregions wherestatisticsis small.

The maingoal ofthe NNparametrisation isthe estimationof thesubprocessfractionsRj.Inthetypicalcaseofsignaland

back-groundseparation, the expectedNN output wouldbe set to one forthe signal andzerofor thebackground. Theoutput value re-turnedbytheNNwouldthencorrespondtothefractionofsignal eventsinthesample atthe givenphasespacepoint oftheinput parametervector.Inthepresentanalysis, thesubprocessfractions were estimated simultaneously. In order to have a closure rela-tiononthesubprocessprobabilities,theirsummustadduptoone andhenceonlytwoindependentoutputvariablesareneededfrom the NN. The estimation of the subprocess fractions Rj fromthe NNoutput is accomplishedby assigning to each eventthe prob-abilities PPGF

NN, P QCDC

NN and PLPNN.The distributionof theNNoutput after training is shownin Fig. 4 inthe “Mandelstam representa-tion”,i.e. as points inan equilateraltrianglewithunit height. As thereare threesubprocesses,idealseparation wouldmeanthat a

Fig. 4. NeuralnetworkoutputaftertrainingtheMCsamplefortheprotonanalysis shownonaequilateraltriangleofunitheight.Verticesrepresenttheperfect identi-ficationofthesubprocesses.Toeachpoint,whichrepresentanexperimentalevent, threeprobabilitiesareassigned.Theprobabilityofeachsubprocess

P

NNj forapoint isgivenbyitsdistancetothesideoppositetothevertex j ofthetriangleasinthe “Mandelstamrepresentation”.Thisisillustratedbythecolouredlinesrepresenting probabilitiesassignedtoanarbitrarypoint.

given event is located in one of the vertices, i.e. belongsto one of the subprocesses. This is not the case in general, instead all information obtained fromthe neural network fora givenevent consists of the three probabilities P_{N N}P G F

,

PQ C DC_{N N}

,

PL P

N N, which add

Fig. 5. Toppanels:Neuralnetworkvalidation.Here

P

NNisthefractionofthesubprocessgivenbytheNNand

P

MCisthetruefractionofeachsubprocessfromMCinagiven PNNbin.Bottompanels:Difference

P

MC–PNNperbin.

Fig. 6. SystematicchangesinthefinalresultcausedbyusingdifferentMCsettings.Besidesthefinalresultshownonthetop,sevenotherresultsareshownthatareobtained withMCsamplesthatdifferbythechoiceofCOMPASSordefaultLEPTOtuning,‘PartonShower’onoroff,

F

L fromLEPTOorfrom

R

=σL/σT,MSTWorCTEQ5LPDFsets,

FLUKAorGHEISHAforsecondaryinteractions.Theresultsfordeuteron(proton)dataareshownintheleft(right)panel. (seecoloured lines in

Fig. 4

).When points fall outsidethe

trian-gle, one estimator was negative. This is possible because in the training the estimators are not bound to be positive. The values of the resulting three probabilities are taken as the subprocess fractions RPGF, RQCDC and RLP in the data analysis described in Section4.

Forthe validation of the NN training, a statistically indepen-dentMCsampleisusedtocheckhowtheNNworksonasample differentfrom theone usedfor the training. In each binof PNN (the value assigned to every MC event by the trained NN), the true fraction obtained from LEPTO based on the subprocess ID,

PMC,iscalculated.TheresultsfortheNNtrainedwithaMC sam-ple for the proton data are presented in Fig. 5. Altogether, the agreementbetweenPNN andPMCforallthreesubprocessesis sat-isfactory.However, forthetwolast binsofPGF, thetwolast bins ofQCDC and the last bin of LP the neural network output does not coincide with the true fraction of the given subprocess. As thisdiscrepancyappliesonlyto asmallpartoftheeventsample, it is included as a part of the MC-dependent systematic uncer-tainty.

