First measurement of the Sivers asymmetry for gluons from SIDIS data

1.1 MB·

by  C. Adolph

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
COMPASS
CERN-EP-2017–xxx
January2017
First measurement of the Sivers asymmetry for gluons from SIDIS data
Abstract
TheSiversfunctiondescribesthecorrelationbetweenthetransversespinofanucleonandthetrans-
7
1 verse motion of its partons. It was extracted from measurements of the azimuthal asymmetry of
0 hadronsproducedinsemi-inclusivedeepinelasticscatteringofleptonsofftransverselypolarisednu-
2 cleon targets, andit turned out tobe non-zero for quarks. In this letter the evaluation of the Sivers

n asymmetry for gluons in the same process is presented. The analysis method is based on a Monte
a Carlosimulationthatincludesthreehardprocesses: photon-gluonfusion,QCDComptonscattering
J
andleading-ordervirtual-photonabsorptionprocess. TheSiversasymmetriesofthethreeprocesses

0 aresimultaneouslyextractedusingtheLEPTOeventgeneratorandaneuralnetworkapproach. The
1
method is applied to samples of events containing at least two hadrons with large transverse mo-

mentum from the COMPASS data taken with a 160 GeV/c muon beam scattered off transversely
]
x polariseddeuteronsandprotons. Withasignificanceofmorethantwostandarddeviationsanegative
e
valueisobtainedforthegluonSiversasymmetry. TheresultofasimilaranalysisforaCollins-like
-
p asymmetryforgluonsisconsistentwithzero.
e
h
[

1
v
3
5
4
Keywords: deepinelasticscattering,gluon,Sivers,TMD,PDF.
2
0
.
1
0
7
1
:
v
i
X
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a
(tobesubmittedtoPhys.Lett.B)
TheCOMPASSCollaboration
C.Adolph9,M.Aghasyan26,R.Akhunzyanov8,M.G.Alexeev28,G.D.Alexeev8,A.Amoroso28,29,
V.Andrieux30,22,N.V.Anfimov8,V.Anosov8,A.Antoshkin8,K.Augsten8,20,W.Augustyniak31,
A.Austregesilo17,C.D.R.Azevedo2,B.Badełek32,F.Balestra28,29,M.Ball4,J.Barth5,R.Beck4,
Y.Bedfer22,J.Bernhard14,11,K.Bicker17,11,E.R.Bielert11,R.Birsa26,M.Bodlak19,P.Bordalo13,a,
F.Bradamante25,26,C.Braun9,A.Bressan25,26,M.Bu¨chele10,W.-C.Chang23,C.Chatterjee7,
M.Chiosso28,29,I.Choi30,S.-U.Chung17,b,A.Cicuttin27,26,M.L.Crespo27,26,Q.Curiel22,S.Dalla
Torre26,S.S.Dasgupta7,S.Dasgupta25,26,O.Yu.Denisov29,#,L.Dhara7,S.V.Donskov21,N.Doshita34,
Ch.Dreisbach17,V.Duic25,W.Du¨nnweberc,M.Dziewiecki33,A.Efremov8,P.D.Eversheim4,
W.Eyrich9,M.Faesslerc,A.Ferrero22,M.Finger19,M.Fingerjr.19,H.Fischer10,C.Franco13,
N.duFresnevonHohenesche14,11,J.M.Friedrich17,V.Frolov8,11,E.Fuchey22,d,F.Gautheron3,
O.P.Gavrichtchouk8,S.Gerassimov16,17,J.Giarra14,F.Giordano30,I.Gnesi28,29,M.Gorzellik10,m,
S.Grabmu¨ller17,A.Grasso28,29,M.GrossePerdekamp30,B.Grube17,T.Grussenmeyer10,A.Guskov8,
F.Haas17,D.Hahne5,G.Hamar25,26,D.vonHarrach14,F.H.Heinsius10,R.Heitz30,F.Herrmann10,
N.Horikawa18,e,N.d’Hose22,C.-Y.Hsieh23,f,S.Huber17,S.Ishimoto34,g,A.Ivanov28,29,
Yu.Ivanshin8,T.Iwata34,V.Jary20,R.Joosten4,P.Jo¨rg10,E.Kabuß14,A.Kerbizi25,26,
B.Ketzer4,G.V.Khaustov21,Yu.A.Khokhlov21,h,i,Yu.Kisselev8,F.Klein5,K.Klimaszewski31,
J.H.Koivuniemi3,30,V.N.Kolosov21,K.Kondo34,K.Ko¨nigsmann10,I.Konorov16,17,
V.F.Konstantinov21,A.M.Kotzinian28,29,O.M.Kouznetsov8,M.Kra¨mer17,P.Kremser10,F.Krinner17,
Z.V.Kroumchtein8,Y.Kulinich30,F.Kunne22,K.Kurek31,R.P.Kurjata33,A.A.Lednev21,*,
A.Lehmann9,M.Levillain22,S.Levorato26,Y.-S.Lian23,k,J.Lichtenstadt24,R.Longo28,29,
A.Maggiora29,A.Magnon30,N.Makins30,N.Makke25,26,G.K.Mallot11,#,B.Marianski31,
