Electroproduction of ϕ(1020) mesons at 1.4≤ Q 2 ≤ GeV 2 measured with the CLAS spectrometer
J.P. Santoro, E.S. Smith, M. Garcon, M. Guidal, J.M. Laget, C. Weiss
aa r X i v : . [ nu c l - e x ] A ug Electroproduction of φ (1020) mesons at . ≤ Q ≤ GeV measured with the CLASspectrometer J.P. Santoro,
1, 2
E.S. Smith, M. Gar¸con, M. Guidal, J.M. Laget, C. Weiss, G. Adams, M.J. Amaryan, M. Anghinolfi, G. Asryan, G. Audit, H. Avakian, H. Bagdasaryan,
N. Baillie, J. Ball, J.P. Ball, N.A. Baltzell, S. Barrow, M. Battaglieri, I. Bedlinskiy, M. Bektasoglu, ∗ M. Bellis, N. Benmouna, B.L. Berman, A.S. Biselli,
35, 15
L. Blaszczyk, B.E. Bonner, C. Bookwalter, S. Bouchigny, S. Boiarinov,
R. Bradford, D. Branford, W.J. Briscoe, W.K. Brooks, S. B¨ultmann, V.D. Burkert, C. Butuceanu, J.R. Calarco, S.L. Careccia, D.S. Carman, L. Casey, A. Cazes, S. Chen, L. Cheng, P.L. Cole,
3, 20
P. Collins, P. Coltharp, D. Cords, † P. Corvisiero, D. Crabb, H. Crannell, V. Crede, J.P. Cummings, D. Dale, N. Dashyan, R. De Masi, E. De Sanctis, R. De Vita, P.V. Degtyarenko, H. Denizli, L. Dennis, A. Deur, S. Dhamija, K.V. Dharmawardane, K.S. Dhuga, R. Dickson, C. Djalali, G.E. Dodge, D. Doughty,
11, 3
M. Dugger, S. Dytman, O.P. Dzyubak, H. Egiyan,
42, 3, ‡ K.S. Egiyan, † L. El Fassi, L. Elouadrhiri, P. Eugenio, R. Fatemi, G. Fedotov, R.J. Feuerbach, J. Ficenec, T.A. Forest, A. Fradi, H. Funsten, † G. Gavalian,
30, 33
N. Gevorgyan, G.P. Gilfoyle, K.L. Giovanetti, F.X. Girod, J.T. Goetz, W. Gohn, C.I.O. Gordon, R.W. Gothe, L. Graham, K.A. Griffioen, M. Guillo, N. Guler, L. Guo, V. Gyurjyan, C. Hadjidakis, K. Hafidi, H. Hakobyan, C. Hanretty, J. Hardie,
11, 3
N. Hassall, D. Heddle, § F.W. Hersman, K. Hicks, I. Hleiqawi, M. Holtrop, C.E. Hyde-Wright, Y. Ilieva, D.G. Ireland, B.S. Ishkhanov, E.L. Isupov, M.M. Ito, D. Jenkins, H.S. Jo, J.R. Johnstone, K. Joo,
3, 12
H.G. Juengst, N. Kalantarians, D. Keller, J.D. Kellie, M. Khandaker, W. Kim, A. Klein, F.J. Klein, A.V. Klimenko, M. Kossov, Z. Krahn, L.H. Kramer,
16, 3
V. Kubarovsky, J. Kuhn,
35, 10
S.E. Kuhn, S.V. Kuleshov, V. Kuznetsov, J. Lachniet,
10, 33
J. Langheinrich, D. Lawrence, Ji Li, K. Livingston, H.Y. Lu, M. MacCormick, C. Marchand, N. Markov, P. Mattione, S. McAleer, B. McKinnon, J.W.C. McNabb, B.A. Mecking, S. Mehrabyan, J.J. Melone, M.D. Mestayer, C.A. Meyer, T. Mibe, K. Mikhailov, R. Minehart, M. Mirazita, R. Miskimen, V. Mokeev,
29, 3
L. Morand, B. Moreno, K. Moriya, S.A. Morrow,
5, 4
M. Moteabbed, J. Mueller, E. Munevar, G.S. Mutchler, P. Nadel-Turonski, R. Nasseripour,
16, 39, ¶ S. Niccolai,
18, 5
G. Niculescu,
32, 25
I. Niculescu,
18, 3, 25
B.B. Niczyporuk, M.R. Niroula, R.A. Niyazov,
33, 35
M. Nozar, G.V. O’Rielly, M. Osipenko, A.I. Ostrovidov, K. Park,
26, 39
S. Park, E. Pasyuk, C. Paterson, S. Anefalos Pereira, S.A. Philips, J. Pierce, N. Pivnyuk, D. Pocanic, O. Pogorelko, I. Popa, S. Pozdniakov, B.M. Preedom, J.W. Price, S. Procureur, Y. Prok, ∗∗ D. Protopopescu,
30, 19
L.M. Qin, B.A. Raue,
16, 3
G. Riccardi, G. Ricco, M. Ripani, B.G. Ritchie, G. Rosner, P. Rossi, F. Sabati´e, M.S. Saini, J. Salamanca, C. Salgado, V. Sapunenko, D. Schott, R.A. Schumacher, V.S. Serov, Y.G. Sharabian, D. Sharov, N.V. Shvedunov, A.V. Skabelin, L.C. Smith, D.I. Sober, D. Sokhan, A. Stavinsky, S.S. Stepanyan, S. Stepanyan, B.E. Stokes, P. Stoler, I.I. Strakovsky, S. Strauch,
18, 39
M. Taiuti, D.J. Tedeschi, A. Tkabladze, ∗ S. Tkachenko, L. Todor, †† C. Tur, M. Ungaro,
35, 12
M.F. Vineyard,
40, 37
A.V. Vlassov, D.P. Watts, ‡‡ L.B. Weinstein, D.P. Weygand, M. Williams, E. Wolin, M.H. Wood, §§ A. Yegneswaran, M. Yurov, L. Zana, J. Zhang, B. Zhao, and Z.W. Zhao (The CLAS Collaboration) Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061-0435 Catholic University of America, Washington, D.C. 20064 Thomas Jefferson National Accelerator Facility, Newport News, Virginia 23606 CEA-Saclay, Service de Physique Nucl´eaire, 91191 Gif-sur-Yvette, France Institut de Physique Nucleaire ORSAY, Orsay, France Argonne National Laboratory Arizona State University, Tempe, Arizona 85287-1504 University of California at Los Angeles, Los Angeles, California 90095-1547 California State University, Dominguez Hills, Carson, CA 90747 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213 Christopher Newport University, Newport News, Virginia 23606 University of Connecticut, Storrs, Connecticut 06269 Edinburgh University, Edinburgh EH9 3JZ, United Kingdom Emmy-Noether Foundation, Germany Fairfield University, Fairfield CT 06824 Florida International University, Miami, Florida 33199 Florida State University, Tallahassee, Florida 32306 The George Washington University, Washington, DC 20052 University of Glasgow, Glasgow G12 8QQ, United Kingdom Idaho State University, Pocatello, Idaho 83209 INFN, Laboratori Nazionali di Frascati, 00044 Frascati, Italy INFN, Sezione di Genova, 16146 Genova, Italy Institute f¨ur Strahlen und Kernphysik, Universit¨at Bonn, Germany Institute of Theoretical and Experimental Physics, Moscow, 117259, Russia James Madison University, Harrisonburg, Virginia 22807 Kyungpook National University, Daegu 702-701, Republic of Korea Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307 University of Massachusetts, Amherst, Massachusetts 01003 Moscow State University, General Nuclear Physics Institute, 119899 Moscow, Russia University of New Hampshire, Durham, New Hampshire 03824-3568 Norfolk State University, Norfolk, Virginia 23504 Ohio University, Athens, Ohio 45701 Old Dominion University, Norfolk, Virginia 23529 University of Pittsburgh, Pittsburgh, Pennsylvania 15260 Rensselaer Polytechnic Institute, Troy, New York 12180-3590 Rice University, Houston, Texas 77005-1892 University of Richmond, Richmond, Virginia 23173 Universidad T´ecnica Federico Santa Mar´ıa, Casilla 110-V, Valpara´ıso, Chile University of South Carolina, Columbia, South Carolina 29208 Union College, Schenectady, NY 12308 University of Virginia, Charlottesville, Virginia 22901 College of William and Mary, Williamsburg, Virginia 23187-8795 Yerevan Physics Institute, 375036 Yerevan, Armenia
Electroproduction of exclusive φ vector mesons has been studied with the CLAS detector in thekinematical range 1 . ≤ Q ≤ . , 0 . ≤ t ′ ≤ . , and 2 . ≤ W ≤ . / ( Q + M φ ) n was determined to be n = 2 . ± .
