Azimuthal asymmetries of charged hadrons produced in high-energy muon scattering off longitudinally polarised deuterons
C. Adolph, M. Aghasyan, R. Akhunzyanov, M.G. Alexeev, G.D. Alexeev, A. Amoroso, V. Andrieux, N.V. Anfimov, V. Anosov, K. Augsten, W. Augustyniak, A. Austregesilo, C.D.R. Azevedo, B. Badelek, F. Balestra, M. Ball, J. Barth, R. Beck, Y. Bedfer, J. Bernhard, K. Bicker, E. R. Bielert, R. Birsa, M. Bodlak, P. Bordalo, F. Bradamante, C. Braun, A. Bressan, M. Buechele, W.-C. Chang, C. Chatterjee, M. Chiosso, I. Choi, S.-U. Chung, A. Cicuttin, M.L. Crespo, Q. Curiel, S. Dalla Torre, S.S. Dasgupta, S. Dasgupta, O.Yu. Denisov, L. Dhara, S.V. Donskov, N. Doshita, Ch. Dreisbach, V. Duic, W. Duennweber, M. Dziewiecki, A. Efremov, P.D. Eversheim, W. Eyrich, M. Faessler, A. Ferrero, M. Finger, M. Finger jr., H. Fischer, C. Franco, N. du Fresne von Hohenesche, J.M. Friedrich, V. Frolov, E. Fuchey, F. Gautheron, O.P. Gavrichtchouk, S. Gerassimov, J. Giarra, F. Giordano, I. Gnesi, M. Gorzellik, S. Grabmueller, A. Grasso, M. Grosse Perdekamp, B. Grube, T. Grussenmeyer, A. Guskov, F. Haas, D. Hahne, G. Hamar, D. von Harrach, F.H. Heinsius, R. Heitz, F. Herrmann, N. Horikawa, N. d'Hose, C.-Y. Hsieh, S. Huber, S. Ishimoto, A. Ivanov, Yu. Ivanshin, T. Iwata, V. Jary, R. Joosten, P. Joerg, E. Kabuss, B. Ketzer, G.V. Khaustov, Yu.A. Khokhlov, Yu. Kisselev, F. Klein, K. Klimaszewski, J.H. Koivuniemi, et al. (125 additional authors not shown)
EEUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
COMPASS
CERN-EP-2016–245.R5December 6, 2018
Azimuthal asymmetries of charged hadrons produced in high-energymuon scattering off longitudinally polarised deuterons
The COMPASS Collaboration
Abstract
Single hadron azimuthal asymmetries of positive and negative hadrons produced in muon semi-inclusive deep inelastic scattering off longitudinally polarised deuterons are determined using the2006 COMPASS data and also combined all deuteron COMPASS data. For each hadron charge, thedependence of the azimuthal asymmetry on the hadron azimuthal angle φ is obtained by means of afive-parameter fitting function that besides a φ -independent term includes four modulations predictedby theory: sin φ , sin 2 φ , sin 3 φ and cos φ . The amplitudes of the five terms have been extracted, first,for the hadrons in the whole available kinematic region. In further fits, performed for hadrons froma restricted kinematic region, the φ -dependence is determined as a function of one of three variables(Bjorken- x , fractional energy of virtual photon taken by the outgoing hadron and hadron transversemomentum), while disregarding the others. Except the φ -independent term, all the modulation am-plitudes are very small, and no clear kinematic dependence could be observed within experimentaluncertainties. PACS: 13.60.Hb, 13.85.Hd, 13.85.Ni, 13.88.+eKeywords: lepton deep inelastic scattering, polarisation, spin asymmetry, parton distribution functions (To be submitted to the European Physical Journal C) a r X i v : . [ h e p - e x ] D ec he COMPASS Collaboration C. Adolph , M. Aghasyan , R. Akhunzyanov , M.G. Alexeev , G.D. Alexeev , A. Amoroso ,V. Andrieux , N.V. Anfimov , V. Anosov , K. Augsten , W. Augustyniak , A. Austregesilo ,C.D.R. Azevedo , B. Badełek , F. Balestra , M. Ball , J. Barth , R. Beck , Y. Bedfer ,J. Bernhard , K. Bicker , E. R. Bielert , R. Birsa , M. Bodlak , P. Bordalo ,F. Bradamante , C. Braun , A. Bressan , M. B¨uchele , W.-C. Chang , C. Chatterjee ,M. Chiosso , I. Choi , S.-U. Chung , A. Cicuttin , M.L. Crespo , Q. Curiel , S. DallaTorre , S.S. Dasgupta , S. Dasgupta , O.Yu. Denisov , L. Dhara , S.V. Donskov , N. Doshita ,Ch. Dreisbach , V. Duic , W. D¨unnweber c , M. Dziewiecki , A. Efremov , P.D. Eversheim ,W. Eyrich , M. Faessler c , A. Ferrero , M. Finger , M. Finger jr. , H. Fischer , C. Franco ,N. du Fresne von Hohenesche , J.M. Friedrich , V. Frolov , E. Fuchey , F. Gautheron ,O.P. Gavrichtchouk , S. Gerassimov , J. Giarra , F. Giordano , I. Gnesi , M. Gorzellik ,S. Grabm¨uller , A. Grasso , M. Grosse Perdekamp , B. Grube , T. Grussenmeyer , A. Guskov ,F. Haas , D. Hahne , G. Hamar , D. von Harrach , F.H. Heinsius , R. Heitz , F. Herrmann ,N. Horikawa , N. d’Hose , C.-Y. Hsieh , S. Huber , S. Ishimoto , A. Ivanov ,Yu. Ivanshin , T. Iwata , V. Jary , R. Joosten , P. J¨org , E. Kabuß , B. Ketzer ,G.V. Khaustov ,Yu.A. Khokhlov , Yu. Kisselev , F. Klein , K. Klimaszewski , J.H. Koivuniemi , V.N. Kolosov ,K. Kondo , K. K¨onigsmann , I. Konorov , V.F. Konstantinov , A.M. Kotzinian ,O.M. Kouznetsov , M. Kr¨amer , P. Kremser , F. Krinner , Z.V. Kroumchtein , Y. Kulinich ,F. Kunne , K. Kurek , R.P. Kurjata , A.A. Lednev , A. Lehmann , M. Levillain , S. Levorato ,Y.-S. Lian , J. Lichtenstadt , R. Longo , A. Maggiora , A. Magnon , N. Makins ,N. Makke , G.K. Mallot , B. Marianski , A. Martin , J. Marzec , J. Matouˇsek ,H. Matsuda , T. Matsuda , G.V. Meshcheryakov , M. Meyer , W. Meyer , Yu.V. Mikhailov ,M. Mikhasenko , E. Mitrofanov , N. Mitrofanov , Y. Miyachi , A. Nagaytsev , F. Nerling ,D. Neyret , J. Nov´y , W.-D. Nowak , G. Nukazuka , A.S. Nunes , A.G. Olshevsky , I. Orlov ,M. Ostrick , D. Panzieri , B. Parsamyan , S. Paul , J.