A.Martin25,26,J.Marzec33,J.Matousˇek19,26,H.Matsuda34,T.Matsuda15,G.V.Meshcheryakov8,
M.Meyer30,22,W.Meyer3,Yu.V.Mikhailov21,M.Mikhasenko4,E.Mitrofanov8,N.Mitrofanov8,
Y.Miyachi34,A.Nagaytsev8,F.Nerling14,D.Neyret22,J.Novy´20,11,W.-D.Nowak14,G.Nukazuka34,
A.S.Nunes13,A.G.Olshevsky8,I.Orlov8,M.Ostrick14,D.Panzieri1,29,B.Parsamyan28,29,S.Paul17,
J.-C.Peng30,F.Pereira2,M.Pesˇek19,D.V.Peshekhonov8,N.Pierre14,22,S.Platchkov22,
J.Pochodzalla14,V.A.Polyakov21,J.Pretz5,j,M.Quaresma13,C.Quintans13,S.Ramos13,a,C.Regali10,
G.Reicherz3,C.Riedl30,M.Roskot19,N.S.Rogacheva8,D.I.Ryabchikov21,i,A.Rybnikov8,
A.Rychter33,R.Salac20,V.D.Samoylenko21,A.Sandacz31,C.Santos26,S.Sarkar7,I.A.Savin8,
T.Sawada23 G.Sbrizzai25,26,P.Schiavon25,26,K.Schmidt10,m,H.Schmieden5,K.Scho¨nning11,l,
E.Seder22,26,A.Selyunin8,L.Silva13,L.Sinha7,S.Sirtl10,M.Slunecka8,J.Smolik8,A.Srnka6,
D.Steffen11,17,M.Stolarski13,O.Subrt11,20,M.Sulc12,H.Suzuki34,e,A.Szabelski31,26,#,
T.Szameitat10,m,P.Sznajder31,S.Takekawa28,29,M.Tasevsky8,S.Tessaro26,F.Tessarotto26,
F.Thibaud22,A.Thiel4,F.Tosello29,V.Tskhay16,S.Uhl17,A.Vauth11,J.Veloso2,M.Virius20,
J.Vondra20,S.Wallner17,T.Weisrock14,M.Wilfert14,J.terWolbeek10,m,K.Zaremba33,P.Zavada8,
M.Zavertyaev16,E.Zemlyanichkina8,N.Zhuravlev8,M.Ziembicki33 andA.Zink9
1 UniversityofEasternPiedmont,15100Alessandria,Italy
2 UniversityofAveiro,Dept.ofPhysics,3810-193Aveiro,Portugal
3 Universita¨tBochum,Institutfu¨rExperimentalphysik,44780Bochum,Germanyno
4 Universita¨tBonn,Helmholtz-Institutfu¨rStrahlen-undKernphysik,53115Bonn,Germanyn
5 Universita¨tBonn,PhysikalischesInstitut,53115Bonn,Germanyn
6 InstituteofScientificInstruments,ASCR,61264Brno,CzechRepublicp
7 MatrivaniInstituteofExperimentalResearch&Education,Calcutta-700030,Indiaq
8 JointInstituteforNuclearResearch,141980Dubna,Moscowregion,Russiar
9 Universita¨tErlangen–Nu¨rnberg,PhysikalischesInstitut,91054Erlangen,Germanyn
10 Universita¨tFreiburg,PhysikalischesInstitut,79104Freiburg,Germanyno
11 CERN,1211Geneva23,Switzerland
12 TechnicalUniversityinLiberec,46117Liberec,CzechRepublicp
13 LIP,1000-149Lisbon,Portugals
14 Universita¨tMainz,Institutfu¨rKernphysik,55099Mainz,Germanyn
15 UniversityofMiyazaki,Miyazaki889-2192,Japant
16 LebedevPhysicalInstitute,119991Moscow,Russia
17 TechnischeUniversita¨tMu¨nchen,PhysikDept.,85748Garching,Germanync
18 NagoyaUniversity,464Nagoya,Japant
19 CharlesUniversityinPrague,FacultyofMathematicsandPhysics,18000Prague,CzechRepublicp
20 CzechTechnicalUniversityinPrague,16636Prague,CzechRepublicp
21 StateScientificCenterInstituteforHighEnergyPhysicsofNationalResearchCenter‘Kurchatov
Institute’,142281Protvino,Russia
22 IRFU,CEA,Universite´ Paris-Saclay,91191Gif-sur-Yvette,Franceo
23 AcademiaSinica,InstituteofPhysics,Taipei11529,Taiwan,u
24 TelAvivUniversity,SchoolofPhysicsandAstronomy,69978TelAviv,Israelv
25 UniversityofTrieste,Dept.ofPhysics,34127Trieste,Italy
26 TriesteSectionofINFN,34127Trieste,Italy
27 AbdusSalamICTP,34151Trieste,Italy
28 UniversityofTurin,Dept.ofPhysics,10125Turin,Italy
29 TorinoSectionofINFN,10125Turin,Italy
30 UniversityofIllinoisatUrbana-Champaign,Dept.ofPhysics,Urbana,IL61801-3080,USA,w
31 NationalCentreforNuclearResearch,00-681Warsaw,Polandx
32 UniversityofWarsaw,FacultyofPhysics,02-093Warsaw,Polandx
33 WarsawUniversityofTechnology,InstituteofRadioelectronics,00-665Warsaw,Polandx
34 YamagataUniversity,Yamagata992-8510,Japant
#Correspondingauthors
*Deceased
