EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

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1 EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH COMPASS CERN-EP CERN-EP-208 xxx 07 September 208 2, 208 Measurement of P T -weigted ers asymmetries in leptoproduction of adrons Te COMPASS Collaboration Abstract Te transverse spin asymmetries measured in semi-inclusive leptoproduction of adrons, wen weigted wit te adron transverse momentum P T, allow for te extraction of important transverse-momentum-dependent distribution functions. In particular, te weigted ers asymmetries provide direct information on te ers function, wic is a leadingtwist distribution tat arises from a correlation between te transverse momentum of an unpolarised quark in a transversely polarised nucleon and te spin of te nucleon. Using te ig-statistics data collected by te COMPASS Collaboration in 20 wit a transversely polarised proton target, we ave evaluated two types of P T -weigted ers asymmetries, wic are bot proportional to te product of te first transverse moment of te ers function and of te fragmentation function. Te results are compared to te standard unweigted ers asymmetries and used to extract te first transverse moments of te ers distributions for u and d quarks. (to be submitted to Nuclear Pysics B)

2 Te COMPASS Collaboration M.G. Alexeev 26, G.D. Alexeev 8, A. Amoroso 26,27, V. Andrieux 29,2, N.V. Anfimov 8, V. Anosov 8, A. Antoskin 8, K. Augsten 8,9, W. Augustyniak 30, C.D.R. Azevedo 2, B. Badełek 3, F. Balestra 26,27, M. Ball 4, J. Bart 5, V. Barone,27, R. Beck 4, Y. Bedfer 2, J. Bernard 3,, K. Bicker 6,, E. R. Bielert, M. Bodlak 8, P. Bordalo 2,a, F. Bradamante 24,25, A. Bressan 24,25, M. Bücele 9, V.E. Burtsev 28, W.-C. Cang 22, C. Catterjee 7, M. Ciosso 26,27, A.G. Cumakov 28, S.-U. Cung 6,b, A. Cicuttin 25,c, M.L. Crespo 25,c, S. Dalla Torre 25, S.S. Dasgupta 7, S. Dasgupta 24,25, O.Yu. Denisov 27,#, L. Dara 7, S.V. Donskov 20, N. Dosita 33, C. Dreisbac 6, W. Dünnweber d, R.R. Dusaev 28, M. Dziewiecki 32, A. Efremov 8,w, C. Elia 24,25, P.D. Everseim 4, M. Faessler d, A. Ferrero 2, M. Finger 8, M. Finger jr. 8, H. Fiscer 9, C. Franco 2, N. du Fresne von Hoenesce 3,, J.M. Friedric 6,#, V. Frolov 8,, F. Gauteron 3,29, O.P. Gavrictcouk 8, S. Gerassimov 5,6, J. Giarra 3, I. Gnesi 26,27, M. Gorzellik 9,r, A. Grasso 26,27, A. Gridin 8, M. Grosse Perdekamp 29, B. Grube 6, A. Guskov 8, D. Hane 5, G. Hamar 25, D. von Harrac 3, R. Heitz 29, F. Herrmann 9, N. Horikawa 7,, N. d Hose 2, C.-Y. Hsie 22,i, S. Huber 6, S. Isimoto 33,j, A. Ivanov 26,27, T. Iwata 33, M. Jandek 9, V. Jary 9, R. Joosten 4, P. Jörg 9,g, K. Juraskova 9 ; E. Kabuß 3, F. Kaspar 6, A. Kerbizi 24,25, B. Ketzer 4,G.V. Kaustov 20, Yu.A. Koklov 20,k, Yu. Kisselev 8, F. Klein 5, J.H. Koivuniemi 3,29, V.N. Kolosov 20, K. Kondo 33, I. Konorov 5,6, V.F. Konstantinov 20, A.M. Kotzinian 27,m, O.M. Kouznetsov 8, Z. Kral 9, M. Krämer 6, F. Krinner 6, Z.V. Kroumctein 8,*, Y. Kulinic 29, F. Kunne 2, K. Kurek 30, R.P. Kurjata 32, A. Kveton 9, A.A. Lednev 20,*, S. Levorato 25, Y.-S. Lian 22,n, J. Lictenstadt 23, R. Longo 26,27, V.E. Lyubovitskij 28,o, A. Maggiora 27, A. Magnon 29, N. Makins 29, N. Makke 25,c, G.K. Mallot, S.A. Mamon 28, B. Marianski 30, A. Martin 24,25, J. Marzec 32, J. Matoušek 24,25,8, T. Matsuda 4, G.V. Mesceryakov 8, M. Meyer 29,2, W. Meyer 3, Yu.V. Mikailov 20, M. Mikasenko 4, E. Mitrofanov 8, N. Mitrofanov 8, Y. Miyaci 33, A. Moretti 24, A. Nagaytsev 8, D. Neyret 2, J. Nový 9,, W.-D. Nowak 3, G. Nukazuka 33, A.S. Nunes 2, A.G. Olsevsky 8, I. Orlov 8, M. Ostrick 3, D. Panzieri 27,p, B. Parsamyan 26,27, S. Paul 6, J.-C. Peng 29, F. Pereira 2, M. Pešek 8, M. Pešková 8, D.V. Pesekonov 8, N. Pierre 3,2, S. Platckov 2, J. Pocodzalla 3, V.A. Polyakov 20, J. Pretz 5,l, M. Quaresma 2, C. Quintans 2, S. Ramos 2,a, C. Regali 9, G. Reicerz 3, C. Riedl 29, D.I. Ryabcikov 20,6, A. Rybnikov 8, A. Rycter 32, R. Salac 9, V.D. Samoylenko 20, A. Sandacz 30, S. Sarkar 7, I.A. Savin 8,w, T. Sawada 22, G. Sbrizzai 24,25,#, P. Sciavon 24,25, H. Scmieden 5, E. Seder 2, A. Selyunin 8, L. Silva 2, L. Sina 7, S. Sirtl 9, M. Slunecka 8, J. Smolik 8, F. Sozzi 25, A. Srnka 6, D. Steffen,6, M. Stolarski 2, O. Subrt,9, M. Sulc, H. Suzuki 33,, A. Szabelski 24,25,30 T. Szameitat 9,r, P. Sznajder 30, M. Tasevsky 8, S. Tessaro 25, F. Tessarotto 25, A. Tiel 4, J. Tomsa 8, F. Tosello 27, V. Tskay 5, S. Ul 6, B.I. Vasilisin 28, A. Vaut, B.M. Veit 3, J. Veloso 2, A. Vidon 2, M. Virius 9, M. Wagner 4, S. Wallner 6, M. Wilfert 3, K. Zaremba 32, P. Zavada 8, M. Zavertyaev 5, Y. Zao 25, E. Zemlyanickina 8,w, M. Ziembicki 32 Universita degli Studi del Piemonte Orientale A. Avogadro, Di.S.I.T., 52 Alessandria, Italy 2 University of Aveiro, Dept. of Pysics, Aveiro, Portugal 3 Universität Bocum, Institut für Experimentalpysik, Bocum, Germany s,t 4 Universität Bonn, Helmoltz-Institut für Stralen- und Kernpysik, 535 Bonn, Germany s