-C. Peng , F. Pereira , M. Peˇsek ,D.V. Peshekhonov , N. Pierre , S. Platchkov , J. Pochodzalla , V.A. Polyakov , J. Pretz ,M. Quaresma , C. Quintans , S. Ramos , C. Regali , G. Reicherz , C. Riedl , M. Roskot ,N.S. Rossiyskaya , D.I. Ryabchikov , A. Rybnikov , A. Rychter , R. Salac , V.D. Samoylenko ,A. Sandacz , C. Santos , S. Sarkar , I.A. Savin , T. Sawada G. Sbrizzai , P. Schiavon ,K. Schmidt , H. Schmieden , K. Sch¨onning , E. Seder , A. Selyunin , L. Silva , L. Sinha ,S. Sirtl , M. Slunecka , J. Smolik , A. Srnka , D. Steffen , M. Stolarski , O. Subrt , M. Sulc ,H. Suzuki , A. Szabelski , T. Szameitat , P. Sznajder , S. Takekawa , M. Tasevsky ,S. Tessaro , F. Tessarotto , F. Thibaud , A. Thiel , F. Tosello , V. Tskhay , S. Uhl , J. Veloso ,M. Virius , J. Vondra , S. Wallner , T. Weisrock , M. Wilfert , J. ter Wolbeek , K. Zaremba ,P. Zavada , M. Zavertyaev , E. Zemlyanichkina , N. Zhuravlev , M. Ziembicki and A. Zink University of Eastern Piedmont, 15100 Alessandria, Italy University of Aveiro, Department of Physics, 3810-193 Aveiro, Portugal Universit¨at Bochum, Institut f¨ur Experimentalphysik, 44780 Bochum, Germany mn4
Universit¨at Bonn, Helmholtz-Institut f¨ur Strahlen- und Kernphysik, 53115 Bonn, Germany m5 Universit¨at Bonn, Physikalisches Institut, 53115 Bonn, Germany m6 Institute of Scientific Instruments, AS CR, 61264 Brno, Czech Republic o7 Matrivani Institute of Experimental Research & Education, Calcutta-700 030, India p8 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia q9 Universit¨at Erlangen–N¨urnberg, Physikalisches Institut, 91054 Erlangen, Germany m10
Universit¨at Freiburg, Physikalisches Institut, 79104 Freiburg, Germany mn11
CERN, 1211 Geneva 23, Switzerland Technical University in Liberec, 46117 Liberec, Czech Republic o3 LIP, 1000-149 Lisbon, Portugal r14
Universit¨at Mainz, Institut f¨ur Kernphysik, 55099 Mainz, Germany m15
University of Miyazaki, Miyazaki 889-2192, Japan s16
Lebedev Physical Institute, 119991 Moscow, Russia Technische Universit¨at M¨unchen, Physik Department, 85748 Garching, Germany mc18
Nagoya University, 464 Nagoya, Japan s19
Charles University in Prague, Faculty of Mathematics and Physics, 18000 Prague, Czech Republic o20
Czech Technical University in Prague, 16636 Prague, Czech Republic o21
State Scientific Center Institute for High Energy Physics of National Research Center ‘KurchatovInstitute’, 142281 Protvino, Russia IRFU, CEA, Universit´e Paris-Saclay, 91191 Gif-sur-Yvette, France n23
Academia Sinica, Institute of Physics, Taipei 11529, Taiwan Tel Aviv University, School of Physics and Astronomy, 69978 Tel Aviv, Israel t25
University of Trieste, Department of Physics, 34127 Trieste, Italy Trieste Section of INFN, 34127 Trieste, Italy Abdus Salam ICTP, 34151 Trieste, Italy University of Turin, Department of Physics, 10125 Turin, Italy Torino Section of INFN, 10125 Turin, Italy University of Illinois at Urbana-Champaign, Department of Physics, Urbana, IL 61801-3080, USA National Centre for Nuclear Research, 00-681 Warsaw, Poland u32
University of Warsaw, Faculty of Physics, 02-093 Warsaw, Poland u33
Warsaw University of Technology, Institute of Radioelectronics, 00-665 Warsaw, Poland u34
Yamagata University, Yamagata 992-8510, Japan s* Deceased
Corresponding authors a Also at Instituto Superior T´ecnico, Universidade de Lisboa, Lisbon, Portugal b Also at Department of Physics, Pusan National University, Busan 609-735, Republic of Korea andat Physics Department, Brookhaven National Laboratory, Upton, NY 11973, USA c d Also at Chubu University, Kasugai, Aichi 487-8501, Japan se Also at Department of Physics, National Central University, 300 Jhongda Road, Jhongli 32001,Taiwan f Also at KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan g Also at Moscow Institute of Physics and Technology, Moscow Region, 141700, Russia h Supported by Presidential grant NSh–999.2014.2 i Present address: RWTH Aachen University, III. Physikalisches Institut, 52056 Aachen, Germany j Also at Department of Physics, National Kaohsiung Normal University, Kaohsiung County 824,Taiwan k Present address: Uppsala University, Box 516, 75120 Uppsala, Sweden l Supported by the DFG Research Training Group Programmes 1102 and 2044 m Supported by the German Bundesministerium f¨ur Bildung und Forschung n Supported by EU FP7 (HadronPhysics3, Grant Agreement number 283286) o Supported by Czech Republic MEYS Grant LG13031 p Supported by SAIL (CSR), Govt. of India q Supported by CERN-RFBR Grant 12-02-91500 r Supported by the Portuguese FCT - Fundac¸ ˜ao para a Ciˆencia e Tecnologia, COMPETE and QREN,Grants CERN/FP 109323/2009, 116376/2010, 123600/2011 and CERN/FIS-NUC/0017/2015 s Supported by the MEXT and the JSPS under the Grants No.18002006, No.20540299 and No.18540281;Daiko Foundation and Yamada Foundation
Supported by the Israel Academy of Sciences and Humanities u Supported by the Polish NCN Grant 2015/18/M/ST2/00550zimuthal asymmetries . . . 1
