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DESY 05-161 ISSN 0418-9833October 2005

Elastic J/ψ Production at HERA

H1 Collaboration

Abstract

Cross sections for elastic production ofJ/ψ mesons in photoproduction and electropro-duction are measured in electron proton collisions at HERA using an integrated lumi-nosity of 55 pb−1. Results are presented for photon virtualitiesQ2 up to 80GeV2. Thedependence on the photon-proton centre of mass energyWγp is analysed in the range40 ≤ Wγp ≤ 305GeV in photoproduction and40 ≤ Wγp ≤ 160GeV in electroproduc-tion. TheWγp dependences of the cross sections do not change significantly with Q2 andcan be described by models based on perturbative QCD. Withinsuch models, the data showa high sensitivity to the gluon density of the proton in the domain of low Bjorkenx and lowQ2 . Differential cross sections dσ/dt, wheret is the squared four-momentum transfer atthe proton vertex, are measured in the range|t| < 1.2GeV2 as functions ofWγp andQ2.Effective Pomeron trajectories are determined for photoproduction and electroproduction.TheJ/ψ production and decay angular distributions are consistentwith s-channel helicityconservation. The ratio of the cross sections for longitudinally and transversely polarisedphotons is measured as a function ofQ2 and is found to be described by perturbative QCDbased models.

Submitted toEur. Phys. J.C

A. Aktas10, V. Andreev26, T. Anthonis4, B. Antunovic27, S. Aplin10, A. Asmone34,A. Astvatsatourov4, A. Babaev25, S. Backovic31, J. Bahr39, A. Baghdasaryan38, P. Baranov26,E. Barrelet30, W. Bartel10, S. Baudrand28, S. Baumgartner40, J. Becker41, M. Beckingham10,O. Behnke13, O. Behrendt7, A. Belousov26, Ch. Berger1, N. Berger40, J.C. Bizot28,M.-O. Boenig7, V. Boudry29, J. Bracinik27, G. Brandt13, V. Brisson28, D. Bruncko16,F.W. Busser11, A. Bunyatyan12,38, G. Buschhorn27, L. Bystritskaya25, A.J. Campbell10,S. Caron1, F. Cassol-Brunner22, K. Cerny33, V. Cerny16,47, V. Chekelian27, J.G. Contreras23,J.A. Coughlan5, B.E. Cox21, G. Cozzika9, J. Cvach32, J.B. Dainton18, W.D. Dau15,K. Daum37,43, Y. de Boer25, B. Delcourt28, M. Del Degan40, A. De Roeck10,45, K. Desch11,E.A. De Wolf4, C. Diaconu22, V. Dodonov12, A. Dubak31,46, G. Eckerlin10, V. Efremenko25,S. Egli36, R. Eichler36, F. Eisele13, M. Ellerbrock13, E. Elsen10, W. Erdmann40, S. Essenov25,A. Falkewicz6, P.J.W. Faulkner3, L. Favart4, A. Fedotov25, R. Felst10, J. Ferencei16, L. Finke11,M. Fleischer10, P. Fleischmann10, G. Flucke10, A. Fomenko26, I. Foresti41, G. Franke10,T. Frisson29, E. Gabathuler18, E. Garutti10, J. Gayler10, C. Gerlich13, S. Ghazaryan38,S. Ginzburgskaya25, A. Glazov10, I. Glushkov39, L. Goerlich6, M. Goettlich10, N. Gogitidze26,S. Gorbounov39, C. Goyon22, C. Grab40, T. Greenshaw18, M. Gregori19, B.R. Grell10,G. Grindhammer27, C. Gwilliam21, D. Haidt10, L. Hajduk6, M. Hansson20, G. Heinzelmann11,R.C.W. Henderson17, H. Henschel39, O. Henshaw3, G. Herrera24, M. Hildebrandt36,K.H. Hiller39, D. Hoffmann22, R. Horisberger36, A. Hovhannisyan38, T. Hreus16, S. Hussain19,M. Ibbotson21, M. Ismail21, M. Jacquet28, L. Janauschek27, X. Janssen10, V. Jemanov11,L. Jonsson20, D.P. Johnson4, A.W. Jung14, H. Jung20,10, M. Kapichine8, J. Katzy10,I.R. Kenyon3, C. Kiesling27, M. Klein39, C. Kleinwort10, T. Klimkovich10, T. Kluge10,G. Knies10, A. Knutsson20, V. Korbel10, P. Kostka39, K. Krastev10, J. Kretzschmar39,A. Kropivnitskaya25, K. Kruger14, J. Kuckens10, M.P.J. Landon19, W. Lange39,T. Lastovicka39,33, G. Lastovicka-Medin31, P. Laycock18, A. Lebedev26, G. Leibenguth40,V. Lendermann14, S. Levonian10, L. Lindfeld41, K. Lipka39, A. Liptaj27, B. List40, J. List11,E. Lobodzinska39,6, N. Loktionova26, R. Lopez-Fernandez10, V. Lubimov25,A.-I. Lucaci-Timoce10, H. Lueders11, D. Luke7,10, T. Lux11, L. Lytkin12, A. Makankine8,N. Malden21, E. Malinovski26, S. Mangano40, P. Marage4, R. Marshall21, M. Martisikova10,H.-U. Martyn1, S.J. Maxfield18, D. Meer40, A. Mehta18, K. Meier14, A.B. Meyer11,H. Meyer37, J. Meyer10, S. Mikocki6, I. Milcewicz-Mika6, D. Milstead18, D. Mladenov35,A. Mohamed18, F. Moreau29, A. Morozov8, J.V. Morris5, M.U. Mozer13, K. Muller41,P. Murın16,44, K. Nankov35, B. Naroska11, Th. Naumann39, P.R. Newman3, C. Niebuhr10,A. Nikiforov27, D. Nikitin8, G. Nowak6, M. Nozicka33, R. Oganezov38, B. Olivier3,J.E. Olsson10, S. Osman20, D. Ozerov25, V. Palichik8, I. Panagoulias10, T. Papadopoulou10,C. Pascaud28, G.D. Patel18, M. Peez29, E. Perez9, D. Perez-Astudillo23, A. Perieanu10,A. Petrukhin25, D. Pitzl10, R. Placakyte27, B. Portheault28, B. Povh12, P. Prideaux18,A.J. Rahmat18, N. Raicevic31, P. Reimer32, A. Rimmer18, C. Risler10, E. Rizvi19,P. Robmann41, B. Roland4, R. Roosen4, A. Rostovtsev25, Z. Rurikova27, S. Rusakov26,F. Salvaire11, D.P.C. Sankey5, E. Sauvan22, S. Schatzel10, F.-P. Schilling10, D. Schmidt10,S. Schmidt10, S. Schmitt10, C. Schmitz41, L. Schoeffel9, A. Schoning40,H.-C. Schultz-Coulon14, K. Sedlak32, F. Sefkow10, R.N. Shaw-West3, I. Sheviakov26,L.N. Shtarkov26, T. Sloan17, P. Smirnov26, Y. Soloviev26, D. South10, V. Spaskov8,A. Specka29, B. Stella34, J. Stiewe14, I. Strauch10, U. Straumann41, V. Tchoulakov8,G. Thompson19, P.D. Thompson3, F. Tomasz16, D. Traynor19, P. Truol41, I. Tsakov35,G. Tsipolitis10,42, I. Tsurin10, J. Turnau6, E. Tzamariudaki27, M. Urban41, A. Usik26,

1

D. Utkin25, A. Valkarova33, C. Vallee22, P. Van Mechelen4, A. Vargas Trevino7, Y. Vazdik26,C. Veelken18, A. Vest1, S. Vinokurova10, V. Volchinski38, K. Wacker7, J. Wagner10,G. Weber11, R. Weber40, D. Wegener7, C. Werner13, M. Wessels10, B. Wessling10,C. Wigmore3, Ch. Wissing7, R. Wolf13, E. Wunsch10, S. Xella41, W. Yan10, V. Yeganov38,J. Zacek33, J. Zalesak32, Z. Zhang28, A. Zhelezov25, A. Zhokin25, Y.C. Zhu10,J. Zimmermann27, T. Zimmermann40, H. Zohrabyan38, and F. Zomer28

1 I. Physikalisches Institut der RWTH, Aachen, Germanya

2 III. Physikalisches Institut der RWTH, Aachen, Germanya

3 School of Physics and Astronomy, University of Birmingham,Birmingham, UKb4 Inter-University Institute for High Energies ULB-VUB, Brussels; Universiteit Antwerpen,Antwerpen; Belgiumc5 Rutherford Appleton Laboratory, Chilton, Didcot, UKb6 Institute for Nuclear Physics, Cracow, Polandd

7 Institut fur Physik, Universitat Dortmund, Dortmund, Germanya8 Joint Institute for Nuclear Research, Dubna, Russia9 CEA, DSM/DAPNIA, CE-Saclay, Gif-sur-Yvette, France10 DESY, Hamburg, Germany11 Institut fur Experimentalphysik, Universitat Hamburg, Hamburg, Germanya12 Max-Planck-Institut fur Kernphysik, Heidelberg, Germany13 Physikalisches Institut, Universitat Heidelberg, Heidelberg, Germanya

14 Kirchhoff-Institut fur Physik, Universitat Heidelberg, Heidelberg, Germanya

15 Institut fur Experimentelle und Angewandte Physik, Universitat Kiel, Kiel, Germany16 Institute of Experimental Physics, Slovak Academy of Sciences, Kosice, Slovak Republicf

17 Department of Physics, University of Lancaster, Lancaster, UKb18 Department of Physics, University of Liverpool, Liverpool, UKb19 Queen Mary and Westfield College, London, UKb

20 Physics Department, University of Lund, Lund, Swedeng

21 Physics Department, University of Manchester, Manchester, UKb22 CPPM, CNRS/IN2P3 - Univ. Mediterranee, Marseille - France23 Departamento de Fisica Aplicada, CINVESTAV, Merida, Yucatan, Mexicok24 Departamento de Fisica, CINVESTAV, Mexicok25 Institute for Theoretical and Experimental Physics, Moscow, Russial26 Lebedev Physical Institute, Moscow, Russiae