33. The slopeof the four-momentum transfer t ′ distribution is b φ = 0 . ± .
17 GeV − . Under the assumption ofs-channel helicity conservation (SCHC), we determine the ratio of longitudinal to transverse crosssections to be R = 0 . ± .
24. A 2-gluon exchange model is able to reproduce the main features ofthe data.
PACS numbers: 13.60.Le, 12.40.Nn, 12.40.Vv, 25.30.Rw
I. INTRODUCTION
Exclusive electroproduction of vector mesons is an es-sential tool for exploring the structure of the nucleon andthe exchange mechanisms governing high–energy scat-tering. For low photon virtualities relative to the vec-tor meson mass, Q . m V , or in the case of photopro- ∗ Current address:Ohio University, Athens, Ohio 45701 † Deceased ‡ Current address:University of New Hampshire, Durham, NewHampshire 03824-3568 § Current address:Christopher Newport University, Newport News,Virginia 23606 ¶ Current address:The George Washington University, Washington,DC 20052 ∗∗ Current address:Massachusetts Institute of Technology, Cam-bridge, Massachusetts 02139-4307 †† Current address:University of Richmond, Richmond, Virginia23173 ‡‡ Current address:Edinburgh University, Edinburgh EH9 3JZ,United Kingdom §§ Current address:University of Massachusetts, Amherst, Mas-sachusetts 01003 duction, Q = 0, these processes are well described by t –channel exchange of Regge poles (Pomeron, Reggeon)— extended objects whose properties can be related tothe observed hadron spectrum [1]. At high virtualities, Q ≫ m V , a QCD factorization theorem [2] states thatvector meson production from longitudinally polarizedphotons proceeds by exchange of a small–size system ofquarks or gluons, whose coupling to the nucleon is de-scribed by the generalized parton distributions (GPDs).By studying the dependence of exclusive electroproduc-tion on Q , one can thus “resolve” the Pomeron andReggeon into their quark and gluon constituents. Ad-ditional information comes from the comparison of the ρ , ω and φ channels, which couple differently to quarksand gluons. The self-analyzing decays of the spin–1mesons allow one to study also the helicity structure ofthe γ ∗ N interaction and, assuming helicity conservation,to separate longitudinal and transverse photon polariza-tions.This article presents data for exclusive φ vector me-son electroproduction off the proton above the resonanceregion, taken with a 5.754 GeV electron beam of the CE-BAF accelerator and the CLAS detector at Jefferson Lab[3]. The measurement was performed as part of a seriesof experiments aimed at studying vector meson produc-tion in the valence quark region at the highest availablephoton virtualities. The analysis of ω production hasbeen completed [4], and the analysis of ρ production isin progress [5]. The analysis of φ -meson production re-ported here is based in part on the work of Ref.[6].The φ -meson is unique in that its quark composition ismostly ¯ ss containing little, if any, u and d flavors whichpopulate the valence quarks in the nucleon. Thus, φ pro-duction primarily probes the gluon degrees of freedom inthe target. High–energy photoproduction of φ proceedsmainly by Pomeron exchange. At large Q , calculationsbased on current GPD models show that the φ produc-tion cross section is dominated by the gluon GPD, withonly small contributions arising from intrinsic strangequarks in the nucleon [7, 8]. At intermediate Q , a de-scription of φ production based on effective two–gluonexchange has been proposed [9], which effectively inter-polates between the “soft” and “hard” regimes. Thus, φ production provides us with a clean method of probingthe gluon field in the nucleon, even at JLab energies.A natural framework for discussing exclusive vectormeson production is the space–time picture in the tar-get rest frame ( i.e. , the laboratory frame) [10]. At highenergies, the interaction of the virtual photon with thetarget proton proceeds by way of fluctuation of the pho-ton into virtual hadronic (or quark–antiquark) configu-rations that subsequently scatter diffractively off the tar-get. This process occurs over a characteristic time givenby the lifetime of the fluctuation as dictated by the un-certainty principle and is given by∆ τ = 2 ν ( Q + M ) , (1)where ν is the photon laboratory energy and M fluct isthe mass of the virtual hadronic state. This interval alsodetermines the coherence length in the longitudinal direc-tion, l coh = c ∆ τ . In photoproduction or electroproduc-tion at Q . m V , this picture is the basis for the success-ful vector dominance model (VDM), where the dominanthadronic fluctuations are assumed to be the observedground–state vector mesons ( ρ , ω, φ ). Their interactionwith the target can be described by Pomeron exchange.As Q increases, higher–mass states become important.Eventually, at Q ≫ m V , the fluctuations of the pho-ton can appropriately be described as quark–antiquarkpairs (“dipoles”) with transverse momenta k ⊥ ∼ Q , ortransverse size r ⊥ ∼ /Q ≪ /m V . Their interactionwith the nucleon is described by the gluon GPD, whichcan be interpreted as the “color dipole moment” of thetarget.In the context of the space–time picture, measuringthe Q –dependence of exclusive electroproduction up to Q ∼ few GeV allows one to vary the transverse sizeof the projectile from “hadronic size” ( r ⊥ ∼ /m V ) to“small size” ( r ⊥ ∼ /Q ), thus resolving the structure ofthe target at very different distance scales. At HERA en-ergies, where l coh ≫ Q ∼ few GeV , one can neglect the variation of the coherence length with Q and associate the Q –dependence entirely with a changeof the transverse size of the projectile. The predictionsderived in this approximation are nicely confirmed by thedata, e.g. the decrease of the t –slope with Q , and theincrease with Q of the exponent governing the energydependence (for a review see Ref. [11]). At JLab en-ergies, where the coherence length in electroproductionis l coh . Q , i.e. , the “shrinkage” of the longitudinalsize of the virtual photon with increasing Q . Anothereffect modifying the space–time interpretation is the non-negligible longitudinal momentum transfer to the target,which increases with Q . Nevertheless, the space–timepicture remains a very useful framework for discussingvector meson electroproduction even at JLab energies.In the present φ -meson production experiment, the t –dependence of the differential cross section was measuredover a wide range, from the kinematic minimum at t ∼ t (small CM scattering angle) to t ∼ s/ t is related to the transversemomentum transfer to the target, ∆ ⊥ , and thus deter-mines the effective impact parameters in the cross sec-tion, b ⊥ ∼ / ∆ ⊥ . Exclusive meson production at large − ( t − t ) probes configurations of small transverse sizein the target. The possibility to vary both Q and t inelectroproduction allows one to control both the size ofthe projectile and the size of the target configurationscontributing to the process, and to study their interplay[12].Quantitative predictions for the production of vectormesons in our kinematic regime have been