Spin Density Matrix Elements in Exclusive ω Meson Muoproduction ∗
M.G. Alexeev, G.D. Alexeev, A. Amoroso, V. Andrieux, V. Anosov, A. Antoshkin, K. Augsten, W. Augustyniak, C.D.R. Azevedo, B. Badelek, F. Balestra, M. Ball, J. Barth, R. Beck, Y. Bedfer, J. Berenguer Antequera, J. Bernhard, M. Bodlak, F. Bradamante, A. Bressan, V.E. Burtsev, W.-C. Chang, C. Chatterjee, M. Chiosso, A.G. Chumakov, S.-U. Chung, A. Cicuttin, P.M.M. Correia, M.L. Crespo, D. D'Ago, S. Dalla Torre, S.S. Dasgupta, S. Dasgupta, I. Denisenko, O.Yu. Denisov, S.V. Donskov, N. Doshita, Ch. Dreisbach, W. Duennweber, R.R. Dusaev, A. Efremov, P.D. Eversheim, P. Faccioli, M. Faessler, A. Ferrero, M. Finger, M. Finger jr., H. Fischer, C. Franco, J.M. Friedrich, V. Frolov, F. Gautheron, O.P. Gavrichtchouk, S. Gerassimov, J. Giarra, I. Gnesi, M. Gorzellik, A. Grasso, A. Gridin, M. Grosse Perdekamp, B. Grube, A. Guskov, D. von Harrach, R. Heitz, F. Herrmann, N. Horikawa, N. d'Hose, C.-Y. Hsieh, S. Huber, S. Ishimoto, A. Ivanov, T. Iwata, M. Jandek, T. Jary, R. Joosten, P. Joerg, E. Kabuss, F. Kaspar, A. Kerbizi, B. Ketzer, G.V. Khaustov, Yu.A. Khokhlov, Yu. Kisselev, F. Klein, J.H. Koivuniemi, V.N. Kolosov, K. Kondo Horikawa, I. Konorov, V.F. Konstantinov, A.M. Kotzinian, O.M. Kouznetsov, A. Koval, Z. Kral, F. Krinner, Y. Kulinich, F. Kunne, K. Kurek, R.P. Kurjata, A. Kveton, K. Lavickova, et al. (101 additional authors not shown)
EEUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH
COMPASS
CERN-EP-2020–xxxSeptember 8, 2020
Spin Density Matrix Elements in Exclusive ω Meson Muoproduction ∗ The COMPASS Collaboration
Abstract
We report on a measurement of Spin Density Matrix Elements (SDMEs) in hard exclusive ω mesonmuoproduction on the proton at COMPASS using 160 GeV/ c polarised µ + and µ − beams impingingon a liquid hydrogen target. The measurement covers the range 5.0 GeV/ c < W < c ,with the average kinematics (cid:104) Q (cid:105) = c ) , (cid:104) W (cid:105) = . c , and (cid:104) p (cid:105) = .
16 (GeV/ c ) .Here, Q denotes the virtuality of the exchanged photon, W the mass of the final hadronic systemand p T the transverse momentum of the ω meson with respect to the virtual-photon direction. Themeasured non-zero SDMEs for the transitions of transversely polarised virtual photons to longitu-dinally polarised vector mesons ( γ ∗ T → V L ) indicate a violation of s -channel helicity conservation.Additionally, we observe a sizeable contribution of unnatural-parity-exchange (UPE) transitions thatdecreases with increasing W . The results provide important input for modelling Generalised PartonDistributions (GPDs). In particular, they may allow to evaluate in a model-dependent way the contri-bution of UPE transitions and assess the role of parton helicity-flip GPDs in exclusive ω production. (to be submitted to Eur. Phys. J. C) a r X i v : . [ h e p - e x ] S e p he COMPASS Collaboration G.D. Alexeev , M.G. Alexeev , A. Amoroso , V. Andrieux , V. Anosov , A. Antoshkin ,K. Augsten , W. Augustyniak , C.D.R. Azevedo , B. Badełek , F. Balestra , M. Ball , J. Barth ,R. Beck , Y. Bedfer , J. Berenguer Antequera , J. Bernhard , M. Bodlak , F. Bradamante ,A. Bressan , V.E. Burtsev , W.-C. Chang , C. Chatterjee , M. Chiosso , A.G. Chumakov ,S.-U. Chung A. Cicuttin
P.M.M. Correia , M.L. Crespo , D. D’Ago , S. Dalla Torre ,S.S. Dasgupta , S. Dasgupta , I. Denisenko , O.Yu. Denisov , S.V. Donskov , N. Doshita ,Ch. Dreisbach , W. Dünnweber d , R.R. Dusaev , A. Efremov , P.D. Eversheim , P. Faccioli ,M. Faessler d , A. Ferrero , M. Finger , M. Finger Jr. , H. Fischer , C. Franco , J.M. Friedrich ,V. Frolov , F. Gautheron , O.P. Gavrichtchouk , S. Gerassimov , J. Giarra , I. Gnesi ,M. Gorzellik , A. Grasso , A. Gridin , M. Grosse Perdekamp , B. Grube , A. Guskov ,D. von Harrach , R. Heitz , F. Herrmann , N. Horikawa , N. d’Hose , C.-Y. Hsieh , S. Huber ,S. Ishimoto , A. Ivanov , T. Iwata , M. Jandek , V. Jary , P. Jörg , R. Joosten , E. Kabuß ,F. Kaspar , A. Kerbizi , B. Ketzer , G.V. Khaustov , Yu.A. Khokhlov ,Yu. Kisselev ,F. Klein , J.H. Koivuniemi , V.N. Kolosov , K. Kondo Horikawa , I. Konorov ,V.F. Konstantinov , A.M. Kotzinian , O.M. Kouznetsov , A. Koval , Z. Kral , F. Krinner ,Y. Kulinich , F. Kunne , K. Kurek , R.P. Kurjata , A. Kveton , K. Lavickova , S. Levorato ,Y.-S. Lian , J. Lichtenstadt , P.-J. Lin , R. Longo , V. E. Lyubovitskij , A. Maggiora ,A. Magnon p , N. Makins , N. Makke , G.K. Mallot , A. Maltsev , S. A. Mamon ,B. Marianski , A. Martin , J. Marzec , J. Matoušek , T. Matsuda , G. Mattson ,G.V. Meshcheryakov , M. Meyer , W. Meyer , Yu.V. Mikhailov , M. Mikhasenko ,E. Mitrofanov , N. Mitrofanov , Y. Miyachi , A. Moretti , A. Nagaytsev , C. Naim , D. Neyret ,J. Nový , W.-D. Nowak , G. Nukazuka , A.S. Nunes , A.G. Olshevsky , M. Ostrick ,D. Panzieri , B. Parsamyan , S. Paul , H. Pekeler , J.-C. Peng , M. Pešek , D.V. Peshekhonov ,M. Pešková , N. Pierre , S. Platchkov , J. Pochodzalla , V.A. Polyakov , J. Pretz ,M. Quaresma , C. Quintans , C. Regali , G. Reicherz , C. Riedl , T. Rudnicki ,D.I. Ryabchikov , A. Rybnikov , A. Rychter , V.D. Samoylenko , A. Sandacz , S. Sarkar ,I.A. Savin , G. Sbrizzai , H. Schmieden , A. Selyunin , L. Sinha , M. Slunecka , J. Smolik ,A. Srnka , D. Steffen , M. Stolarski , O. Subrt , M. Sulc , H. Suzuki , T. Szameitat ,P. Sznajder , S. Tessaro , F. Tessarotto , A. Thiel , J. Tomsa , F. Tosello , A. Townsend ,V. Tskhay , S. Uhl , B. I. Vasilishin , A. Vauth , B. M. Veit , J. Veloso , B. Ventura ,A. Vidon , M. Virius , M. Wagner , S. Wallner , K. Zaremba , P. Zavada , M. Zavertyaev ,M. Zemko , E. Zemlyanichkina , Y. Zhao and M. Ziembicki University of Aveiro, I3N, Dept. of Physics, 3810-193 Aveiro, Portugal Universität Bochum, Institut für Experimentalphysik, 44780 Bochum, Germany u,v3
Universität Bonn, Helmholtz-Institut für Strahlen- und Kernphysik, 53115 Bonn, Germany u4 Universität Bonn, Physikalisches Institut, 53115 Bonn, Germany u5 Institute of Scientific Instruments of the CAS, 61264 Brno, Czech Republic w6 Matrivani Institute of Experimental Research & Education, Calcutta-700 030, India x7 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia y8 Universität Freiburg, Physikalisches Institut, 79104 Freiburg, Germany u,v9
CERN, 1211 Geneva 23, Switzerland Technical University in Liberec, 46117 Liberec, Czech Republic w11
LIP, 1649-003 Lisbon, Portugal z12
Universität Mainz, Institut für Kernphysik, 55099 Mainz, Germany u13
University of Miyazaki, Miyazaki 889-2192, Japan aa14
Lebedev Physical Institute, 119991 Moscow, Russia Technische Universität München, Physik Dept., 85748 Garching, Germany u,d6
Nagoya University, 464 Nagoya, Japan aa17
Charles University, Faculty of Mathematics and Physics, 18000 Prague, Czech Republic w18
Czech Technical University in Prague, 16636 Prague, Czech Republic w19
State Scientific Center Institute for High Energy Physics of National Research Center ‘KurchatovInstitute’, 142281 Protvino, Russia IRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette, France v21
Academia Sinica, Institute of Physics, Taipei 11529, Taiwan ab22
Tel Aviv University, School of Physics and Astronomy, 69978 Tel Aviv, Israel ac23
Tomsk Polytechnic University, 634050 Tomsk, Russia ad24
University of Trieste, Dept. of Physics, 34127 Trieste, Italy Trieste Section of INFN, 34127 Trieste, Italy University of Turin, Dept. of Physics, 10125 Turin, Italy Torino Section of INFN, 10125 Turin, Italy University of Illinois at Urbana-Champaign, Dept. of Physics, Urbana, IL 61801-3080, USA ae29
National Centre for Nuclear Research, 02-093 Warsaw, Poland af30
University of Warsaw, Faculty of Physics, 02-093 Warsaw, Poland af31
Warsaw University of Technology, Institute of Radioelectronics, 00-665 Warsaw, Poland af32
Yamagata University, Yamagata 992-8510, Japan aa*
Dedicated to the memory of Bohdan Marianski
Corresponding authors