27 Max-Planck-Institut fur Physik, Munchen, Germany28 LAL, Universite de Paris-Sud, IN2P3-CNRS, Orsay, France29 LLR, Ecole Polytechnique, IN2P3-CNRS, Palaiseau, France30 LPNHE, Universites Paris VI and VII, IN2P3-CNRS, Paris, France31 Faculty of Science, University of Montenegro, Podgorica, Serbia and Montenegroe32 Institute of Physics, Academy of Sciences of the Czech Republic, Praha, Czech Republice,i33 Faculty of Mathematics and Physics, Charles University, Praha, Czech Republice,i34 Dipartimento di Fisica Universita di Roma Tre and INFN Roma 3, Roma, Italy35 Institute for Nuclear Research and Nuclear Energy, Sofia, Bulgariae36 Paul Scherrer Institut, Villigen, Switzerland37 Fachbereich C, Universitat Wuppertal, Wuppertal, Germany38 Yerevan Physics Institute, Yerevan, Armenia

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39 DESY, Zeuthen, Germany40 Institut fur Teilchenphysik, ETH, Zurich, Switzerlandj41 Physik-Institut der Universitat Zurich, Zurich, Switzerlandj

42 Also at Physics Department, National Technical University, Zografou Campus, GR-15773Athens, Greece43 Also at Rechenzentrum, Universitat Wuppertal, Wuppertal, Germany44 Also at University of P.J.Safarik, Kosice, Slovak Republic45 Also at CERN, Geneva, Switzerland46 Also at Max-Planck-Institut fur Physik, Munchen, Germany47 Also at Comenius University, Bratislava, Slovak Republic

a Supported by the Bundesministerium fur Bildung und Forschung, FRG, under contractnumbers 05 H1 1GUA /1, 05 H1 1PAA /1, 05 H1 1PAB /9, 05 H1 1PEA /6,05 H1 1VHA /7 and05 H1 1VHB /5b Supported by the UK Particle Physics and Astronomy ResearchCouncil, and formerly by theUK Science and Engineering Research Councilc Supported by FNRS-FWO-Vlaanderen, IISN-IIKW and IWT and byInteruniversity AttractionPoles Programme, Belgian Science Policyd Partially Supported by the Polish State Committee for Scientific Research,SPUB/DESY/P003/DZ 118/2003/2005e Supported by the Deutsche Forschungsgemeinschaftf Supported by VEGA SR grant no. 2/4067/ 24g Supported by the Swedish Natural Science Research Councili Supported by the Ministry of Education of the Czech Republicunder the projects LC527 andINGO-1P05LA259j Supported by the Swiss National Science Foundationk Supported by CONACYT, Mexico, grant 400073-Fl Partly Supported by Russian Foundation for Basic Research,grants 03-02-17291 and04-02-16445

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1 Introduction

Quantum Chromodynamics (QCD), the field theory of quark and gluon interactions, is expectedto describe the strong force between hadrons. QCD is a successful theory in the limit of shortdistances, corresponding to small values of the strong coupling constantαs, where perturbativemethods can be applied (perturbative QCD, pQCD). The bulk ofthe scattering cross section ofhadrons however, is dominated by long-range forces (“soft interactions”), where a satisfactoryunderstanding of QCD still remains a challenge. A large fraction of these soft interactions ismediated by vacuum quantum number exchange and is termed “diffractive”. In hadronic inter-actions, diffraction is well described by Regge theory, where it is due to thet-channel exchangeof a leading trajectory with vacuum quantum numbers, calledthe “Pomeron” trajectory. In thehigh energy limit, Pomeron exchange dominates over all other contributions to the scatteringamplitude and leads to an almost energy-independent total cross section. Elastic photoproduc-tion of vector mesons,γp → VM p, is a particular example for a diffractive process. Measure-ments of the cross sections for the elastic production of light vector mesons (ρ, ω, andφ) in lowQ2 electron-proton collisions at HERA as function of the photon-proton centre of mass energyWγp [1,2] have verified the expected universal Regge behaviour.

The cross section for elastic photoproduction ofJ/ψ mesons,γp → J/ψ p, on the contrary,rises steeply withWγp [3–6], incompatible with a universal Pomeron. Due to the large massof the J/ψ meson, which provides a “hard” scale (equivalent to a short range of the forcesinvolved), the elastic photoproduction ofJ/ψ mesons is expected to be described by pQCD. Inelectroproduction the photon virtualityQ2 can provide a second hard scale in addition to theJ/ψ mass, allowing the interplay between these two scales to be studied.

a)

Wγp

IP

b)J/

γ

ψ

(q)

Q2

p(P )’

e(k )’

e(k)

p(P)t

c

J/

e e

γ

ψ

gg

c

pp

Figure 1: ElasticJ/ψ production, a) in an approach based on Pomeron (IP) exchangeand b) ina pQCD approach via two gluon exchange. The kinematic variables are indicated in a).

The elastic production ofJ/ψ mesons is illustrated in figure 1. In QCD at lowest order theprocess is mediated by a colour-singlet state of two gluons (figure 1b) and the cross section isrelated to the square of the gluon density in the proton. The gluon’s momentum fractionx iskinematically related toWγp : the steep rise in the gluon density towards low values ofx thusexplains the steep rise of the cross section with increasingWγp observed in the data. Beyondthis approximation correlations between the gluons have tobe taken into account and the cross

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section for elasticJ/ψ production involves the generalised gluon density (see forexample [7]for a review).

The dependence of the elasticJ/ψ cross sections on the squared four-momentum transfert atthe proton vertex shows a fast fall with increasing|t|. This dependence can be parameterised asan exponential functionebt at low values of|t|, although other shapes have also been proposed(for example [8]). In Regge theory with at dependent Pomeron trajectory, thet dependence ofthe cross section varies withWγp, the slope parameterb increasing logarithmically withWγp

(“shrinkage” of the diffractive peak). In QCD-based models, on the other hand, the dependenceof b on Wγp is expected to be weak [9, 10]. In addition to the elastic process, in which theproton remains intact, diffractiveJ/ψ production can lead to proton dissociation,γp→ J/ψ Y ,in which a low mass baryonic stateY is produced. This process is expected to be important atlarge values of|t|.

In the past, diffractiveJ/ψ cross sections have been measured using photon and electronbeamsin fixed target experiments up to centre of mass energies of about 20 GeV. At HERA the kine-matic range is extended up to photon-proton centre of mass energies ofWγp ∼ 290GeV inphotoproduction, and in electroproduction up to photon virtualities ofQ2 . 100GeV2 [1, 3–6, 11–16]. In this paper new data are presented on theQ2, Wγp andt dependence of the crosssection for elasticJ/ψ production. The data correspond to a factor of three more integratedluminosity than our previous publication for photoproduction [4] and a factor of two more forelectroproduction [11]. The kinematic range is extended tovalues ofWγp up to 305GeV inphotoproduction, while in electroproduction the range covered is40 < Wγp < 160GeV. Fur-thermore, the angular distributions for production and decay of theJ/ψ mesons are determinedin order to extract the cross sections of longitudinally andtransversely polarised photons and totest the hypothesis ofs-channel helicity conservation (SCHC), which predicts that the helicityof theJ/ψ meson in the final state is the same as that of the initial (virtual) photon.

2 Models for Elastic J/ψ Production

Within the Regge framework (see for example [17] for a review) the cross section for diffractivephotoproduction of vector mesons at low values of|t| approximately follows a power lawσγp ∝W δγp with δ ≃ 0.2 [18]. The power is related to the Pomeron trajectoryδ ∼ 4 (αIP (t)−1) where

αIP (t) = αIP (0)+α′IP t. The existing measurements forJ/ψ mesons, however, indicate a much

steeper dependence onWγp (δ ≃ 0.8) than is predicted by the universal (“soft”) Pomeron. Thereare also indications that the slopeα′

IP , responsible for the shrinkage of the diffractive peak ofelasticJ/ψ photoproduction, is smaller [4] than the value of 0.25 GeV−2 expected from thesoft Pomeron trajectory. To overcome this difficulty, Donnachie and Landshoff have suggestedan additional “hard” Pomeron trajectory [19] for processeswhich involve a hard scale, such asthe vector meson mass or a large momentum transferQ2. With this conjecture the concept ofa single universal Pomeron trajectory becomes obsolete forhard scattering processes. It hasbecome customary, however, to introduce an effective Pomeron trajectoryα(t) = α0 + α′t,where the interceptα0 can be calculated within certain QCD models (see for example[20] fora review).

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In photoproduction ofJ/ψ mesons the massMJ/ψ may serve as a hard scale and in electro-production bothMJ/ψ andQ2. A third hard scale may be provided by a sufficiently largemomentum transfer|t| at the proton vertex [13, 14]. In the presence of a hard scale QCD fac-torisation methods (e.g. collinear factorisation,kT factorisation) may be applied. Factorisationallows the separation of the scattering amplitude into a perturbative hard scattering coefficientfunction and non-perturbative quantities, such as the input gluon density for the proton and thevector meson wave function.

Early pQCD predictions, for example [21], assume that the two exchanged gluons have the samelongitudinal momentum fractionx with respect to the proton, wherex ≃ (Q2 +M2

ψ)/(Q2 +

W 2γp), and that each of the quarks making up theJ/ψ meson carries half of the photon mo-

mentum. Such models correspond to a leading approximation in log 1/x, and at high energiesthe cross section depends on the square of the gluon density within the proton. More recentlygeneralised or “skewed” parton distributions have been considered, where the two gluons havedifferent fractional momenta [22–26].

The data presented here are compared with a pQCD model by Martin, Ryskin and Teubner(MRT [26]) which is based onkT factorisation and uses a parton-hadron duality ansatz avoid-ing the large uncertainties from the poorly knownJ/ψ meson wave function. In this model,effects beyond the leading logarithmic approximation inlogQ2 are included at the amplitudelevel, requiring an integration over the transverse momenta of the two gluons and hence the useof unintegrated gluon distributions. In the MRT calculations these distributions are derived [25]from the conventional integrated parton distributions1, as extracted from inclusive deep-inelasticscattering. The skewing effects are estimated independently by applying a factor to the ampli-tude [27]. Since these calculations only apply to the imaginary part of the scattering amplitude,dispersion relations are used to estimate the effects of thereal part. In the parton-hadron dualityapproach the correct spin-parity state (JP = 1−−) of the cc pair is projected out by using theappropriate rotation matrices in the integrals over the resonance mass region. Since the choiceof the mass range in the integration is arbitrary to some extent, the normalisation of the crosssections is predicted with limited accuracy. The overall normalisation contains additional uncer-tainties due to missing higher order corrections. The approximations used are however believedto have little influence on theWγp andQ2 dependences of the cross sections [28]. Predictionsare provided both in the photoproduction and electroproduction regimes.