6. Systematicuncertainties

Themainsourceofsystematicuncertaintiesisthedependence ofthefinalresultsonsettingsandtuningoftheMonteCarlo sim-ulation.Inordertoestimatethisuncertainty,differentMCsettings wereusedintheprocessofneuralnetworktraining.Twodifferent combinations offragmentation parameters were used,i.e. default

LEPTOtuningorCOMPASStuningforthehigh-pT selectedsample.

The event generation was done withand without‘Parton Show-er’[51].TwoPDFsets wereused(MSTW08orCTEQ5L[52]).Two different parametrisations of the longitudinal structure function

FL are used,either the one fromLEPTOor theone used for the R

=

σ

L

/

σ

T parametrisationofRef.[53].Forsecondaryinteractions,

eithertheFLUKAortheGHEISHA[54]packagewereused.

Fig. 7. Siverstwo-hadronasymmetryextractedforPhoton–Gluonfusion(PGF),QCDCompton(QCDC)andLeadingProcess(LP)fromtheCOMPASShigh-pTdeuteron(left)and

proton(right)data.The

x range

istheRMSofthelogarithmicdistributionof

x in

theMCsimulation.Theredbandsindicatethesystematicuncertainties.Notethedifferent ordinatescaleusedinthethirdrowofpanels.(Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.) and MSTW08 PDFs. The systematic uncertainty originating from

differentMCtuningsiscalculatedas

(

APGF

max

−

APGFmin

)/2.

Thesystematicuncertaintyduetofalseasymmetrieswas

stud-ied by extracting asymmetries between two parts of the same

targetcell.Theresultsare foundtobecompatiblewithzero. Fur-thermore,itwas checkedhowa smallartificialfalseamplitudeof thesin

φ

g

− φ

S modulationintheMCproductioninfluencesthe

fi-nalresult. When afalse asymmetry of1% isintroduced, forboth protonand deuteron datathe final result changes by 25% ofthe statisticaluncertainty.Nosystematicuncertaintyisassignedto ac-countforfalseasymmetries.

Thefinal state ofthephoton–gluon-fusion processisa quark– antiquark pair. Thus for most hadron pairs produced from this subprocessthetwohadronsshouldhaveoppositecharge.Although a selection q1q2

= −

1 slightlyincreases the (φg,

φ

P) correlation,

it also reduces statistics. The results with and without this re-quirementarestatisticallyconsistent.Therequirementofopposite chargesofthetwo hadronsis hencenotincludedinthe data se-lection.

Radiativecorrectionswere not included inthe MC production thatisusedinthemainanalysisofthisletter.Inordertoestimate the systematic uncertainty introduced by this omission, a sepa-rateMCsampleisusedthatwasproducedforthe2006COMPASS set-upincludingradiativecorrectionsbasedonRADGEN[55].The difference in the final value for the gluon Sivers asymmetry for the proton is 0.018, which corresponds to 21% ofthe statistical uncertainty.Acorrespondingsystematicuncertaintyisassignedto accountforthe factthat radiativecorrectionsare notincludedin theMCsimulationsandhenceintheNNtraining.

Ourresults are obtained in only one xg binfor ASivPGF, one xC

bin for ASiv_QCDC andone xB j bin for ASiv_LP. As the asymmetries are

stronglycorrelated,abinninginxB j affectsthevaluesofASivPGF and

ASiv_QCDC,whichareextractedinasinglebinasbefore.Theresulton

ASiv

PGF changesby0.07 fordeuterondataand0.011 forprotondata whentwo xB j binsare introduced,andthesevaluesare takenas

anestimateoftherelatedsystematicuncertainty(see

Table 1

). Theasymmetries ASiv_j ofEq.(5)were alsoextractedusingthe unbinnedmaximumlikelihoodmethod,whichyieldsasexpected, thesameresultsforAPGFastheabovedescribedanalysis. Concern-ingtheorthogonalityofdifferentmodulationsofthecrosssection,

it was checked by what amount the Sivers asymmetry changes,

when also the other seven asymmetries were includedin the fit (seeabove).ThechangeinthefinalvalueofthePGFasymmetryis negligibleforbothdeuteronandprotondata.