E-mail addresses : [email protected], [email protected], [email protected] a Also at Dept. of Physics, Pusan National University, Busan 609-735, Republic of Korea b Also at Physics Dept., Brookhaven National Laboratory, Upton, NY 11973, USA c Also at Abdus Salam ICTP, 34151 Trieste, Italy d e Supported by the DFG Research Training Group Programmes 1102 and 2044 (Germany) f Also at Chubu University, Kasugai, Aichi 487-8501, Japan g Also at Dept. of Physics, National Central University, 300 Jhongda Road, Jhongli 32001, Taiwan h Also at KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan i Present address: Universität Bonn, Physikalisches Institut, 53115 Bonn, Germany j Also at Moscow Institute of Physics and Technology, Moscow Region, 141700, Russia k Deceased l Also at Yerevan Physics Institute, Alikhanian Br. Street, Yerevan, Armenia, 0036 m Also at Dept. of Physics, National Kaohsiung Normal University, Kaohsiung County 824, Taiwan n Supported by ANR, France with P2IO LabEx (ANR-10-LBX-0038) in the framework “Investisse-ments d’Avenir” (ANR-11-IDEX-003-01) o Also at Institut für Theoretische Physik, Universität Tübingen, 72076 Tübingen, Germany p Retired q Present address: Brookhaven National Laboratory, Brookhaven, USA r Also at University of Eastern Piedmont, 15100 Alessandria, Italy s Present address: RWTH Aachen University, III. Physikalisches Institut, 52056 Aachen, Germany t Present address: Universität Hamburg, 20146 Hamburg, Germany u Supported by BMBF - Bundesministerium für Bildung und Forschung (Germany) v Supported by FP7, HadronPhysics3, Grant 283286 (European Union) w Supported by MEYS, Grant LM20150581 (Czech Republic) x Supported by B. Sen fund (India) y Supported by CERN-RFBR Grant 12-02-91500 z Supported by FCT, Grants CERN/FIS-PAR/0007/2017 and CERN/FIS-PAR/0022/2019 (Portugal) a Supported by MEXT and JSPS, Grants 18002006, 20540299, 18540281 and 26247032, the Daikoand Yamada Foundations (Japan) ab Supported by the Ministry of Science and Technology (Taiwan) ac Supported by the Israel Academy of Sciences and Humanities (Israel) ad Supported by the Tomsk Polytechnic University Competitiveness Enhancement Program (Russia) ae Supported by the National Science Foundation, Grant no. PHY-1506416 (USA) af Supported by NCN, Grant 2017/26/M/ST2/00498 (Poland)pin Density Matrix Elements in Exclusive ω Meson Muoproduction ∗ In this paper, exclusive ω meson muoproduction is studied in the process µ + p → µ (cid:48) + p (cid:48) + ω , (1)which in the one-photon-exchange approximation is described by the interaction of a virtual photon γ ∗ with the target proton p : γ ∗ + p → p (cid:48) + ω . (2)This process, which at high virtuality Q of the photon is known as Hard Exclusive Meson Production(HEMP), serves at low values of the squared four-momentum transfer t as an important tool to accessGeneralised Parton Distributions (GPDs) [1–5] that contain a wealth of new information on the partonstructure of the nucleon.The amplitude for Hard Exclusive Meson Production by longitudinally polarised virtual photons wasproven to factorise into a hard-scattering part, which is calculable in perturbative QCD (pQCD), and asoft part [4, 6]. The soft part contains GPDs that describe the structure of the probed nucleon and a dis-tribution amplitude that accounts for the structure of the produced meson. The factorisation is referred toas collinear because parton transverse momenta are neglected. No similar proof of collinear factorisationexists for transversely polarised virtual photons. However, phenomenological pQCD-inspired modelshave been proposed [7–10] that go beyond the collinear factorisation by postulating the so called k ⊥ factorisation, where k ⊥ denotes parton transverse momentum. In the Goloskokov-Kroll model [8–12],hereafter referred to as GK model, cross sections, Spin Density Matrix Elements (SDMEs) as well astarget and beam-spin asymmetries for HEMP by both longitudinally and transversely polarised virtualphotons can be described simultaneously.At leading twist, longitudinally polarised vector-meson production by longitudinally polarised virtualphotons is described by the chiral-even GPDs H f and E f , where f denotes a quark of a given flavour or agluon. When higher-twist effects are included in the three-dimensional light-cone wave function, the pro-duction of longitudinally polarised vector mesons by transversely polarised virtual photons is describedby the chiral-odd GPDs H fT and ¯ E fT , which allow a helicity flip of the “active” quark. Unnatural-parityexchange (UPE) contributes also to transitions from transversely polarised virtual photons to transverselypolarised vector mesons or from longitudinally polarised virtual photons to transversely polarised vectormesons. These contributions are described by the GPDs (cid:101) H f and (cid:101) E f . Besides these UPE contributions,there is a sizeable pion-pole contribution that is treated as a one-boson exchange in the GK model.The SDMEs describe the spin structure of the reaction shown in Eq. (1). They are related to helicity am-plitudes that describe transitions between specified spin states of virtual photon, target proton, producedvector meson and recoil proton. For an unpolarised nucleon target, after summing over initial and finalspin states of the proton, SDMEs only depend on the helicities of virtual photon and produced meson.The measured SDME values can be used to establish a hierarchy of helicity amplitudes, to test the hypoth-esis of s -channel helicity conservation (SCHC), to evaluate the contribution of unnatural-parity-exchangetransitions and to assess the role of chiral-odd, i.e. parton helicity-flip GPDs in exclusive ω production.They also allow to determine the phase difference between helicity amplitudes as well as the longitudinal-to-transverse cross-section ratio. The measurements of SDMEs can provide further constraints on GPDparameterisations beyond those from measurements of cross sections and spin asymmetries for HEMP.The HERMES measurements of SDMEs for exclusive electroproduction of ω mesons [13] in the kine-matic region 1.0 (GeV/ c ) < Q <
10 (GeV/ c ) , 3.0 GeV/ c < W < . c and | t | < . c ) ,where t is the squared four-momentum transfer to the target, indicate a sizeable contribution of UPEtransitions that can be described by GPDs (cid:101) H f and (cid:101) E f related to quark helicity distributions. Here, Q denotes the virtuality of the exchanged virtual photon and W is the mass of the final hadronic system. In The COMPASS Collaborationthe framework of the GK model it turns out [12] that the pion-pole exchange, which is treated as one-boson exchange, is an important contribution required to reproduce the HERMES results. The effect ofsuch a t -channel π exchange decreases with W while it is predicted still to be measurable at COMPASS.The HERMES results on SDMEs for exclusive ω production, as well as those for exclusive ρ produc-tion [13, 14], indicate a violation of the SCHC hypothesis, which in the framework of the GK model isattributed to a contribution of chiral-odd GPDs.Early papers on exclusive ω electroproduction are summarised in Ref. [15], which also contains resultson SDMEs obtained at DESY for 0.3 (GeV/ c ) < Q < . c ) and 0.3 GeV/ c < W < . c .The SDMEs in exclusive ω electroproduction were also studied [16] at CLAS in the range 1.6 (GeV/ c ) < Q < c ) and 1.9 GeV/ c < W < c . It was found that the exchange of the pionRegge trajectory dominates exclusive ω production, even for Q values as large as 5 (GeV/ c ) .The present COMPASS results on SDMEs for exclusive ω muoproduction, which supplement the pub-lished COMPASS results on azimuthal asymmetries for transversely polarised protons [17], have thepotential to further constrain GPDs. In the framework of the GK model it may become possible to assessthe role of chiral-odd GPDs in the mechanism of SCHC violation and to shed light onto the mechanismof UPE transitions. Adopting the notation from Ref. [13], the theoretical formalism of SDMEs and helicity amplitudes in-troduced by K. Schilling and G. Wolf [18] is used throughout this paper.