3 5 Universität Bonn, Pysikalisces Institut, 535 Bonn, Germany s 6 Institute of Scientific Instruments, AS CR, 6264 Brno, Czec Republic u 7 Matrivani Institute of Experimental Researc & Education, Calcutta , India v 8 Joint Institute for Nuclear Researc, 4980 Dubna, Moscow region, Russia w 9 Universität Freiburg, Pysikalisces Institut, 794 Freiburg, Germany s,t CERN, 2 Geneva 23, Switzerland Tecnical University in Liberec, 467 Liberec, Czec Republic u 2 LIP, Lisbon, Portugal x 3 Universität Mainz, Institut für Kernpysik, Mainz, Germany s 4 University of Miyazaki, Miyazaki , Japan y 5 Lebedev Pysical Institute, 999 Moscow, Russia 6 Tecnisce Universität Müncen, Pysik Dept., Garcing, Germany s,d 7 Nagoya University, 464 Nagoya, Japan y 8 Carles University in Prague, Faculty of Matematics and Pysics, 8000 Prague, Czec Republic u 9 Czec Tecnical University in Prague, 6636 Prague, Czec Republic u 20 State Scientific Center Institute for Hig Energy Pysics of National Researc Center Kurcatov Institute, 4228 Protvino, Russia 2 IRFU, CEA, Université Paris-Saclay, 99 Gif-sur-Yvette, France t 22 Academia Sinica, Institute of Pysics, Taipei 529, Taiwan z 23 Tel Aviv University, Scool of Pysics and Astronomy, Tel Aviv, Israel aa 24 University of Trieste, Dept. of Pysics, 3427 Trieste, Italy 25 Trieste Section of INFN, 3427 Trieste, Italy 26 University of Turin, Dept. of Pysics, 25 Turin, Italy 27 Torino Section of INFN, 25 Turin, Italy 28 Tomsk Polytecnic University, Tomsk, Russia ab 29 University of Illinois at Urbana-Campaign, Dept. of Pysics, Urbana, IL , USA ac 30 National Centre for Nuclear Researc, Warsaw, Poland ad 3 University of Warsaw, Faculty of Pysics, Warsaw, Poland ad 32 Warsaw University of Tecnology, Institute of Radioelectronics, Warsaw, Poland ad 33 Yamagata University, Yamagata , Japan y # Corresponding autors * Deceased a Also at Instituto Superior Técnico, Universidade de Lisboa, Lisbon, Portugal b Also at Dept. of Pysics, Pusan National University, Busan , Republic of Korea and at Pysics Dept., Brookaven National Laboratory, Upton, NY 973, USA c Also at Abdus Salam ICTP, 345 Trieste, Italy d Supported by te DFG cluster of excellence Origin and Structure of te Universe ( (Germany) e Supported by te Laboratoire d excellence P2IO (France) f Present address: University of Connecticut, Storrs, Connecticut 06269, US g Present address: Universität Bonn, Pysikalisces Institut, 535 Bonn, Germany