The systematic multiplicative uncertainties on target polarisa-tionanddilutionfactorareestimatedtobeabout5% and2% ofthe statisticaluncertainty,respectively.Thefinalsystematicuncertainty is obtainedby summing all componentsin quadrature.All above mentioned contributions andthe final systematicuncertaintyare listedin

Table 1

.

7. Resultsanddiscussion

The method of weighted asymmetries described in Section 4

wasappliedtothedeuteronandprotondatasetsdescribedin Sec-tion 2, using trained neural networks as described in Section 5. TheresultsonthegluonSivers asymmetryasextractedfrom lep-ton nucleonSIDIS,withatleasttwo detectedhigh-pT hadrons,is

shownin

Fig. 7

andpresentedin

Table 2

togetherwiththe contri-butionsofthetwootherhardsubprocesses,i.e. QCDComptonand leading process.Theresultoftheanalysisofthedeuterondatais

ASiv_{P G F},d

= −

0.14

±

0.15(stat.)

±

0.10(syst.)measuredat



xg

=

0.13.

The proton result, ASiv_{P G F},p

= −

0.26

±

0.09(stat.)

±

0.06(syst.) ob-tainedat



xg

=

0.15,isconsistentwiththedeuteronresultwithin

lessthanonestandarddeviationofthecombinedstatistical uncer-tainty.The two resultsare expectedto be consistent,as presum-ably thetransverse motionofgluonsis thesame inneutronand proton.Combiningtheprotonanddeuteronresults,themeasured effect is negative, ASiv

P G F

= −

0.23

±

0.08(stat)

±

0.05(syst), which

is away fromzero by more than two standard deviations of the quadraticallycombineduncertainty.Thisresultappearsparticularly interestinginviewofthegluoncontributiontotheprotonspin,as a non-zero gluon Sivers effectis a signature of gluonorbital an-gular momentumintheproton[25].Whiletherecentanalysisof the PHENIX data [33] yields a gluon Sivers effect forthe proton thatiscompatiblewithzero,theCOMPASSresultfortheprotonis negative andmorethan two standarddeviations belowzero.The two resultsmaynotbe directlycomparable,asthey areobtained atdifferentvaluesofcentreofmassenergyandxg.Itmayalsobe

Table 2

SummaryofSiversasymmetries,

A

Si v

P G F,ASi vQ C DC,

A

Si vL P,obtainedfordeuteronandprotondata.

Subprocess Deuteron data Proton data

Asymmetry Statistical error Systematic uncertainty Asymmetry Statistical error Systematic uncertainty

PGF −0.14 0.15 0.10 −0.26 0.09 0.06

QCDC 0.12 0.11 0.08 0.13 0.05 0.03

LP −0.03 0.02 0.01 0.03 0.01 0.01

Fig. 8. Collins-liketwo-hadronasymmetryextractedforPhoton–Gluonfusion(PGF),QCDCompton(QCDC)andLeadingProcess(LP)fromtheCOMPASStwo-hadronhigh-pT

deuteron(left)andproton(right)data.The

x range

istheRMSofthelogarithmicdistributionof

x in

theMCsimulation.Theredbandsindicatethesystematicuncertainties. (Forinterpretationofthereferencestocolourinthisfigurelegend,thereaderisreferredtothewebversionofthisarticle.)

function that appears in one process may be different from the one appearing in a different process, so that and assessment of compatibilitymayrequireadeepertheoreticalanalysis.

Forthe asymmetry of the leading process, the high-pT

sam-ple of the COMPASS proton data has provided a positive value (see Fig. 7 right-bottom panel). It can be compared with the

COMPASS results on the Sivers asymmetry for charged hadrons

produced in SIDIS



p

→ 

h±X single-hadron production [17],

which for negative hadrons was found to be about zero and

forpositive hadrons differentfromzero andpositive, so that for the two-hadron final state a positive value may indeed be ex-pected.

The same analysis method was also applied to extract the

Collins-like asymmetry for charged hadrons, i.e. the cross sec-tion dependence on the sine of the Collins angle

(φ

P

+ φ

S

−

π

).