The helicity amplitudes F λ V λ (cid:48) N λ γ λ N describe the transition of a virtual photon with helicity λ γ to a vectormeson with helicity λ V , where λ N ( λ (cid:48) N ) is the nucleon helicity in the initial (final) state. The helicityamplitudes depend on W , Q , and t (cid:48) ≡ | t | − t ≈ p , where t represents the smallest kinematicallyallowed value of | t | for given Q and meson mass, and p is the square of the vector-meson transversemomentum with respect to the direction of the virtual photon. In the centre-of-mass (CM) system ofvirtual photon and nucleon, the vector-meson spin density matrix ρ λ V λ (cid:48) V is related to the helicity amplitude F λ V λ (cid:48) N λ γ λ N as [18] ρ λ V λ (cid:48) V = N ∑ λ γ λ (cid:48) γ λ N λ (cid:48) N F λ V λ (cid:48) N λ γ λ N ρ U + L λ γ λ (cid:48) γ F ∗ λ (cid:48) V λ (cid:48) N λ (cid:48) γ λ N , (3)where N is a normalisation factor [14, 18]. The virtual-photon spin density matrix ρ U + L λ γ λ (cid:48) γ [14] describesthe subprocess µ → µ (cid:48) + γ ∗ , which is calculable in quantum electrodynamics. It can be decomposed as ρ U + L λ γ λ (cid:48) γ = ρ U λ γ λ (cid:48) γ + P b ρ L λ γ λ (cid:48) γ , (4)where the matrix with superscript L ( U ) contains elements that are coupled (not coupled) to the beampolarisation P b . In the following the corresponding vector-meson SDMEs, which are related to theseelements, will be referred to as “polarised” (“unpolarised”).The vector-meson spin density matrix ρ U + L λ γ λ (cid:48) γ can be decomposed into a set of nine matrices ρ αλ V λ (cid:48) V corre-sponding to different virtual-photon polarisation states: transverse polarisation ( α =0, ..., 3), longitudinalpolarisation ( α =4), and their interference ( α =5, ..., 8) [18]. If it is experimentally not possible to separatethe contributions from longitudinally and transversely polarised photons, SDMEs are usually defined asfollows: r λ V λ (cid:48) V = ( ρ λ V λ (cid:48) V + ε R ρ λ V λ (cid:48) V )( + ε R ) − , pin Density Matrix Elements in Exclusive ω Meson Muoproduction ∗ r αλ V λ (cid:48) V = (cid:40) ρ αλ V λ (cid:48) V ( + ε R ) − , α = , , , √ R ρ αλ V λ (cid:48) V ( + ε R ) − , α = , , , . (5)Here, R = d σ L / d σ T is the differential longitudinal-to-transverse cross-section ratio of virtual photonsand ε is the virtual-photon polarisation parameter, see Eq. (15). The relations between the 23 SDMEsdefined in Eq. (5) and the helicity amplitudes are given in Appendix A of Ref. [14]. As detailed in Refs. [14, 18], each helicity amplitude can be linearly decomposed into a natural-parity-exchange (NPE) amplitude T and an unnatural-parity-exchange (UPE) amplitude U , F λ V λ (cid:48) N λ γ λ N = T λ V λ (cid:48) N λ γ λ N + U λ V λ (cid:48) N λ γ λ N , (6)with the following relations [18]: T λ V λ (cid:48) N λ γ λ N = [ F λ V λ (cid:48) N λ γ λ N + ( − ) λ V − λ γ F − λ V λ (cid:48) N − λ γ λ N ] , (7) U λ V λ (cid:48) N λ γ λ N = [ F λ V λ (cid:48) N λ γ λ N − ( − ) λ V − λ γ F − λ V λ (cid:48) N − λ γ λ N ] . (8)Using the notation (cid:102) ∑ T λ V λ γ T ∗ λ (cid:48) V λ (cid:48) γ ≡ ∑ λ N λ (cid:48) N T λ V λ (cid:48) N λ γ λ N T ∗ λ (cid:48) V λ (cid:48) N λ (cid:48) γ λ N . (9)and the symmetry properties [14, 18] of the amplitudes T , Eq. (9) becomes (cid:102) ∑ T λ V λ γ T ∗ λ (cid:48) V λ (cid:48) γ = T λ V λ γ T ∗ λ (cid:48) V λ (cid:48) γ + T λ V − λ γ T ∗ λ (cid:48) V − λ (cid:48) γ . (10)Here, both products on the right-hand side represent the contribution of NPE amplitudes, the first withoutand the second with nucleon-helicity flip. The relations for the UPE amplitudes can be written in ananalogous way. In the abbreviated notation used in the text, the nucleon-helicity indices will be omittedfor amplitudes with λ N = λ (cid:48) N , i.e. T λ V λ γ ≡ T λ V λ γ = T λ V − λ γ − , U λ V λ γ ≡ U λ V λ γ = − U λ V − λ γ − . (11)The hypothesis of s -channel helicity conservation implies that there exist only diagonal γ ∗ → V transi-tions ( λ V = λ γ ). Spin density matrix elements are extracted from experimental data on exclusive muoproduction of ω mesons. The SDMEs are fitted as parameters of the three-dimensional angular distribution W U + L ( Φ , φ , cos Θ ) to the corresponding experimental distribution. Here, Φ is the azimuthal angle of the produced ω meson, while the polar angle Θ and the azimuthal angle φ describe the three-pion decay of the ω meson,see Eqs. (16 - 21). The angular distribution W U + L is decomposed into contributions that are not coupled( W U ) or coupled ( W L ) to the beam polarisation: W U + L ( Φ , φ , cos Θ ) = W U ( Φ , φ , cos Θ ) + P b W L ( Φ , φ , cos Θ ) . (12) The COMPASS CollaborationUsing the data, which were collected with a longitudinally polarised beam, 15 “unpolarised” SDMEs areextracted from W U : W U ( Φ , φ , cos Θ ) = π (cid:34) ( − r ) + ( r − ) cos Θ − √ { r } sin 2 Θ cos φ − r − sin Θ cos 2 φ − ε cos 2 Φ (cid:16) r sin Θ + r cos Θ − √ { r } sin 2 Θ cos φ − r − sin Θ cos 2 φ (cid:17) − ε sin 2 Φ (cid:16) √ { r } sin 2 Θ sin φ + Im { r − } sin Θ sin 2 φ (cid:17) + (cid:112) ε ( + ε ) cos Φ (cid:16) r sin Θ + r cos Θ − √ { r } sin 2 Θ cos φ − r − sin Θ cos 2 φ (cid:17) + (cid:112) ε ( + ε ) sin Φ (cid:16) √ { r } sin 2 Θ sin φ + Im { r − } sin Θ sin 2 φ (cid:17)(cid:35) , (13)and 8 “polarised” SDMEs from W L : W L ( Φ , φ , cos Θ ) = π (cid:34)(cid:112) − ε (cid:16) √ { r } sin 2 Θ sin φ + Im { r − } sin Θ sin 2 φ (cid:17) + (cid:112) ε ( − ε ) cos Φ (cid:16) √ { r } sin 2 Θ sin φ + Im { r − } sin Θ sin 2 φ (cid:17) + (cid:112) ε ( − ε ) sin Φ (cid:16) r sin Θ + r cos Θ − √ { r } sin 2 Θ cos φ − r − sin Θ cos 2 φ (cid:17)(cid:35) . (14)Here, the virtual-photon polarisation parameter ε represents the ratio of fluxes of longitudinal and trans-verse virtual photons, ε = − y − y Q ν − y + y ( Q ν + ) , (15)where y = p · q / p · k lab = ν / E . The symbols p , q and k denote the four-momenta of target proton, virtualphoton and incident lepton respectively. The energy of virtual photon and incident lepton in the targetrest frame is denoted by ν and E , respectively.Angles and reference frames are defined in Fig. 1.The directions of axes of the “hadronic CM system” and the ω -meson rest frame coincide with thedirections of axes of the helicity frame [14, 18, 20]. Following Ref. [18], the right-handed hadronic CMsystem of virtual photon and target nucleon, with coordinates XY Z , is defined such that the Z -axis isaligned along the virtual-photon three-momentum q and the Y -axis is parallel to q × v , where v is thethree-momentum of the ω meson.For the convenience of the reader, we give in the following the explicit definitions of angles [13]. Theangle Φ between ω production plane and lepton scattering plane in the hadronic CM system is given bycos Φ = ( q × v ) · ( k × k (cid:48) ) | q × v | · | k × k (cid:48) | (16)pin Density Matrix Elements in Exclusive ω Meson Muoproduction ∗ n π + π - π Θϕ X Y Z ω p γ * ω production plane ω d eca y p l a n e helicity frame( ω at rest) Y . Fig. 1:
Definition of angles in the process µ N → µ N ω with ω → π + π − π . Here, Φ is the angle between the ω production plane and the lepton scattering plane in the centre-of-mass system of the virtual photon and the targetnucleon. The variables Θ and φ are respectively the polar and azimuthal angles of the unit vector normal to thedecay plane in the ω meson rest frame. and sin Φ = [( q × v ) × ( k × k (cid:48) )] · q | q × v | · | k × k (cid:48) | · | q | . (17)Here k , k (cid:48) , q = k − k (cid:48) and v are the three-momenta of the incoming and outgoing lepton, the virtualphoton and the ω meson respectively.The unit vector normal to the decay plane in the ω rest frame is defined by n = p π + × p π − | p π + × p π − | , (18)where p π + and p π − are the three-momenta of the positive and negative decay pion in the ω rest frame,respectively.The polar angle Θ of the unit vector n in the ω meson rest frame, with the z -axis aligned opposite to theoutgoing nucleon momentum p (cid:48) and the y -axis directed along p (cid:48) × q , is defined bycos Θ = − p (cid:48) · n | p (cid:48) | . (19)The azimuthal angle φ of the unit vector n is given as follows:cos φ = ( q × p (cid:48) ) · ( p (cid:48) × n ) | q × p (cid:48) | · | p (cid:48) × n | , (20)sin φ = − [( q × p (cid:48) ) × p (cid:48) ] · ( n × p (cid:48) ) | ( q × p (cid:48) ) × p (cid:48) | · | n × p (cid:48) | . (21) The main component of the COMPASS setup is the two-stage magnetic spectrometer. Each spectrometerstage comprises a dipole magnet complemented by a variety of tracking detectors, a muon filter for muon The COMPASS Collaborationidentification and an electromagnetic as well as a hadron calorimeter. A detailed description of the setupcan be found in Refs. [21–23].The data used for this analysis were collected within four weeks in 2012. In this period the COMPASSspectrometer was complemented by a 2.5 m long liquid-hydrogen target surrounded by a time-of-flight(TOF) system for the detection of recoiling protons and by a third electromagnetic calorimeter placeddirectly downstream of the target.Data with µ + and µ − beams were taken separately. The natural polarisation of the muon beam providedby the CERN SPS originates from the parity-violating decay-in-flight of the parent meson, which impliesopposite polarisations for µ + and µ − beams. Within regular time intervals during data taking, chargeand polarisation of the muon beam were swapped simultaneously. In order to equalise the spectrometeracceptance for the two beam charges, also the polarities of the two spectrometer magnets were changedaccordingly. For both beams, the absolute value of the average beam polarisation is about 0.8 with anuncertainty of about 0.04.An event to be accepted for analysis is required to have a topology as that of the observed process µ p → µ (cid:48) p (cid:48) ω π + π − π γγ BR ≈ . BR ≈ π reconstruction A neutral pion is reconstructed via its dominant decay into two photons that are registered as neutralclusters in the electromagnetic calorimeters. As neutral cluster we denote a reconstructed calorimetercluster that is not associated to a charged track, thereby including any cluster for the most upstreamcalorimeter that had no tracking system upstream of it. The method of π reconstruction is similar to theone used in the analysis of azimuthal asymmetries for exclusive ω production on a transversely polarisedtarget [24].In Fig. 2 the distribution of the reconstructed two-photon invariant mass is shown. The π peak isprominent. The distribution is fitted by a superposition of the signal, which is described by a Gaussianfunction, and a linear background. After selection of an event with a π , the energies of the decay photonsare scaled by the factor M PDG π / M γγ , where M PDG π ≈ / c is the nominal π mass. This scalingdoes not affect the angular distributions of neutral pions, albeit it improves the experimental resolutionof the reconstructed three-pion invariant mass. The following kinematic selections are applied to select exclusively produced ω mesons:– 1.0 (GeV/ c ) < Q < . c ) , where the lower limit ensures applicability of pQCD and theupper one suppresses background due to hadrons produced in DIS, which hereafter is referred toas “SIDIS background”.pin Density Matrix Elements in Exclusive ω Meson Muoproduction ∗ ) (MeV/ c γγ M