made by Lagetand collaborators based on the interactions between con-stituent partons (JML model). The high- t behavior ofthe photoproduction cross section of φ -mesons [13] hasbeen reproduced using dressed gluon propagators andcorrelated quark wave functions in the proton [14]. Quarkexchange processes, which contribute also to the pho-toproduction of ρ and ω mesons, have been modeledin terms of saturating Regge trajectories. The modeluses electromagnetic form factors in the Regge amplitude[12, 15] to describe electroproduction data. However, the Q dependence of the 2-gluon amplitude is an intrinsicpart of its construction, and no additional electromag-netic form factors are needed. Therefore, the predicted φ -meson electroproduction cross section is parameter freeand constitutes a strong test of the partonic descriptionthat underlies the model. The full form for the ampli-tudes are given in Refs. [1, 9, 14]. Thus far, comparisonsof the JML model for electroproduction have been madewith ω [4, 16], and ρ [5, 17] electroproduction data from The transverse momentum transfer to the target is given by∆ ⊥ = (1 − ξ )( t − t ), where ξ is the fractional longitudinalmomentum transfer to the target, which in turn is related to theBjorken variable in the kinematics of deep–inelastic scattering, ξ = x B / (2 − x B ). JLab, and ρ electroproduction data from HERMES [18].One of the leading motivations for the present work isthe sparse amount of existing φ electroproduction data.The body of φ -meson electroprodution data at similarkinematics consists of early data from Cornell [19, 20, 21],and some data from CLAS at lower energy [22]. Re-cent data on φ electroproduction comes from HERMES[18, 23] and HERA [24, 25, 26, 27] at much higher center-of-mass energy ( W ). A summary of the world data in-dicating their kinematic range is given in Table I. Thedata from this experiment are complementary to mea-surements at collider energies which cover a higher W and higher Q range where diffraction mechanisms areprobed. TABLE I: Summary of φ electroproduction data and kine-matic range.Experiment Q (GeV ) W (GeV)Cornell Dixon [19, 20] 0.23 - 0.97 2.9Cornell Cassel [21] 0.80 - 4.00 2.0 - 3.7HERMES [18, 23] 0.70 - 5.00 4.0 - 6.0CLAS[22] 0.70 - 2.20 2.0 - 2.6H1 [24] > ∼ We have measured φ -meson electroproduction at thehighest possible Q accessible at CEBAF energies in thevalence quark regime. The data set covers the kinemat-ical regime 1 . ≤ Q ≤ . , 0 . ≤ t ′ ≤ . ,and 2 . ≤ W ≤ . − t , theazimuthal angle Φ between the electron and hadron scat-tering planes, as well as the angular decay distributionsin the rest frame of the φ -meson. Although limitations ofthe statistical sample will preclude determining correla-tions between different kinematic variables, the distribu-tions will provide insights into the distance scale of theinteraction and explore kinematics that begin to probepartonic degrees of freedom. II. KINEMATICS AND NOTATION
The kinematic variables in exclusive φ production (seeFig. 1) described by e ( k ) p ( P ) → e ( k ′ ) φ ( υ ) p ( P ′ ) , (2)are k , k ′ , P , P ′ and υ which are, respectively, the four-momenta of the incident electron, scattered electron, tar-get proton, scattered proton and the φ -meson: e * g ’ e pp’ f LAB Z F + K HEL Y HEL
ZElectron Scattering Plane (Lab)Hadron Production Plane (c.m.) f HEL X H f H q Decay Plane (Helicity Frame) f FIG. 1: (Color online) Graphical representation of φ -mesonelectroproduction. Shown from left to right then above arethe electron scattering plane, the hadron production planeand helicity rest frame of the φ respectively. • Q = − q = − ( k − k ′ ) , the negative four-momentum squared of the virtual photon; • W = ( q + P ) , the squared invariant mass of thephoton-proton system; • x B = Q / (2 P · q ), the Bjorken scaling variable; • ν = P · q/M p , energy of the virtual photon; • t = ( P − P ′ ) , the squared four-momentum transferat the proton vertex, is given by t = t − p γ ∗ cm p φcm sin ( θ cm / , where t = ( E γ ∗ cm − E φcm ) − ( p γ ∗ cm − p φcm ) and the above formulas are calculated using theenergy and momenta of the virtual photon and φ in the γ ∗ p center-of-mass; • t ′ = | t − t | , momentum transfer relative to thekinematic limit − t , which increases with Q anddecreases with increasing W; • The coordinate system in the γ ∗ p center-of-massis defined with the z-axis along the direction ofthe virtual photon, and the y-axis normal to thehadronic production plane along ~p γ ∗ cm × ~p φcm ; • Φ, the angle between the hadron production ( γ ∗ φp )plane and the electron scattering ( ee ′ γ ∗ ) plane fol-lowing the convention in Ref. [28] ; The azimuthal angle Φ used here is − φ from the “Trento con-vention” [29]. • cos θ H and φ H , decay angles of the K + in the he-licity frame [28], which is defined in the rest frameof the φ -meson with the z-axis along the directionof the φ -meson in the γ ∗ p center-of-mass system; • ψ = φ H − Φ, azimuthal angle that simplifies theangular decay distributions when s-channel helicityis conserved (SCHC).The electroproduction reaction integrated over the de-cay angles of the φ -meson can be described by the follow-ing set of four independent variables: Q , − t , Φ and W .For the analysis of the decay distribution, the additionalvariables cos θ H and ψ are required. In total there are sixindependent variables in the approximation of negligible φ width. III. EXPERIMENT
The experiment was conducted with the CEBAF LargeAcceptance Spectrometer (CLAS) [3] located in Hall Bof the Thomas Jefferson National Accelerator Facility.The CLAS spectrometer is built around six independentsuperconducting coils that generate a toroidal magneticfield azimuthally around the beam direction. The az-imuthal coverage is limited by the magnetic coils andis approximately 90% at large angles and narrows to50% at forward angles. Each sector is equipped withthree regions of multi-wire drift chambers and time-of-flight counters that cover the angular range from 8 ◦ to143 ◦ . Charged-particle trajectories are tracked throughthe field with the drift chambers, and the scintillatorsprovide a precise determination of the particle flight time.In the forward region (8 ◦ to 45 ◦ ), each sector is fur-thermore equipped with gas-filled threshold Cerenkovcounters (CC) and electromagnetic calorimeters (EC).The Cerenkov counters are used to discriminate electronsfrom pions, and the calorimeters are used to measure theenergy of electrons and photons.The data were collected between October 2001 andJanuary 2002 with a 5.754 GeV electron beam incidenton a 5 cm-long liquid hydrogen target. The typical beamcurrent was 7 nA. The CLAS torus magnet was set to3375 A with a polarity that caused negatively chargedparticles to bend in towards the beamline. The inclusiveelectron trigger fired when signals in the forward elec-tromagnetic calorimeter exceeded a predefined thresholdin coincidence with a hit in the Cerenkov counters. Thekinematical domain of the selected sample correspondsapproximately to Q from 1.5 to 5.5 GeV and W be-tween 2 and 3 GeV. The typical experimental dead timewas about 8% with a trigger rate of about 1.5 kHz. IV. EVENT RECONSTRUCTION