a AlsoatInstitutoSuperiorTe´cnico,UniversidadedeLisboa,Lisbon,Portugal
b Also at Dept. of Physics, Pusan National University, Busan 609-735, Republic of Korea and at
PhysicsDept.,BrookhavenNationalLaboratory,Upton,NY11973,USA
c SupportedbytheDFGclusterofexcellence‘OriginandStructureoftheUniverse’(www.universe-
cluster.de)(Germany)
d SupportedbytheLaboratoired’excellenceP2IO(France)
e AlsoatChubuUniversity,Kasugai,Aichi487-8501,Japant
f AlsoatDept.ofPhysics,NationalCentralUniversity,300JhongdaRoad,Jhongli32001,Taiwan
g AlsoatKEK,1-1Oho,Tsukuba,Ibaraki305-0801,Japan
h AlsoatMoscowInstituteofPhysicsandTechnology,MoscowRegion,141700,Russia
i SupportedbyPresidentialGrantNSh–999.2014.2(Russia)
j Presentaddress: RWTHAachenUniversity,III.PhysikalischesInstitut,52056Aachen,Germany
k AlsoatDept.ofPhysics,NationalKaohsiungNormalUniversity,KaohsiungCounty824,Taiwan
l Presentaddress: UppsalaUniversity,Box516,75120Uppsala,Sweden
m SupportedbytheDFGResearchTrainingGroupProgrammes1102and2044(Germany)
n SupportedbyBMBF-Bundesministeriumfu¨rBildungundForschung(Germany)
o SupportedbyFP7,HadronPhysics3,Grant283286(EuropeanUnion)
p SupportedbyMEYS,GrantLG13031(CzechRepublic)
q SupportedbySAIL(CSR)andB.Senfund(India)
r SupportedbyCERN-RFBRGrant12-02-91500
s SupportedbyFCT-Fundac¸a˜oparaaCieˆnciaeTecnologia,COMPETEandQREN,GrantsCERN/FP
116376/2010,123600/2011andCERN/FIS-NUC/0017/2015(Portugal)
t Supported by MEXT and JSPS, Grants 18002006, 20540299 and 18540281, the Daiko and Ya-
madaFoundations(Japan)
u SupportedbytheMinistryofScienceandTechnology(Taiwan)
v SupportedbytheIsraelAcademyofSciencesandHumanities(Israel)
w SupportedbyNSF-NationalScienceFoundation(USA)
x SupportedbyNCN,Grant2015/18/M/ST2/00550(Poland)
FirstmeasurementoftheSiversasymmetryforgluonsfromSIDISdata 1
1 Introduction
An interesting and recently examined property of the quark distribution in a nucleon that is polarised
transversely to its momentum is the fact that it is not left-right symmetric with respect to the plane
defined by the directions of nucleon spin and momentum. This asymmetry of the distribution function
is called the Sivers effect and was first suggested [1] as an explanation for the large left-right single
transversespinasymmetriesobservedforpionsproducedinthereaction p↑p→πX [2–4]. Onthebasis
ofT-invarianceargumentstheexistenceofsuchanasymmetricdistribution,knownasSiversdistribution
function, was originally excluded [5]. Ten years later it was recognised however that it was indeed
possible [6]. At that time it was also predicted that the Sivers function in semi-inclusive measurements
of hadron production in DIS (SIDIS) and in the Drell-Yan process have opposite sign [7], a property
referred to as “restricted universality”. A few years later the Sivers effect was experimentally observed
in SIDIS experiments on transversely polarised proton targets, first by the HERMES Collaboration [8]
and then by the COMPASS Collaboration [9]. Using the first HERMES data and the early COMPASS
datatakenwithatransverselypolariseddeuterontarget[10],acombinedanalysissoonallowedforfirst
extractionsoftheSiversfunctionforuandd-quarks[11–13]. MoreprecisemeasurementsoftheSivers
effect were performed since by the HERMES [14] and COMPASS [15–17] Collaborations, and new
measurements with a transversely polarised 3He target were also carried out at JLab [18,19]. More
informationcanbefoundinrecentreviews[20–22].