4 Also at Cubu University, Kasugai, Aici , Japan y i Also at Dept. of Pysics, National Central University, 300 Jongda Road, Jongli 3200, Taiwan j Also at KEK, - Oo, Tsukuba, Ibaraki , Japan k Also at Moscow Institute of Pysics and Tecnology, Moscow Region, 4700, Russia l Present address: RWTH Aacen University, III. Pysikalisces Institut, Aacen, Germany m Also at Yerevan Pysics Institute, Alikanian Br. Street, Yerevan, Armenia, 0036 n Also at Dept. of Pysics, National Kaosiung Normal University, Kaosiung County 824, Taiwan o Also at Institut für Teoretisce Pysik, Universität Tübingen, Tübingen, Germany p Also at University of Eastern Piedmont, 50 Alessandria, Italy q Present address: Uppsala University, Box 56, 7520 Uppsala, Sweden r Supported by te DFG Researc Training Group Programmes 2 and 2044 (Germany) s Supported by BMBF - Bundesministerium für Bildung und Forscung (Germany) t Supported by FP7, HadronPysics3, Grant (European Union) u Supported by MEYS, Grant LG303 (Czec Republic) v Supported by B.Sen fund (India) w Supported by CERN-RFBR Grant x Supported by FCT - Fundação para a Ciência e Tecnologia, COMPETE and QREN, Grants CERN/FP 6376/20, 23600/20 and CERN/FIS-NUC/007/205 (Portugal) y Supported by MEXT and JSPS, Grants , , and , te Daiko and Yamada Foundations (Japan) z Supported by te Ministry of Science and Tecnology (Taiwan) aa Supported by te Israel Academy of Sciences and Humanities (Israel) ab Supported by te Russian Federation program Nauka (Contract No GZB.207) (Russia) ac Supported by te National Science Foundation, Grant no. PHY (USA) ad Supported by NCN, Grant 207/26/M/ST2/00498 (Poland)

5 4 Te COMPASS Collaboration Introduction Te traditional description of te nucleon structure in ard inclusive processes in terms of collinear parton distributions functions, wic depend on te parton ligt-cone momentum fraction x and on a caracteristic ard scale Q 2, was recently generalised to take into account te transverse momentum k T of te parton wit respect to te nucleon direction (for reviews, see [ 3]). A complete picture of te nucleon at leading twist requires a total of eigt transverse-momentum-dependent distributions (TMDs). Tey provide important information on te dynamics of te partons in te transverse plane in momentum space. Upon integration over te transverse momentum, tree of tem reduce to te number density, te elicity and te transversity collinear distributions. Te oter five TMDs contain prefactors tat are sensitive to te direction of te quark transverse momentum vector k T, and teir contribution to te adronic tensor vanises wen integrating over k T. Among te TMDs, an important rôle is played by te ers distribution function f q [4 7], wic for an unpolarised quark of flavour q describes te correlation between its transverse momentum and te transverse polarisation of te nucleon. In semi-inclusive measurements of deep-inelastic scattering (SIDIS) off a transversely polarised nucleon, te ers TMD embodies in te cross section a sine modulation on te difference between te azimutal angle φ of te produced adron and tat of te target nucleon spin, φ S. Te ers effect was experimentally observed in SIDIS using transversely polarised proton targets, first by te HERMES Collaboration [8,9] and ten, at iger energy, by te COMPASS Collaboration [, ]. Te COMPASS measurements on te deuteron [2, 3] sowed asymmetries compatible wit zero witin te experimental accuracy. More recently, data on pion production off a transversely polarised 3 He target were made available by te Hall A Collaboration at JLab [4]. Combined analyses of tese measurements [5 25] allowed for extractions of te ers functions and of teir first transverse moments f ()q : f ()q (x) = d 2 k T k 2 T 2M 2 f q (x,k2 T ), () wic are found to be different from zero, a very important result in TMD pysics. In particular, te u and te d distributions turn out to ave similar magnitude, but opposite sign. In Eq. (), M is te target nucleon mass. Wile in most penomenological studies te first transverse moments of te ers distributions are extracted by fitting te data using a given functional form for te x dependence of f, in Ref. [26] a different approac was adopted: te COMPASS measurements on proton and deuteron targets in te same kinematics were used to extract point-by-point te first transverse moments of te ers distributions f () directly from te data by combining te various asymmetries. Te main problem in all extractions performed up to now is tat te standard ers asymmetries involve transverse-momentum convolutions of TMDs and fragmentation functions, from wic te first transverse moments of te ers functions can be obtained analytically only by assuming a specific form, typically a Gaussian, for te transverse-momentum dependence of all involved quantities.