The φ -mesons were detected using the charged-particledecay mode into K + and a K − . Events corresponding P (GeV/c)1 2 3 4 5 E ( G e V ) FIG. 2: Energy deposited by the electron candidates in theelectromagnetic calorimeter versus momentum. The linesshow the selection cuts for good electrons as described in thetext. to ep → epK + ( K − ) were classified initially by requiringat least one negative track and two positive tracks. Nor-mally the K − remained undetected due to the limitedacceptance for negative particles at this high magneticfield setting. After calibration of the spectrometer, themomentum of each particle was determined with a frac-tional resolution of about a percent using the track seg-ments in the drift chambers. The momentum resolutionis sufficient to identify the missing particle as a K − .The identification of good electrons is the crucial firststep and is accomplished through energy and momentumcuts [6]. After selection of tracks within the fiducial vol-ume of the detector, the momentum of the electron can-didate track in each event was required to correspond tothe energy deposition in the electromagnetic calorimeterand the visible energy be greater than 0.2 GeV (Fig. 2).Pions were rejected by requiring a minimum energy of0.06 GeV in the inner layer of the calorimeter and a pulseheight in the Cerenkov counter corresponding to at least2.5 photoelectrons [30, 31].The two positive tracks in the fiducial volume wereidentified as a proton and K + using the measured flighttime ( δT ∼
160 ps) from the target to the time-of-flightcounters [32], a typical distance of about 5 m. Fiducialvolume cuts were made to cut out tracks in inefficientparts of the detector and small momentum correctionswere applied to compensate for uncertainties in the mag-netic field and detector positioning. The time of the in-teraction was determined using the vertex time of theelectron corrected to the time of the bunch crossing ofthe machine. Using the known momenta of each of thetracks, the vertex time was computed making assump-tions for the mass of the particle and comparing to thetime of the bunch crossing. Events were kept where thetwo positive tracks were consistent with the assignmentof one proton and one K + . Tracks were identified as pro-tons when the projected vertex time assuming a proton P (GeV/c)0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 M ass ( G e V ) FIG. 3: Mass computed from the flight time versus momen-tum for positive particles. The top band corresponds to pro-tons, the middle band corresponds to K + ’s and the lowerenhancement at 1.5 GeV/c momentum is due to pion con-tamination. mass differed from the interaction time by less than 0.75ns, and as a K + when the projected time assuming akaon mass differed from the interaction time by less than0.6 ns. In cases where one track satisfied both criteria,the ambiguity was resolved using the second track. Thenumber of events where both tracks satisfied both crite-ria was less than 1% and were eliminated. The calculatedmass versus momentum, shown in Fig. 3, indicates thatat high momenta, there remain a number of pions thatare identified as kaons in the sample.Once the electron, proton and K + tracks were iden-tified, the missing mass was computed and is plotted inFig. 4. A clear peak is found at the mass of the K − , whichcorresponds to the exclusive reaction ep → epK + K − . A2 σ cut was applied to the epK + X events to select thesample of interest. For those events, the four-vector forthe K − was constructed by setting the three-momentumequal to the missing momentum of the ep → epK + X re-action, and the energy was then calculated using the K − mass recommended by the Particle Data Group [33]. Thefraction of events where the K − was detected in the de-tector was so small that they were not treated differentlythan the rest of the sample. A. φ Event Identification
The sample satisfying the epK + ( K − ) criteria contains27,950 events out of 947,300 epK + X candidates. Thesample includes all physical processes that contribute tothis final state, as well as real φ ’s and background frommisidentified pions. Fig. 5 shows the K + K − invariantmass ( M KK ) for the entire data set with a clear φ -mesonpeak. This distribution is simultaneously fit to a Gaus-sian plus an empirical phase space function for the back- (X) (GeV) + Missing Mass epK0.4 0.5 0.6 0.7050010001500200025003000
FIG. 4: (Color online) Distribution of epK + X missing mass.The vertical lines indicate the cuts placed to select events witha missing K − . (GeV) K+K- M – =792 f N FIG. 5: (Color online) K + K − invariant mass including alldata cuts and a fit to φ peak with Eq. 3. ground, FIT = A G ( σ, µ ) + B q M kk − M th + B (cid:16) M kk − M th (cid:17) , (3)where G ( σ, µ ) is a Gaussian distribution, M th =0.986 GeVis the threshold for two kaon production and A, B , and B are parameters of the fit. This fit yields N φ = 792 ± µ = 1 . ± . σ = 6 . ± . φ signal in various kinematic bins(see subsequent sections for details). The average χ perdegree of freedom for all the fits was 1.07, indicating thatdeviations from the fit function are statistical in nature.There are competing physics channels that also lead to v g p p’ P f + K _ K v g p p’ _ K + K + K*) S *( L FIG. 6: (Color online) Sketch of exclusive φ (1020) productionvia Pomeron exchange and of excited hyperon production, ofwhich Λ(1520) is an example. This is the primary physicsbackground for φ (1020) production. the same final state. The majority of these backgroundscome from the production and subsequent decay of high-mass hyperons produced via ep → e ′ K + Λ ∗ (Σ ∗ ) as illus-trated in Fig. 6. The Dalitz plot in Fig. 7 clearly showsthe dominant Λ(1520) background contribution (horizon-tal strip), as well as the φ (1020) (vertical strip). Thereare additional contributions from the higher-mass statessuch as Λ ∗ (1600), Λ ∗ (1800), Λ ∗ (1820), Σ ∗ (1660), andΣ ∗ (1750) but they cannot be separately identified. In or-der to avoid the introduction of holes in the acceptance,no cuts are made to remove these hyperon backgrounds.Instead they are taken into account during the fittingprocedure by assuming they contribute to the smoothbackground under the φ -meson peak. Nevertheless, manydifferent fits were performed removing events in the peakof the Λ ∗ (1520) to study this systematic with no indica-tion that they changed the results significantly. Thesestudies focused on the t-distributions, since the effectivemomentum transfer in Λ ∗ reactions is very flat comparedto that expected from φ -meson production. V. ACCEPTANCE CORRECTIONS
Particle interactions and event reconstruction in thedetector were simulated using a GEANT-based MonteCarlo called GSIM [34]. The events were generated ac-cording to a VDM-inspired cross section [10] with the ) (GeV K+K-2