The calculations by Frankfurt, McDermott and Strikman (FMS[9]) are based on the dipoleapproach. Here the exchanged photon turns into aqq pair long before the interaction with theproton. A leading logarithmic approximation for the interaction of thisqq pair, described as asmall transverse-size dipole, is used. For the interactionwith the proton two-gluon exchangeis assumed. In addition the effect of a running quark mass, aWγp dependent slope of theexponentialt distribution, and generalised gluon distributions are considered in this calculation.Similarly to the MRT calculations the model does not providean accurate normalisation of thecross section. Predictions are only available for photoproduction.

1In this model the unintegrated gluon distribution is determined from the derivative of the standard gluondistribution with respect tologQ2, i.e. essentially from the second derivative of the proton structure functionF2(x,Q

2) with respect tologQ2, which in the kinematic region of theJ/ψ analysis, at lowx andQ2, is not wellmeasured at HERA yet.

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3 Data Analysis

The data were recorded with the H1 detector in the years 1999 and 2000 when HERA wasoperated with electrons or predominantly with positrons2 of 27.5GeV and protons of920GeV.TheJ/ψ mesons are detected via their decays intoµ+µ− or e+e− pairs. They are selected fromdata corresponding to an integrated luminosity of 55 pb−1.

3.1 The H1 Detector

The H1 detector is described in detail in [29]. Charged particles are detected in the central andforward3 tracking detectors (CTD and FTD), which consist of drift andproportional chambersthat provide a polar angle coverage between7◦ and165◦. Tracks at largeθ are detected in thebackward silicon tracker (BST [30],165◦ < θ < 175◦). The central liquid argon (LAr) [29]and backward lead scintillator (SpaCal) calorimeters [31,32] cover the polar angle regions4◦ <θ < 153◦ and153◦ < θ < 177.5◦, respectively. ForQ2 & 2 GeV2 the scattered positron isdetected in the SpaCal, while the decay electrons from theJ/ψ meson are identified in theLAr and SpaCal calorimeters. Muons are identified as minimumionising particles in the LArcalorimeter or in the instrumented iron return yoke of the solenoidal magnet which surroundsthe central detector (central muon detector, CMD,4◦ < θ < 171◦).

Dissociated proton statesY with massesMY & 1.6GeV may, after a secondary interaction, bemeasured in a set of detectors in the forward direction. These are the proton remnant tagger(PRT), an array of scintillators covering0.06◦ < θ < 0.17◦, the drift chambers of the forwardmuon detector (FMD) [33] closest to the beam interaction region in the angular range3◦ < θ <17◦ and the forward region of the LAr calorimeter (θ < 10◦).

H1 uses a multi-stage trigger system. At level 1 signals fromthe CTD, SpaCal, and CMD areused to obtain the present data sets. At level 2 information from these detectors and the LArcalorimeter is used in neural network algorithms [34].

The luminosity is determined from the rate of Bethe Heitler events.

3.2 Kinematics

The kinematics of the processep→ epJ/ψ are described by the following variables: the squareof the ep centre of mass energys = (p + k)2; the negative four-momentum transfer squaredat the lepton vertexQ2 = −q2 = −(k − k′)2; the four-momentum transfer squared at theproton vertext = (p − p′)2 and the inelasticityy = (p · q)/(p · k). The four-momentak, k′,p, p′ andq refer to the incident and scattered positron, the incoming and outgoing proton (ordissociated systemY ) and the exchanged photon, respectively. The centre of massenergy ofthe photon-proton systemWγp is given byW 2

γp = (p + q)2 = ys − Q2 neglecting the protonmass.

2Hereafter the term ‘positron’ is used for all lepton beam particles, whereas ‘electron’ is used for both electronsand positrons fromJ/ψ decays.

3The positivez-axis is defined by the proton beam direction. The polar angleθ is measured with respect to thez axis andθ < 90◦ is called the ‘forward’ direction.

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In electroproduction the event kinematics are reconstructed using the double angle method [35],

y =sin θe(1− cos θψ)

sin θψ + sin θe − sin(θe + θψ)

Q2 = 4E2e

sin θψ(1 + cos θe)

sin θψ + sin θe − sin(θe + θψ).

HereEe is the energy of the incident positron andθψ andθe are the polar angles of theJ/ψmeson and the scattered positron, respectively. The variable t is calculated ast ≃ −(~pt,ψ+~pt,e)

2,where~pt,ψ is the transverse momentum of theJ/ψ meson candidate and~pt,e that of the outgoingpositron.

In photoproduction, where the positron is not observed in the central detector,y is reconstructedvia y = (E − pz)ψ/(2Ee) [36], whereE andpz denote the energy and the longitudinal com-ponent of the momentum of theJ/ψ meson. The variablet is approximated ast ≃ −p2t,ψ (seealso the section on dσ/dt below).

3.3 Monte Carlo Simulation

Monte Carlo simulations are used to calculate acceptances and the efficiencies for triggering,track reconstruction, event selection and lepton identification.

The elasticJ/ψ signal events are generated using the program DIFFVM [37] which is basedon the Vector Dominance Model and permits separate variation of the dependence onWγp, tandQ2 . The parameters are iteratively adjusted to those of the present measurements. DIF-FVM is also used to generateJ/ψ production with proton dissociation. A mass dependence ofdσ/dM2

Y ∝ f(M2Y )M

−βY is implemented, wheref(M2

Y ) = 1 for M2Y > 3.6GeV2. At lower

M2Y the functionf(M2

Y ) takes into account the production of excited nucleon states. The decayangular distributions of theJ/ψ meson are simulated assumings-channel helicity conservation.For electroproduction, radiative corrections are included using the generator HERACLES [38],where contributions up to orderα3

QED are taken into account.

The non-resonant background is estimated using the generators LPAIR [39], which simulatesthe processγγ → ℓ+ℓ− and COMPTON [40] for the QED Compton processep → eγp. Crosschecks with the generator GRAPE [41] did not show significantdeviations from the results ofLPAIR in the region of the present analysis.

For all processes, detector effects are simulated in detailwith the GEANT program [42]. Thedetector response including trigger efficiencies is tuned using independent data. Remainingdifferences are included in the systematic errors. The simulated events are passed through thesame reconstruction software as the data.

3.4 Event Selection

ElasticJ/ψ events are selected by requiring two muons or two electrons and, in the case ofelectroproduction, a scattered positron candidate. For photoproduction the absence of any such

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candidate is required. As described in table 1 four data setsare defined covering differentregions ofQ2 andWγp and corresponding to different signatures of theJ/ψ decay leptons.

For data set I (electroproduction) the scattered positron must be detected with an energy ofat least12GeV in the SpaCal and the reconstructed value ofQ2 must be within2 < Q2 <80GeV2. To suppress photoproduction background and to reduce the fraction of events withinitial state QED radiation, events are rejected if

∑

(E − pz) < 45GeV, where the sum runsover all final state particles including the scattered positron. Neglecting radiative effects thisvariable is expected to be twice the incident positron energy due to longitudinal momentumconservation.

For the selection of photoproduction events (data sets II–IV) the absence of any candidate forthe scattered positron is required, restricting the accepted range of negative four-momentumtransfer squaredQ2 to below about 1 GeV2, with 〈Q2〉 = 0.05 GeV2.

Data sets I and II (40 < Wγp < 160GeV) containJ/ψ → µ+µ− events. Exactly two oppositelycharged particles must be present in the CTD, with transverse momenta (with respect to thebeamline)pt > 0.8GeV. A reconstructed vertex within±40 cm of thez coordinate of thenominal beam interaction point is required. At least one particle must be identified as a muonin the central calorimeter or in the CMD. For data set II background from cosmic ray muons isrejected using an acollinearity cut as well as timing information from the CTD. Further detailsof this analysis may be found in [43].

Data sets III and IV are selected to cover photoproduction athigh values ofWγp, which arerelated to large polar angles of theJ/ψ decay leptons. TheJ/ψ decay intoe+e− pairs isused. Data set III (135 < Wγp < 235GeV) requires one decay electron to be measured inthe CTD coming from within±40 cm of the nominal beam interaction point and one in the

Data set I II III IVKinematic region Electroproduction Photoproduction

Q2 range[ GeV2] 2− 80 < 1

〈Q2〉 [ GeV2] 8.9 0.05Wγp [ GeV] 40− 160 135− 235 205− 305

|t| [ GeV2] < 1.2Decay channel J/ψ → µ+µ− J/ψ → e+e−

Lepton signature Track-Track Track-Cluster Cluster-ClusterLepton polarangle region[◦]

20− 160θ1 : 80− 155θ2 : 160− 177

θ1 : 160− 174θ2 : 160− 175.5

Lepton energy[ GeV]

pt > 0.8pt,1 > 0.7, p1 > 0.8E2 > 4.2

E1,2 > 4.2max(E1, E2) > 6

Elastic selection No signal in forward detectors∫

Ldt [pb−1] 54.79 30.26 26.90

Table 1: Summary of the most important event selection criteria for the four different data setstogether with the corresponding integrated luminosities.

9

backward calorimeter SpaCal. The selected polar angle regions are given in table 1. The elec-tron measured in the CTD must have a momentump1 > 0.8GeV and a transverse momentumpt,1 > 0.7GeV and must be identified by a matching electromagnetic energy deposition in thecentral calorimeter. The other electron is selected by requiring a cluster in the SpaCal with anenergyE2 > 4.2GeV. Data set IV (185 < Wγp < 305GeV) requires both electrons to bedetected as energy clusters in the SpaCal with energiesE1,2 > 4.2GeV and the more energeticcluster to be above6GeV. At least one electron must be in the acceptance region of theBSTand every electron in the BST acceptance region must be validated by a BST track from thenominal interaction point. This requirement rejects most of the non-resonant background fromCompton scattering. In both data sets III and IV the energy inthe SpaCal outside the selectedelectron cluster(s) must be negligible. Further details ofthis analysis may be found in [44].