Atthispointthequestionariseswhetherthegluondistributioninatransverselypolarisednucleonisleft-
rightsymmetricorexhibitsaSiverseffectsimilartothequarkdistributions. Recently,theissuehasbeen
discussedrepeatedlyintheliteratureandthepropertiesofthegluonSiversdistributionshavebeenstudied
ingreatdetail[23,24]. Whileitwasfoundthatanon-zeroSiversfunctionimpliesmotionofpartonsinthe
nucleon,presentlytheconnectionbetweentheSiversfunctionandthepartonorbitalangularmomentum
in the nucleon can only be described in a model-dependent way [25]. The correspondence between the
SiverseffectandthetransversemotionofpartonshasbeenoriginallyproposedbyM.Burkardt[26–28].
HenceitisofgreatinteresttoknowwhetherthereexistsagluonSiverseffectornot.
Presently, little is known on the gluon Sivers function. An important theoretical constraint comes from
theso-calledBurkardtsumrule[29]. Itstates,basedonthepresenceofQCDcolour-gaugelinks,thatthe
totaltransversemomentumofallpartonsinsideatransverselypolarisedprotonshouldvanish. Fitstothe
SiversasymmetryusingSIDISdata[13]almostfulfil,withinuncertainties,theBurkardtsumrule,leav-
inglittlespaceforagluoncontribution. FromthenullresultoftheCOMPASSexperimentfortheSivers
asymmetry of positive and negative hadrons produced on a transversely polarised deuteron target [10],
togetherwithadditionaltheoreticalconsiderations,BrodskyandGardner[30]statedthatthegluoncon-
tribution to the parton orbital angular momentum should be negligible, and consequently that the gluon
Siverseffectshouldbesmall. Also,usingtheso-calledtransversemomentumdependent(TMD)gener-
alisedpartonmodelandthemostrecentphenomenologicalinformationonthequarkSiversdistributions
coming from SIDIS data, interesting constraints on our knowledge of the gluon Sivers function were
derived[31]fromtherecentprecisedataonthetransversesinglespinasymmetryA (p↑p→π0X)mea-
N
suredatcentralrapiditybythePHENIXCollaborationatRHIC[32].
In DIS, the leading-order virtual-photon absorption process (LP) does not provide direct access to the
gluon distribution since the virtual-photon does not couple to the gluon, so that higher-order processes
have to be studied, i.e. QCD Compton scattering (QCDC) and Photon-Gluon Fusion (PGF). It is well
known that in lepton-proton scattering one of the most promising processes to directly probe the gluon
is open charm production, (cid:96)p↑ → (cid:96)(cid:48)cc¯X. This is the channel that the COMPASS Collaboration has
investigated at length in order to measure ∆g/g , the gluon polarisation in a longitudinally polarised
nucleon[33]. TaggingthecharmquarkbyidentifyingD-mesonsinthefinalstatehastheadvantagethat
in the lowest order of the strong coupling constant there are no other contributions to the cross section
and one becomes essentially sensitive to the gluon distribution function. An alternative method to tag
2
the gluon in DIS, which has the advantage of higher statistics, has also been developed and used by
COMPASS,i.e. theproductionofhigh-p hadrons[34,35]. IntheLP,thehadrontransversemomentum
T
p withrespecttothevirtualphotondirection(intheframewherethenucleonmomentumisparallelto
T
this direction) originates from the intrinsic transverse momentum k of the struck quark in the nucleon
T
anditsfragmentation,whichbothleadtoasmalltransversecomponent. Onthecontrary,boththeQCDC
andPGFhardprocessescanprovidehadronswithhightransversemomentum. Therefore,taggingevents
with hadrons of high transverse momentum p enhances the contribution of higher-order processes.
T
Nevertheless, although in the high-p sample the PGF fraction is enriched, in order to single out the
T
contributionofthePGFprocesstothemeasuredasymmetrythecontributionsfromLPandQCDChave
tobesubtracted[36].