6 Measurement of P T -weigted ers asymmetries in leptoproduction of adrons 5 Already twenty years ago an alternative metod was proposed [27 29] to determine f () witout making any assumption on te functional form of te transverse-momentum dependence, neiter for te distribution functions nor for te fragmentation functions. Te metod, wic consists in measuring asymmetries weigted by te measurable transverse momentum P T of te adron, was not pursued; te only and still preliminary results came from HERMES [30]. It is wort to mention tat te first transverse moment of te ers function is directly entering in te Burkardt sum rule [3], wic allows to constrain te gluon ers function using te measured ers functions for quarks [22]. Recently, muc interest as been dedicated again to te weigted asymmetries (see e.g. [32, 33]). In tis paper, we present te first measurements of two types of P T -weigted ers asymmetries performed by te COMPASS collaboration using te ig statistics data collected in 20 wit a 60 GeV muon beam impinging on a transversely polarised proton target. Te results are compared to te standard unweigted ers asymmetries and used to extract te first transverse moments of te ers functions for u and d quarks. 2 Te ers asymmetries Te ers asymmetry is associated to a sinφ sin(φ φ S ) modulation of te SIDIS cross section in a reference frame were te momentum vectors of virtual poton and nucleon are collinear, te z axis is taken along te virtual-poton momentum and te x axis along te lepton transverse momentum. Te relevant part of te fully differential cross section is dσ = dσ U + S T dσ S sinφ, (2) were S T is te target nucleon polarisation, and dσ U and dσ S are te spin-independent and spindependent parts of te cross section, respectively. In te standard, i.e. unweigted case, te ers asymmetry is defined as dφ dφ sinφ dσ = 2. (3) dφ dφ dσ At leading twist and leading order in QCD, is given [29,34] in terms of te ers function f and te transverse-momentum-dependent unpolarised distribution and fragmentation functions f and D by [ ] (x,z,p T ) = q e 2 PT k T qxc MP T f q (x,k2 T )Dq (z, p2 T ) q e 2 qxc [ f q (x,k2 T )Dq (z,, (4) p2 T )] were te sums are over quark and antiquark flavours, e q are te quark carges, and te transverse momentum convolutions are given by and [ PT k T C MP T d 2 k T C [ f q ] Dq f q Dq ] d 2 p T δ 2 (zk T + p T P T ) P T k T MP T f q (x,k2 T )D q (z, p2 T ), (5) d 2 k T d 2 p T δ 2 (zk T + p T P T ) f q (x,k2 T )D q (z, p2 T ). (6)

7 6 Te COMPASS Collaboration In Eqs. (4,5,6), z is te fraction of te longitudinal momentum of te fragmenting quark carried by te produced adron, p T is te transverse momentum of te produced adron wit respect to te direction of te fragmenting quark momentum. For simplicity, we ave omitted te Q 2 dependence of parton distributions, fragmentation functions and ers asymmetry. Wen integrating over P T, te denominator of Eq. (4) is easily computed yielding te familiar collinear expression e 2 qx d 2 P T C [ f q ] Dq = e 2 qx f q (x)dq (z), (7) q q were f q (x) and Dq (z) are te above defined partonic functions integrated over te transverse momentum, wile, in te general case, te numerator of Eq. (4) cannot be analytically evaluated. Hence, in order to disentangle f and D and to extract te ers function, some functional form must be assumed for te transverse-momentum dependence of te distribution and fragmentation functions. Assuming tis form to be a Gaussian, te ers asymmetry becomes [5, 6, 29],G (x,z) = a G q e 2 qx f ()q (x)zd q (z) q e 2 qx f q (x)dq (z). (8) Te factor a G in Eq. (8) is πm a G =, (9) p 2 T + z2 kt 2 S were p 2 T and k2 T S are te Gaussian widts of te fragmentation function and of te ers function, respectively. In te Gaussian model, te average transverse momentum of te produced adrons (integrated over its azimutal angle) is written as π P T = p 2 T 2 + z2 kt 2, () were k 2 T is te widt of te transverse-momentum-dependent number density f, wic in principle differs from k 2 T S. Taking approximately k 2 T S k 2 T, we can write a G as a G πm 2 P T. () Te Gaussian ansatz clearly introduces a bias into te extraction of te ers function. In order to avoid tis problem one can consider, instead of Eq. (3), an asymmetry tat is weigted by te transverse momentum of te produced adron. In particular, wen coosing w = P T /zm as weigt, te weigted ers asymmetry becomes dφ A w = sinφ d 2 ( P PT ) T zm dσ dφ d 2. (2) P T dσ In terms of quark distribution and fragmentation functions, it reads A w (x,z) = q e 2 qx [ ] d 2 P P T T zm C PT k T MP T f q (x,k2 T )Dq (z, p2 T ) q e 2 qx f q (x)dq (z), (3)