In order to suppress background from proton dissociative orinelasticJ/ψ production, no ad-ditional tracks are allowed in the CTD or FTD and the selectedevents are required to have nosignificant signals in the forward detectors (PRT, FMD and LAr). The fraction of proton disso-ciation is further suppressed by limitingt to the range|t| < 1.2GeV2, where elastic processesare dominant. These requirements reject most of the proton dissociative background. The re-maining fraction is 14% on average, ranging from 8% at|t| ≈ 0 to 35% at|t| ≈ 1.2 GeV2.It is corrected for using the MC simulation, which is tuned togive a good description of theforward detectors. A further correction is applied to account for ψ(2S) decays intoJ/ψ andneutral mesons. This correction is estimated to be 4% for data sets I and II and approximately2% in sets III and IV, where the neutral decays are partly rejected by the cut on the energy inthe SpaCal.

Triggers based on muon and track signatures from the decay leptons are used for data sets I andII. For data set I a trigger signal is also derived from the scattered positron. The triggers fordata sets III and IV are based on signals due to theJ/ψ decay electrons from the SpaCal andthe CTD (set III) . In addition the triggers for data sets II-IV use second level triggers based onneural network algorithms.

Figure 2 shows the two-lepton invariant mass distributionsfor the four data sets. The shapesof theJ/ψ signal peaks reflect the usage of different detectors with different resolutions anda different response to electrons, muons and photons. The signal of data set III shows a tailtowards low masses due to radiative energy losses of the electron reconstructed in the trackingdetector. In all data sets the non resonant background belowtheJ/ψ signal peaks is dominatedby γγ → ℓ+ℓ−, where one photon originates from each of the positron and the proton. AthighWγp a potential source of background is Compton scatteringep → eγp where the finalstate electron and photon can form an invariant mass of the same order as theJ/ψ mass. Itis efficiently suppressed by the BST track requirements explained above. In addition the BSTtracks lead to an improved mass resolution in data set IV.

The number ofJ/ψ events is determined in each analysis bin by a fit of the sum of asignaland a background function to the dilepton mass distribution. For data sets I and II (J/ψ →µ+µ−) the signal shape is a Gaussian function, and the backgroundis fitted using a powerlaw distribution. For the signal in data set III (J/ψ → e+e−) the radiative tail is taken intoaccount by fitting a modified Gaussian distribution,f(Mee) ∝ 1

σ′exp (−(Mee − µ)2/(2 σ′))

whereσ′ = (σ + r (|Mee − µ| − Mee + µ))2. Hereµ andσ denote the peak position andthe standard deviation andr parameterises the contribution of the radiative tail. In data set IV

10

(J/ψ → e+e−) a single Gaussian function is adequate to describe the signal. In data sets III andIV the shape of the background is found to depend strongly onWγp. The shapes are reasonablywell described by the predicted shapes of the Monte Carlo simulations LPAIR and COMPTON,which are therefore used in the fit.

Data and MC simulation are compared in figure 3. Each row corresponds to one of the fourdata sets. The selected events from a mass window around the nominalJ/ψ mass (±0.2GeVin data sets I and II,±0.3GeV in data set IV and2.6 < Mee < 3.4GeV for data set III)are shown before applying the cuts on the forward detectors.For data sets I and II the non-resonant background, which is small in this kinematic region, has been subtracted and the dataare described by a combination of DIFFVM for elastic and proton dissociativeJ/ψ production.For data sets III and IV the selected events are shown but without subtracting the non resonantbackground. The data are seen to be reasonably well described by a sum of simulations forelastic and proton dissociativeJ/ψ production (DIFFVM),γγ → e+e− (LPAIR) andep→ eγp(COMPTON).

3.5 Systematic Uncertainties

The systematic uncertainties on the cross sections are dominated by detector effects whichare not perfectly modelled in the Monte Carlo simulation. Most uncertainties are obtainedby comparisons of data with simulation after tuning the detector simulation with independentdata sets. The uncertainties on the measured cross sectionsare then estimated by variationsof the simulation. In the following the main sources of the uncertainties are summarised andtypical values are given for the uncertainty on the total cross section.

• The uncertainty due to the track reconstruction efficiency in the CTD is1% per track.The track information from the BST has two sources of uncertainty: coherent signallosses (3.0%) and track reconstruction efficiency (1.5%).

• The uncertainty on the lepton identification efficiency leads to a cross section uncertaintyof 1.5% for muons and 2% for electrons measured in the CTD. The uncertainty on theenergy measurement of the decay electrons in the backward calorimeter is estimated tovary linearly from 2.7% at 3 GeV to 0.5% at 27.5 GeV from an analysis of Comptonscattering [45]. The resulting uncertainties on the cross sections vary from 1% to 7%,depending onWγp. A small additional uncertainty for data set IV arises due toan uncer-tainty of 0.3 mrad in the reconstruction of the polar angle ofthe decay electrons in theBST, leading to aWγp dependent cross section uncertainty of1− 3%.

• The uncertainties of the trigger efficiencies are determined to be1.6%, 5%, 6.5% and5%for data sets I to IV, respectively.

• The separation of elastic events from proton dissociation leads to a systematic uncertaintyof 4 − 6% due to the modelling of the response of the forward detectors, with a smalldependence onWγp and |t|. The error due to the simulation of the dependence of thecross section onMY was found to be negligible by comparison.

11

• The uncertainty in the modelling of thez position of the interaction region affects theWγp dependence of the cross section and is found to be1% on average for data sets I andII, 0.5− 2.6% for III and2.0% for IV.

• Varying the methods of determination of the number of signalevents (e.g. by using acounting method instead of fits, or by changing the shapes of the background functions),results in a1% uncertainty for data sets I and II (µ+µ−). For data sets III and IV (elec-trons) an uncertainty between 3% and 6% is estimated, which is due to the uncertaintiesin the signal and background shapes.

• For the electroproduction sample, an additional uncertainty of 4% is estimated whichcovers uncertainties in the reconstruction of the energy and angle of the scattered positron.

• Other sources of systematic uncertainties are the luminosity measurement (1.5%), theJ/ψ branching ratio (1.7%) and theψ(2S) background (0.5% for sets I and II, 1.5% onaverage for III and IV).

The systematic uncertainties are calculated in each analysis bin and the total uncertainty isobtained by adding all individual contributions in quadrature. The average values for the totalsystematic uncertainties on the cross sections are8%, 9%, 10% and11% for the data sets I toIV, respectively. The correlated part of the error, which affects all bins equally, is estimated tobe approximately 5% and is not included in subsequent fits unless mentioned otherwise.

4 Results

Cross sections are calculated for the individual data sets I–IV using the numberN of selectedevents after correcting for non resonant, proton dissociative andψ(2S) backgrounds as de-scribed in the previous section. The efficienciesA for the event selection are in general deter-mined from the MC simulation. In the equivalent photon approximation theγp cross section isgiven by:

σ(γp→ J/ψp) =N

A ·BR · L · Φγ. (1)

HereΦγ [46] denotes the photon flux in theQ2 andWγp range considered,L the integratedluminosity andBR the branching ratio for the decay of theJ/ψ mesons4.

Note that this cross section corresponds toσγp = σTγp + εσLγp, whereσTγp andσLγp are the crosssections for transversely and longitudinally polarised photons, respectively, andε is the po-larisation parameter of the virtual photon5. The parameterε depends only on the kinematics,ε = (1− y)/(1− y + 1

2y2). In the kinematic range of the present analysisε is generally above

0.95 with 〈ε〉 = 0.993. Cross sections are given at ‘bin centres’,〈Wγp〉, 〈Q2〉 and〈t〉, which aredetermined taking into account the measuredWγp ,Q2 andt dependences.

4Branching fractions(5.88± 0.10)% and(5.93± 0.10)% [47] are used forJ/ψ → µ+µ− ande+e−, respec-tively.

5The present results will be compared with results from the ZEUS collaboration [16], whereσTγp + σLγp isextracted. In the present kinematic region the difference is however small compared with the measurement errors.

12

4.1 Q2 Dependence

The cross sections for elasticJ/ψ production as a function ofQ2 atWγp = 90 GeV are listedin table 2 and shown in figure 4a. The photoproduction point isobtained from the fit describedin the next section.

A phenomenological fit of the formσγp ∝ (M2ψ + Q2)−n to the H1 data yields a value of

n = 2.486 ± 0.080(stat.) ± 0.068(syst.). This result confirms, with smaller errors, theQ2

dependence observed previously by H1 [11]. The quality of the fit is good (χ2/ndf= 0.5).Recent results from the ZEUS collaboration [6, 16] are also shown in figure 4a, which agreewell with the present data in the entire range ofQ2.

In figure 4b the pQCD calculations ‘MRT’ of Martin et al. [26] are compared with the fit resultquoted above. Results with four different gluon distributions (CTEQ6M [48], MRST02 [49],H1QCDFIT [50] and ZEUS-S [51]) derived from global fits to current inclusiveF2 measure-ments and other data are shown. A normalisation factor is determined individually for eachprediction by comparing with the data across the completeQ2 range. The different factors,which are mainly given by the photoproduction measurement,are between1.5 and2.8. Thetheoretical predictions of the shape of theQ2 dependence are consistent with the fit to the datawithin the experimental uncertainties, which are shown as agrey band in figure 4b.

4.2 Wγp Dependence

Theγp cross section for elasticJ/ψ production is presented as a function ofWγp in figures 5aand 6a and in tables 3 and 4 for photoproduction and electroproduction, respectively.

In figure 5a the photoproduction data are shown with the result of a fit of the formσγp ∝ W δγp.

Separate relative normalisation factors for the three datasets are additional fit parameters whichtake into account the correlated systematic uncertainties. The fit yields a value ofδ = 0.75 ±0.03± 0.03. The first error is obtained using only the statistical uncertainties in the fit while thesecond one reflects the systematic uncertainties. The fit result is in agreement with our previousresult [4]. Similar data from the ZEUS collaboration [6] (also shown in figure 5a) agree wellwith the present data.