In this letter, the gluon Sivers effect is investigated using COMPASS data collected by scattering a
160 GeV/c muon beam off transversely polarised deuterons and protons. The experimental set-up and
the data selection are described in Section 2. In Section 3 the measurement is described. The details of
theanalysis aregivenin Section4. Theprocedureof neuralnetwork(NN) trainingwitha MonteCarlo
datasampleisshowninSection5. Section6containstheoverviewofthesystematicstudies. InSection7
theresultsarepresented. SummaryandconclusionsaregiveninSection8.
2 Experimentalset-upanddatasamples
The COMPASS experiment uses a fixed target set-up and a polarised muon beam delivered by the M2
beamlineoftheCERNSPS.Thetransverselypolariseddeuterontargetusedforthe2003and2004data
taking consisted of two oppositely polarised cylindrical cells situated along the beam, each 60 cm long
with a 10 cm gap in between. In 2010 the transversely polarised proton target consisted of three cells:
30cm,60cmand30cmlongwiththecentralcelloppositelypolarisedtothedownstreamandupstream
celland5cmgapsbetweenthecells. Duringalldatatakingperiodsthepolarisationwasreversedonceper
week,inthiswaysystematiceffectsduetoacceptancearecancelled. Forthedeuteronrunsthetargetwas
filledwith6LiD.The6Linucleuscanberegardedasonequasi-freedeuteronanda4Hecore. Theaverage
dilution factor f , defined as the ratio of the DIS cross section on polarisable nucleons in the target to
d
the cross section on all target nucleons, amounts to 0.36 and includes also electromagnetic radiative
corrections. The average polarisation of the deuteron was 0.50. For the asymmetry measurements on
the proton, NH was used as a target. Its average dilution factor f amounts to 0.15 and the proton
3 p
polarisationto0.80. Inbothcases,thenaturallypolarisedmuonbeamof160GeV/cwasused. Thebasic
featuresoftheCOMPASSspectrometer,asdescribedinRef.[37],arethesamefor2003-4and2010data
taking. Several upgrades were performed in 2005, the main one being the installation of a new target
magnet,whichallowedtoincreasethepolarangleacceptancefrom70mradto180mrad.
A crucial point of this analysis is the search for an observable that is strongly correlated with the gluon
azimuthal angle φ . In the LEPTO generator [38], gluons are accessed via PGF with a quark-antiquark
g
pairinthefinalstateandthefragmentationprocessisdescribedbytheLundmodel[39]. Asaresultof
MCstudies,thebestcorrelationisfoundbetweenφ andφ ,wherethelatterdenotestheazimuthalangle
g P
ofthevectorsumPofthetwohadronmomenta. Forthepresentanalysis,twochargedhadronsforeach
eventareselected. Ifmorethantwochargedhadronsarereconstructedinanevent,onlythehadronwith
the largest transverse momentum, p , and the one with the second-largest transverse momentum, p ,
T1 T2
aretakenintoaccount. InordertoenhancethePGFfractioninthesampleandatthesametimethecorre-
lationbetweenφ andφ ,afurtherrequirementisappliedtothetransversemomentaofthetwohadrons:
g P
p >0.7GeV/cand p >0.4GeV/c. Moreover,thefractionalenergiesofthetwohadronsmustfulfil
T1 T2
thefollowingconditions: z >0.1(i=1,2)andz +z <0.9,wherethelastrequirementrejectsevents
i 1 2
fromdiffractivevectormesonproduction. Hadronpairsareselectedwithnochargeconstraint. Withthis
choicethecorrelationcoefficientis0.54. TheSiversasymmetryisthenobtainedasthesinemodulation
intheSiversangle,φ =φ −φ . Hereφ istheazimuthalangleofthenucleonspinvector.
Xiv P S S
FirstmeasurementoftheSiversasymmetryforgluonsfromSIDISdata 3
Thesamekinematicdataselectionisusedforbothdeuteronandprotondata. Therequirementonphoton
virtuality, Q2>1 (GeV/c)2, selects events in the perturbative region and the one on the mass of the
hadronicfinalstate, W >5GeV/c2,removestheregionoftheexclusivenucleonresonanceproduction.
The Bjorken-x variable covers the range 0.003<x <0.7. For the fractional energy of the virtual
Bj
photon, y, the limit y > 0.1 removes a region sensitive to experimental biases and the requirement
y<0.9rejectseventswithlargeelectromagneticradiativecorrections.