50 100 150 200 250 300 E v en t s / ( . M e V / c ) Gauss backgroundsum
Fig. 2:
Distribution of the two-photon invariant mass fitted by a Gaussian function and a linear background. Thedashed vertical line denotes the PDG value of the π mass. The black arrows indicate the selection window. – 0.1 < y < 0.9, where the lower limit suppresses events with poorly reconstructed kinematics andthe upper one removes events with large radiative corrections.– W > 5.0 GeV/ c to remove the kinematic region where the cross section changes rapidly due to theproduction of resonances.– 0.01 (GeV/ c ) < p < . c ) , where p T is the transverse momentum of the ω meson withrespect to the virtual-photon direction. The lower limit removes events with a poorly determinedazimuthal angle of the produced meson and the upper one suppresses SIDIS background.– 0.1 GeV/ c < M γγ < c , where M γγ is the two-photon invariant mass, in order to select π mesons.– 0.71 GeV/ c < M π + π − π < 0.86 GeV/ c , where M π + π − π is the three-pion invariant mass, in order toselect ω mesons. In Fig. 3, the ω signal is clearly visible above a small background. The invariantmass of the three-pion system is fitted by a Breit-Wigner function and a linear background.– E ω > 14 GeV to reduce the SIDIS background contribution.In order to enhance the fraction of events with exclusively produced ω mesons, the missing energy E miss = M − M M (22)is constrained by − < E miss < M is the proton mass, M = ( p + q − p π + − p π − − p π ) is the missing mass squared, and p π + , p π − and p π are the four-momenta of the three pions. The E miss distribution for the experimental data is shown inFig. 4 as open histogram. The exclusive peak is apparent.After applying all kinematic selection criteria, 3060 events are available for further analysis. The E miss distribution is used to determine the fraction of SIDIS background under the exclusive peak,following the procedure described in Refs. [17, 24]. For the simulation of background, the LEPTO 6.5.1 The COMPASS Collaboration ) (MeV/ c π − π + π M
650 700 750 800 850 900 E v en t s / ( . M e V / c ) B−Wbackgroundsum
Fig. 3:
Distribution of the π + π − π invariant mass fitted by a Breit-Wigner function and linear background. Thedashed vertical line denotes the PDG value of the ω mass. The black arrows denote the applied limits. generator is used with the COMPASS tuning of parameters [25]. In order to achieve the best possibleagreement between experimental and simulated E miss distributions, the simulated data are reweighted ona bin-by-bin basis using the weight w ( E miss ) = N scrd ( E miss ) N scMC ( E miss ) . (23)Here N scrd ( E miss ) and N scMC ( E miss ) are numbers of events containing same-charge hadron pairs, h + h + γγ and h − h − γγ , in the three-pion system for experimental and simulated data, respectively. In order to improvethe statistical significance, the constraint on the ω invariant mass is not used for the purpose of estimatingthe weight w . The shaded histogram in Fig. 4 represents the simulated SIDIS background, which isgenerated by LEPTO and processed through the full simulation of the COMPASS setup [26], followedby the same event reconstruction and selection procedure as for the real data, and then reweighted inthe way described above. The distribution is normalised to the experimental data in the region 7 GeV < E miss <
20 GeV. The fraction of background in the signal window − E miss < 3.0 GeV isfound to be f bg = 0.28 for the total kinematic range. The fraction of SIDIS background increases withincreasing Q and p , and it decreases with increasing W . For the results on kinematic dependences ofSDMEs, which are presented in the following, the background fraction f bg is evaluated separately foreach kinematic bin, the values ranging between 0.20 and 0.41. The SDMEs are determined by an Unbinned Maximum Likelihood fit of the function W ( R ; Φ , φ , cos Θ ) to the experimental three-dimensional angular distribution of ω production and decay. The explicitexpression for the dependence of W on SDMEs was given in Sec. 3 by Eqs. (12, 13, 14). Here R denotes the set of 23 SDMEs r αλ V λ (cid:48) V . The negative log-likelihood function to be minimised reads − ln L ( R ) = − N ∑ i = ln W U + L ( R ; Φ i , φ i , cos Θ i ) (cid:102) N ( R ) , (24)pin Density Matrix Elements in Exclusive ω Meson Muoproduction ∗ E miss (GeV) E v en t s / ( . G e V ) Fig. 4:
The missing-energy distribution from experimental data (open histogram) compared to the distribution ofSIDIS events from a LEPTO MC simulation (shaded histogram). The MC distribution is normalised to the datain the region 7 GeV < E miss <
20 GeV. The vertical dashed line denotes the upper limit of the exclusive region.Each LEPTO MC event is reweighted by a E miss -dependent weight that is calculated using both experimental andsimulated data with same-charge hadron pairs. See text for a detailed explanation. where N is the number of selected events.The likelihood normalisation factor (cid:102) N ( R ) = N MC ∑ j = W U + L ( R ; Φ j , φ j , cos Θ j ) (25)is calculated numerically using the sample of MC events generated with the HEPGEN++ ω generator, inthe following denoted by HEPGEN [27, 28]. This generator is used to model the kinematics of exclusive ω production. For the purpose of the present analysis, the option with an isotropic three-dimensionalangular distribution of ω production and decay is chosen. The generated events are passed through acomplete description of the COMPASS setup and the resulting data are treated in the same way as it wasdone for experimental data. The number of HEPGEN events is denoted N MC in Eq. (25). In order to determine SDMEs that are corrected for SIDIS background, a two-step procedure is used.First, the parameterisation of the background angular distributions is obtained by applying the abovedescribed maximum likelihood method to selected SIDIS events simulated with the LEPTO generator.These events are required to pass the same selection criteria as experimental data. Performing an un-binned likelihood fit according to Eq. (24) using simulated events in the range − < E miss < B of 23 “background SDMEs”.In the second step, the set B of background SDMEs is used to extract the set R of background-correctedSDMEs by applying the unbinned maximum likelihood fit to the experimental data. For this purpose thefollowing negative log-likelihood function is fitted: − ln L ( R ) = − N ∑ i = ln (cid:104) ( − f bg ) W U + L ( R ; Φ i , φ i , cos Θ i ) (cid:102) N ( R , B ) + f bg W U + L ( B ; Φ i , φ i , cos Θ i ) (cid:102) N ( R , B ) (cid:105) . (26)Here, f bg is the fraction of background events in the selected experimental data as determined in Sec. 4.20 The COMPASS Collaborationand (cid:102) N is the normalisation factor: (cid:102) N ( R , B ) = N MC ∑ j = [( − f bg ) W U + L ( R ; Φ j , φ j , cos Θ j ) + f bg W U + L ( B ; Φ j , φ j , cos Θ j )] . (27) The following sources of systematic uncertainties are considered:i)
Difference between results for µ + and µ − beams The µ + beam intensity was about 2.7 times higher than that of the µ − beam. A possible impactof this difference on the determination of SDMEs is checked by comparing the SDMEs extractedseparately for the µ + beam (negative polarisation) and the µ − beam (positive polarisation). Foreach SDME, half of the difference between the SDME values determined with opposite beampolarisations is taken as systematic uncertainty.ii) Influence of shifted E miss peak position
It was observed in Ref. [29] that certain SDME values depend on the position of the E miss peak.The E miss distribution shown in Fig. 4 is not precisely centred at zero, but slightly shifted towardsnegative values. This results from an imbalance between the energy measured for the incomingmuon and the energies of the final-state particles measured in the forward spectrometer. The effectof this shift on the extracted SDMEs is investigated by applying the small kinematic correction(+0.7 GeV/ c ) to the beam momentum that is needed to centre the E miss peak at zero. The differ-ence between the values of final SDMEs and those obtained with corrected kinematics is taken assystematic uncertainty.iii) Effect of background subtraction
As detailed in Sec. 5.2, the background-corrected SDMEs are obtained with background SDMEsobtained from LEPTO events in the exclusive region − < E miss < < E miss < p -range is covered by the RPD, the same limited kinematic region is used tocompare the SDMEs obtained with and without RPD. The results are consistent within statisticaluncertainties, hence no systematic uncertainty is assigned here.iv) Comparison of unbinned and binned maximum likelihood methods
The two fitting methods are expected to yield consistent results for sufficiently large statistics. Inthis analysis however, when using the unbinned method the background treatment is different fromthat when using the binned method. In the former case, the angular dependence of the backgroundis parameterised, while in the latter case the background is subtracted in each angular bin on abin-by-bin basis. Comparing the results from the two methods hence probes a possible systematicuncertainty due to the background-correction procedure. For each SDME, the systematic uncer-tainty is given by the difference between the final value as obtained using the unbinned methodand the value obtained with the binned method.pin Density Matrix Elements in Exclusive ω Meson Muoproduction ∗ Sensitivity to the shapes of the kinematic distributions generated by HEPGEN
As no experimental data exists on the differential cross section for exclusive ω production at COM-PASS energies, a model is used to simulate the process in HEPGEN. In order to check the sen-sitivity of SDMEs to the shapes of kinematic distributions in the HEPGEN generator, the SDMEextraction was repeated by reweighting the MC events with weights depending on Q and ν . Theweights are tuned such that the Q and ν distributions from the experimental data match thosefrom the reweighted simulated data. The effect of this reweighting on the extracted SDMEs issmall in most cases, and the difference between final SDMEs and those obtained with reweightedMC events is taken as systematic uncertainty.The total systematic uncertainties are obtained by adding the above described components in quadrature.Table 1 gives the values for the total kinematic region. The individual contributions i) - v) to the sys-tematic uncertainty for each SDME are compiled in Table A.1 in the Appendix. When averaged over allSDMEs it appears that the group i) systematics dominates by contributing almost half of the systematicuncertainties, while about one-fifth contributions arise from both group ii) and group iv) systematics. Inmost cases, the statistical uncertainty is comparable to or smaller than the total systematic one. The SDMEs extracted in the total kinematic region 1.0 (GeV/ c ) < Q < . c ) , 5.0 GeV/ c < W < . c and 0.01 (GeV/ c ) < p < . c ) , with mean values (cid:104) Q (cid:105) = .