M1 1.2 1.4 1.6 1.8 2 ) ( G e V p K - M (1020) f (1520) L FIG. 7: Dalitz plot of M pK versus M KK . The well-definedhorizontal strip is the Λ(1520) band. The vertical strip is the φ (1020) band. following form: σ V DMφ ( Q , W ) = σ φ (0 , W ) [1 + Rǫ ](1 + Q /M φ ) × ( W − M p ) exp( − bt ′ ) q ( W − M p − Q ) + 4 W Q (4) ǫ = 4 E e ( E e − ν ) − Q E e ( E e − ν ) + 2 ν + Q , (5)where σ φ (0 , W ) is the (transverse) photoproduction crosssection, E e is the incident electron beam energy, ǫ is thevirtual photon polarization parameter and R is the ra-tio of the longitudinal to transverse cross section. Theparameters of the model were tuned during preliminaryanalysis and found to reproduce the general features ofthe data. The main variation from the conventionalVDM model was in the propagator, where preliminarydata seemed to indicate a stronger dependence on Q and an exponent of 3 was used instead of 2.The acceptance function is a combination of the geo-metrical acceptance of CLAS, the detector efficiencies ofthe scintillators and drift chambers, the track reconstruc-tion efficiency, and the event selection efficiency. TheCerenkov detector [30] is not well modeled in GSIM, andits efficiency was determined separately using the data.The acceptance was defined in each bin of a 6-dimensional table as the ratio of reconstructed to gener-ated Monte Carlo events. In order to account for corre-lations between all kinematic variables, a total of 33,600acceptance bins are defined in the kinematic variables Q , − t , W , Φ, cos θ H and ψ . The binning selection isgiven in Table II for the first three variables and uniformbinning was used for Φ (6 bins), cos θ H (5 bins) and ψ (8bins). The projected 2-D acceptance surface in Q and − t and the 1-D projections in Q , t , and W are shown inFig. 8 . The projected 2-D acceptance surface in cos θ H and ψ is shown in Fig. 9, as well as the 1-D projectionsin cos θ H , ψ , and Φ. The variation of the acceptance is ) - t ( G e V ) ( G e V Q (a) ) (GeV Q1.5 2 2.5 30.010.020.03 (b)
W (GeV)2 2.2 2.4 2.6 2.800.010.020.03 (c) ) -t (GeV0 0.5 1 1.5 2 2.5 30.010.020.03 (d) FIG. 8: (Color online) 2-D Acceptance in Q and t , as wellas the 1-D acceptance in Q , W , and − t . Error bars are notshown; the lines are present to guide the eye. The axes in the2-D plot in a) have the same range as that of the axis of thetwo 1-D plots in b) and d). relatively smooth as a function of these variables (exceptfor Φ, which is a reflection of the CLAS torus coils) andis of the order of 1–3%.Events that fell into bins with extremely small accep-tances ( ≤ No. Bin Definition Q W − t t ′ Q (GeV ), − t (GeV ), and W (GeV). An additional acceptancetable was also generated for t ′ in the place of t , but it is notan independent variable. VI. RADIATIVE CORRECTIONS
The radiative effects were calculated in two distinctsteps. The external radiative process, which is the finiteprobability that the incoming or scattered electron willradiate a hard photon in the presence of a nucleon in thetarget other than the one associated with the event, istaken into account during the Monte Carlo acceptancecalculation. The internal radiative corrections includethe Bremsstrahlung process for the incoming or scat-tered electron in the presence of the nucleon associatedwith the event, as well as diagrams such as vacuum po-larization, which are not accounted for during the ac- ( d e g r e e s ) y ) H q c o s ( (a) (degrees) y
100 200 30000.0050.010.0150.02 (b) H q cos-0.5 0 0.50.0050.010.015 (c) (degrees) F (d) FIG. 9: (Color online) 2-D Acceptance in cos θ H and ψ , aswell as the 1-D acceptance in cos θ H , Φ, and ψ . Error barsare not shown; the lines are present to guide the eye. Theaxes in the 2-D plot in a) have the same range as that of theaxis of the two 1-D plots in b) and c). ceptance calculation. These are included in the correc-tion factor F rad using the radiative correction code EX-CLURAD setting the controlling parameter v cut = 0 . [35]. F rad is calculated in each W and Φ bin as theratio σ rad /σ norad (variable δ in Eq. 75 from Ref. [35]),where σ norad is the cross section calculated without anyradiative effects (i.e. the Born cross section) and σ rad is the cross section calculated with radiative effects in-cluded. The correction factor for various W bins is shownas a function of Φ in Fig. 10. The correction was com-puted in bins of W and Φ for average values of Q andcos θ CM (directly related to − t ) because the correctionwas found to change less than 2% over the range of Q and cos θ CM [36]. VII. CROSS SECTIONS
The reduced γ ∗ p → φp electroproduction cross sectionis given by σ ( Q , W ) = 1Γ( Q , W, E e ) dσdQ dW , Γ( Q , W, E e ) = α π W ( W − M p ) M p E e Q − ǫ , (6)where Γ( Q , W, E e ) is the virtual photon flux factor. Wecan extract the φ cross section from the data via dσdQ dW = 1 Br n W L int ∆ Q ∆ W , (7)where Br is the branching fraction ratio of φ → K + K − = 0 . ± .
009 [33], L int =2 . × cm − isthe live-time-corrected integrated luminosity, ∆ Q and (degrees) F r a d F W=2.9 GeVW=2.8 GeVW=2.7 GeVW=2.6 GeV W=2.5 GeVW=2.4 GeVW=2.3 GeVW=2.2 GeVW=2.1 GeVW=2.0 GeV
FIG. 10: (Color online) Plot of radiative correction F rad as afunction of Φ for assorted values of W from 2.0 to 3.0 GeV.The correction for each W value was computed for < Q > =2 .
47 GeV and < cos θ c.m. > = 0 . ∆ W are the corresponding bin widths modified appro-priately when not completely filled due to kinematics,and n W is the result of a fit to the M KK distributionweighted by acceptance, CC efficiency correction and ra-diative corrections. The binning in Q , − t , t ′ , and W for the extraction of the cross section in those variablesis shown in Table II. We emphasize here that we haveperformed a fit to Eq. 3 to determine the signal n W andthe estimated background under the peak for each entryin the table. The differential cross section in a variable X is given as dσdX = σ ( Q )∆ X , (8)where σ ( Q ) is the total cross section in a bin in X . Thecross sections presented in this paper have been correctedfor the bin size and are quoted at the center of each bin. A. Total Cross Section σ ( Q ) The cross section σ ( Q ) as a function of Q is ob-tained by integrating over W due to the limited statistics.Each event was weighted for acceptance, radiative effects,the CC efficiency, as well as the virtual photon flux fac-tor. The invariant mass distribution ( M KK ) of weightedevents in each Q bin was then fit to Eq. 3. The binsused in the analysis are given in Table III. The range in W was restricted at the low end where acceptance correc-tions change rapidly and are large, and at the high end tomatch the high end of the kinematically accessible range.The cross section for each of the bins was calculated ac-cording to Eqs. 6 and 7. A small correction ( ∼ Q bin are shown in Table IV. The total Q range W range1 . ≤ Q ≤ . . ≤ W ≤ . . ≤ Q ≤ . . ≤ W ≤ . . ≤ Q ≤ . . ≤ W ≤ . . ≤ Q ≤ . . ≤ W ≤ . . ≤ Q ≤ . . ≤ W ≤ . W range for each Q bin. cross section was fit to the function A ( Q + M φ ) n (9)to determine the scaling behavior. For this data we de-termined the parameter n = 1 . ± .