3 Siversasymmetryintwohadronproduction
InordertoextractthegluonSiversasymmetry, µ+N →µ(cid:48)+2h+X eventsareselectedasdescribedin
Section2. Bylabellingwiththesymbol↑thecrosssectionassociatedtoatargetcellpolarisedupwards
in the laboratory and by ↓ the cross section of a target cell polarised downwards in the laboratory, the
Siversasymmetrycanbewrittenas
∆σ((cid:126)x,φ )
A2h((cid:126)x,φ )= Siv , (1)
T Siv σ((cid:126)x)
where(cid:126)x=(x ,Q2,p ,p ,z ,z ), ∆σ ≡d7σ↑−d7σ↓ and σ ≡d7σ↑+d7σ↓. All cross sections are
Bj T1 T2 1 2
integrated over the two azimuthal angles φ and φ , where φ is the azimuthal angle of the relative
S R R
momentumofthetwohadrons,R=P −P . Thenumberofeventsinaφ binisgivenby
1 2 Siv
N((cid:126)x,φ )=α((cid:126)x,φ )(cid:0)1+ fP ASiv((cid:126)x)sinφ (cid:1). (2)
Siv Siv T Siv
Here f isthedilutionfactor, P thetargetpolarisationandα =anΦσ anacceptance-dependentfactor,
T 0
whereaisthetotalspectrometeracceptance,nthedensityofscatteringcentres,Φthebeamfluxandσ
0
thespin-averagedpartofthecrosssection. Fromhereon,theSiversasymmetryA2h((cid:126)x,φ )isfactorised
T Siv
intotheazimuth-independentamplitudeASiv((cid:126)x)andthemodulationsinφ .
Siv
InordertoextracttheSiversasymmetryofthegluon,theamplitudeofthesinφ modulationisextracted
Siv
fromdata. ThegeneralexpressionforthecrosssectionofSIDISproductionwithatleastonehadronin
the final state is well known [40]. It contains eight azimuthal modulations, which are functions of the
single-hadronazimuthalangleandφ . Intheabsenceofcorrelationspossiblyintroducedbyexperimental
S
effects,theyareallorthogonal,sothattheSiversasymmetrycaneitherbeextractedastheamplitudeof
the sinφ modulation or one can perform a simultaneous fit of all eight amplitudes. For the case of
Siv
heavy-quarkpairanddijetproductioninlepton-nucleoncollisions,allazimuthalasymmetriesassociated
tothegluondistributionfunctionhavebeenrecentlyworkedoutinRef. 41. There,theSiversasymmetry
is defined as the amplitude of the sin(φ −φ ) modulation, where φ is the azimuthal angle of the
T S T
transverse-momentumvectorofthequark-antiquarkpair, q . Inouranalysis, φ isreplacedbyφ , due
T T P
to its correlation with the gluon azimuthal angle φ , and the Sivers asymmetry is extracted taking into
g
account only the sin(φ −φ ) modulation in the cross section. It has been verified that including in the
P S
crosssectionthesameeighttransverse-spinmodulationsasinSIDISsingle-hadronproduction[40]and
extractingsimultaneouslyallasymmetriesgivesthesameresultonthegluonSiversasymmetry.
In order to determine the Sivers asymmetry for gluons from two-hadron production in SIDIS, it is nec-
essary to assume that the main contributors to muon-nucleon DIS are the three processes (Fig. 1) as
presentedinRef.[38]. Thismodelissuccessfulindescribingtheunpolariseddata. AtCOMPASSkine-
matics, the leading process appears at zero-order QCD in the total DIS cross section and it is the dom-
inant process, while the other two processes, photon-gluon fusion and QCD Compton, are first-order
QCDprocessesandhencesuppressed. However,theircontributioncanbeenhancedbyconstrainingthe
transversemomentumoftheproducedhadrons,asmentionedabove.
IntroducingtheprocessfractionsR =σ /σ (j∈{PGF,QCDC,LP}),theamplitudeoftheSiversasym-
j j
4
(a) (b) (c)
Fig.1: Feynmandiagramsconsideredforγ∗N scattering: a)photon-gluonfusion(PGF),b)gluonradia-
tion(QCDComptonscattering),c)Leadingorderprocess(LP).
metrycanbeexpressedintermsoftheamplitudesofthethreecontributingprocesses:
∆σ σ ∆σ σ ∆σ σ ∆σ
fP ASivsinφ = = PGF PGF + QCDC QCDC + LP LP
T Siv
σ σ σPGF σ σQCDC σ σLP (3)
= fP (R ASiv +R ASiv +R ASiv)sinφ ,
T PGF PGF QCDC QCDC LP LP Siv
with σ =∑jσj, ∆σ =∑j∆σj and fPTASjivsinφSiv =∆σj/σj. The determination of Rj is done on an
event-by-event basis by using the neural networks (NN) trained on Monte Carlo data as described in
Section5.