Measurements of Semi-Inclusive Deep-Inelastic Scattering (SIDIS) µ + N → µ (cid:48) + nh + X , n = , , ... (1)of high-energy polarised muons µ off nucleons N in the initial state and scattered muons µ (cid:48) , n measuredhadrons h and unobserved particles X in the final state are sensitive to the spin-dependent Parton Distri-bution Functions (PDFs) of nucleons. The SIDIS cross section depends, in particular, on the azimuthalangle of each produced and measured hadron (see e.g. Ref. [1]), which leads to azimuthal asymmetriesrelated to convolutions of the nucleon Transverse-Momentum-Dependent (TMD) PDFs and parton-to-hadron Fragmentation Functions (FFs). These asymmetries can appear in SIDIS off unpolarised, longi-tudinally or transversely polarised nucleons.The TMD PDFs were studied in a number of experiments. The short overview of earlier results obtainedby the HERMES, CLAS and COMPASS collaborations on azimuthal asymmetries in SIDIS productionof charged hadrons was given in Ref. [2]. The COMPASS collaboration has published results on asym-metries off unpolarised LiD (referred to as ”deuteron”) target [4], transversely polarised deuterons [5]and transversely polarised NH (referred to as ”proton”) target [6]. The common analysis of transverselypolarised deuteron and proton data is included in Ref. [6] also. The updated overview of the TMD PDFsincluding the COMPASS results can be found in Ref. [7]. The COMPASS results on azimuthal asym-metries off longitudinally polarised deuterons based on the data collected in 2002, 2003 and 2004 werepublished in Ref. [2] for so called “integrated” asymmetries and asymmetries as functions of kinematicvariables extracted in the restricted kinematic region. Similar data have been collected in 2006 but notpublished yet. The results on the integrated asymmetries for 2006 and for the combined 2002 – 2006data which are presented in this Paper are extracted using hadrons from the whole available kinematicregion, at variance with Ref. [2], while asymmetries as functions of the kinematic variables are extractedusing the hadrons from a restricted kinematic region, similar to Ref. [2].The Paper is organised as follows. The SIDIS kinematics, basic formulae and a brief theoretical overvieware given in Section 2. The analysis of the 2006 data is described in Section 3. The results on theasymmetries of the combined 2002 – 2006 data are presented in Section 4. Systematic uncertainties arediscussed in Section 5 and conclusions are given in Section 6. The SIDIS kinematics is illustrated in Fig. 1. p p P L|| θ φ γ S φ q P T P Production planeScattering plane h T l’l h Fig. 1:
The SIDIS kinematics shown for target deuteron polarisation
PPP (cid:107) antiparallel to the beam direction.
The 4-momenta of the incident and scattered muons are denoted by l and l (cid:48) , respectively. The 4-momentum of the virtual photon is given by q = l − l (cid:48) with Q = − q . The angle of the momentum The COMPASS Collaborationvector qqq of the virtual photon with respect to the incident muon is denoted by θ γ . The vectors ppp h and PPP (cid:107) denote the hadron momentum and the longitudinal target deuteron polarisation, respectively. Theirtransverse components ppp hT and PPP T are defined with respect to the virtual-photon momentum. The longi-tudinal component | PPP L | = | PPP (cid:107) | cos θ γ is approximately equal to | PPP (cid:107) | due to the smallness of the angle θ γ .The small transverse component is equal to | PPP T | = | PPP (cid:107) | sin θ γ where sin θ γ ≈ ( Mx / Q ) √ − y . Here, M is the nucleon mass and y = ( qp ) / ( pl ) is the fractional energy of the virtual photon, where p is the 4-momentum of the target nucleon. The angle φ denotes the azimuthal angle between the lepton scatteringplane and the hadron production plane, while φ S denotes the angle of the deuteron polarisation vectorwith respect to the scattering plane: φ S = ◦ or 180 ◦ for deuteron polarisation parallel or antiparallel tothe beam direction, respectively. Furthermore, the Bjorken variable, x B j ≡ x = Q / ( pq ) , the fraction ofthe virtual-photon energy taken by a hadron, z = ( pp h ) / ( pq ) , the transverse momentum of a hadron, p hT ,and the invariant mass of the photon-nucleon system, W = ( p + q ) , that, together with Q > c ) and 0 < y <
1, characterise SIDIS under study.The general expression for the differential SIDIS cross section (see Ref. [1] and references therein) is alinear function of the incident muon polarisation P µ and of the longitudinal and transverse components PPP L and PPP T of the target deuteron polarisation PPP (cid:107) :d σ = d σ + PPP µ d σ L + PPP L (cid:0) d σ L + PPP µ d σ LL (cid:1) + PPP T (cid:0) d σ T + PPP µ d σ LT (cid:1) . (2)Here, the first (second) subscript of the partial cross sections refers to the beam (target) polarisation: 0 , L or T denote unpolarised, longitudinally or transversely polarised.The azimuthal asymmetries of charged hadron production a h ± ( φ ) are defined as follows: a h ± ( φ ) = d σ ←⇒ − d σ ←⇐ | P L | ( d σ ←⇒ + d σ ←⇐ ) , (3)where all cross sections are functions of the angle φ . The Eq. (3) represents a definition of the experimen-tally measured asymmetries common for this Paper and for Refs. [2, 3]. The first (second) superscriptdenotes the beam (target) spin orientation. The symbol ← denotes the incident muon spin orientationthat, in the case of a positive charge of the incident muons, is mainly opposite to the beam direction. Forthe CERN muon beam, the average value of | PPP µ | is equal to 0 .