8 Measurement of P T -weigted ers asymmetries in leptoproduction of adrons 7 Te convolution in te numerator can now be carried out in a straigtforward way (see Appendix A) and te final expression is A w (x,z) = 2 q e 2 qx f ()q (x)d q (z) q e 2 qx f q (x)dq (z), (4) wic sows tat te asymmetry contains te product of te first kt 2 moment of te ers function and te unpolarised fragmentation function. Wen using w = P T /M as weigt, te resulting ers asymmetry reads A w = dφ sinφ d 2 ( P PTM ) T dσ dφ d 2. (5) P T dσ Tis asymmetry is of interest because it sould exibit a z dependence close to tat of te unweigted asymmetries. Its expression in te parton model, (x,z) = 2 q e 2 qx f ()q (x)zd q (z) q e 2 qx f q (x)dq (z), (6) A w is indeed very similar to tat of te unweigted asymmetry in te Gaussian model, Eq. (8). In particular, from Eqs. (8,,6) one sees tat te ratio A w /,G is related to te average value of te adron transverse momentum: A w 4 P T,G πm. (7) 3 Experimental set-up and data analysis Te COMPASS spectrometer [35,36] is in operation in te SPS Nort Area of CERN since Te data used in tis analysis were collected in 20 by scattering a 60 GeV µ + beam on a transversely polarised target. Te.2 m long NH 3 target was kept at 50 mk in a dilution refrigerator cryostat and segmented in tree cells, 30 cm, 60 cm and 30 cm long respectively. Te proton polarisation of about 80% was oriented vertically by a 0.63 T magnetic field tat was provided by te saddle coils of te polarised target magnet [37]. Te data were taken at a mean beam intensity of µ/spill, for a spill lengt of about s every 40 s. About 37 9 events, corresponding to.9 PB of data, were collected in twelve separate periods. In order to minimize systematic errors, during eac period of data taking te orientation of te proton polarisation in te tree target cells was eiter up-down-up or down-up-down in te first subperiod, and reversed in te second one. By suitably combining te data, instrumental asymmetries could be limited to negligible values. Te principles of te measurement and te data analysis were already described in several publications [ 2] and will not be repeated ere. In order to allow for a comparison of te weigted ers asymmetries wit te unweigted asymmetries, all constraints to select DIS events and final-state adrons are te same as for te publised data []. Here we only recall tat in order to ensure te DIS regime only events wit poton virtuality Q 2 > (GeV/c) 2, fractional energy of te virtual poton 0. < y < 0.9, and mass of te

9 8 Te COMPASS Collaboration adronic final-state system W > 5 GeV/c 2 are considered. A carged adron is required to ave a transverse momentum P T 0. GeV/c and a fraction of te available energy z > 0.2. Wit tese constraints, about 8 7 adrons are left and used for te extraction of te asymmetries. Tis sample consists mainly of pions (about 70% for positive adrons, 75% for negative adrons [38]). In addition, te analysis was also done for carged adrons in te region 0. < z < 0.2. Te weigted asymmetries are measured separately for positive and negative adrons as a function of x or z. For eac bin in x or z and for eac period of data taking, te asymmetries are extracted from te number of adrons produced in eac cell for te two directions of te target polarisation, and te mean of te results from te twelve periods is taken as final result. Te unweigted asymmetries were extracted using bot an extended unbinned maximum likeliood metod and te so-called double ratio metod (DRM). Te two metods led to very similar results and te small differences were added to te systematic uncertainties. In bot cases, te adrons produced in te two data-taking subperiods and in te tree target cells are combined in order to ensure cancellation of te azimutal acceptance and of te beam flux. Since only te counts in te numerator of te expression of A w are weigted, a modified DRM is used in tis analysis. In eac kinematic bin, we divide te Φ range in 2 bins, and in eac of tem we calculate te quantity R(Φ ) = w Σ w Σ, (8) were w = N w +N w + N w N w, Σ w = N w +N w + + N w N w, Σ = N + N + + N N. (9) Here N and N w are te number of counts and te sum of weigts, respectively, and N (N ) refers to te first (second) subperiod. Te numbers of adrons produced in te first and in te tird target cell, wic are always polarised in te same direction, are added up. Te subscripts + and indicate te up and down orientation of te target polarisation. Bot azimutal acceptance and beam flux cancel in te ratio of Eq. (8), so tat R(Φ ) 4 S T A w sinφ, (20) were S T is te mean transverse polarisation of te target protons. Cancellation of azimutal acceptance is guaranteed as long as te ratios of te acceptances of te oppositely polarised cells in te two data taking subperiod are te same, wic is te so-called reasonable assumption [3]. Several tests were performed to assess te correctness of te results and te size of possible systematic uncertainties. Two alternative estimators were used, wic are not expected to guarantee an as good cancellation of te azimutal acceptance as te modified DRM but are muc simpler, one of tem being te mean value of sinφ S P T /zm. It turned out tat te results are essentially identical. Te effect of te P T /z acceptance was also investigated. Tis acceptance is about 60% and rater flat in te range < x < 0.7 bot for positive and negative adrons. At smaller x it increases