4 Asymmetryextractionusingthemethodsofweights
The method adopted in the present analysis was already applied to extract the gluon polarisation from
the longitudinal double-spin asymmetry in the SIDIS measurement of single-hadron production [36].
Both for the deuteron data (two target cells) and the proton data (three target cells), four target con-
figurations can be introduced. In the case of the two-cell target: 1 - upstream, 2 - downstream, 3 -
upstream(cid:48), 4 - downstream(cid:48). In the case of the three-cell target: 1 - (upstream+downstream), 2 - centre,
3 - (upstream(cid:48)+downstream(cid:48)), 4 - centre(cid:48). Here upstream(cid:48), centre(cid:48) and downstream(cid:48) denote the cells af-
ter the polarisation reversal and configuration 1 has the polarisation pointing upwards in the laboratory
frame. DecomposingtheSiversasymmetryintotheasymmetriesofthecontributingprocesses(Eq.(3))
andintroducingtheSiversmodulationβt((cid:126)x,φ )=R ((cid:126)x)f((cid:126)x)Pt sinφ ,whichisspecificforprocess j,
j Siv j T Siv
onecanrewriteEq.(2):
(cid:16)
Nt((cid:126)x,φ )=αt((cid:126)x,φ ) 1+βt ((cid:126)x,φ )ASiv ((cid:126)x)
Siv Siv PGF Siv PGF
(4)
(cid:17)
+βt ((cid:126)x,φ )ASiv ((cid:126)x)+βt ((cid:126)x,φ )ASiv((cid:126)x) ,
QCDC Siv QCDC LP Siv LP
wheret =1,2,3,4denotesthetargetconfiguration.
In order to minimise statistical uncertainties for each process, a weighting factor is introduced. It is
known [42] that the choice ω =β for the weight optimises the statistical uncertainty but variations of
j j
thetargetpolarisationP intimemayintroduceabiastothefinalresult. Therefore,theweightingfactor
T
ω ≡β /P isusedinstead. Eachofthefourequations(4)isweightedthreetimeswithω dependingon
j j T j
the process j ∈{PGF,QCDC,LP} and integrated over φ and(cid:126)x, yielding twelve observed quantities
Siv
FirstmeasurementoftheSiversasymmetryforgluonsfromSIDISdata 5
qt:
j
(cid:90)
qt = d(cid:126)xdφ ω ((cid:126)x,φ )Nt((cid:126)x,φ )
j Siv j Siv Siv
(cid:16) (cid:17)
=α˜t 1+{βt } (cid:8)ASiv (cid:9) +{βt } (cid:8)ASiv (cid:9) +{βt } (cid:8)ASiv(cid:9) , (5)
j PGF ωj PGF βPGFωj QCDC ωj QCDC βQCDCωj LP ωj LP βLPωj
where α˜tj is the ωj-weighted acceptance-dependent factor. The quantities {βit}ωj and {ASiiv}βitωj are
weightedaverages,wheretheweightfactorisdenotedinthesubscript.1
Theacceptancefactorsα˜t cancelwhenforasymmetryextractiononeusesthedoubleratio
j
q1q4
j j
r := (6)
j q2q3
j j
asthedatatakingwasperformedsuchthatα˜1α˜4/α˜2α˜3=1. Ifthisconditionisnotfulfilled,falseasym-
j j j j
metriesmayoccur. Itischeckedthatthisisnotthecase(seeSection6).
Intheanalysis,thequantitiesq and{βt} areapproximatedasfollows:
j i ωj
Nt
qt ≈ ∑ωk, (7)
j j
k=1
Nt
∑ βt,kωk
i j
{βt} ≈ k=1 . (8)
i ωj Nt
∑ ωk
j
k=1
The latter approximation holds for small observed raw asymmetries, i.e. ωA (cid:28) 1. In order to avoid
numericalinconsistenciesinEq.(8)duetoazero-polewhenintegratingoverthefullrangeofφ ,2 two
Siv
bins in φ ([0;π],[π;2π]) are introduced. In the aforementioned three double ratios given in Eq. (6)
Siv
onlyasymmetriesareunknown. However,inordertosolvethesystemofequationsoneneedstoassume
that the weighted asymmetry for a given process i is the same for the three different weights ω β, i.e.
j i
{A} ={A} ={A} . Thismeansthatthevaluesofω andA mustbeuncorrelated. For
i βiωPGF i βiωQCDC i βiωLP j i
example, since ω is proportional to R , which strongly depends on the hadron transverse momentum,
j j
onehastouseakinematicregionwheretheasymmetriesA areexpectedtobeindependentof p . Under
i T
theseassumptions,thenumberofunknownweightedasymmetriesisthree,whichexactlycorrespondsto
thenumberofequationsoftype(6). Theseequationsaresolvedbya χ2 fitthatincludessimultaneously
bothbinsinφ .