84. The measuredexponent spans the range expected for the dependenceon Q due to VDM ( n = 2) to hard scattering ( n = 3 forfixed momentum transfer t ). Q (GeV ) < ǫ > σ (nb)1.6 0.488 9.9 ± ± ± ± ± σ ( Q ) and kinematics of eachdata point, along with the center of each Q bin. < ǫ > is theaverage virtual photon polarization in each bin. B. Differential Cross Section in t ′ , dσ/dt ′ The differential cross section in t ′ was extracted inseven bins in t ′ by fitting Eq. 3 to the K + K − mass distri-bution to determine the φ signal and background in thatparticular bin. The average χ per degree of freedom forthese fits was 1.2. The signal-to-background ratio variedfrom bin to bin, ranging from 0.33 to 0.86. The lowestsignal-to-background ratio occurred in the mid range of t ′ . The resulting values for the cross section in each t ′ bin are shown in Table V.In cases of limited statistics, dσ/dt ′ is often used in-stead of dσ/dt in order to eliminate kinematic correctionsdue to − t , which varies with Q and W . This procedureis most useful when the cross section factorizes into termsthat depend only on t and terms that depend on Q and W , aside from the threshold dependence, as in the VDMmodel. Indeed, our measured differential cross section in t ′ show very similar trends as previous data, namely theyare consistent with diffractive production ( e − b φ | t ′ | ) [22].Fig. 11 shows an exponential fit to the measured differen-tial cross section, which yields a b φ = 0 . ± .
17 GeV − .At high energies, the slope can be directly interpreted in0 ) t’ (GeV0 0.5 1 1.5 2 2.5 ) / d t ’ ( nb / G e V s d -2 – =0.98 f b FIG. 11: (Color online) Plot of dσ/dt ′ along with an expo-nential fit. terms of the transverse size of the interacting configura-tion, as described later when presenting results. In thatlimit, the small value of the exponential slope implies theinteraction takes place at very short distances inside thenucleon. t ′ (GeV ) dσ/dt ′ (nb/GeV )0.1 9.4 ± ± ± ± ± ± ± dσ/dt ′ and kinematics ofeach data point. t ′ is the center of the bin, and correspondsto an average value of ǫ = 0 . C. Differential Cross Section in t , dσ/dt The differential cross section is easiest to compare withtheory if it is computed in terms of the Mandelstam vari-able t . The cross section is given as dσdt = σ ( Q )∆ t · Corr ( t ) , (10)where ∆ t is the bin size and Corr ( t ) is a correction factorto account for the fact that the kinematic limit t ( Q , W )varies across the bin. The yield was extracted over theranges of Q and W given in Table III in six bins in − t .The kinematic threshold t varies between -0.09 and -1.14 GeV for extreme values of Q and W . For thebin corresponding to 0 ≤ − t ≤ . , the thresholdvaries so much that corrections could not be modeled reliably, so that bin was dropped. The first bin reportedcontains a significant correction, but was included withan increased systematic error. Subsequent bins had smallor no corrections. The values for the cross section in each − t bin are given in Table VI. − t (GeV ) dσ/dt ( nb/GeV )0.6 10.7 ± ± ± ± ± ± dσ/dt and kinematics ofeach data point. − t is the center of each bin at an averagevalue of ǫ = 0 . D. Differential Cross Section dσ/d Φ and test ofSCHC The cross section dependence on the angle Φ betweenthe electron and hadron scattering planes takes the fol-lowing form: dσd
Φ = 12 π σ + ǫσ T T cos 2Φ + p ǫ (1 + ǫ ) σ LT cos Φ ! , (11)where σ LT and σ T T are the interference terms betweenthe longitudinal and transverse contributions to the crosssection. If helicity is conserved in the s-channel (SCHC),then both of these terms will vanish. The magnitude ofthese interference terms can therefore be used as a testfor the validity of SCHC.The differential cross sections in Φ were extracted inthe same manner as the other differential cross sections(Eq. 8) after integrating over Q , − t and W . The crosssection dσ/d Φ was extracted in six bins in Φ. The crosssections, along with a fit to Eq. 11, are shown in Fig. 12.The fit yields a value of σ T T = − . ± . σ LT =2 . ± . χ /D.F. = 1 .
3. A fit of the dσ/d
Φ distribution to aconstant, constraining the interference terms to be zero,yields a χ /D.F. = 1 .
6. The small change in the goodnessof fit between the two cases leads us to conclude that theprecision of this experiment is insensitive to violations ofSCHC for φ -meson production in our kinematic domain. VIII. ANGULAR DECAY DISTRIBUTIONS
The angular decay distribution of the K + in the φ rest frame describes the polarization properties of the φ -meson. The scattering amplitude for vector mesonelectroproduction γ ∗ + N → P + V can be expressed in1 (degrees) F
50 100 150 200 250 300 ( nb / d e g r ees ) F / d s d FIG. 12: (Color online) dσ/d
Φ vs Φ. The curve shows a fit toEq. 11 which is used to determine σ TT and σ LT . The dottedline is a fit to a constant function which is expected fromSCHC. terms of the helicity amplitudes T λ V λ P λ γ ∗ λ N , where λ i isthe helicity of each particle (i=V, P, γ , N). The vectormeson spin density matrix is derived from these helicityamplitudes by exploiting the von Neumann formula ρ ( V ) = 12 T ρ ( γ ∗ ) T † , (12)where ρ ( γ ∗ ) is the spin-density matrix of the virtualphoton. The details of this derivation can be found inRef. [28]. The density matrix element is denoted ρ αij ,where the index α can be related to the virtual photonpolarization. α = 0 − α = 4 for purely longitudinal photons, while other valuescorrespond to longitudinal-transverse interference terms.The indices ij correspond to the helicity state of the vec-tor meson [38]. In cases where the data do not allow fora σ L /σ T separation, the unseparated matrix elements r αij can be parameterized as: r ij = ρ ij + ǫRρ ij ǫR (13) r αij = ρ αij ǫR ; α = 0 − r αij = √ R ρ αij ǫR ; α = 5 − , (15)Recall that R is the ratio of longitudinal to transversecross section. The angular distribution of the K + is usu-ally described in the helicity frame, defined in the restframe of the φ -meson with the z-axis oriented along the φ -meson in the γ ∗ p center-of-mass. The full decay dis-tribution, which we denote by W F (cos θ H , Φ , φ H ), can befound in the literature [39], but will only be given here insimplified forms. In particular, further analysis of angu-lar distributions is done under the assumption of SCHC,which leads to considerable simplifications with the in- troduction of ψ = φ H − Φ and the following constraints: − Im r = Re r = √ R cos δ √ ǫR ) ; (16) r − = − Im r − = 12(1 + ǫR ) ; (17) r = ǫR ǫR . (18)All other r αij ’s are 0, and √ Re iδ is the ratio of the longitu-dinal to transverse amplitudes. The angular distributionbecomes a function of two variables only and is given by: W (cos θ H , ψ ) = 38 π
11 + ǫR " sin θ H + 2 ǫR cos θ H − ǫR ) ǫ ( r − ) sin θ H cos 2 ψ + 4(1 + ǫR ) p ǫ (1 + ǫ )( Re r ) sin 2 θ H cos ψ i . (19)In order to extract the r αij parameters from the mea-sured angular distribution, we use two 1-dimensional pro-jections of the full angular distribution. A. Polar Angular distribution projection
To obtain the polar angular distribution, an integra-tion of the full angular distribution W F over φ H yields W (cos θ H ) = 34 "(cid:16) − r (cid:17) + (cid:16) r − (cid:17) cos θ H , (20)which is independent of SCHC. In order to obtain thisprojection from the data, the K + K − invariant mass dis-tribution is plotted in five bins in cos θ H (0.40 units ofcos θ H each). The same fit to a Gaussian plus a polyno-mial background was made to extract the weighted yieldsin each of these bins. The fit to dσ/d cos θ H in Fig. 13yields a value r = 0 . ± .