Theasymmetries Asin(φP+φS−π)

PGF , A

sin(φP+φS−π)

QCDC , A

sin(φP+φS−π)

LP

weredeterminedusingthesameCOMPASShigh-pT deuteronand

proton data as used for main analysis in this work. The results areshownin Fig. 8. Theamplitude of theCollins modulationfor gluons is found to be consistent with zero, in agreement with the naive expectation that is based on the fact that there is no gluontransversitydistribution[57].Recentlyitwassuggestedthat a transversity-like TMD gluon distribution h₁g could generate a sin

(φ

S

+ φ

T

)

modulationin leptoproductionoftwo jetsor heavy

quarks [44]. In this case the results shown in Fig. 8 provide a boundon thesize ofh₁g. Theresults giveninthisletter canalso beinterpreted such that nofalse Collins-likeasymmetry is intro-ducedby the rather complex analysis method used, so that the non-zeroresult obtainedfor the gluon Sivers asymmetry is sup-ported. In addition it is noted that the Collins-like asymmetry of the leading process for the proton is found to be consistent withzeroforhigh-pT hadronpairs,inqualitativeagreementwith

themeasurementoftheCollinsasymmetryinsingle-hadronSIDIS

measurement[58],whereoppositevaluesofaboutequalsizewere observedforpositiveandnegativehadrons.

8. Summaryandconclusions

TheSiversasymmetryforgluonsisextractedfromthe measure-mentofhigh-pT hadronpairsinSIDISatCOMPASSofftransversely

polarised deuterons and protons. The analysis is very similar to the one already used by the COMPASS collaboration to measure



g

/

g,thegluonpolarisationinalongitudinallypolarisednucleon. ThelargekinematicacceptanceoftheCOMPASSapparatusandthe

highenergyof themuon beammake thesample containing two

high-pT hadronssufficientlylargeforthe presentanalysis, which

islimitedtoasmallpartoftheaccessiblephasespace.Thecriteria appliedtoselecthadronpairsallowustoenhancethecontribution ofthephoton–gluonfusionsubprocesswithrespecttothe leading-ordervirtual-photonabsorptionsubprocess.TheSiversasymmetry isobtainedasthe amplitudeofthesine modulationintheSivers angle,

φ

Siv

= φ

P

− φ

S.

The selected high-pT hadron-pair sample contains already an

enriched fractionofeventsproduced inthe PGF subprocess.Still, itisnecessarytosubtractthecontributionsoftheothertwo sub-processes,LPandQCDC.Inthisanalysis,thefractionsofthethree subprocesses are determined usingMC algorithms, andthethree correspondingasymmetriesareextractedfromthedatausingaNN technique. Sincetheresultsderived fromaNNapproach strongly

depend on the Monte Carlo sample on which the network was

Averaging the results obtainedfrom the deuteron andproton data,themeasuredgluonSiversasymmetryisobtainedas

−

0.23

±

0.08(stat.)

±

0.05(syst.), which is away from zero by more than twostandarddeviationsofthetotalexperimentaluncertainty.This resultsupportsthepossibleexistenceofanon-zeroSiversfunction, andhence of gluon orbital angular momentum in a transversely polarisednucleon.

In addition, another result obtained in this work is the ex-tractionoftheCollins-like gluonasymmetry,i.e. theamplitude of thesine modulationof theCollins angle

φ

Col

= φ

P

+ φ

S

−

π

.

Re-centdevelopmentshavehypothesisedanon-zeroCollins-likegluon asymmetry,whichis howevernot relatedtotransversity.Our re-sult on the Collins-like asymmetry, which is obtained from the samehadron-pair datathat was usedto extract thenon-zero re-sultonthegluonSiversasymmetry,isfoundtobecompatiblewith zero.

Acknowledgements

This work was made possible thanks to the financial support ofourfundingagencies.We alsoacknowledgethesupportofthe CERNmanagementandstaff,aswellastheskillsandeffortsofthe techniciansofthecollaboratinginstitutes.

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