Siv
Assuming that A can be approximated by a linear function of x and that x is not correlated with ω ,
i i i j
resultsin
{A} =A({x} ). (9)
i βiωi i i βiωi
This approximation allows to interpret the obtained results as an asymmetry value measured at the
weighted value of x. For each process, the weighted value of x is obtained from MC using the rela-
i i
1{β}ω = (cid:82)(cid:82)ααβωωdd(cid:126)x(cid:126)xddφφSSiviv, {A}βω = (cid:82)(cid:82)Aααββωωdd(cid:126)x(cid:126)xddφφSSiviv
2Notethatωk,whichcontainssinφ ,isintegratedintheregion0to2π.
j Siv
6
tion
Ni
∑ xkβkωk
i i i
{x} = k=1 . (10)
i βiωi Ni
∑ βkωk
i i
k=1
Here, N is the number of events of type i in MC data. The assumption that the values of x are not
i i
correlatedwithω ,whichallowsustoconsideronly{x} ,wasverifiedusingMCdata. Thedetailsof
j i ωiβi
theanalysisaregivenin[43].
5 MonteCarlooptimisationandNeuralNetworktraining
The present analysis is very similar to the one used for the ∆g/g extraction from high-p hadron
T
pairs [35] and single hadrons [36]. For the NN training with custom input, output and target vector
the package NetMaker [44] is used. The NN is trained with a Monte Carlo sample with process iden-
tification. As input vector the following six kinematic variables are chosen: x ,Q2,p ,p ,p ,p .
Bj T1 T2 L1 L2
The latter two are the longitudinal components of the hadron momenta. The trained neural network is
appliedtothedatabytakingthevectoroftheaforementionedsixvariables,anditsoutputisinterpreted
as probabilities that the given event is a result of one of the three contributing processes. Hence the
simulateddistributionsofthesevariablesneedtobeinagreementwiththecorrespondingdistributionsin
thedatasamples.
UsingtheLEPTOgenerator(version6.5)[38],twoseparateMCdatasampleswereproducedtosimulate
the deuteron and proton data. The generator is tuned to the COMPASS data sample obtained with
the high-p hadron-pair selection as described in Ref. [35]. The MSTW08 parameterisation of input
T
PDFs [45] was chosen as it gives a good description of the F structure function in the COMPASS
2
kinematicrangeandisvaliddowntoQ2=1(GeV/c)2. Electromagneticradiativecorrections[46]were
appliedasaweightingfactortotheMCdistributionsshowninFigures2and3butnotintheMCsamples
usedinNNtraining. Thisdifferencewasstudiedanditwasestimatedtobenegligible.
The generated events were processed by COMGEANT, the COMPASS detector simulation program
basedonGEANT3. TheMCsamplesfortheprotonanddeuterondatadifferinthetargetmaterialandin
the spectrometer set-up. The FLUKA package [47] is used in order to simulate secondary interactions.
Asthenextstep,theCOMPASSreconstructionprogramCORALwasapplied. Thesamedataselection
as for real events was used for MC data. Figures 2 and 3 show the comparison between experimental
andMCdataforthecaseofthedeuteronandprotondata,respectively.
The main goal of the NN parameterisation is the estimation of the process fractions R . In the typical
j
caseofsignalandbackgroundseparation,theexpectedNNoutputwouldbesettooneforthesignaland
zeroforthebackground. TheoutputvaluereturnedbytheNNwouldthencorrespondtothefractionof
signal events in the sample in the given phase space point of the input parameter vector. In the present
analysis, theprocessfractionswereestimatedsimultaneously. Inordertohaveaclosurerelationonthe
processprobabilities,thesumofthemmustadduptoone,henceonlytwoindependentoutputvariables
fromtheNNareneeded. TheestimationoftheprocessfractionsRj fromtheNNoutputisaccomplished
by assigning to each event the probabilities PPGF, PQCDC and PLP. The distribution of the NN output
NN NN NN
after training is shown in Fig. 4 on the “Mandelstam representation”, i.e. as points in an equilateral
triangle with unit height. Points outside of the triangle refer to one estimator being negative, which is
possiblebecauseinthetrainingtheestimatorsarenotboundtobepositive. Thedirectseparationofthe
PGF process using this distribution is statistically less efficient than weighting each event by the three
probabilities obtained from the NN output values. These probabilities are used as values of the process
fractions(R ,R andR )inthedataanalysisdescribedinSection4.
PGF QCDC LP
ForthevalidationoftheNNtraining,astatisticallyindependentMCsampleisusedtocheckhowtheNN