8. The beam polarisation does not enterin the definition of measured asymmetries. The symbols ⇒ and ⇐ denote the target deuteron spin ori-entations (polarisations) the first of which is parallel, considered further as positive (+) , and the secondone is antiparallel ( − ) to the beam direction (see Section 3.1).Substituting the general expression for d σ (Eq. (2)) in the cross sections of the Eq. (3), one can obtainthe expected contributions of the partial cross sections to the azimuthal asymmetries. As the result,when taking into account the signs of the target polarizations, one can see that only four partial crosssections contribute to the numerator of Eq. (3) and two to its denominator. In the numerator, we expectto have contributions from d σ L , P µ d σ LL and tan θ γ ( d σ T + P µ d σ LT ) , while in the denominator fromd σ and P µ d σ L . The explicit expression for these partial cross sections in terms of the PDFs and theirdependences on the hadron azimuthal angle have been given in Ref. [2] and briefly commented below.Following phenomenological considerations based on the QCD parton model of the nucleon and SIDISin one-photon exchange approximation, the squared modulus of the matrix element, defining the crosssections, is represented by a number of diagrams. As an example , the diagram accounting for thecontribution to the SIDIS cross section of the chiral-odd transversity PDF h ( x ) convoluted with thechiral-odd Collins FF H ⊥ ( z ) is shown in Fig. 2. The diagram contributes to the asymmetry Eq. (3) viathe term d σ T . Other PDFs convoluted with corresponding FFs can also contribute to the cross sectionof spinless or unpolarised hadron production off longitudinally polarised deuterons and their expected In this Paper we follow the Amsterdam notations for PDFs and PFFs, see e.g. Ref. [1]. zimuthal asymmetries . . . 3 l’l (x) h p q p h H (z) Fig. 2:
The diagram describing the contribution to the SIDIS cross section of PDF h ( x ) convoluted with FF H ⊥ ( z ) . azimuthal modulations. As motivated in Ref. [2], out of predicted terms the φ -independent term a h ± andfour modulation terms up to the order of M / Q are retained for the analysis: a h ± ( φ ) = a h ± + a sin φ h ± sin φ + a sin2 φ h ± sin 2 φ + a sin3 φ h ± sin 3 φ + a cos φ h ± cos φ . (4)The sign and the amplitude of each modulation is a subject of the a h ± ( φ ) data analysis (see Section 3.6).In Eq. (4), the term a h ± is related to the well known helicity PDF g L contributing to asymmetries via d σ LL . The two terms with amplitudes a sin2 φ h ± and a sin φ h ± reported in Ref. [10] are related to the worm-gear-L PDF h ⊥ L , and to the PDFs h L and f L , respectively, contributing to the asymmetries via d σ L . Thetransversity PDF h and Sivers PDF f ⊥ T can also contribute to the term with amplitude a sin φ h ± via d σ T witha small factor tan θ γ . Other terms in Eq. (4), not seen yet in experiments with longitudinally polariseddeuterons, are the terms with amplitudes a sin3 φ h ± and a cos φ h ± . They are related to the pretzelosity PDF h ⊥ T and the worm-gear-T PDF g T contributing to asymmetries via d σ T and d σ LT , respectively, suppressedby the small factor tan θ γ . The COMPASS results [2] obtained from the 2002 – 2004 data suggested someindications for a possible x -dependence of terms with amplitudes a sin2 φ h ± and a cos φ h ± . The contribution ofthe term with amplitude a cos2 φ h ± which could have appeared from d σ in the denominator of Eq. (3) isdisregarded. This amplitude is expected [11,12] to be of the order of 0.1 and would enter Eq. (4) with thefactor a h ± , that is of the order of 10 − for integrated asymmetries (see Table 2), or with a h ± ( x ) ≤ .
05 forasymmetries as functions of kinematic variables (see Fig. 6). This is beyond our experimental accuracy.The same comments apply to possible contributions of terms with amplitudes a cos φ h ± and a sin φ h ± which alsocould originate from d σ and d σ L of the denominator of Eq. (3), respectively. The negligible impactof the disregarded modulations on the amplitudes in Eq. (4) is confirmed by the 2006 data (see Section3.6). All modulation amplitudes obtained in this Paper refer to the average value of the beam polarisationequal to − a h ± ( φ ) as manifestation ofTMD PDFs describing the nucleons in the deuteron and to investigate the x , z and p hT dependences of thecorresponding modulation amplitudes. For these purposes, we used first the 2006 deuteron data and thenthe combination of all 2002 – 2006 COMPASS deuteron data with longitudinal target polarisation. The COMPASS set-up is a two-stage forward spectrometer with the world’s largest polarised targetand various types of tracking and particle identification detectors (PID) in front and behind of two large-aperture magnets SM1 and SM2. These detectors provide data for reconstruction of corresponding tracks.The spectrometer was operated in the high energy (160 GeV) muon beam at CERN. Its initial configu- The COMPASS Collaborationration (see Ref. [8]) was used for data taking in 2002 – 2004. During the long accelerator shutdown in2005, the set-up was modified (see Ref. [9]). The major modifications influencing the present analysiswere as follows: (i) the replacement of the two 60 cm long target cells (denoted as U and D ) by three cells U , M and D of lengths 30 cm, 60 cm and 30 cm, (ii) the replacement of the target solenoid magnet bythe new one with a wider aperture and (iii) the installation of the electromagnetic calorimeter ECAL1 infront of the hadron calorimeter HCAL1. The ECAL1 is not used in the analysis because it was not fullyoperational yet in 2006 and partially acted as a hadron absorber. These modifications of the apparatuswhere aimed at further reduction of systematic uncertainties, enlargement of the spectrometer acceptanceand improvement of e / γ PID capabilities. These modifications have required reconsidering the Ref. [2]methods of data stability tests and asymmetry calculations (see Sections 3.3 and 3.6.)The data in 2006 were taken in two groups of periods. Each group is characterised by its initial set ofpolarisations in the target cells which are obtained by using different frequencies of the microwave fieldto polarise the target material (deuterons) in different cells at the certain direction of the target magnetsolenoid field. The solenoid fields holds the polarisation. The field direction is denoted as f = + , if itcoincides with the beam direction, or f = − , if opposite. The first group of the periods is denoted byG1 and other one by G2. Each period includes a certain number of intervals of continuous data taking(referred to as runs). The G1 data taking periods started with the initial setting of positive deuteronpolarisation in target cells U and D and the negative one in cell M , both corresponding to f = + . Aftertaking some number of runs, the field was reversed to f = − causing the reversal of the target cellpolarisations, so that the data were taken with opposite deuteron polarisations in the cells. The periodicreversal of polarisations continued up to the end of G1 periods. Within the periods, the cell polarisations,needed for asymmetry calculations (see Section 3.5), were measured for each run in order to make surethat they are stable at the level of about 55%. If polarisations dropped below this limit, they wererestored by the microwave field before the beginning of the next period. For G2 periods, the procedurewas analogous but the initial setting of polarisation