A comparison with theoretical predictions is shown in figure5b, where the ratio of theory tothe fit result is shown. The uncertainty of the fit result is indicated by the grey band. The MRTpredictions are normalised using the factors obtained fromtheQ2 distributions. The same fourgluon distributions are used to calculate the respective unintegrated skewed gluon distributionswhich are required by MRT. TheWγp dependence is observed to be quite sensitive to the shapeof the gluon distribution6. While the results based on the gluon distributions CTEQ6M andZEUS-S describe the shape of the data well, the gluon distribution from the H1 fit to inclusivedata leads to a steeperWγp dependence and the one from MRST02 to a flatterWγp dependencethan is observed. The dipole model result FMS [9] based on theCTEQ4L [52] gluon densityis somewhat too steep. Note, however, that these observations are based on the central val-ues of the respective gluon distributions and do not take into account their uncertainties. Thekinematic range used in the MRT calculations extends to lower values of Bjorkenx andQ2

6For a detailed discussion of the sensitivity and the uncertainties of the model assumptions see [28].

13

than was available in the inclusive data used for the determination of the gluon densities and anextrapolation to very low values ofQ2 is performed.

In figure 6a the electroproduction cross section is shown in three bins ofQ2 (2 < Q2 < 5GeV2,5 < Q2 < 10GeV2 and10 < Q2 < 80GeV2). Data from the ZEUS experiment [16], whichare shifted to the present bin centres using theQ2 dependence measured by ZEUS, are in agree-ment. In figure 6a the results from MRT based on the gluon density CTEQ6M using the samenormalisation factor as above are also shown and give a reasonable description of the data.

TheWγp dependence is found to be similar to that obtained in photoproduction. When param-eterised in the formW δ

γp, the fits to the H1 electroproduction data yieldδ values which arecompatible with photoproduction within the rather large experimental errors (see table 5). Thefitted values forδ describing theWγp dependence of elasticJ/ψ production from this analysisand from [6,16] are displayed in figure 6b as a function ofQ2. Within the present experimentalaccuracy no dependence onQ2 is observed.

4.3 Differential Cross Sections dσ/dt

Thet dependence of the elastic cross section forJ/ψ meson production is studied in the range40 < Wγp < 160GeV for differentQ2 bins. The differential cross sections dσ/dt as derivedfrom data sets I and II are listed in table 6 and shown in figure 7a with fits of the form dσ/dt ∝ebt. The resultingb values (table 5) for electroproduction are systematicallylower than the valuefor photoproduction but are compatible within the errors.

In the context of developing the calculations using generalised parton densities, Frankfurt andStrikman [8] have proposed an alternativet dependence. It is based on a dipole function witha t dependent two-gluon form factor, leading to dσ/dt ∝ (1 − t/m2

2g)−4. In a fit to the pho-

toproduction data the two-gluon invariant massm2g is left as a free parameter. A value ofm2g = (0.679±0.006±0.011)GeV is obtained withχ2/ndf= 5.5 compared toχ2/ndf= 0.25for the exponential function. The dipole form is thus strongly disfavoured by the data.

For photoproduction, the measurement of thet dependence has been extended to significantlyhigherWγp than in our previous publication [4] using data sets III (135 < Wγp < 235 GeV) andIV (205 < Wγp < 305 GeV). Due to the reconstruction of theJ/ψ electrons via calorimetersignals the resolution inp2t,ψ, which is used to approximatet, is worse than in the track basedmeasurements. The differential cross sections dσ/dt are obtained using an unfolding proce-dure [53]. The results (last two lines in table 7) are shown infigures 7b and c with exponentialfits, which describe the data well. The resultingb values are listed in table 10 and are discussedfurther in the following section.

4.4 Effective Pomeron Trajectories

In models based on Regge phenomenology and Pomeron exchange, the energy dependence ofthe elastic cross section follows a power law:

dσdt

=dσdt

∣

∣

∣

∣

t=0,Wγp=W0

· eb0t(

Wγp

W0

)4(α(t)−1)

, (2)

14

whereα(t) = α0 + α′t describes the exchanged trajectory andb0 andW0 are constants. Equa-tion 2 relates the dependence of the differential cross section ont to that onWγp by

dσdt

(t) ∝ e(b0+4α′ ln(Wγp/W0))t. (3)

Here onlyt dependent terms are kept. In hard interactions, where Reggephenomenology with asingle universal Pomeron may no longer be applicable, an ‘effective Pomeron trajectory’ [20] isnevertheless often extracted in order to describe the dependence of the differential cross sectionsonWγp andt. For the determination of this effective trajectory, a double differential analysisis performed in which the differential cross section dσ/dt is measured in bins ofWγp and t.The measurements are displayed in figures 8a and b for photoproduction and electroproduction,respectively (tables 7 and 8). First, one-dimensional fits of the formW

4(α(〈t〉)−1)γp to the cross

sections in each|〈t〉| bin are performed. The results, which are listed in table 9 and displayed assolid lines in figures 8a and b, describe the data well. In figures 9b and c the one-dimensionalfit results forα(t) are compared with recent results [6,16] from the ZEUS collaboration, whichare in good agreement.

A two-dimensional fit of the function given in equation 2 to the data yields values forb0, α0

andα′. The parameterW0 is arbitrarily chosen to be90GeV; the fit result does not dependon this choice. As described before, different normalisations are allowed for the different datasets in the fit. Figure 9a shows the result of the two-dimensional fit forα(t) as solid and dashedlines for photoproduction and electroproduction, respectively. Error bands corresponding to onestandard deviation are shown, taking the correlation betweenα0 andα′ into account. The resultsfor α(〈t〉) from the one-dimensional fits are shown as points with error bars for comparison.Good internal consistency is observed.

The results of the two-dimensional fits are listed in the following table.

Q2 [ GeV2] b0 [ GeV−2] α0 α′ [ GeV−2]

. 1 4.630± 0.060+0.043−0.163 1.224± 0.010± 0.012 0.164± 0.028± 0.030

2− 80 3.86± 0.13± 0.31 1.183± 0.054± 0.030 0.019± 0.139± 0.076

Here the first errors are statistical and the second reflect the systematic uncertainties.

TheWγp dependence of the cross section is predominantly determined byα0 and the fit valueslead to aWγp dependence very similar to the parameterisation withδ discussed above. Theparameterα′ relates thet andWγp dependences and if non-zero leads to the ‘shrinkage’ of thediffractive peak. For photoproductionα′ is larger than zero by four standard deviations and istwo standard deviations below the value of0.25 GeV−2 obtained for the soft Pomeron in [54].For electroproductionα′ is compatible with 0, which matches the expectation in [20],but dueto the errorsα′ is also compatible with the value measured for photoproduction.

Alternatively the value ofα′ can be measured using the dependence of thet slope parameter onWγp , usingb(Wγp) = b0 + 4α′ ln(Wγp/W0). Exponential fits of the formebt to the measureddifferential cross sections dσ/dt in bins ofWγp are performed and the resulting values forb aredisplayed in figure 10a and b and listed in table 10 for photoproduction and electroproduction.

15

For photoproduction theb values are seen to increase withWγp. Theseb values are independentof normalisation uncertainties between data sets. The curves in figure 10a and b show thecorresponding resultb(Wγp) from the two-dimensional fit described above.

In figure 10a photoproduction results for the slope parameter from the ZEUS experiment [6] ina similar kinematic region are also shown. They show a similar dependence onWγp but are onaverage0.5 GeV−2 lower. This difference in the absolute size ofb may be due to differences inthe handling of the background from proton dissociative events, which has a much shallowerbslope than for the elastic case (1.6 GeV−2 [13]).

4.5 Helicity Studies

The assumption that theJ/ψ meson observed in the final state keeps the helicity of the photonis referred to ass-channel helicity conservation (SCHC). This assumption can be tested bymeasurements of the angles in the production and decay of theJ/ψ meson. If SCHC holds, theangular analysis leads to a separation of the cross sectionsdue to longitudinally and transverselypolarised photons which are both predicted in the MRT calculations.

µ+

µ−

µ− µ+− production

plane

ψJ/

− decayplane

θ

ψJ/

ψJ/

ψJ/

*

e

e’ p

p’

γ

γ

e − scatteringplane

p centre of mass frame

Φ

φ*ψ rest frameJ/

p frameγin

direction

Three angles are defined, which are illustrated in the figure above. θ∗ is the polar angle of thedecay muon with the charge of the beam lepton in theJ/ψ rest frame.θ∗ = 0◦ corresponds tothe flight direction of theJ/ψ in theγp centre of mass frame.φ∗ is the angle between theJ/ψproduction plane, defined by the exchanged photon and theJ/ψ meson, and the decay planein theγp centre of mass frame.Φ is the angle between the scattering plane of the beam leptonand theJ/ψ production plane. The angleΦ can only be measured when the scattered electronis observed, i.e. for electroproduction. In the case of SCHCand natural parity exchange, theangular distributions of theJ/ψ production and decay are functions ofcos θ∗ andΨ = φ∗ − Φonly [55].

The angular distributions are expected to be similar for elastic and proton dissociative pro-cesses. This expectation is verified within the present statistical accuracy. Therefore, in orderto increase the statistics, the cross sections differential in the angles are derived using a dataset, which includes proton dissociative events in additionto data sets I and II. The selection ofproton dissociative events is similar to that for data sets Iand II (section 3.4 and table 1), butnow a signal in one of the forward detectors is required and one additional track is allowed with

16

θ < 20◦. Non-diffractive events are rejected by requiringz > 0.95, wherez = EJ/ψ/Eγ in theproton rest frame. Furthermore, an increasedt range,|t| < 5GeV2, is allowed, since the protondissociativeJ/ψ cross section shows a flattert dependence, with an exponential slope of about1.6 GeV−2 [13].