13 (GeV/ c ) , (cid:104) W (cid:105) = . c and (cid:104) p (cid:105) = .
16 (GeV/ c ) are presented in Fig. 5 and Table 1. These SDMEs arepresented in five classes corresponding to different helicity transitions. For the SDMEs in class A, thedominant contributions are related to the squared amplitudes for transitions from longitudinal virtualphotons to longitudinal vector mesons, γ ∗ L → V L , and from transverse virtual photons to transverse vectormesons, γ ∗ T → V T . In class B, the dominant terms correspond to the interference between amplitudes forthe two aforementioned transitions. The main terms in the SDMEs for classes C, D and E are proportionalto the products of small amplitudes describing γ ∗ T → V L , γ ∗ L → V T and γ ∗ T → V − T transitions respectively.In Fig. 5, polarised SDMEs are shown in shaded areas. The experimental uncertainties of these SDMEsare larger than those of the unpolarised SDMEs because the lepton-beam polarisation is smaller thanunity ( | P b | ≈ | P b |√ − ε , where ε ≈ .
96. In the calculation of the statistical uncertainty,the correlations between the various SDMEs are taken into account. Q , p and W The kinematic dependences of the SDMEs on Q , p and W , which have been determined in three binsfor each of the variables, are shown in Figs. 6, 7 and 8. In Table 2, the limits of the kinematic bins andthe mean values of kinematic variables in the bins are given. The values of SDMEs in bins of Q , p and W are given in Table A.2, A.3 and A.4 respectively, in the Appendix. In case of SCHC, only the seven SDMEs of classes A and B are not restricted to vanish, while all SDMEsfrom classes C, D, and E should be equal to zero. Six of the SDMEs in classes A and B have to fulfil thefollowing relations [18]:2 The COMPASS Collaboration
Im r r r r r Im r
Im r r r r Im r r r Im r
Re r
Re r
Re r
Im r
Im r
Re r
Im r r r −0.2 −0.1 0 0.1 0.2 0.3 0.4Im r r r r r Im r
Im r r r r Im r r r Im r
Re r
Re r
Re r
Im r
Im r
Re r
Im r r r −0.2 −0.1 0 0.1 0.2 0.3 0.4 SDME valueA : γ * L → ω L γ * T → ω T B : Interference γ * L → ω L & γ * T → ω T C : γ * T → ω L D : γ * L → ω T E : γ * T → ω −T Fig. 5:
The 23 SDMEs for exclusive ω leptoproduction extracted in the total COMPASS kinematic region with (cid:104) Q (cid:105) = .
13 (GeV/ c ) , (cid:104) W (cid:105) = . c , (cid:104) p (cid:105) = .
16 (GeV/ c ) . Inner error bars represent statistical uncertaintiesand outer ones statistical and systematic uncertainties added in quadrature. Unpolarised (polarised) SDMEs aredisplayed in unshaded (shaded) areas. r − = − Im { r − } , Re { r } = − Im { r } , Im { r } = Re { r } . Within uncertainties, the extracted SDMEs are consistent with these relations: r − + Im { r − } = − . ± . ± . , Re { r } + Im { r } = . ± . ± . , Im { r } − Re { r } = − . ± . ± . . However, for the transitions γ ∗ T → V L of class C the non-zero values of SDMEs r and Re { r } showSCHC violation at the level of three standard deviations of the statistical uncertainty. In the GK model[10], these SDMEs are related to the chiral-odd GPDs H T and ¯ E T coupled to the higher-twist wavefunction of the meson. The kinematic dependences of these SDMEs, as presented in Section 6, may helpto further constrain the model.pin Density Matrix Elements in Exclusive ω Meson Muoproduction ∗ Table 1:
The 23 unpolarised and polarised SDMEs for the total COMPASS kinematic region, shown in the sameorder as in Fig. 5 for classes A to E. The first uncertainties are statistical, the second systematic.
SDME r . ± . ± . r − − . ± . ± . r − . ± . ± . r . ± . ± . r − . ± . ± . r . ± . ± . r . ± . ± . r . ± . ± . r − . ± . ± . r . ± . ± . r . ± . ± . r − . ± . ± . r . ± . ± . r . ± . ± . r − . ± . ± . r − − . ± . ± . r − . ± . ± . r − . ± . ± . r − . ± . ± . r − − . ± . ± . r − . ± . ± . r − . ± . ± . r − . ± . ± . ω meson production The existence of UPE transitions in exclusive ω production can be tested by examining linear combina-tions of SDMEs such as u = − r + r − − r − r − . (28)The quantity u can be expressed in terms of helicity amplitudes as u = (cid:102) ∑ ε | U | + | U + U − | N . (29)Since the numerator depends only on UPE amplitudes, a u value different from zero indicates non-zerocontribution from UPE transitions. For the total kinematic region of COMPASS u is equal to 0.830 ± ± u = r + r − (30)and u = r + r − , (31)4 The COMPASS Collaboration A: r −0.250.0250.3 2 4
A: r
1 1−1
A: Im r
2 1−1
B: Re r
5 10 −0.15−0.1−0.05 2 4
B: Im r
6 10 −0.35−0.050.25 2 4
B: Im r
7 10
B: Re r
8 10
C: Re r −0.200.2 2 4
C: Re r
1 10
C: Im r
2 10
C: r
5 00 −0.250 2 4
C: r
1 00
C: Im r
3 10 −0.350.050.45 2 4
C: r
8 00 −0.10 2 4
D: r
5 11 −0.10 2 4
D: r
5 1−1
D: Im r
6 1−1
D: Im r
7 1−1 −0.45−0.050.35 2 4
D: r
8 11 −0.4500.45 2 4
D: r
8 1−1
E: r −0.100.1 2 4
E: r
1 11
E: Im r
3 1−1 Q ((GeV/ c ) ) Fig. 6: Q dependence of the measured 23 SDMEs. The capital letters A to E denote the class, to which the SDMEbelongs. Inner error bars represent statistical uncertainties and outer ones statistical and systematic uncertaintiesadded in quadrature. which in terms of helicity amplitudes can be combined into u + iu = √ (cid:102) ∑ ( U + U − ) U ∗ N . (32)For COMPASS, u = − ± ± u = − ± ± u , u , u on Q , p , and W is presented. The quantity u decreases with increasing W and p , which indicates that the UPE contribution becomes smaller, while u , u fluctuate around zero.More detailed information on the W dependence of certain UPE transitions in terms of helicity amplitudespin Density Matrix Elements in Exclusive ω Meson Muoproduction ∗ Table 2:
Kinematic binning and mean values for kinematic variables. bin (cid:104) Q (cid:105) c ) < Q < 1.35 (GeV/ c ) c ) c ) < Q < 2.05 (GeV/ c ) c ) c ) < Q < 10.0 (GeV/ c ) c ) bin (cid:104) p (cid:105) c ) < p < 0.07 (GeV/ c ) c ) c ) < p < 0.19 (GeV/ c ) c ) c ) < p < 0.5 (GeV/ c ) c ) bin (cid:104) W (cid:105) c < W < 6.4 GeV/ c c c < W < 7.9 GeV/ c c c < W < 17.0 GeV/ c c can be obtained by considering the difference of the following two class-A SDMEs [14]: r − = N (cid:102) ∑ {| T | + | T − | −| U | − | U − | } , (33)Im { r − } = N (cid:102) ∑ {−| T | + | T − | + | U | − | U − | } , (34)which reads: Im { r − } − r − = N (cid:102) ∑ ( −| T | + | U | ) . (35)For the total kinematic region, both SDMEs and their difference are close to zero. For the present data,Im { r − } − r − = 0.07 ± (cid:101) ∑ | U | ≈ (cid:101) ∑ | T | . By applying Eq. (10), Eq. (35) canbe rewritten as follows: Im { r − } − r − = N ( −| T | − | T − | + | U | + | U − | ) . (36)Bilinear contributions of nucleon helicity-flip amplitudes are suppressed by a factor ( √− t (cid:48) / M ) , where t (cid:48) is a measure of the transverse momentum of the vector meson with respect to the direction of the virtualphoton. Neglecting these bilinear contributions yields:Im { r − } − r − ≈ N ( | U | − | T | ) . (37)In Table 3, the values of the SDMEs r − and Im { r − } and their difference are shown as a function of (cid:104) W (cid:105) . The difference is large and positive at (cid:104) W (cid:105) = . c , i.e. | U | > | T | . For (cid:104) W (cid:105) = . c , | U | ≈ | T | holds and for (cid:104) W (cid:105) = . c the situation is reversed: | U | < | T | .A substantial contribution of UPE transitions in hard exclusive ω meson electroproduction was ob-served at HERMES [13]. In their total kinematic range, with mean values (cid:104) Q (cid:105) = . c ) , (cid:104) W (cid:105) = A: r −0.20.2 0 0.2 0.4
A: r
1 1−1 −0.20.2 0 0.2 0.4
A: Im r
2 1−1
B: Re r −0.15−0.05 0 0.2 0.4
B: Im r −0.40.4 0 0.2 0.4
B: Im r
7 10 −0.250.4 0 0.2 0.4
B: Re r
8 10 −0.10.15 0 0.2 0.4
C: Re r −0.30.1 0 0.2 0.4
C: Re r
1 10 −0.050.2 0 0.2 0.4
C: Im r
2 10 −0.050.3 0 0.2 0.4
C: r
5 00 −0.20.15 0 0.2 0.4
C: r
1 00 −0.250.25 0 0.2 0.4
C: Im r
3 10 −0.250.6 0 0.2 0.4
C: r
8 00 −0.050.05 0 0.2 0.4
D: r
5 11 −0.10.05 0 0.2 0.4
D: r
5 1−1 −0.10.1 0 0.2 0.4
D: Im r
6 1−1 −0.450.55 0 0.2 0.4
D: Im r
7 1−1 −0.60.3 0 0.2 0.4
D: r
8 11 −0.60.6 0 0.2 0.4
D: r
8 1−1 −0.050.1 0 0.2 0.4
E: r −0.10.1 0 0.2 0.4
E: r
1 11 −0.40.45 0 0.2 0.4
E: Im r
3 1−1 p T2 ( (GeV/ c ) ) Fig. 7: p dependence of the measured 23 SDMEs. The capital letters A to E denote the class, to which the SDMEbelongs. Inner error bars represent statistical uncertainties and outer ones statistical and systematic uncertaintiesadded in quadrature. . c and (cid:104) t (cid:48) (cid:105) = .