in the cells was opposite to the one in G1 at thesame field f = + . The periodic reversal of the cell polarisations within each group of periods was usedto estimate a possible time-dependent systematics of the data. The change of the initial setting of thecell polarisations was used to estimate a possible systematic change of the spectrometer acceptance dueto superposition of the solenoid field and the field of SM1. If there is no such systematic change, theacceptance in G1 and G2 periods must be the same for stable performance of the spectrometer. Let us call as ”SIDIS event” an event determined by Eq. (1) and reconstructed with tracks using the datarecorded by the tracking and PID detectors.The overall statistics of 2006 is about 44 . × of preselected candidates for inclusive DIS and SIDISevent with Q > c ) . The sample was obtained after rejection of runs that did not pass the datastability tests (see Section 3.3) and events that did not pass the reconstruction tests. The latter ones wererejected if the Z -coordinate (along the beam) of the interaction point (vertex) was determined with anuncertainty larger than 3 σ of average which varied within 1.5-2 cm for different target cells.The selection of SIDIS events from the preselected sample was done as described in Ref. [2]. Foreach SIDIS event, a reconstructed vertex with incident ( µ ) and scattered ( µ (cid:48) ) muons and one or moreadditional tracks were required. Trajectories of the incident muons were required to traverse all targetcells in order to have the same beam intensity for each of them. The track crossing more than 30 radiativelengths along the reconstructed trajectory was associated with µ (cid:48) . The cuts were applied on the quality ofthe reconstructed tracks forming vertices, the effective lengths of the target cells (28 cm, 56 cm, 28 cm),the momentum of incident muons (140 GeV/ c −
180 GeV/ c ), the fractional energy carried by all tracksfrom the event ( z < ) and the fractional virtual-photon energy ( . < y < . ) . About 36 . × SIDISevent candidates remained after cuts.zimuthal asymmetries . . . 5The distribution of track multiplicities per SIDIS candidates peaks at four. These tracks include scattered µ (cid:48) and hadron candidates. For a track to be identified as hadron, it was required that: its transversemomentum was larger than 0.05 GeV/ c , it was produced in the current fragmentation region, as definedby the c.m. Feynman variable x F ≈ z − ( E hT ) / ( zW ) >
0, and it was associated with a cluster in oneof the hadron calorimeters HCAL1 or HCAL2 with an energy deposit greater than 5 GeV in HCAL1 or7 GeV in HCAL2. The efficiencies of the calorimetries above these energies are close to 100%. Theenergy of hadrons extended up to 120 GeV in the former and up to 140 GeV in the latter. All hadronsof the SIDIS candidates were included in the analysis of asymmetries. For the final selection of theSIDIS events and hadrons, the SIDIS candidates have to pass stability tests, as described in Section 3.3and in Section 3.4. The total number of hadrons in 2006 after afore-mentioned selections is 15 . × including 8 . × h + and 7 . × h − .To summarise, the SIDIS events and hadrons have been selected from preselected candidates requiring:140 GeV/c < p µ <
180 GeV/c, Q > , 0.1 < y < < z < p hT > x F > E HCAL > E HCAL > p hT and z kinematic cutson the “integrated” azimuthal asymmetries, they were calculated summing up all selected hadrons (seeSection 3.6) at variance with Ref. [2]. Azimuthal asymmetries as functions of the kinematic variables x , z or p hT were calculated in a restricted region following Ref. [2], i.e. summing up hadrons within theintervals given in Table 1 below. The number of hadrons within these intervals is reduced by a factor ofabout two compared to the total number. Table 1:
Intervals of x , z , p hT and their weighted mean values for which asymmetries as functions of kinematic variables werecalculated. The Q -intervals corresponding to the x -intervals are shown for reference. x intervals mean Q (GeV/ c ) intervals mean z intervals mean p hT (GeV/ c )intervals mean0.004 – 0.012, 0.0100.012 – 0.022, 0.0200.022 – 0.035, 0.0310.035 – 0.076, 0.0530.076 – 0.132, 0.0980.132 – 0.700, 0.190 1.0 – 3.0, 1.451.0 – 6.0, 2.071.0 – 9.5, 2.891.0 – 20.0, 4.822.0 – 35.0, 9.203.0 – 100.0, 21.26 0.200 – 0.234, 0.2160.234 – 0.275, 0.2530.275 – 0.327, 0.2990.327 – 0.400, 0.3610.400 – 0.523, 0.4550.523 – 0.900, 0.661 0.100 – 0.239, 0.1770.239 – 0.337, 0.2890.337 – 0.433, 0.3850.433 – 0.542, 0.4850.542 – 0.689, 0.6100.689 – 1.000, 0.814The distributions of selected SIDIS events as a function of Q and y and of charged hadrons as a functionof z and p hT for different data samples are presented in Fig. 3. ) GeV/c ( Q E n t r i e s y z ) c / GeV ( hT p Fig. 3:
Kinematic distributions of selected SIDIS events vs. Q and y and of charged hadrons vs. z and p hT withinthe region shown in Table 1: 2006 (lower, red), 2002 – 2004 (middle, green) and 2002 – 2006 (upper, blue). Taking advantage of the three-cell polarised target, stability tests for the 2006 data were performed byinvestigating variations of events from run to run for certain observables via ratios R i , where i is the run The COMPASS Collaborationnumber, using the combined information from cells U and D denoted by ( U + D ) , and that of cell M .One expects the ratio R i = ( U i + D i ) / M i to be independent of luminosity, close to unity and stable fromrun to run. In order to confirm this expectation, the ratios R i per run were obtained for the followingobservables that are relevant to the selection of SIDIS events and hadrons: number of SIDIS events,number of tracks per SIDIS event, number of clusters in HCAL1 (HCAL2) with E > ( ) GeV, averageenergy of clusters in HCAL1 (HCAL2), average energy of the associated clusters per event in HCAL1(HCAL2) and average angle (cid:104) φ (cid:105) . The R i values as a function of the run number were fitted by constants R for all runs. It was found that most of these R i were stable within the ± σ limits around the averagevalues R ≈ .
05, except for some runs and one of the periods.The stability of the measurement of the hadron azimuthal angle φ in the range ± ◦ is essential fordetermination of asymmetries. Distributions of φ -values in this range were obtained for each run of datataking and average values (cid:104) φ (cid:105) i per run determined. The distribution of (cid:104) φ (cid:105) i had a Gaussian shape aroundthe mean value equal to zero for almost all runs. The modified ”acceptance-cancelling” double ratio method was used to calculate the ratios of the SIDIScross sections for positive and negative target polarisations denoted as σ + / σ − and the 2006 asymmetries.In this Paper, the modified double ratio method is applied in three forms: first, in a form of ”acceptance-cancelling”, second, in a form of ”cross section-cancelling” and, third (see Section 5), again in the”acceptance-cancelling” form to test the hadron yield stability.In order to cancel acceptances, the method utilises double ratios, i.e. the product of two ratios of events.For the three-cell target the method was modified as follows. The target cell M was artificially dividedin two sub-cells M M
2, each 28 cm long, and two pairs of cells ( U and M