In the present analysis, differential cross sections for the four angles are used to measure com-binations of seven of the 15 spin-density matrix elements, which describe the spin structure ofthe interaction completely7. The measured angular distributions and their dependence on thespin-density matrix elementsrikλ(γ)λ(ψ) are [55]:

dσdcos θ∗

∝ 1 + r0400 +(

1− 3r0400)

cos2 θ∗ (4)

dσdφ∗

∝ 1 + r041−1 cos(2φ∗) (5)

dσdΨ

∝ 1− εr11−1 cos(2Ψ) (6)

dσdΦ

∝ 1− ε(

r100 + 2r111)

cos 2Φ +√

2ε (1 + ε)(

r500 + 2r511)

cosΦ. (7)

Hereε is the polarisation parameter of the virtual photon.

Figures 11a and b show the differentialγp cross sections dσ/dcos θ∗ and dσ/dφ∗ in four binsof Q2. Figures 11c and d show the differential cross sections dσ/dΨ and dσ/dΦ in three binsof Q2. The results of fits of equations 4, 5, 6 and 7 are shown as full lines. In the fits, thespin-density matrix elements or the combinationsr100 + 2r111 andr500 + 2r511 for equation 7 arefree parameters. In figure 11b and d results from a fit assumingSCHC are also shown.

The spin density matrix elements, which are determined by the fits, are shown in figures 12a–e as functions ofQ2 (tables 11 and 12). The analysis is also performed in bins of|t| andthe resulting spin density matrix elements are displayed infigures 12f–j (tables 13 and 14) asfunctions of|t|.

Results from the ZEUS experiment [5,6,15,16] are also shownin figures 12a, b, c and f, whichare in good agreement with the present results. In figures 12b, d, e, g, i and j the expectationfrom SCHC, namely 0, matches the data well. SCHC yields a relation between two spin densitymatrix elements:r11−1 = (1− r0400)/2. This is observed to be fulfilled within errors.

In the case of SCHC, the matrix elementr0400 provides a direct measurement ofR, the ratio ofthe cross sections for longitudinal and transverse polarised photons,σL andσT , respectively:

R =σL

σT=

1

ε

r04001− r0400

.

The values ofR are presented in figure 13a and in table 11. For comparison thepredictionfrom MRT [26] is shown, which depends only weakly on the gluondensity. In figure 13a the

7Spin density matrix elementsriλ(γ)λ(ψ) or rik

λ(γ)λ(ψ) are linear combinations of the transition amplitudes

T i,kλ(γ)λ(ψ) from a photon of helicityλ(γ) to aJ/ψ of helicity λ(ψ).

17

gluon density from CTEQ6M is chosen with the normalisation as before, which gives the bestdescription of theQ2 andWγp dependences of the cross sections. The prediction is somewhatabove the data but still describes theQ2 dependence reasonably well. Similar results from [6,16]agree also with the present data.

The values ofR can be used to derive the cross sectionsσL andσT using the relationshipσγp =σTγp + εσLγp. The results are shown in figure 13b as a function ofQ2. σT dominates at lowQ2,while atQ2 ∼M2

ψ bothσT andσL are of similar magnitude. The MRT predictions are comparedwith the data using different gluon density parameterisations. The differences between thepredictions are not very large. All gluon density parameterisations give a reasonable descriptionof the data, althoughσL is somewhat above the data forQ2 & 3 GeV2.

In brief, the helicity studies show consistency with SCHC within experimental errors. The ratioof cross sections for longitudinally and transversely polarised photons is extracted and itsQ2

dependence is found to be reasonably described by the MRT calculations.

5 Summary

New measurements are presented of elasticJ/ψ photoproduction and electroproduction in theranges40 < Wγp < 305GeV and40 < Wγp < 160GeV, respectively8.

The cross sectionσ(γp → J/ψp) is measured as a function ofQ2 in the range0 < Q2 <80GeV2, and a fit of the formσγp ∝ (M2

ψ+Q2)−n yields a value ofn = 2.486±0.080(stat.)±

0.068(syst.). The shape of theQ2 distribution is well described by a perturbative QCD cal-culation by Martin, Ryskin and Teubner (MRT), almost independently of the gluon densitydistribution used.

The photoproduction cross section is measured as a functionof the photon-proton centre ofmass energyWγp in the range40 < Wγp < 305GeV, and can be parameterised asσγp ∝ W δ

γp

with δ = 0.754± 0.033(stat.)± 0.032(syst.). The results forδ in electroproduction, measuredin the range40 < Wγp < 160GeV, are consistent with those in photoproduction and noQ2

dependence is observed within experimental errors. Predictions of theWγp dependence of thecross section in pQCD-based models depend strongly on the gluon distribution, as can be seenexplicitly in the MRT model. A good description of the shape of the data can currently beachieved only with some gluon parameterisations. This demonstrates the potential to constrainthe gluon distribution with the elasticJ/ψ data in a kinematic region (lowx, low Q2 ) wherefits from inclusive data yield gluon distributions with large uncertainties.

The differential cross section dσ/dt for elasticJ/ψ photoproduction for|t| ≤ 1.2GeV2 is mea-sured in the extended range of40 ≤ Wγp ≤ 305 GeV. A single exponential function yieldsa good description of dσ/dt in this range, while a functional form based on a dipole functionis strongly disfavoured. The slope parameterb of the exponential shows a dependence onWγp

which is weaker than expected from soft Pomeron phenomenology, but is clearly positive, lead-ing to shrinkage of the diffractive peak. The slope parameter b in electroproduction agrees withthe photoproduction values within errors, but has a tendency to decrease with increasingQ2 .

8The results of the present analysis agree within errors withour previous results. We consider the new data tosupersede them due to improved statistics and better understanding of the detector efficiencies.

18

Effective Pomeron trajectoriesα0 + α′t for elasticJ/ψ photoproduction and electroproductionare determined from a simultaneous analysis of dσ/dt as a function ofWγp and|t|. The elec-troproduction and photoproduction results are consistentwith each other within errors. Thetrajectory for photoproduction has at slope which is two standard deviations below the softPomeron value but four standard deviations above zero.

Finally, the helicity structure of diffractiveJ/ψ production is analysed as a function ofQ2 and|t|. No evidence is found for a violation ofs-channel helicity conservation (SCHC). AssumingSCHC, the ratio of the longitudinal to the transverse polarised photon cross sections is deter-mined as a function ofQ2 and is found to be consistent with QCD calculations.

Acknowledgments

We are grateful to the HERA machine group whose outstanding efforts have made this experi-ment possible. We thank the engineers and technicians for their work in constructing and main-taining the H1 detector, our funding agencies for financial support, the DESY technical staff forcontinual assistance and the DESY directorate for support and for the hospitality which theyextend to the non DESY members of the collaboration. We thankM. Strikman and T. Teubnerfor valuable discussions and for making their theoretical predictions available.

References

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22

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Figure 2: The dilepton invariant mass distributions (data and fits) in the four kinematic regionsdefined in table 1.

23

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Figure 3: Observed event distributions for the four data sets (points) defined in table 1, omit-ting the forward detector cuts against proton dissociativeevents. The first two rows correspondto the selectedJ/ψ → µ+µ− candidates where the small non-resonant background has beensubtracted. The data are shown with the elastic (signal) simulations (DIFFVM el., white area)and proton dissociation MC (DIFFVM pdiss., shaded area). Rows three and four correspond toJ/ψ → e+e− candidates, where the non-resonant background is not subtracted. Here, in addi-tion to the elastic and proton dissociativeJ/ψ simulations the contributions fromγγ → e+e−

(LPAIR) and Compton scattering (COMPTON) are shown. The normalisations are obtainedfrom a fit of the overall mass peak of each data set. The variablesWγp , Q2, t andp2t,ψ aredefined in the text.θµ refers to the decay muons of data set II. In row threeθ1 andθ2 refer tothe decay electrons which are selected in different polar angular regions. In row fourE1 andE2

refer to the energies of the decay electrons.

24

σ(γp

→ J

/ψp)

[nb]

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Wγp= 90 GeV

H1

a)

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σ / F

it

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Fit + fit uncertaintyMRT (MRST02)MRT (CTEQ6M)MRT (ZEUS-S)MRT (H1 QCD Fit)

b)

Figure 4: a) Total cross section for elasticJ/ψ production as a function ofQ2 in the range|t| < 1.2GeV2 at Wγp = 90GeV. The inner error bars show the statistical errors, whilethe outer error bars show the statistical and systematic uncertainties added in quadrature. Thesolid line is a fit to the H1 data of the formσγp ∝ (M2

ψ + Q2)−n. Data from the ZEUSexperiment [6, 16] are also shown. b) The ratio of the MRT calculations [26] to the fit froma). The MRT QCD predictions are based on different gluon distributions [48–51]. The curvesare individually normalised to the measurements across thecompleteQ2 range yielding factorsbetween 1.5 and 2.8. The shaded band represents the uncertainty of the fit result.

25

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it

Fit + fit uncertaintyMRT (H1 QCD Fit)FMS (CTEQ4L, λ=4)MRT (ZEUS-S)MRT (CTEQ6M)MRT (MRST02)

b)

Figure 5: a) Total cross sections for elasticJ/ψ production as a function ofWγp in the range|t| < 1.2GeV2 in photoproduction. The inner error bars show the statistical errors, while theouter error bars show the statistical and systematic uncertainties added in quadrature. The solidline shows a fit to the H1 data of the formσ ∝ W δ

γp. Results from the ZEUS experiment [6]in a similar kinematic range are also shown. b) The ratio of theoretical predictions to the fitto the H1 data in a). The shaded band represents the uncertainties of the fit result. Predictionsfrom MRT QCD calculations [26] and a dipole model (FMS, [9]) based on different gluondistributions [48–52] are shown. For the MRT curves the normalisation factors determinedfrom theQ2 distributions are used. The FMS prediction is normalised tothe fit result atWγp =90GeV.

26

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σ(γp

→ J

/ψp)

[nb]

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[GeV2]

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b)

Figure 6: a) Total cross sections for elasticJ/ψ production as a function ofWγp in the range|t| < 1.2GeV2 in electroproduction in three bins ofQ2. 〈Q2〉 indicates the bin centre value intheQ2 range considered. The solid lines show fits to the H1 data of the formσ ∝ W δ

γp. Thedashed curves show the MRT QCD prediction based on the gluon distribution CTEQ6M [48]with the normalisation factors from the fit to theQ2 distribution. Results from the ZEUS exper-iment [16] in a similar kinematic range are also shown. They have been scaled to the given〈Q2〉values using theQ2 dependence measured by ZEUS. b) The fit parameterδ as a function ofQ2.The inner error bars show the statistical error, while the outer error bars show the statistical andsystematic uncertainties added in quadrature.