12 with a χ /D.F. = 1 . R = r ǫ (1 − r ) = 1 . ± . , (21)where we have used the average value of < ǫ > = 0 . B. Angular distribution projection in ψ After an integration of W F in cos θ H , a substitutionof φ H = ψ + Φ, and an integration in Φ, the projectedangular distribution in ψ is given as W ( ψ ) = 12 π " ǫ ( r − ) cos 2 ψ , (22)2 H q cos-0.5 0 0.5 ) H q W ( c o s – = 0.33 r FIG. 13: (Color online) Unnormalized polar angular decaydistribution of the K + integrated over all Q values plus a fitto Eq. 20. Also shown is the extracted r parameter. (degrees) y ) y W ( FIG. 14: (Color online) Unnormalized azimuthal angular dis-tribution extracted for all Q values plus a fit to Eq. 22. Thevalue of r − can also be used to determine R . which assumes SCHC. The factor of 1 / π is a normal-ization factor. A fit of dσ/dψ to Eq. 22 is shown inFig. 14. The fit yields a value r − = 0 . ± .
23 with a χ /D.F. = 1 .
3. The ratio of longitudinal to transversecross sections can also be computed from r − (Eq. 17)and gives R = 0 . ± .
3, in agreement with the valueobtained previously.
IX. SYSTEMATIC UNCERTAINTIES
The relatively low number of measured φ events causesstatistical errors to dominate. The sources of systematicerrors in this experiment are summarized in Table VII.The major sources of systematic errors are due to accep-tance corrections and estimation of backgrounds. Studies of backgrounds and their uncertainties were also limitedby the finite sample size. The total systematic error of18.6% was added in quadrature with the statistical errorsin all quoted cross sections.The acceptance correction contributes to the system-atic error in two distinct ways. The uncertainty of 6%introduced by eliminating events with very large weights(i.e. very low acceptance) was estimated by changing themaximum weight allowed and recomputing the extractedcross section. The uncertainties introduced by the use ofour acceptance table (12%) were estimated by combiningbins and comparing the extracted result to the averageof the constituent bins.To estimate the systematic uncertainty due to the un-known distribution of backgrounds, the functional formof the background (see Eq. 3) was modified by adding aterm proportional to ( M KK − M th ) and refitting the − t and Q distributions. The new fits were less constrained,but the average change in cross section was 9%. We foundthat the extraction of the slope parameter b φ was fairlyrobust to these changes. In addition, the fitted invari-ant mass distributions included some background due tomisidentified pions. The estimated uncertainty due tothis contamination under the peak was estimated to be7%.The systematic uncertainty in the placement of the cutto select the K − from the epK + missing mass (5%) wasinvestigated by varying the cut and observing the effecton the cross sections. The systematic error associatedwith the bin centering correction is almost negligible ( ∼ ∼ ≤ t kinematic cutoff in the first t -bin introduces a 25% systematic error in that bin. Source ∆ σ %Acceptance correction 13.4Background functional form 9.1Misidentified pion background 7.0 epK + ( X ) cut 5.0Bin-centering correction 1.0Radiative correction 3.0Cerenkov efficiency correction 1.0Total 18.6TABLE VII: Table of systematic errors. X. DISCUSSION
The measurements of σ ( Q ) from the present analysisare shown along with other data on φ electroproduction3 ) (GeV Q0 1 2 3 4 5 ) nb ( Q s CLAS/Santoro W=2.5 GeVCLAS/Lukashin W=2.5 GeVCornell/Cassel W=2.7 GeVHERMES/Borissov W=4-6 GeVH1/Adloff W=75 GeV
FIG. 15: (Color online) Total cross sections as a function of Q for our data (red full circles), previous JLab data (opencircles) [22], Cornell data (stars) for W between 2 and 3.7GeV [21], HERMES data (triangles) for W between 4 and6 GeV [23], and HERA data (squares) at high W [26]. Thecurves show the predictions of the JML model at W =2.9, 2.45and 2.1 GeV (top to bottom). [20, 21, 22, 23, 26] in Fig. 15. The one overlap point at Q = 1 . GeV is in good agreement with the previousCLAS measurement [22]. The data sets span the rangefrom threshold at W =2 GeV up to HERA energies.The data sets have a similar trend as a function of Q and increase monotonically as a function of W . Thethree curves using the JML model at W = 2.1, 2.45 and2.9 GeV are also plotted for Q greater than 1.5 GeV .The calculation for W =2.45 GeV, which is close to theaverage of our data, seems to overestimate our data byabout a factor of two, although it does reproduce theexisting Cornell data from Ref. [21]. The Cornell datahas a much wider acceptance range in W between 2.0and 3.7 GeV, so in fact it could be representative of thecross section at higher W . The new data from CLAS,together with the existing world data, in particular thedata from HERA, indicate that the qualitative behavioras a function of Q does not change between thresholdand a W of about 100 GeV.Of interest is the applicability of factorization and theformalism of GPDs to meson production in general, and φ production in particular. QCD factorization makescertain asymptotic predictions about the cross section,namely that the longitudinal part of the cross section, σ L , becomes dominant as Q increases, and that the dif-ferential cross section will scale as 1 / ( Q ) at fixed t and x B . For a slow variation of the cross section over therange of x B of the data (0.2–0.5), this prediction can becompared to the Q dependence integrated over W and t , although quantitative estimates are modified by powercorrections as well as kinematics near threshold. On theother hand, the VDM model predicts the cross section toscale as 1 / ( Q + M φ ) n with n = 2. The Q range of our ) (GeV Q ( nb ) s n ) f +M – n = 2.49 FIG. 16: (Color online) Fit to the cross section as a functionof Q distribution to determine scaling using data from thepresent experiment and CLAS data from Ref. [22]. data is limited, but in combination with previous CLASdata at lower Q [22] (see Fig. 16) we can determine thescaling exponent of 1 / ( Q + M φ ) n to be n = 2 . ± . φ productioncross section based on GPD models suffer from consid-erable quantitative uncertainties when applied to fixed–target energies. At HERA energies the approach takenin Ref. [40], which relies on the equivalence of leading-order QCD factorization with the dipole picture of high–energy scattering, gives a good description of the abso-lute cross section, as well as of subtle features such as thechange of the W – and t –dependence with Q . Essentialfor the success of this approach is the fact that the effec-tive scale of the gluon GPD, Q , is considerably smallerthan the external photon virtuality, Q , as has been con-firmed by detailed quantitative studies [41]. The sameis expected in vector meson production at fixed–targetenergies; however, implementing it in a consistent man-ner in these kinematics has so far proven to be difficult.Leading-twist, leading-order QCD calculations of the φ production cross section at JLab and HERMES energiesdone with the assumption that Q = Q [7] overestimatethe measured cross section by a factor 5–10 and predicttoo steep an energy dependence. A satisfactory solutionto this problem likely requires a comprehensive approachthat combines contributions from small–size ( ∼ /Q )and hadronic–size configurations in the virtual photonat moderate coherence lengths ( cτ . Q and W .The four-momentum transfer distribution probes thesize of the interaction volume. At high energies, theexponential slope (see Fig.11) is directly related to thetransverse size b φ ∼ R int in analogy to the classical4 (fm) tD c -1