1) and ( M D )are considered below. The cells in each pair have equal lengths, i.e. equal densities of deuterons, butopposite polarisations p ( + or − ) at a given solenoid field direction f ( + or − ). For each pair of cellsat a given f , one can construct the double ratio using the number of selected SIDIS events or hadrons.These numbers obtained from the cell i and denoted as N ip f are usually expressed via a product of a cellluminosity ( L if ) given by the beam intensity times the target cell material density a target cell acceptance( A if ) and the corresponding cross section ( σ p ): N ip f = L if × A if × σ p , i.e. luminosity, acceptance and crosssection are folded in the number of events. Taking this relation into account as well as the COMPASSprocedure of measurements divided in two groups of runs G1 and G2, one can construct for each pairof the target cells the ”acceptance-cancelling” double ratio of events (hadrons) that provides a way tounfold the ( σ + / σ − ) . Particularly, for the polarisation settings at f = + , the two double ratios of eventnumbers constructed for the ( U , M
1) and for the ( M , D ) pairs have the forms given by Eqs. (5), whereevents for the first (second) ratio of each pair are taken from the G1 (G2) runs. (cid:20) N U ++ N M − + (cid:21) G1 × (cid:20) N M ++ N U − + (cid:21) G2 = (cid:18) σ + σ − (cid:19) , (cid:20) N D ++ N M − + (cid:21) G1 × (cid:20) N M ++ N D − + (cid:21) G2 = (cid:18) σ + σ − (cid:19) . (5)Substituting in the left parts of Eqs. (5) the above expressions for N ip f one can see that, after ”cancella-tions” of L if and A if , the double ratios of events are directly related to the cross section ratios squared.Because the luminosities of cells are equal, they contribute equally to the numerators and denominatorsand their cancellations are expected in each of the Eqs. (5) ratios. If the acceptances A if of the cells aresimilar at the same f in the G1 and G2 groups of runs (it is subject for tests below), they are also foldedequally in the corresponding number of events and “cancel” in the double ratios, i.e. it is not necessary tocalculate them (see Section 5). In the numerator of each ratio, the number of events (hadrons) are takenfrom the runs with the positive target cell polarisation, while in the denominator they are taken from theruns with negative target polarisation. Thus under above conditions each ratio of events (hadrons) in theleft parts the Eqs. (5) is equal to the ratio σ + / σ − , which is known to be close to unity (it is subject for testzimuthal asymmetries . . . 7below). Hence, each double ratio in Eqs. (5), which is equal to ( σ + / σ − ) , also have to be equal withinstatistical uncertainties and is expected to be close to unity. The stability of acceptances during the G1and G2 runs have been checked using the cross sections canceling double ratios of events (hadrons) in theforms similar to ones given in Eqs. (11), (12) of Ref. [2] which are related to the ratios of acceptances.Similarly, at f = − the two double ratios constructed for the same target pairs are: (cid:20) N U + − N M −− (cid:21) G1 × (cid:20) N M + − N U −− (cid:21) G2 = (cid:18) σ + σ − (cid:19) , (cid:20) N D + − N M −− (cid:21) G1 × (cid:20) N M + − N D −− (cid:21) G2 = (cid:18) σ + σ − (cid:19) . (6)Because no requirements except time stabilities are imposed on the data, the Eqs. (5, 6) are valid andcan be used for calculations of either the cross section ratios in the restricted kinematic regions or in thewhole available region (see Section 3.6).Thus each of four double ratios of the events (hadrons) in Eqs. (5, 6) calculated with SIDIS eventsare related to the squared ratio of the SIDIS cross sections for positive and negative target polarisationsdetermined with a part of the data. When statistically averaged, they can be used to calculate asymmetrieswith the whole data provided that (i) acceptances in the G1 and G2 periods are indeed stable and equal,(ii) the values of the double ratios calculated for SIDIS events and for hadrons with polarisation settings at f = + and f = − are stable and equal within statistical uncertainties. These requirements were checkedwith SIDIS event candidates and hadrons and final selections of them were determined.Altogether, the stability tests have shown that (i) acceptances are stable and equal during the G1 and G2groups of runs, (ii) the double ratios in Eqs. (5, 6) calculated with SIDIS events or hadrons are stableover the data taking periods and contained inside the ± σ corridors around the average values which areclose to unity. In order to be accepted for analysis the average value per a given run of the acceptances,the angles (cid:104) φ (cid:105) i and the double ratio values defined by Eqs. (5, 6) have to be within ± σ limits of thecorresponding mean value for all runs. Otherwise the run was rejected. The rejected runs containedabout 10% of 2006 hadrons. For the extraction of the azimuthal asymmetries a h ± ( φ ) off the cross section ratios, the distributions ofthe charged hadrons h + and h − were separately analysed as a function of the azimuthal angle φ in theregion from − ◦ to + ◦ divided into 10 φ -bins. For both h + and h − , the double ratios of the hadronsdefined by Eqs. (5, 6), ( σ + / σ − ) k ( φ ) , k = , ...
4, were calculated and combined as follows: (cid:18) σ + σ − (cid:19) h ± ( φ )= (cid:34)(cid:18) σ + σ − (cid:19) ⊕ (cid:18) σ + σ − (cid:19) ⊕ (cid:18) σ + σ − (cid:19) ⊕ (cid:18) σ + σ − (cid:19) (cid:35) h ± ( φ ) ∼ = + a h ± ( φ ) ∑ k (cid:34) ∑ i , f , p ∈ k P ip f ( x ) (cid:35) h ± · W k , (7)where the symbol ⊕ means statistically weighted averaging. As it was shown in Ref. [2], in first ap-proximation, the squared ratios of cross sections ( σ + / σ − ) h ± ( φ ) are related to the asymmetries a h ± ( φ ) multiplied by polarisation terms. For each hadron charge, the polarisation term is given by the sum ofthe P ip f ( x ) values, each of them being the product of target cell polarisations | P ip f | and dilution factorf i ( x ) , as defined in Refs. [2, 3], where i , p and f are those used to calculate the ratio ( σ + / σ − ) k ( φ ) , i.e.four polarisation values at each k . The weight W k is equal to the ratio of the number of hadrons, N k , tothe total number of hadrons, N tot . Therefore, the a h ± ( φ ) , referred to as single-hadron asymmetries, are: a h ± ( φ ) ∼ = ( σ + σ − ) h ± ( φ ) − ∑ k [ ∑ i , f , p ∈ k P ip f ( x )] h ± · W k . (8) The COMPASS Collaboration Following Section 3.2, the asymmetries a h ± ( φ ) were calculated in this Paper (i) as the “integrated”asymmetries using the total number of h + or h − , and (ii) as the asymmetries vs. one of kinematicvariables x , p hT or z disregarding the others and using the numbers of h + or h − within the intervalsdefined in Table 1. In each case, the asymmetries were fitted by the function from Eq. (4) using thestandard least-square-method and extracting all asymmetry modulation amplitudes simultaneously. Integrated asymmetries.
These asymmetries for the 2006 data as a function of the azimuthal angle φ are shown in Fig. 4 together with results of the fits given in Table 2. -150 -100 -50 0 50 100 150-0.02-0.015-0.01-0.00500.0050.010.0150.02 a f + h -150 -100 -50 0 50 100 150-0.02-0.015-0.01-0.00500.0050.010.0150.02 a f - h Fig. 4:
The 2006 “integrated” asymmetries a as functions of the azimuthal angle φ . Curves show the corresponding fits. Table 2:
The h + and h − modulation amplitudes of the integrated azimuthal asymmetries obtained from the statisticallycombined amplitudes of the 2002–2006 (left), those of 2002–2004 (middle) and those of 2006 (right).Modulation 2002–2006 amplitudes in 10 − − − amplitudes h + h − h + h − h + h − a . ± .