27

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Figure 7: Differential cross section dσ/dt for elasticJ/ψ production as a function of|t| a) infour bins ofQ2 in the range40 < Wγp < 160GeV. 〈Q2〉 indicates the bin centre value in theQ2 range considered. The inner error bars show the statisticalerror, while the outer error barsshow the statistical and systematic uncertainties added inquadrature. The solid lines show fitsto the data of the form dσ/dt ∝ ebt. The dashed curve shows the result of a fit proposed byFrankfurt and Strikman [8]. Figures b) and c) show the photoproduction measurements in theranges135 < Wγp < 235 GeV and205 < Wγp < 305 GeV.

28

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Figure 8: Differential cross section dσ/dt for elasticJ/ψ production as a function ofWγp inbins of |t| a) in photoproduction and b) in electroproduction.〈Q2〉 and〈|t|〉 indicate the bincentre values in the ranges considered. The inner error barsshow the statistical error, while theouter error bars show the statistical and systematic uncertainties added in quadrature. The solidlines show the results of one-dimensional fits of the formσ ∝W

4(α(〈t〉)−1)γp in eacht bin.

29

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Figure 9: a) The effective trajectoryα(t) as a function of|t| in the range40 < Wγp < 305GeVfor photoproduction (〈Q2〉 = 0.05 GeV2) and40 < Wγp < 160GeV for electroproduction(〈Q2〉 = 8.9 GeV2). The data points are the results of the one-dimensional fitsshown infigure 8. The inner error bars show the statistical error, while the outer error bars show thestatistical and systematic uncertainties added in quadrature. The solid and dashed lines showthe results of two-dimensional fits (equation 2) together with 1σ-error bands, which take thecorrelation between the fit parameters into account. A comparison with the results of the ZEUScollaboration [6, 16] is shown in b) and c) for photoproduction and electroproduction respec-tively. The data in [16] are derived at slightly different values of〈Q2〉. The lines are resultsfrom the two-dimensional fits.

30

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4

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γp)

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Figure 10: The values of thet slope parameterb(Wγp) as a function ofWγp in the range|t| <1.2GeV2 for a) photoproduction and b) electroproduction.〈Q2〉 indicates the bin centre valuein theQ2 range considered. The data points are the results of one-dimensional fits of the formdσ/dt ∝ ebt in Wγp bins. The inner error bars show the statistical errors, while the outer errorbars show the statistical and systematic uncertainties added in quadrature. The solid lines showthe results of the two-dimensional fits (equation 2) as in figure 9. In a) the data are comparedwith results from the ZEUS collaboration [6].

31

-1 -0.5 0 0.5 1cosθ*

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cosθ

* [n

b]

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Fit ∝ 1-ε(r00+2r11)cos2Φ 1 1 ______ +√2ε(1+ε)(r00+2r11)cosΦ 5 5

SCHC

d)

Figure 11: The differential cross sections for diffractive(elastic and proton dissociative)J/ψproduction as functions of the four anglesθ∗, φ∗, Ψ andΦ defined in the text. The data areshown in bins ofQ2 for the range40 < Wγp < 160GeV and |t| < 5GeV2. 〈Q2〉 indicatesthe bin centre value in theQ2 range considered. The inner error bars show the statisticalerror,while the outer error bars show the statistical and systematic uncertainties added in quadrature.The solid lines represent the results of fits to the data. The dashed lines in b) and d) are fitsassumings-channel helicity conservation.

32

0

0.5 H1 a)r00 04

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r1-1 04SCHC

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r1-1 1

H1 PhotoproductionH1 ElectroproductionZEUS PhotoproductionZEUS Electroproduction

0

0.5d)

r00+2r11 5 5

Q2 [GeV2]

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e)r00+2r11 1 1

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0.5f)

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j)r00+2r11 1 1

Figure 12: Spin-density matrix elements as functions ofQ2 (a-e) and|t| (f-j) for the range40 < Wγp < 160GeV. The data points are the results of fits of equations 4- 7 to thedata shownin figure 11. The inner error bars show the fit result includingonly the statistical error, whilethe outer error bars also include the systematic uncertainties. The expectations from SCHC areshown as solid lines. The results from the ZEUS collaboration are also shown, ( a), c) andf) [6,16] and b) [5,15]).

33

-0.5

0

0.5

1

1.5

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R=

σ L/σ T

0.1 1 10

H1ZEUSMRT(CTEQ6M)

H1a)

0

20

40

60

80

Q2 [GeV2]

σ [n

b]

0.1 1 10

MRT(CTEQ6M)MRT(MRST02)MRT(H1 QCD Fit)

σT

σL

H1

b)

Figure 13: a) RatioR = σL/σT as a function ofQ2 for the range40 < Wγp < 160GeV and|t| < 5GeV2. The data are compared with the result of a MRT calculation [26] based on theCTEQ6M [48] gluon distribution. Also shown are results fromthe ZEUS collaboration [6,16].b) The cross sections for longitudinally and transversely polarised photonsσL and σT as afunction ofQ2. The MRT QCD calculations based on different gluon distributions ( [48–50])are also shown with the same normalisation factors as derived from figure 4. The inner errorbars show the statistical errors, while the outer error barsshow the total errors.

34

Q2 〈Q2〉 σ[ GeV2] [ GeV2] [ nb]

. 1 0.05 73.1± 1.1± 6.42 − 3.2 2.5 37.3± 3.9± 3.6

3.2 − 5.0 4.0 31.7± 2.7± 3.05.0 − 8.0 6.3 21.8± 2.4± 2.18.0 − 12.7 10.0 13.3± 1.8± 1.312.7 − 20.1 15.8 7.53± 1.24± 0.7220.1 − 31.8 25.0 3.43± 0.81± 0.3331.8 − 80.0 47.3 0.60± 0.24± 0.06

Table 2: Cross section for the elastic processγp → J/ψp measured in bins ofQ2 for Wγp =90GeV and for|t| < 1.2GeV2. 〈Q2〉 indicates the bin centre value in theQ2 range considered.The first error is statistical and the second the total systematic uncertainty.

Data set Wγp 〈Wγp〉 σ[ GeV] [ GeV] [ nb]

II 40 − 50 44.8 46.0± 2.4± 4.050 − 60 54.8 48.5± 2.3± 4.360 − 70 64.8 59.7± 2.8± 5.370 − 80 74.8 62.7± 3.2± 5.580 − 90 84.9 72.6± 3.4± 6.490 − 100 94.9 78.6± 3.7± 6.9100 − 110 104.9 82.6± 4.0± 7.3110 − 130 119.5 91.5± 3.5± 8.1130 − 160 144.1 98.3± 4.4± 8.7

III 135 − 155 144.9 98.6± 6.6± 9.6155 − 170 162.5 114± 8± 11170 − 185 177.3 126± 8± 12185 − 205 194.8 143± 10± 15205 − 235 219.6 187± 14± 25

IV 205 − 235 219.6 133± 10± 18235 − 255 244.8 171± 13± 17255 − 280 267.2 173± 13± 18280 − 305 292.3 194± 19± 23

Table 3: Photoproduction cross section for the elastic processγp → J/ψp in bins ofWγp for|t| < 1.2GeV2 using the data sets II-IV (table 1).〈Wγp〉 indicates the bin centre value in theWγp range considered. The first error on the cross section is statistical and the second the totalsystematic uncertainty. Note that there is an overlapping bin between data sets III and IV at〈Wγp〉 = 219.6 GeV, which is averaged for figure 5 toσ = 151± 8± 20 nb.

35

Q2 〈Q2〉 Wγp 〈Wγp〉 σ[ GeV2] [ GeV2] [ GeV] [ GeV] [ nb]

2− 5 3.2 40 − 70 53.3 25.1± 2.9± 2.470 − 100 83.9 30.0± 3.4± 2.9100 − 130 114.1 41.5± 5.1± 4.0130 − 160 144.2 45.0± 8.8± 4.5

5− 10 7.0 40 − 70 53.3 12.9± 2.5± 1.270 − 100 83.9 14.5± 2.5± 1.4100 − 130 114.1 24.7± 4.1± 2.4130 − 160 144.2 24.1± 6.2± 2.5

10− 80 22.4 40 − 70 53.4 3.19± 0.69± 0.3170 − 100 83.9 4.04± 0.70± 0.39100 − 130 114.1 5.29± 1.0± 0.5130 − 160 144.2 6.10± 1.6± 0.6

Table 4: Total cross section for the elastic processγp→ J/ψpmeasured in bins ofQ2 andWγp

for |t| < 1.2GeV2. 〈Q2〉 and〈Wγp〉 are the bin centre values in the indicated ranges. The firsterror on the cross section is statistical and the second the total systematic uncertainty.

Q2 [ GeV2] 〈Q2〉 [ GeV2] δ b [ GeV−2]

. 1 0.05 0.75± 0.03± 0.03 4.57± 0.06+0.11−0.18

2 − 5 3.2 0.67± 0.20± 0.14 4.11± 0.26± 0.375 − 10 7.0 0.83± 0.31± 0.15 3.50± 0.50± 0.4910 − 80 22.4 0.69± 0.32± 0.14 3.49± 0.45± 0.33

Table 5: The parametersδ (σ ∝ W δγp) andb (dσ

dt∝ ebt) measured in bins ofQ2 in the range

40 < Wγp < 160GeV and|t| < 1.2GeV2. The values〈Q2〉 indicate the bin centre value in theQ2 range considered. The first error is statistical and the second systematic.