10 1 10 ) - ( G e V f b CLAS/SantoroCLAS/LukashinZEUSCornell/DixonCornell/Cassel
FIG. 17: (Color online) Exponential slope b φ plotted as afunction of the fluctuation parameters c ∆ τ for the world data.The data at high W measure an asymptotic slope correspond-ing to long fluctuation times. At low W and relatively large Q , the fluctuation times becomes small and constrain thesize of the interaction volume. scattering of light through an aperture of radius R int ∼ τ ofthe virtual meson, which can be estimated through un-certainty principle arguments, and is given by Eq. 1. Thenature of the interaction becomes more point-like as Q increases and the fluctuation time decreases. This tran-sition should be observed as a decrease in the measuredslope parameter. Since the differential cross section in t ′ was extracted for all Q , the value for b φ corresponds tothe average value of c ∆ τ =0.46 fm. The slope parame-ters for various experiments are shown in Fig. 17 for theworld data on φ electroproduction. The measured slowerfall-off of the t -distribution, corresponding to the smallslope parameter, is consistent with the expectation thatshort interaction time probe small ss dipoles.The differential cross section in − t is compared to theJML model in Fig. 18. The data covers 1 . ≤ Q ≤ . and the JML model predictions [42] are plotted forfixed values of Q from 1.6 to 5 GeV . The data tend tohave a shallower slope than the calculation, but there isgeneral agreement. This agreement is highly non-trivialsince the few parameters of the model have been fixedat the real photon point and kept frozen in the virtualphoton sector. Our data confirm both the Q and − t dependence of the cross section that are built into thedynamics of the ss loop and the 2-gluon loop.The angular decay distributions provide informationon the longitudinal part of the production cross section.We have extracted values of σ T T and σ LT from the cross ) -t (GeV0 1 2 3 ) / d t ( nb / G e V s d -1 FIG. 18: (Color online) dσ/dt vs − t for the entire Q rangeand the JML predictions for W =2.5 GeV at five values of Q = 1.6, 2.1, 2.6, 3.8 and 5 GeV , top to bottom. section dependence on the angle Φ between the electronand hadron scattering planes. The value of the σ LT isconsistent with zero and the assumption that SCHC isvalid for φ production in this kinematic regime. How-ever, small deviations are still possible as shown by moreaccurate measurements of these parameters at HERA en-ergies [25].The ratio R = σ L /σ T has been determined from twoprojections of the angular decay distribution of the K + in the φ -meson rest frame and under the hypothesis ofSCHC. The measurement of r gives R = 1 . ± .
38 andthe measurement of r − gives a value of R = 0 . ± . R = 0 . ± .
24. This mea-surement can be compared to the value of R=1.25 pre-dicted by the JML 2-gluon exchange model. We note thatthese extractions, at least from r , are relatively insen-sitive to the assumption of SCHC as shown in Ref. [25].The measurements of R from this analysis and otherworld data are plotted as a function of Q in Fig. 19 .The data show that the ratio R is increasing as a func-tion of Q , but σ L is still not dominant at these kine-matics. Using our measurement of R , we can computethe average longitudinal cross section for our data. Theaverage cross section is given by σ ( Q = 2 . GeV ) =6 . ± . σ L = 4 . ± . XI. SUMMARY φ -meson electroproduction was examined in the kine-matical regime 1 . ≤ Q ≤ . , 0 . ≤ t ′ ≤ . The W -dependence of R has been studied at HERA [26], whichcovers a very large range in W . ) (GeV Q -1
10 1 10 R -2 -1 CLAS/SantoroCLAS/LukashinHERMESCornell/DixonZEUS/ChekanovHERA/Adloff
FIG. 19: (Color online) R = σ L /σ T vs. Q for our data (solidcircles), previous CLAS result (open circle), HERMES results(triangles) Cornell data (stars), ZEUS data (open diamonds)and HERA data (squares). The two determinations from thepresent analysis are separated for ease of viewing about theactual Q value of 2.21 GeV . GeV , and 2 . ≤ W ≤ . Q previously reported at JLab energies[22], accruing approximately four times the luminosity re-quired for sensitivity to smaller cross sections. We havepresented distributions as a function of the momentumtransfer − t , the azimuthal angle Φ between the electronand hadron scattering planes, as well as angular decaydistributions in the rest frame of the φ -meson.We have analyzed the angular distributions under theassumption of SCHC to extract the ratio of longitudinalto transverse cross sections of R = 0 . ± .
24, which isconsistent with the world trend. The longitudinal com-ponent is comparable to the transverse one, which sug- gests that we have not yet reached the asymptotic regimewhere QCD factorization can be applied without sub-stantial corrections.The cross sections have a weak dependence on − t ,which indicates that at this Q , the photons couple toconfigurations of substantially smaller size than the tar-get. Our data provide a very precise measurement of theexponential slope b φ at small c ∆ τ ∼ φ production is dominated by thescattering of small size ss virtual pairs off the target.This conclusion is supported by the good agreement be-tween our data and the extension of the JML model fromthe real photon point (where it has been calibrated) tothe virtual photon sector. It describes the interactionbetween this ss pair and the nucleon by the exchange oftwo dressed gluons. We conclude that these constituentdegrees of freedom are appropriate for the description of φ -meson production at low W and Q ∼ . XII. ACKNOWLEDGMENTS
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