10 Measurement of P T -weigted ers asymmetries in leptoproduction of adrons 9 smootly from 0.4 to about 0.8 as P T /z increases from 0. GeV/c to GeV/c. In order to evaluate te effect of te acceptance in te results, we ave re-evaluated A w after aving corrected for te P T /z acceptance. Te difference between te results obtained wit and witout te corrections is at most one tent of a standard deviation, and tus negligible. Te stability of te results was cecked paying particular attention to te P T limits. Te effect of te lower P T cut, wic is expected to be negligible, was investigated by extracting te weigted ers asymmetries using tree different lower cuts, P T > 0.5 GeV/c, P T > 0.20 GeV/c and P T > 0.25 GeV/c. Also, te effect of a cut on te upper value of P T was investigated by extracting te asymmetries using te limits: P T <.5 GeV/c, P T <.25 GeV/c and P T <.0 GeV/c. In all te cases te differences to te results obtained wit te standard cuts are negligibly small in all x bins. Te contributions from iger-order processes, i.e. QCD Compton and poton-gluon fusion, wic are more relevant at ig P T [39], ave neiter been taken into account nor corrected for. Altogeter, no evidence was found for additional relevant systematic uncertainties. Te systematic uncertainties are estimated to be alf of te statistical uncertainties, as in te analysis of te standard ers asymmetries of te same data []. 4 ers asymmetries weigted by P T /zm Te distributions of te weigts w = P T /zm are very similar for all nine x bins. As an example, te distribution for positive adrons in te bin < x < 0.30 is sown in te left panel of Fig.. Te mean values of w in te nine x bins are given in te rigt panel of te same figure and in Table. For negative adrons te distributions are very muc te same. Te distributions of w in te nine z bins ave also similar sapes but different slopes. Te distribution for 0.50 < z < 0.65 and te mean values of w as function of z are sown in Fig. 2 for positive adrons. Again, for negative adrons te distributions are very muc te same. Te measured weigted asymmetries are presented as a function of x in Fig. 3. Te unweigted ers asymmetries [] are also sown for comparison. As expected, te trends of te weigted + N 5 P / zm =.58 T 4 /zm T P <x< P T / zm Fig. : Left panel: Distribution of te weigt w = P T /zm for positive adrons in te bin < x < 0.3. Rigt panel: Mean value of w as function of x. No acceptance correction applied x

11 Te COMPASS Collaboration + N 6 P / zm =. T 5 4 /zm T P <z< P T / zm z Fig. 2: Left panel: Distribution of te weigt w = P T /zm for positive adrons in te bin 0.50 < z < Rigt panel: Mean value of w as function of z. No acceptance correction applied w A w A x Fig. 3: Full points: A w in te nine x bins for positive (left panel) and negative (rigt panel) adrons. Te open crosses are te unweigted ers asymmetries [], wic are sligtly sifted towards smaller x values for clarity. and unweigted asymmetries are similar bot for positive and negative adrons. Te asymmetry for positive adrons is clearly different from zero, in particular at large x. In tis range, te ratios A w / are very close to te mean value of te weigt. Te statistical uncertainties are scaled by about te same ratio. Assuming u-quark dominance for positive adrons produced on a proton target, one as and te results on A w 2 A w 2 f ()u (x,q 2 ) f u, (2) (x,q2 ) represent te first direct measurement of f ()u / f u. In Fig. 4, te weigted ers asymmetries measured in our standard range z > 0.2 are compared wit te corresponding ones in te range 0. < z < 0.2. It is interesting to note tat te positiveadron asymmetries are basically uncanged, wic empasizes u-quark dominance and supports te idea tat factorisation works already at small values of z in te COMPASS kinematic range. At low z, te difference between favoured and unfavoured fragmentation functions decreases, x

12 Measurement of P T -weigted ers asymmetries in leptoproduction of adrons w w A A (0.<z<0.2) (0.2<z<.0) 0.05 w w A A (0.<z<0.2) (0.2<z<.0) 2 x Fig. 4: Comparison of te weigted asymmetries vs. x measured in te range (0. < z < 0.2) for positive (left) and negative (rigt) adrons and te corresponding ones in te standard range z > 0.2, wic are sligtly sifted towards smaller x values for clarity. 2 x w A w A z Fig. 5: Full points: A w in te nine z bins for positive (left panel) and negative (rigt panel) adrons. Te open crosses are te corresponding unweigted ers asymmetries [], wic are sligtly sifted towards smaller x values for clarity. z tus it is expected tat te u-quark contribution to te negative-adron asymmetry increases. Te asymmetry itself is ten expected to become larger and similar to te positive-adron asymmetries, as observed in Fig. 4. In order to furter investigate te z dependence, it is of interest to look at A w as a function of z, after integration over x. Te results in te range 0. < z < are sown in Fig. 5. For positive adrons, te values are almost constant witin statistical uncertainties, as it is expected in te case of u-quark dominance if te measurement is performed in te current-fragmentation region and factorisation olds. Te values of te measured P T /zm-weigted asymmetries are given in Tables 2 and 3. From te P T /zm-weigted asymmetries it is straigtforward to extract te first transverse moment of te ers function. Tis will be done in Section 6.