96 2 . ± .
98 3 . ± .
10 2 . ± .
10 0 . ± .
92 0 . ± . a sin φ − . ± . − . ± . − . ± . − . ± . − . ± . − . ± . a sin2 φ − . ± .
33 1 . ± . − . ± .
50 2 . ± .
60 3 . ± . − . ± . a sin3 φ . ± . − . ± .
42 0 . ± .
50 0 . ± .
60 0 . ± . − . ± . a cos φ . ± .
32 0 . ± .
42 2 . ± .
50 1 . ± . − . ± . − . ± . In order to compare the 2006 integrated asymmetries to those of the 2002, 2003 and 2004, the latterones were recalculated using the total number of hadrons. The results of the fit for the 2006 integratedasymmetries together with those of the 2002, 2003 and 2004 calculated similarly are shown in Fig. 5.The modulation amplitudes obtained for each year are in agreement with one another, as confirmed bythe compatibility tests (see Section 5). -0.0100.01 a h a + h - h -0.0100.01 f sin h a -0.0100.01 f sin2 h a -0.0100.01 f sin3 h a -0.0100.01 f cos h a Fig. 5:
The values of modulation amplitudes a together with their uncertainties obtained from the fits of the integrated asymme-tries a h ± ( φ ) by the function from Eq. (4) separately for the data of 2002, 2003, 2004 and 2006 as well as statistically combinedmodulation amplitudes for four years denoted by AV (see Section 4.1). In order to check the impact of the disregarded modulations, which could have appeared from SIDISpartial cross sections d σ and d σ L , on the modulation amplitudes in Eq. (4), we have performed fits ofthe 2006 integrated asymmetries by a new fitting function which contains a numerator and denominator.zimuthal asymmetries . . . 9In the numerator we have used the same modulations as in Eq. (4), but in the denominator we includedthe disregarded modulation with average amplitudes determined in Ref. [4]. For the asymmetry a h − ( φ ) ,the fitting function is as follows: a h − ( φ ) = a h − + a sin φ h − sin φ + a sin2 φ h − sin 2 φ + a sin3 φ h − sin 3 φ + a cos φ h − cos φ + . φ + . φ + . φ . (9)By comparing results of this fit with results of the standard fit of the 2006 data shown in Fig. 5, it wasfound that the differences between values of modulation amplitudes in the numerator are smaller than 1%of the fit uncertainties. Similar results are obtained for a h + ( φ ) replacing amplitudes in the denominatorof Eq. (9) by corresponding values from Ref. [4]. Thus the contributions of the disregarded modulationsto the integrated asymmetries in Eq. (4) are indeed negligible. Asymmetries as functions of kinematic variables.
The 2006 modulation amplitudes as functions ofkinematic variables were compared to those from the combined 2002 – 2004 data and found to be inagreement within uncertainties of the fits. They are used for calculations of the combined 2002– 2006modulation amplitudes (see Section 4.2).
For the integrated asymmetries, the values of the combined 2002 – 2006 modulation amplitudes, whichare shown in Fig. 5 and denoted by AV, were obtained using a statistical combination of four corre-sponding amplitudes. They are given in Table 2. The combined 2002 – 2004 modulation amplitudes,calculated as for 2006, are also shown in Table 2 in order to allow comparison to those of Ref. [2], whichwere calculated in the restricted kinematic region. As expected for the iso-scalar deuteron target, con-sistent results are obtained for the φ -independent terms a h + and a h − . All φ -modulation amplitudes areconsistent with zero within uncertainties of fits. Comparing results for the 2002 – 2006 combined datapresented in Table 2 to the results for 2002 – 2004 and to the results of Ref. [2], one can see that theyare in agreement between themselves within the quoted uncertainties. This indicates that the integratedasymmetries with and without kinematic cuts of Table 1 are consistent, i.e. these cuts reduce statisticsbut do not change the values of the asymmetries within experimental uncertainties. Due to increasedstatistics of each year, the statistical uncertainties of the combined 2002 – 2006 amplitudes are reducedby a factor of about 1/1.6 compared to those of Ref. [2]. The final 2002 – 2006 results on the modulation amplitudes of asymmetries a h ± ( φ ) calculated as thefunction of one of the variables x , z and p hT while disregarding the others were obtained from the statis-tically averaged 2002, 2003, 2004 and 2006 modulation amplitudes. The results are presented in Fig. 6.Except for the a h ± ( x ) , all amplitudes when fitted by constants are found to be consistent with zero withinstatistical uncertainties ( χ / NDF (cid:39) a h ± ( x ) for deuteron target have the same x -dependence for positive and negative hadrons. Additionally, the x -dependence of the a h ± ( x ) / D ( x , y ) values are presented in Fig. 7, where D ( x , y ) is the virtual-photon depolarisation factor for each x intervalmultiplied by the average beam polarisation | P µ | , as defined in Ref. [2]. If the a h ± ( x ) represent the maincontributions to the asymmetries of Eq. (3), the values of a h ± ( x ) / D ( x , y ) by definition (see e.g. Ref. [13])should be equal to the asymmetries A h ± d ( x ) . Within experimental uncertainties, there is a good agreementbetween our data on a h ± ( x ) / D ( x , y ) and the data of Ref. [14] on A h ± d ( x ) , which confirms the correctnessof the results on the asymmetries calculated by the modified acceptance-cancelling method. The valuesof A h ± d ( x ) were obtained with the 2002 – 2004 data. A similar x -dependence was also observed with 2002– 2006 data for the asymmetries A π ± d ( x ) and A K + d ( x ) obtained with the identified hadrons (see Ref. [9]).0 The COMPASS Collaboration h a − h + h −0.0200.02 φ s i n h a −0.0200.02 −0.0200.02 φ s i n2 h a −0.0200.02 −0.0200.02 φ s i n3 h a −0.0200.02 −2 −1 x φ c o s h a z −0.0200.02 ) c GeV/( hT p Fig. 6:
The modulation amplitudes a of the h + and h − azimuthal asymmetries as the function of x , z and p hT obtained fromthe combined 2002 – 2006 data on the muon SIDIS off longitudinally polarised deuterons. Only uncertainties of fits are shown. -2 -1 + h - h x h A D / h a - h + h D / h a Fig. 7:
The x -dependence of the values a h ± ( x ) / D ( x , y ) for 2002 – 2006 data compared to the data of Ref. [14]on the asymmetries A h ± d ( x ) . zimuthal asymmetries . . . 11 The global compatibility test of the results on the asymmetries a h ± ( φ ) , that were obtained separately for2002, 2003, 2004 and 2006 years, was performed by building the pull distribution: pull i = ( a i − (cid:104) a (cid:105) ) ×| σ a i − σ (cid:104) a (cid:105) | − / , where a i is the asymmetry for a given year, hadron charge and kinematic interval, (cid:104) a (cid:105) is the corresponding weighted mean value over four years and σ denotes the corresponding standarddeviation. The pull distribution had in total 750 entries compared to 540 for 2002 - 2004 years. Theasymmetries reported in this Paper, in principle, could have the non-cancelled-acceptance or luminositytime-dependent effects folded in the event numbers of one of the Eqs. (5, 6)-ratio and, consequently, inone or several values of a i distorting the pull distribution. In the absence of such effects, as expected, thepull distribution follows the Gaussian distribution with the mean value consistent with zero ( − . ± .