08 (GeV/ c ) , for the proton target they found u = . ± . ± .
12 andIm { r − } − r − = . ± . ± .
05. The latter value indicates that | U | > | T | in their kinematicrange. Also they observed that the quantity u , when averaged over the total range of W , increases(decreases) with increasing values of Q ( t (cid:48) ).A quantitative comparison of COMPASS and HERMES results is not straightforward, because the cov-ered kinematic regions only partially overlap, and COMPASS covers significantly wider ranges of W and p . It is important to note here that, when studying the kinematic dependences of the measured observ-ables, results are extracted in one-dimensional intervals of a given kinematic variable, while averagingover the full ranges of the other two variables. As the two experiments have only partially overlappingkinematic ranges, the results after averaging cannot be directly compared.When neglecting the observed Q and t (cid:48) ( p ) dependences, which exhibit opposite trends, one can com-pin Density Matrix Elements in Exclusive ω Meson Muoproduction ∗ A: r −0.250.1 6 8 10
A: r
1 1−1 −0.150.2 6 8 10
A: Im r
2 1−1
B: Re r
5 10 −0.15−0.05 6 8 10
B: Im r
6 10 −0.20.55 6 8 10
B: Im r
7 10 −0.10.8 6 8 10
B: Re r
8 10 −0.050.1 6 8 10
C: Re r −0.20.1 6 8 10
C: Re r
1 10 −0.050.2 6 8 10
C: Im r
2 10 −0.050.25 6 8 10
C: r
5 00 −0.250.2 6 8 10
C: r
1 00 −0.10.55 6 8 10
C: Im r
3 10 −1.40.85 6 8 10
C: r
8 00 −0.050.05 6 8 10
D: r
5 11 −0.10.05 6 8 10
D: r
5 1−1 −0.050.1 6 8 10
D: Im r
6 1−1 −0.50.4 6 8 10
D: Im r
7 1−1 −0.550.55 6 8 10
D: r
8 11 −0.450.45 6 8 10
D: r
8 1−1 −0.050.1 6 8 10
E: r −0.10.1 6 8 10
E: r
1 11 −0.250.8 6 8 10
E: Im r
3 1−1 W (GeV/ c ) Fig. 8: W dependence of the measured 23 SDMEs. The capital letters A to E denote the class, to which the SDMEbelongs. Inner error bars represent statistical uncertainties and outer ones statistical and systematic uncertaintiesadded in quadrature. pare the HERMES result on u for their total kinematic range to the COMPASS result shown at the lowest W value in Fig. 9. Similarly, HERMES result on Im { r − } − r − can be compared to the correspondingvalue from COMPASS at (cid:104) W (cid:105) = . c , which is shown in Table 3. Within uncertainties the resultsfrom the two experiments are consistent for both observables.Altogether, the main COMPASS results presented in this subsection, i.e. the W dependence of u aswell as that of Im { r − } − r − indicate that the UPE contribution decreases with increasing W withoutvanishing towards largest W values accessible at COMPASS. In the GK model, UPE is described by theGPDs (cid:101) H f and (cid:101) E f (non-pole), and by the pion-pole contribution treated as a one-boson exchange [12].The latter one, which is a sizeable contribution, results in a significantly faster decrease of the predictedUPE contribution with increasing W than that measured at COMPASS.8 The COMPASS Collaboration Q ((GeV/ c ) ) Q ((GeV/ c ) ) u −0.1−0.0500.05 1 2 3 4 average value Q ((GeV/ c ) ) u −0.500.5 1 2 3 4 Q ((GeV/ c ) ) u p T2 ((GeV/ c ) ) u −0.1−0.0500.05 0 0.2 p T2 ((GeV/ c ) ) u −1−0.500.5 0 0.2 p T2 ((GeV/ c ) ) u W (GeV/ c ) u −0.10 6 8 10 12 W (GeV/ c ) u −101 6 8 10 12 W (GeV/ c ) u Fig. 9: Q , p , and W dependences of u , u , u . The open symbols represent the values over the total kinematicregion. Inner error bars represent statistical uncertainties and outer ones statistical and systematic uncertaintiesadded in quadrature. Table 3: W dependence of SDMEs r − , Im r − and their difference (cid:104) W (cid:105) (GeV/ c ) 5.9 7.1 9.9 r − − . ± . ± . − . ± . ± .
33 0 . ± . ± . r − . ± . ± .
46 0 . ± . ± . − . ± . ± . r − − r − . ± . ± .