13 2 Te COMPASS Collaboration + N 6 5 P /M = 0.53 T /M T P <x< P T /M Fig. 6: Left: distribution of te weigt w = P T /M for positive adrons in te bin < x < 0.3. Rigt: mean value of w as a function of x. No acceptance correction applied x + N 6 5 P /M = 0.63 T /M T P <z< P T /M z Fig. 7: Left: Distribution of te weigt w = P T /M for positive adrons in te bin 0.50 < z < Rigt: mean value of w as a function of z. No acceptance correction applied. 5 ers asymmetries weigted by P T /M Let us now turn to te ers asymmetries weigted wit w = P T /M. Te distributions of w are very similar in all x and z bins. Examples of te distributions and te mean values of w in te x and z bins for positive adrons are given in Figs. 6 and 7, respectively, and in Table 4. Again, for negative adrons te distributions are very muc te same. Te results for A w are sown in Fig. 8 for positive and negative adrons. Te ratio Rw = A w / for positive adrons is sown in Fig. 9. Correlations between numerator and denominator were accounted for. Te ratio R w = A w / is almost constant as function of x wit a mean value of 0.62, not far from tat expected using te Gaussian model [see Eq. (7)], wic is also sown in te figure. In order to better investigate te z dependence, as in te case of te A w asymmetries, te analysis was repeated adding te adrons wit 0. < z < 0.2. Te results for te x-integrated asymmetry A w as a function of z are sown in Fig. for positive and negative adrons. Te values for positive adrons are in qualitative agreement wit te u-quark dominance approximation, i.e.: A w (z) z. (22)

14 Measurement of P T -weigted ers asymmetries in leptoproduction of adrons 3 A w' w' A x Fig. 8: Te weigted asymmetry A w wit w = PT /M, as a function of x for positive (left) and negative (rigt) adrons wit z > x w w' /A A Fig. 9: Ratio A w / as a function of x for positive adrons and z > 0.2. Te black points are te values of 4 z /πm z/p T. x w' 0.04 A w' A z Fig. : Closed points: A w wit w = PT /M in te nine x bins for positive (left panel) and negative (rigt panel) adrons. Te open crosses are te unweigted ers asymmetries [], wic are sligtly sifted towards smaller x values for clarity. z For comparison, te publised ers asymmetries [] are also sown in te same figure. All

15 4 Te COMPASS Collaboration values of te measured P T /M-weigted asymmetries are given in Tables 5 and 6. 6 Point-by-point extraction of te first moments of te ers functions Te final goal of te measurement of te weigted ers asymmetries is te extraction of te first moments of te ers functions. Tus we consider te weigted asymmetry integrated over z (we restore te Q 2 dependence): A w (x,q2 ) = 2 q e 2 qx f ()q (x,q 2 ) D q (Q2 ) q e 2 qx f q (x,q2 ) D q, (23) (Q2 ) were D q zmax (Q2 ) = dzd q (z,q2 ). (24) z min Te denominator of Eq. (23) can be fully evaluated by resorting to global fits of distribution and fragmentation functions. Tere are two sets of asymmetries, i.e. for unidentified positively (superscript +) and negatively (superscript ) carged adrons. In our analysis, we omit te sea-quark ers distributions, wic were sown to be negligible in a previous study [26]. Te asymmetries ten read (for simplicity we omit again te x and Q 2 dependence) = 2 4x f ()u v 9 q e 2 qx f q A w,± Denoting te denominator by δ ± D u,± + x f ()d v D d,± D q,±. (25) δ ± 9 e 2 qx f q q,± D, (26) q te valence ers distributions can be extracted from te asymmetries as follows x f ()u v = 8 x f ()d v = 2 δ + A w,+ d, D D u,+ δ A w, D u,+ δ A w, D d, D u, D d,+ δ + A w,+ D u,+ D d, D d,+ D u, D d,+ D u,, (27). (28) Eqs. (27) and (28) allow for a point-by-point extraction of te ers distributions for valence quarks. For te distribution functions we use te CTEQ5D parametrisation [4] and for te fragmentation functions of unidentified adrons te DSS parametrisation [42]. Te results are displayed in Fig. and tabulated in Table 7 togeter wit te mean values of Q 2 (ranging from.24 (GeV/c) 2 to 25.6 (GeV/c) 2 ). Te extracted values for x f ()u v and x f ()d v are correlated, as tey are linear functions of te same two measured asymmetries, and te computed correlation coefficients are also given in Table 7. All te numerical values for te results presented in tis paper, as well as te covariance matrices are available on HEPDATA [40].