36

Q2 〈Q2〉 |t| 〈|t|〉 dσ/dt[ GeV2] [ GeV2] [ GeV2] [ GeV2] [ nb/GeV2]

. 1 0.05 0 − 0.07 0.03 285± 9± 250.07 − 0.14 0.10 180± 7± 160.14 − 0.21 0.17 130± 6± 110.21 − 0.30 0.25 92.1± 4.0± 8.10.30 − 0.40 0.35 61.2± 3.1± 5.40.40 − 0.60 0.49 32.5± 1.5± 2.90.60 − 0.90 0.73 10.60± 0.60± 0.900.90 − 1.20 1.03 2.70± 0.20± 0.30

2− 5 3.2 0 − 0.08 0.04 107± 14± 100.08 − 0.18 0.13 95.1± 11.0± 9.10.18 − 0.38 0.27 40.2± 5.4± 3.90.38 − 1.20 0.68 8.04± 1.05± 0.77

5− 10 7.0 0 − 0.08 0.04 78.6± 13.2± 7.50.08 − 0.18 0.13 27.7± 5.7± 2.70.18 − 0.38 0.27 18.9± 3.7± 1.80.38 − 1.20 0.68 5.21± 0.96± 0.50

10− 80 22.4 0 − 0.08 0.04 15.0± 3.1± 1.40.08 − 0.18 0.13 8.90± 2.14± 0.850.18 − 0.38 0.27 4.55± 0.93± 0.440.38 − 1.20 0.68 1.36± 0.25± 0.13

Table 6: Differential cross section for the elastic processγp → J/ψp measured in bins ofQ2

and|t| in the range40 < Wγp < 160GeV using the data sets I and II (table 1).〈Q2〉 and〈|t|〉are the bin centre values in the indicated ranges. The first error on the cross section is statisticaland the second is the total systematic uncertainty.

dσ/dt [nb/GeV2]

〈Wγp〉 |t| 0− 0.07 0.07 − 0.14 0.14− 0.30 0.30 − 0.60 0.60 − 1.20[GeV] [GeV2]

45 182± 20± 16 115± 15± 10 64.9± 6.7 ± 5.7 35.5 ± 3.6± 3.1 5.7± 0.8± 0.555 208± 20± 18 118± 14± 10 69.6± 6.9 ± 6.1 35.6 ± 3.4± 3.1 5.5± 0.7± 0.565 225± 23± 20 169± 18± 15 107.1 ± 9.4± 9.4 34.3 ± 3.7± 3.0 6.2± 0.9± 0.575 321± 31± 28 151± 19± 13 93.4± 9.5 ± 8.2 38.9 ± 4.3± 3.4 7.1± 1.1± 0.685 292± 29± 26 178± 20± 16 132± 11± 12 41.4 ± 4.4± 3.6 8.2± 1.1± 0.795 326± 32± 29 224± 24± 20 135± 12± 12 46.3 ± 4.8± 4.1 7.5± 1.1± 0.7105 392± 37± 34 224± 26± 20 125± 12± 11 48.5 ± 5.3± 4.3 7.7± 1.1± 0.7119 376± 31± 33 265± 24± 23 142± 11± 12 60.9 ± 4.9± 5.4 8.2± 1.0± 0.7144 458± 42± 40 267± 29± 23 167± 14± 15 51.5 ± 5.3± 4.5 8.4± 1.1± 0.7181 537± 28± 68 427± 18± 46 202 ± 9± 25 67.0 ± 4.1± 8.8 10.7 ± 1.1± 1.7251 744± 48± 88 573± 28± 75 246± 13± 32 71.6 ± 5.5± 9.8 13.1 ± 1.8± 1.9

Table 7: Differential photoproduction cross sections dσ/dt for the elastic processγp → J/ψpmeasured in bins ofWγp and|t| using data sets II-IV (table 1). The first error is statistical andthe second the total systematic uncertainty.

37

dσ/dt [nb/GeV2]〈Wγp〉 [ GeV] |t| [ GeV2] 0− 0.1 0.1− 0.3 0.3− 1.2

57 33.3± 4.9± 3.2 17.3± 2.2± 1.7 3.4± 0.5± 0.398 51.3± 6.7± 4.9 30.1± 3.5± 2.9 5.5± 0.7± 0.5140 60± 12± 6 31.5± 5.8± 3.0 6.0± 1.1± 0.6

Table 8: Differential electroproduction (〈Q2〉 = 8.9 GeV2) cross section dσ/dt for the elasticprocessγp→ J/ψp measured in bins ofWγp and|t| using data set I (table 1). The first error isstatistical and the second the total systematic uncertainty.

〈Q2〉 |t| 〈|t|〉 α(〈|t|〉)[ GeV2] [ GeV2] [ GeV2]

0.05 0 − 0.07 0.03 1.202± 0.012± 0.0170.07 − 0.14 0.10 1.240± 0.012± 0.0190.14 − 0.30 0.22 1.195± 0.011± 0.0160.30 − 0.60 0.43 1.121± 0.013± 0.0180.60 − 1.20 0.84 1.117± 0.021± 0.019

8.9 0 − 0.1 0.05 1.173± 0.064± 0.0380.1 − 0.3 0.19 1.185± 0.054± 0.0370.3 − 1.2 0.64 1.168± 0.059± 0.037

Table 9: The effective Pomeron trajectoriesα(t) derived from one-dimensional fits of theWγp

dependence in bins oft, in the rangesQ2 . 1GeV2, 40 < Wγp < 305GeV (photoproduction)and2 < Q2 < 80GeV2, 40 < Wγp < 160GeV (electroproduction). The values〈Q2〉 and〈|t|〉are the bin centre values in the indicated ranges. The first error is statistical and the secondsystematic.

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Data set 〈Q2〉 Wγp 〈Wγp〉 b(〈Wγp〉)[ GeV2] [ GeV] [ GeV] [ GeV−2]

I 8.9 40 − 80 57.3 3.77± 0.34± 0.3380 − 120 98.2 3.79± 0.29± 0.32120 − 160 139.6 3.84± 0.45± 0.33

II 0.05 40 − 50 44.8 4.13± 0.20+0.14−0.27

50 − 60 54.8 4.30± 0.19+0.14−0.31

60 − 70 64.8 4.57± 0.20+0.14−0.17

70 − 80 74.8 4.46± 0.24+0.15−0.46

80 − 90 84.9 4.45± 0.20+0.15−0.25

90 − 100 94.9 4.72± 0.21+0.15−0.19

100 − 110 104.9 4.79± 0.22+0.15−0.36

110 − 130 119.5 4.71± 0.16+0.14−0.18

130 − 160 144.1 4.95± 0.19+0.15−0.30

III 0.05 135 − 235 180.6 5.08± 0.14+0.25−0.27

IV 0.05 205 − 305 250.7 5.41± 0.20+0.29−0.40

Table 10: The slope parameterb derived from one-dimensional fits to thet dependence mea-sured in bins ofWγp. The values〈Q2〉 and 〈Wγp〉 are the bin centre values in the indicatedranges. The first error is statistical and the second systematic.

〈Q2〉 r0400 r041−1 R[ GeV2]

0.05 −0.030± 0.016± 0.027 0.020± 0.016± 0.042 −0.030+0.015+0.026−0.015−0.025

3.2 0.049± 0.079± 0.050 −0.129± 0.070± 0.039 0.052+0.096+0.059−0.081−0.053

7.0 0.19± 0.14± 0.06 −0.017± 0.10± 0.04 0.23+0.25+0.09−0.18−0.08

22.4 0.38± 0.16± 0.06 −0.04± 0.12± 0.04 0.62+0.59+0.17−0.34−0.14

Table 11: The spin-density matrix elements,r0400 andr041−1, and the ratio of cross sections oflongitudinally and transversely polarised photonsR as a function ofQ2 in the range|t| <5GeV2 and40 < Wγp < 160GeV. The values〈Q2〉 indicate the bin centre values in theQ2

range considered. The first error is statistical and the second systematic.

〈Q2〉 r11−1 r100 + 2r111 r500 + 2r511[ GeV2]

3.2 0.149± 0.077± 0.064 0.026± 0.035± 0.026 −0.035± 0.072± 0.0557.0 0.43± 0.11± 0.06 0.062± 0.054± 0.028 −0.16± 0.12± 0.0622.4 0.53± 0.13± 0.05 0.026± 0.069± 0.031 0.04± 0.16± 0.06

Table 12: The spin-density matrix elementr11−1 and the combined elementsr100 + 2r111 andr500 +2r511 as a function ofQ2 in the range|t| < 5GeV2 and40 < Wγp < 160GeV. The values〈Q2〉 indicate the bin centre value in theQ2 range considered. The first error is statistical andthe second systematic.

39

〈Q2〉 [ GeV2] 〈|t|〉 [ GeV2] r0400 r041−1

0.05 0.03 0.003± 0.039± 0.028 −0.011± 0.036± 0.0300.10 0.011± 0.043± 0.029 −0.041± 0.042± 0.0300.22 0.026± 0.036± 0.028 0.104± 0.035± 0.0290.43 0.013± 0.037± 0.029 0.025± 0.037± 0.0300.84 0.047± 0.041± 0.029 0.064± 0.047± 0.0341.8 0.066± 0.061± 0.028 −0.010± 0.060± 0.0303.5 0.018± 0.081± 0.028 −0.074± 0.082± 0.032

8.9 0.05 0.32± 0.15± 0.06 −0.13± 0.11± 0.040.19 0.22± 0.13± 0.06 −0.07± 0.10± 0.040.64 0.05± 0.10± 0.05 −0.071± 0.083± 0.0363.0 0.23± 0.19± 0.06 0.06± 0.13± 0.03

Table 13: The spin-density matrix elements,r0400 and r041−1, as a function of|t| in the range40 < Wγp < 160GeV for photoproduction and electroproduction.〈Q2〉 and〈|t|〉 are the bincentre values. The first error is statistical and the second systematic.

〈|t|〉 [ GeV2] r11−1 r100 + 2r111 r500 + 2r5110.05 0.19± 0.11± 0.06 −0.16± 0.12± 0.06 0.031± 0.055± 0.0290.19 0.62± 0.11± 0.06 0.04± 0.10± 0.06 0.004± 0.048± 0.0260.64 0.361± 0.097± 0.061 −0.073± 0.091± 0.057 −0.039± 0.043± 0.0273.0 0.07± 0.15± 0.06 −0.15± 0.15± 0.06 0.083± 0.079± 0.030

Table 14: The spin-density matrix elementr11−1 and the combined elementsr100 + 2r111 andr500 + 2r511 as a function of|t| in the range40 < Wγp < 160GeV and2 < Q2 < 80GeV2. 〈|t|〉indicates the bin centre value. The first error is statistical and the second systematic.

40