046 0 . ± . ± . − . ± . ± . γ ∗ T → V T Another observable that is sensitive to the relative contributions of UPE and NPE amplitudes is the NPE-to-UPE asymmetry of the transverse differential cross section for the transition γ ∗ T → V T . It is defined [12]as P = d σ NT ( γ ∗ T → V T ) − d σ UT ( γ ∗ T → V T ) d σ NT ( γ ∗ T → V T ) + d σ UT ( γ ∗ T → V T )= r − − r − r − , (38)where the superscript N and U denotes the part of the cross section that is fed by NPE and UPE transi-tions, respectively. In Ref. [13] a different definition of the asymmetry is used. pin Density Matrix Elements in Exclusive ω Meson Muoproduction ∗ −1−0.500.51 1 2 3 4 average value Q (GeV/ c ) P −1−0.500.51 0 0.1 0.2 0.3 p T2 (GeV/ c ) P −1−0.500.51 4 6 8 10 12 W (GeV/ c ) P Fig. 10: Q , p and W dependences of the NPE-to-UPE asymmetry of the transverse cross section for thetransition γ ∗ T → V T . The open symbol represents the value over the total kinematic region. Inner error bars representstatistical uncertainties and outer ones statistical and systematic uncertainties added in quadrature. The value of P obtained in the total kinematic region is − . ± . ± . P are shown in Fig. 10. The UPE contribution dominatesat small values of W and p and decreases with increasing values of these kinematic variables. At largevalues of W and p , the NPE contribution becomes dominant, while a non-negligible UPE contributionremains. No significant Q dependence of the asymmetry is observed. In order to evaluate the longitudinal-to-transverse virtual-photon differential cross-section ratio R = d σ L ( γ ∗ L → V ) d σ T ( γ ∗ T → V ) , (39)the quantity R (cid:48) can be used: R (cid:48) = ε r − r . (40)Using expressions defining r and 1 − r in terms of helicity amplitudes [14, 18], one obtains R (cid:48) = ε (cid:101) ∑ ( ε | T | + | T | + | U | ) (cid:101) ∑ {| T | + | U | + | T − | + | U − | + ε ( | T | + | U | ) } . (41)The quantity R (cid:48) may be interpreted as the longitudinal-to-transverse ratio of “effective” cross sectionsfor the production of vector mesons that are polarised longitudinally or transversely irrespective of thevirtual-photon polarisation. In case of SCHC, R (cid:48) is equal to R . In spite of the observed clear violationof SCHC at COMPASS, we use the approximate relation R ≈ R (cid:48) . The accuracy of this approximation isestimated using the GK model [11, 12] and the resulting uni-directional systematic uncertainty is foundto be about +
15% on average, while its magnitude ranges between 3% and 47% with increasing W andbetween 6% and 28% with increasing p T .For the total kinematic region, the ratio R is found to be 0 . ± . stat ± . syst + . − | appr . Here, thethird uncertainty is the systematic one due to the approximation R ≈ R (cid:48) . The kinematic dependences of R are shown in Fig. 11. The ratio appears to increase as Q and p increase, which indicates an increaseof the fraction of longitudinally polarised vector mesons, while it shows no significant change over the W range.0 The COMPASS Collaboration Q ((GeV/ c ) ) R p T2 ((GeV/ c ) ) R W (GeV/ c ) R Fig. 11: Q , p and W dependences of the longitudinal-to-transverse cross-section ratio R . The open symbolrepresents the value obtained for the total kinematic region. Inner error bars represent statistical uncertaintiesand outer ones statistical and systematic uncertainties added in quadrature. Note that the additional positive uni-directional systematic uncertainty due to the approximation R ≈ R (cid:48) is not shown here, see text for details. Using Eq. (42), the phase difference between the UPE amplitudes U and U can be calculated [13]:tan δ U = u / u = r + r − r + r − . (42)The phase difference δ U for the total kinematic region is found to be δ U = ( − . ± . ± . ) degrees.The absolute value of the phase difference δ N between the NPE amplitudes T and T can be calculatedusing Eq. (43) from Ref. [14]:cos δ N = √ ε ( Re { r } − Im { r } ) (cid:113) r ( − r + r − − Im { r − } ) . (43)The phase difference δ N for the total kinematic region is found to be | δ N | = (33.1 ± ± δ N can in principle be determined using the followingequation from Ref. [14]: sin δ N = √ ε ( Re { r } + Im { r } ) (cid:113) r ( − r + r − − Im { r − } ) . (44)However, the large experimental uncertainties of the polarised SDMEs make this presently impossible. In Fig. 12 the 23 SDMEs for exclusive ω production, extracted in the total kinematic region of COM-PASS, are compared with the predictions of the GPD model of Goloskokov and Kroll [11, 12] for hardexclusive vector-meson leptoproduction. In this version of the model, contributions from chiral-oddGPDs as well as from pion-pole exchange are included. The model was tuned to HERMES results onSDMEs and spin asymmetries for exclusive ρ and ω production, which led to a satisfactory agreementbetween the model and the data.pin Density Matrix Elements in Exclusive ω Meson Muoproduction ∗ ω production at Q = . c ) , W = . c and p = .
14 (GeV / c ) , close to the corresponding average kinematicvalues for COMPASS. In the following, we concentrate on the most pronounced differences betweenmodel predictions and experimental results.The most noticeable differences are as follows: i) the predicted value of SDME r , which representsthe fraction of longitudinally polarised mesons in the produced sample, is significantly larger than themeasured one; ii) the SDMEs dominated by the transitions γ ∗ T → ω L (class C) are in general close tozero in the model, while in the data several of them ( r , Re { r } ) indicate a clear violation of the SCHChypothesis.A characteristic prediction of the model is a strong decrease of the UPE contribution with increasingvalues of W . The predicted values of the quantity u at (cid:104) Q (cid:105) = .
13 (GeV/ c ) and (cid:104) p (cid:105) = .
16 (GeV/ c ) are equal to 1.01, 0.39 and 0.07 for W values of 5.0 GeV/ c , 7.1 GeV/ c and 11.0 GeV/ c respectively. Acomparison of these predictions to the results shown in the lower-left panel of Fig. 9 shows that, while atthe smallest accessible value of W the prediction is consistent with the measured u , at large values of W the model predicts a much stronger decrease with increasing W and hence underestimates significantlythe measured UPE contribution. Im r r r r r Im r
Im r r r r Im r r r Im r
Re r
Re r
Re r
Im r
Im r
Re r
Im r r r −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6Im r r r r r Im r
Im r r r r Im r r r Im r
Re r
Re r
Re r
Im r
Im r
Re r
Im r r r −0.5 −0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 SDME valuesCOMPASSGK modelA : γ * L → VM L γ * T → VM T B : Interference γ * L → VM L & γ * T → VM T C : γ * T → VM L D : γ * L → VM T E : γ * T → VM −T Fig. 12:
Comparison of the measured SDMEs with calculations of the GPD model of Goloskokov and Kroll [12].The calculations are obtained for Q = . c ) , W = . c and p = .
14 (GeV / c ) . Inner error barsrepresent statistical uncertainties and outer ones statistical and systematic uncertainties added in quadrature. Using exclusive ω meson muoproduction on protons, we have measured 23 Spin Density Matrix El-ements at the average COMPASS kinematics, (cid:104) Q (cid:105) = c ) , (cid:104) W (cid:105) = . c and (cid:104) p (cid:105) = .
16 (GeV/ c ) . The SDMEs are extracted in the kinematic region 1 . ( GeV / c ) < Q < . ( GeV / c ) ,5.0 GeV/ c < W < . c and 0.01 (GeV/ c ) < p < . c ) , which allows us to study their Q , p and W dependences.Several SDMEs that are dominated by amplitudes describing γ ∗ T → ω L transitions indicate a considerableviolation of the SCHC hypothesis. These SDMEs are expected to be sensitive to the chiral-odd GPDs H T and ¯ E T , which are coupled to the higher-twist wave function of the meson. A particularly prominenteffect is observed for the SDME r , which strongly increases with increasing Q and p , and decreaseswith increasing W .Using specific observables that are constructed to be sensitive to contributions from transitions withunnatural-parity exchanges such as u , Im { r − } − r − and the UPE-to-NPE asymmetry for the trans-verse cross section, a strong W dependence of the UPE contribution is observed. At low values of W ,we confirm the earlier observation by HERMES that the amplitude of the UPE transition γ ∗ T → ω T islarger than the NPE amplitude for the same transition, i.e. | U | > | T | . With increasing W the UPEcontribution decreases and | U | < | T | at large W , still with a non-negligible UPE contribution at thelargest W values accessible at COMPASS.Altogether, the COMPASS results presented in this paper cover a kinematic range that extends con-siderably beyond the ranges of earlier experimental data on SDMEs for exclusive ω leptoproduction.They provide important input for modelling GPDs, in particular they may help to better constrain theamplitudes for UPE transitions and assess the role of chiral-odd GPDs in exclusive ω leptoproduction. Acknowledgements
We are indebted to Sergey Goloskokov and Peter Kroll for numerous fruitful discussions on the interpre-tation of our results and for providing us with predictions of their model. We gratefully acknowledge thesupport of CERN management and staff and the skill and effort of the technicians of our collaboratinginstitutions.pin Density Matrix Elements in Exclusive ω Meson Muoproduction ∗ Appendix
Table A.1 gives the various contributions to the systematic uncertainty of the 23 SDMEs and Tables A.2,A.3, and A.4 list their kinematic dependences.
Table A.1:
Uncertainties for each SDME value: in column 3 the statistical uncertainty (“stat.”), in columns 4–8the individual contributions for each source of systematic uncertainty as defined in Sec. 5.3, in column 9 the totalsystematic uncertainty (“tot. sys.”), and in column 10 the total uncertainty (“tot.”).
SDME value stat. beamcharge E miss back-ground method simu-lation tot.sys. tot.(i) (ii) (iii) (iv) (v) r .
346 0 . − .
001 0 .
000 0 .
001 0 .
003 0 .
007 0 .
008 0 . r − − .
041 0 .
023 0 .
033 0 .
008 0 .
007 0 .
016 0 .
000 0 .
038 0 . r − .
031 0 .
023 0 .
029 0 .
015 0 . − . − .
002 0 .
049 0 . r .
103 0 . − .
004 0 .
003 0 .
005 0 . − .
001 0 .
009 0 . r − .
089 0 .
007 0 .
004 0 .
007 0 . − .
012 0 .
001 0 .
015 0 . r .
005 0 .
081 0 .
115 0 .
024 0 .
000 0 .
009 0 .
002 0 .
118 0 . r .
093 0 . − . − .
004 0 .
000 0 .
016 0 .
001 0 .
025 0 . r .
018 0 .
011 0 .
001 0 .
010 0 .
004 0 .
008 0 .
004 0 .
014 0 . r − .
081 0 . − . − .
019 0 . − . − .
002 0 .
022 0 . r .
061 0 .
015 0 .
000 0 .
010 0 .
006 0 .
017 0 .
004 0 .
021 0 . r .
132 0 .
014 0 .
005 0 . − .
012 0 .
035 0 .
010 0 .
040 0 . r − .
078 0 .
028 0 . − .
031 0 . − . − .
008 0 .
038 0 . r .
057 0 . − .
048 0 .
019 0 .
000 0 . − .
001 0 .
073 0 . r .
125 0 .
130 0 .
125 0 .
077 0 .
000 0 .
016 0 .
001 0 .
148 0 . r − .
016 0 . − .
002 0 . − . − .
013 0 .
006 0 .
032 0 . r − − .
017 0 .
012 0 .
020 0 . − . − .
011 0 .
002 0 .
025 0 . r − .
023 0 .
011 0 . − . − .
005 0 .
012 0 .
001 0 .
018 0 . r − .
150 0 .
111 0 . − .
012 0 .
000 0 .
079 0 .
000 0 .
168 0 . r − .
106 0 . − . − .
001 0 . − .
047 0 .
002 0 .
056 0 . r − − .
009 0 .
101 0 . − .
054 0 . − . − .
002 0 .
124 0 . r − .
022 0 .
016 0 .
005 0 . − .
008 0 .