16 Measurement of P T -weigted ers asymmetries in leptoproduction of adrons 5 x f () u d v v 2 x Fig. : Values of te first moment of te ers function for u (closed red dots) and d (open black dots) quarks from te P T /zm weigted-ers asymmetries for carged adrons wit z > 0.2. Te curves and te uncertainty bands are te results of te fit of Ref. [23]. Te uncertainties are computed from te statistical uncertainties of te measured asymmetries, and no attempt was made to try to assign a systematic uncertainty to te results. Te uncertainties in te extracted d v ers distribution are muc larger tan te corresponding ones for te u v quark. Te u v and d v ers distributions are linear combinations [see Eqs. (27, 28)] of te same ers asymmetries for positive and negative adrons on te proton, tus in principle sufficient for teir determination, but te coefficient of proportionality is four times larger for te d quark, wic makes te uncertainties of te extracted x f ()d v about four times larger tan tose of x f ()u v. In Fig., we also sow for comparison te results, i.e. central values and uncertainty bands, of te fit [23] to te HERMES proton data [9] and te COMPASS proton and deuteron data [38, 43], wic uses DGLAP evolution. Te results are compatible, wit a sligtly different trend of x f ()d v suggested by te present extraction. It is also interesting to compare our present result wit te point-by-point extraction of Ref. [26], were te pion ers asymmetries from te COMPASS proton [38] and deuteron [43] data are used as input. Te data set used in Ref. [26] and te present one ave te dominating pion data on te proton target in common, so tat te results are strongly correlated. As can be seen in Fig. 2, in te present work te uncertainties on te extracted u v and d v ers function moments are on average smaller by a factor of about.5 wit respect to te corresponding quantities in Ref. [26]. Tis is due to te fact tat in te present analysis we ad to assume te ers function of te sea quarks to be zero. Following te metod of Ref. [26] and imposing te sea-quark ers functions to be zero, we ave determined te u v and d v functions from te π + and π proton asymmetries [38] only and verified tat bot te central values and te uncertainties are very similar to te ones presented in tis paper. Tus te differences visible in Fig. 2 can be attributed to te impact of te deuteron data and to te extraction of te sea-quark ers function, rater ten to te use of unweigted asymmetries. Te assumption of a vanising contribution from te sea quarks will be better verified only wen more neutron data will be available.

17 6 Te COMPASS Collaboration () 0. x f u d v v 2 x Fig. 2: Comparison of te values of te first moment of te ers function for u (closed red dots) and d (open black dots) quarks from te P T /zm-weigted ers asymmetries for carged adrons wit z > 0.2, and te corresponding values obtained in Ref. [26] from te unweigted pion ers asymmetries measured by COMPASS on deuteron and proton (closed red and open black squares, respectively). 7 Conclusions and outlook COMPASS as measured te weigted ers asymmetries in SIDIS of 60 GeV muons on transversely polarised protons, extending te standard analysis of unweigted asymmetries. Te weigted asymmetries were determined for positive and negative adrons using as weigt eiter P T /zm or P T /M. In bot cases, te asymmetries were found to be positive for positive adrons in te range x > 0.03 and compatible wit zero for negative adrons wit z > 0.2, very muc as in te case of te standard ers asymmetries. Te z dependence for positive adrons agrees wit te expectation in te case of u-quark dominance and of a measurement performed in te current-fragmentation region. From te P T /zm-weigted ers asymmetries, and under te ypotesis of negligible ers functions for sea quarks, we ave extracted te first moments of te ers functions for u v and d v quarks. In te leading-order pqcd formalism, te obtained values are model independent because of te use of weigted asymmetries and because of te point-by-point extraction. Previous model-dependent extractions tat are based on te Gaussian ansatz compare well wit our results. Te present analysis ints at te validity of te Gaussian parametrisation for te transversemomentum dependence of te ers distribution function and te fragmentation function, at least in te kinematic domain explored by our measurement. As in all oter extractions of te ers functions from SIDIS asymmetries on transversely polarized nucleons, te d-quark ers function turns out to be poorly determined and strongly dependent on te assumptions on te ers functions of te sea quarks. Tis is due to te scarcity of ers asymmetry data taken wit a transversely polarised deuteron target, as compared to te existing data taken wit a transversely polarised proton target. Te recently approved COMPASS run [44] wit a transversely polarised deuteron target in 202 is expected to allow for a muc better extraction of te ers functions

18 Measurement of P T -weigted ers asymmetries in leptoproduction of adrons 7 for bot quarks and antiquarks.

19 8 Te COMPASS Collaboration Table : Mean values of te weigt P T /zm for positive adrons in te nine bins of x for z > 0.2, and in te nine bins of z. x P T /zm z P T /zm Table 2: Measured values of te P T /zm-weigted ers asymmetries in te nine x bins. z > < z < 0.2 x A w, + A w, A w, + A w, ± ± ± 0.03 ± ± ± ± ± ± ± ± ± ± ± ± ± ± 8 4 ± ± ± ± 9 2 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table 3: Measured values of te P T /zm-weigted ers asymmetries in te nine z bins. z A w, + A w, ± ± ± 9 73 ± ± ± ± ± ± 40 ± ± 77 9 ± ± ± ± ± ± ± 87

20 Measurement of P T -weigted ers asymmetries in leptoproduction of adrons 9 Table 4: Mean value of te weigt P T /M for positive adrons in te nine x bins for z > 0.2, and in te nine z bins. x P T /M z P T /M Table 5: Measured values of te P T /M-weigted ers asymmetries in te nine x bins. z > < z < 0.2 x A w, + A w, A w, + A w, ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± 94 6 ± ± ± ± 0.0 ± ± ± ± ± ± ± ± ± ± ± Table 6: Measured values of te P T /M-weigted ers asymmetries in te nine z bins. z A w, + A w, ± 36 ± ± 20 5 ± ± ± ± 3 27 ± ± 37 5 ± ± ± ± ± ± ± ± ± 75

21 20 Te COMPASS Collaboration Table 7: Values of te first moments of te ers functions for u and d quarks. Te last column gives teir correlation coefficient ρ. x Q 2 x f ()u v x f ()d v ρ (GeV/c) ± 5 ± ± 40 4 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ±

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