033 in our case) and σ with unity (0.978 ± a i aresmaller than statistical ones.Quantitative measures of possible systematic effects have been obtained by estimating additive and mul-tiplicative uncertainties. Main contributions to possible additive systematic uncertainties could comefrom the instabilities of the hadron yields. The φ -stability of hadron yields in the 2006 data was checkedfollowing the procedure described in Ref. [2]. For this purpose, the double ratios of hadron numbers asa function of the azimuthal angle φ for different polarisation settings at the field f during the G1 and G2runs were calculated as follows: f = + : F + ( φ ) = N U ++ + N D ++ N M − + · N M ++ N U − + + N D − + , f = − : F − ( φ ) = N U + − + N D + − N M −− · N M + − N U −− + N D −− . (10)Here, N ip f is the number of hadrons per φ -bin from target cell i with polarisation p and field f , asexplained in Section 3.4. The ratios given by Eqs. (10) are modifications of ratios used in Ref. [2]for the case of two target cells. In the above double ratios, we expect the acceptance and luminositycancellations and, as a result, the φ -stability of the hadron yields. If unstable, they could indicate possiblesystematics in the acceptance as well. The fits by constants (see Fig. 8) of the weighted sums F ( φ ) = F + ( φ ) ⊕ F − ( φ ) in the φ -region from − ◦ to + ◦ for h + , h − and h + + h − of the 2006 data gaveresults consistent with unity within statistical uncertainties of the order of 0.001. This means that no φ -instabilities and acceptance-changing (not cancelled) effects have been observed, i.e. there are nolarge additive systematic uncertainties in the 2006 data. The value ∆ a h ± ( φ ) = ± .
001 was chosen as aquantitative measure of possible additive systematic uncertainties in the 2006 asymmetry measurements.It is equal to ± σ of the h + + h − data stability test for F ( φ ) . The same value was obtained for the 2002 –2004 data [2] and hence adopted also for the combined 2002 – 2006 deuteron data. -150 -100 -50 0 50 100 1500.9511.05 f - h F -150 -100 -50 0 50 100 1500.9511.05 + h f F -150 -100 -50 0 50 100 1500.9511.05 f - h + + h F Fig. 8:
The φ -dependence of the weighted sums of double ratios F ( φ ) for 2006 data: h − , h + and h + + h − . The solid (red)lines represent the results of fits by constants. Possible sources of multiplicative systematic uncertainties of the asymmetry evaluation are uncertaintiesin the determination of the beam and target polarisations and estimations of the dilution factor. The mul-tiplicative systematic uncertainties of the extracted asymmetries due to uncertainties in the determination2 The COMPASS Collaborationof the beam and target polarisations were estimated to be less than 5% each and those due to uncertain-ties of the dilution factor to be less than about 2%. When combined in quadrature, overall possiblemultiplicative systematic uncertainty of less than 6% was obtained.
The searches for possible azimuthal modulations in the single-hadron azimuthal asymmetries a h ± ( φ ) ,as a manifestation of TMD PDFs describing the nucleons in the longitudinally polarised deuteron, havebeen performed using all COMPASS deuteron data and the acceptance-cancelling method of analysis.For each hadron charge, beside the φ -independent term, four possible modulations predicted by theory(sin φ , sin 2 φ , sin 3 φ and cos φ ) and their dependence on kinematical variables are considered. The asym-metries have been calculated both for all selected hadrons (”integrated” asymmetries) and for hadrons asfunctions of kinematic variables within the restricted region.For the ”integrated” asymmetries, it was found that results in the restricted range of kinematic variablesare consistent with those of the wider range. In other words, the restricting of the kinematic regionreduces the statistics but does not change the values of the asymmetries beyond the sensitivity of thisexperiment. The same result was obtained in Ref. [2] for the asymmetries as a function of z .The φ -independent terms a h ± ( x ) of the asymmetries a h ± ( φ ) , which are expected to originate mostlyfrom the known helicity PDFs g L ( x ) ≡ g ( x ) , are connected to the virtual photon asymmetry A h ± d ( x ) = a h ± ( x ) / D ( x , y ) . There is good agreement between the COMPASS data on a h ± ( x ) / D ( x , y ) and A h ± d ( x ) from Refs. [9, 14] which confirms this expectation.No statistically significant dependences of φ -modulation amplitudes were observed as functions of x , z or p hT when fitted by constants. Still, there are some hints (statistically not confirmed) for a possible x -dependence of the sin 2 φ , sin 3 φ and cos φ modulation amplitudes. The sin 2 φ amplitude for h − ismostly positive and rises with increasing x , while for h + it is mostly negative and decreases with x . Thisbehaviour agrees with that discussed in Refs. [7, 10, 15], if one takes into account different definitionsof asymmetries by the HERMES and COMPASS Collaborations. The increase with x of the modulusof the cos φ amplitudes, related to the Cahn-effect [16] and predicted in Ref. [17], was already visiblefrom the 2002 – 2004 data [2] and persists for the combined 2002 – 2006 data. Hints for a possible x -dependence of sin 3 φ modulation amplitudes are discussed in Ref. [7]. Quantitative estimates of apossible contribution of the cos φ modulation to the deuteron asymmetries, related to TMD PDFs g ⊥ L and e L , have been obtained in Ref. [18]. They are in agreement with our data.Altogether, one can conclude that contributions of TMD PDFs convoluted with FFs to the azimuthalasymmetries in the cross sections of hadron production in muon SIDIS off longitudinally polariseddeuterons are small. This is either due to possible cancellations of the contributions to the asymme-tries by the deuteron’s up and down quarks, or/and due to the smallness of the transverse component ofthe target polarisation and of the suppression factor that behaves as M / Q . Some of these conclusions canbe checked studying these asymmetries in muon SIDIS off longitudinally polarised protons. Acknowledgements
We gratefully acknowledge the support of our funding agencies and of the CERN management and staffand the skills and efforts of the technicians of our collaborating institutes. Special thanks go to V. Anosovand V. Pesaro for their technical support during the installation and running of this experiment.zimuthal asymmetries . . . 13
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