003 0 .
000 0 .
010 0 . r − .
025 0 . − . − .
005 0 .
000 0 .
002 0 .
000 0 .
016 0 . r − .
095 0 .
071 0 .
109 0 .
002 0 .
000 0 .
010 0 .
002 0 .
110 0 . Table A.2:
The measured 23 unpolarised and polarised ω SDMEs in bins of Q : 1 . − . − . − .
00 (GeV/ c ) . The first uncertainties are statistical, the second systematic. SDME (cid:104) Q (cid:105) = 1.16 (GeV/ c ) (cid:104) Q (cid:105) = 1.64 (GeV/ c ) (cid:104) Q (cid:105) = 3.58 (GeV/ c ) r . ± . ± .
027 0 . ± . ± .
046 0 . ± . ± . r − − . ± . ± . − . ± . ± . − . ± . ± . r − − . ± . ± .
025 0 . ± . ± .
047 0 . ± . ± . r . ± . ± .
008 0 . ± . ± .
013 0 . ± . ± . r − . ± . ± . − . ± . ± . − . ± . ± . r . ± . ± .
140 0 . ± . ± . − . ± . ± . r . ± . ± . − . ± . ± .
116 0 . ± . ± . r . ± . ± .
016 0 . ± . ± .
016 0 . ± . ± . r − . ± . ± . − . ± . ± . − . ± . ± . r . ± . ± .
012 0 . ± . ± .
018 0 . ± . ± . r . ± . ± .
022 0 . ± . ± .
039 0 . ± . ± . r − . ± . ± . − . ± . ± . − . ± . ± . r − . ± . ± .
187 0 . ± . ± . − . ± . ± . r − . ± . ± .
404 0 . ± . ± .
166 0 . ± . ± . r . ± . ± . − . ± . ± . − . ± . ± . r − − . ± . ± . − . ± . ± . − . ± . ± . r − . ± . ± .
016 0 . ± . ± .
024 0 . ± . ± . r − . ± . ± . − . ± . ± .
079 0 . ± . ± . r . ± . ± . − . ± . ± . − . ± . ± . r − − . ± . ± . − . ± . ± .
191 0 . ± . ± . r − . ± . ± .
034 0 . ± . ± .
043 0 . ± . ± . r − . ± . ± . − . ± . ± . − . ± . ± . r − . ± . ± .
229 0 . ± . ± .
162 0 . ± . ± . ω Meson Muoproduction ∗ Table A.3:
The measured 23 unpolarised and polarised ω SDMEs in bins of p : 0 . − . − . − . c ) . The first uncertainties are statistical, the second systematic. SDME (cid:104) p (cid:105) = 0.037 (GeV/ c ) (cid:104) p (cid:105) = 0.124 (GeV/ c ) (cid:104) p (cid:105) = 0.31 (GeV/ c ) r . ± . ± .
016 0 . ± . ± .
025 0 . ± . ± . r − − . ± . ± . − . ± . ± .
034 0 . ± . ± . r − . ± . ± .
020 0 . ± . ± . − . ± . ± . r . ± . ± .
004 0 . ± . ± .
022 0 . ± . ± . r − . ± . ± . − . ± . ± . − . ± . ± . r − . ± . ± .
024 0 . ± . ± . − . ± . ± . r . ± . ± .
052 0 . ± . ± . − . ± . ± . r . ± . ± .
016 0 . ± . ± .
021 0 . ± . ± . r − . ± . ± . − . ± . ± . − . ± . ± . r . ± . ± .
043 0 . ± . ± .
037 0 . ± . ± . r . ± . ± .
024 0 . ± . ± .
039 0 . ± . ± . r − . ± . ± . − . ± . ± . − . ± . ± . r . ± . ± . − . ± . ± .
089 0 . ± . ± . r − . ± . ± .
094 0 . ± . ± .
261 0 . ± . ± . r − . ± . ± . − . ± . ± . − . ± . ± . r − − . ± . ± . − . ± . ± . − . ± . ± . r − . ± . ± .
030 0 . ± . ± .
016 0 . ± . ± . r − . ± . ± . − . ± . ± .
312 0 . ± . ± . r . ± . ± . − . ± . ± . − . ± . ± . r − − . ± . ± . − . ± . ± .
134 0 . ± . ± . r − . ± . ± .
024 0 . ± . ± .
017 0 . ± . ± . r − . ± . ± . − . ± . ± . − . ± . ± . r − . ± . ± . − . ± . ± .
171 0 . ± . ± . Table A.4:
The measured 23 unpolarised and polarised ω SDMEs in bins of W : 5 . − . − . − . c .The first uncertainties are statistical, the second systematic. SDME (cid:104) W (cid:105) = 5.87 GeV/ c (cid:104) W (cid:105) = 7.06 GeV/ c (cid:104) W (cid:105) = 9.90 GeV/ c r . ± . ± .
012 0 . ± . ± .
055 0 . ± . ± . r − − . ± . ± . − . ± . ± .
033 0 . ± . ± . r − . ± . ± .
046 0 . ± . ± . − . ± . ± . r . ± . ± .
021 0 . ± . ± .
010 0 . ± . ± . r − . ± . ± . − . ± . ± . − . ± . ± . r . ± . ± .
122 0 . ± . ± . − . ± . ± . r . ± . ± .
056 0 . ± . ± .
215 0 . ± . ± . r . ± . ± .
010 0 . ± . ± .
022 0 . ± . ± . r − . ± . ± . − . ± . ± . − . ± . ± . r . ± . ± .
038 0 . ± . ± .
017 0 . ± . ± . r . ± . ± .
023 0 . ± . ± .
019 0 . ± . ± . r − . ± . ± . − . ± . ± . − . ± . ± . r . ± . ± .
124 0 . ± . ± . − . ± . ± . r − . ± . ± .
047 0 . ± . ± .
138 0 . ± . ± . r . ± . ± . − . ± . ± . − . ± . ± . r − − . ± . ± . − . ± . ± . − . ± . ± . r − . ± . ± .
007 0 . ± . ± .
020 0 . ± . ± . r − − . ± . ± . − . ± . ± .
082 0 . ± . ± . r . ± . ± . − . ± . ± . − . ± . ± . r − . ± . ± . − . ± . ± . − . ± . ± . r − . ± . ± .
015 0 . ± . ± .
028 0 . ± . ± . r − . ± . ± . − . ± . ± .
011 0 . ± . ± . r − . ± . ± .
100 0 . ± . ± .
042 0 . ± . ± . ω Meson Muoproduction ∗ References [1] D. Müller et al., Fortschr. Phys. , 101 (1994).[2] X. Ji, Phys. Rev. Lett. , 610 (1997).[3] X. Ji, Phys. Rev. D , 7114 (1997).[4] A.V. Radyushkin, Phys. Lett. B , 333 (1996).[5] A.V. Radyushkin, Phys. Rev. D , 5524 (1997).[6] J.C. Collins, L. Frankfurt, M. Strikman, Phys. Rev. D , 2982 (1997).[7] A.D. Martin, M.G. Ryskin, T. Teubner, Phys. Rev. , 4329 (1997).[8] S.V. Goloskokov, P. Kroll, Eur. Phys. J. C , 281 (2005).[9] S.V. Goloskokov, P. Kroll, Eur. Phys. J. C , 367 (2008).[10] S.V. Goloskokov, P. Kroll, Eur. Phys. J. C , 809 (2009).[11] S.V. Goloskokov, P. Kroll, Eur. Phys. J. C , 2725 (2014).[12] S.V. Goloskokov, P. Kroll, Eur. Phys. J. A , 146 (2014).[13] A. Airapetian et al., (HERMES Collaboration), Eur. Phys. J. C , 3110 (2014); Erratum: Eur.Phys. J. C , 162 (2016).[14] A. Airapetian et al., (HERMES Collaboration), Eur. Phys. J. C , 659 (2009).[15] T.H. Bauer, R.D. Spital, D.R. Yennie, F.M. Pipkin, Rev. Mod. Phys. , 261 (1978).[16] L. Morand et al., (CLAS Collaboration), Eur. Phys. J. A , 445 (2005).[17] C. Adolph et al., (COMPASS Collaboration), Nucl. Phys. B , 454 (2017).[18] K. Schilling and G. Wolf, Nucl. Phys. B , 381(1973).[19] M. Diehl, JHEP , 064 (2007).[20] P. Joos et al., Nucl. Phys. B , 365 (1977).[21] P. Abbon et al., (COMPASS Collaboration), Nucl. Instrum. Meth. A , 455 (2007).[22] P. Abbon et al., (COMPASS Collaboration), Nucl. Instrum. Meth. A , 69 (2015).[23] F. Gautheron et al., (COMPASS Collaboration), SPSC-P-340, CERN-SPSC-2019-014.[24] P. Sznajder, PhD thesis, National Centre For Nuclear Research, Otwock â ˘A ¸S ´Swierk, March 2015.[25] C. Adolph et al., (COMPASS Collaboration), Phys. Lett. B , 922 (2013).[26] T. Szameitat, PhD thesis, University of Freiburg (2017), doi:10.6094/UNIFR/11686.[27] A. Sandacz and P. Sznajder, “HEPGEN - generator for hard exclusive leptoproduction”, (2012),arXiv:1207.0333.[28] C. Regali, PhD thesis, University of Freiburg (2016), doi:10.6094/UNIFR/11449.8 The COMPASS Collaboration[29] E. Burtin, N. d’Hose, O.A. Grajek and A. Sandacz, “Angular distributions and R = σ L / σ T forexclusive ρ production”, private communication.[30] R. Akhunzyanov et al., (COMPASS Collaboration), Phys. Lett. B , 188 (2019).[31] M.G. Alexeev et al., (COMPASS Collaboration), Phys. Lett. B805