PHYSICAL REVIEW D 68, 094009 ~2003! Testing color evaporation in photon-photon production ofJÕc at CERN LEP II O. J. P. Éboli,1,* E. M. Gregores,2,† and J. K. Mizukoshi1,‡ 1Instituto de Fı´sica, Universidade de Sa˜o Paulo, Sa˜o Paulo, SP, Brazil 2Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Sa˜o Paulo, SP, Brazil ~Received 18 August 2003; published 11 November 2003! The DELPHI Collaboration has recently reported the measurement ofJ/c production in photon-photon collisions at CERN LEP II. These newly available data provide additional proof of the importance of colored cc̄ pairs for the production of charmonium, because these data can be explained only by considering resolved photon processes. We show here that the inclusion of color octet contributions toJ/c production in the framework of the color evaporation model is able to reproduce these data. In particular, the transverse- momentum distribution of theJ/c mesons is well described by this model. DOI: 10.1103/PhysRevD.68.094009 PACS number~s!: 13.60.Le, 14.40.Gx ry ls lo s be n d b se ai o - a a ce e fe l o ro g p ns. om- to in- t a M pro- ss de- ed te ith uc- se cal- ns. ad- uc- low e- I. INTRODUCTION The DELPHI Collaboration recently released prelimina measurements of the transverse-momentum spectrum ofJ/c mesons produced ingg collisions at the CERNe1e2 col- lider LEP@1,2#. These new data allow further tests of mode for charmonium production. We show here that the co evaporation model~CEM! reproduces these new results u ing the same single nonperturbative parameter that has obtained from previous analysis of charmonium photo- a hadroproduction. These newly available data provide ad tional proof of the importance of coloredcc̄ pairs for the production of charmonium, as the data in this region can explained only by considering resolved photon proces which form coloredcc̄ pairs in the leading order. The CEM for charmonium production incorporates these colored p into the total yield of charmonium in a very simple and ec nomical way. The Fermilab Tevatron data@3,4# on charmonium produc tion at highpT have changed the way we understand ch monium production. The presently successful models based on two key considerations:~i! quarkonium production is a two-step process in which a heavy quark pair is produ first, followed by the nonperturbative formation of th asymptotic states, and~ii ! color octet as well as singletcc̄ states contribute to the production of charmonia. These tures are incorporated in the nonrelativistic QCD~NRQCD! factorization approach@5,6#, the color evaporation mode @7–9# and the soft color interaction model@10#. The color evaporation model simply states that charm nium production is described by the same dynamics asDD̄ production, i.e., by the formation of acc̄ pair in any color configuration. Rather than imposing that thecc̄ pair is in a color singlet state in the short distance perturbative p cesses, it is argued that the appearance of color sin asymptotic states solely depends on the outcome of non *Electronic address: eboli@fma.if.usp.br †Electronic address: gregores@ift.unesp.br ‡Electronic address: mizuka@fma.if.usp.br 0556-2821/2003/68~9!/094009~5!/$20.00 68 0940 r - en d i- e s, rs - r- re d a- - - let er- turbative large distance fluctuations of quarks and gluo These large distance fluctuations are considered to be c plex enough for the occupation of different color states approximately respect statistical counting. In fact, it is deed hard to imagine that a color singlet state formed a rangemc 21 would survive to form ac at a rangeLQCD 21 . Although far more restrictive than other proposals, the CE successfully accommodates all features of charmonium duction @11–13#. The CEM predicts that the sum of the production cro sections of all quarkonium and open charm states is scribed by squarkonium5 1 9E2mc 2mD dMcc̄ dscc̄ dMcc̄ ~1! and sopen5 8 9E2mc 2mD dMcc̄ dscc̄ dMcc̄ 1E 2mD dMcc̄ dscc̄ dMcc̄ , ~2! whereMcc̄ is the invariant mass of thecc̄ pair. The factor 1/9 stands for the probability that a pair of charm quarks form at a typical time scale 1/Mc ends up as a color singlet sta after exchanging an uncountable number of soft gluons w the reaction remnants; for further details see@7#. One attrac- tive feature of this model is the relation between the prod tion of charmonium and open charm, which allows us to u the open charm data to normalize the perturbative QCD culation, and consequently to constrain the CEM predictio Up to this point, the model has no free parameter in dition to the usual QCD ones. In order to predict the prod tion rate of a particular charmonium state, let us say aJ/c meson, we must also know the fractionrc of produced quarkonium states that materialize as this state (J/c), sc5rcsquarkonium. ~3! In its simplest version, the CEM assumes thatrc is energy and process independent, which is in agreement with the energy measurements@14,15#. Notice thatrc is the only free parameter of the CEM, making this a very restrictive fram ©2003 The American Physical Society09-1 e y pa ils - e n s m io on e s d rib- ed ro- m te- e the ula- EM m riza- ÉBOLI, GREGORES, AND MIZUKOSHI PHYSICAL REVIEW D68, 094009 ~2003! work. From charmonium photoproduction, we determin that rc50.43–0.5@8#, a value that can be accounted for b statistical counting of final states@10#: the fraction of a given quarkonium stateX with angular momentumJX can be writ- ten asrX5GX /(YGY with GX5(2JX11)/nX . Notice that a suppression factor for radially excited states with princi quantum numbernx has been introduced; for further deta see Ref.@10#. The fact that allc production data are de scribed in terms of this single parameter, fixed byJ/c pho- toproduction, leads to parameter-free predictions for thZ boson decay rate intoc @16#, and to charmonium productio cross sections at the Tevatron@17# and the DESYepcollider HERA @18,19#, as well as in neutrino initiated reactions@20#. II. RESULTS The differential cross section for the inclusive proce e1e2→e1e2gg→J/cX is d2s dpT 2 5( A,B E E E E dy1dy2dxAdxBf g/e1~y1! 3 f g/e2~y2!FA/g~xA!FB/g~xB! d2ŝ~AB→cY! dpT 2 , ~4! where f g/e6 is the bremsstrahlung photon distribution fro an electron or positron. We denoted the parton distribut function of the photon byFA[B]/g(xA[B] ), wherexA[B] is the fraction of the photon momentum carried by the part A@B#. For direct photon interactions (A@B#[g), we have FA[B]/g(xA[B] )5d(xA[B]21). We considered an averag electron-positron center-of-mass energy 2Ee5197 GeV. We also applied the experimentalJ/c rapidity cut 22,hc ,2, and imposed that thegg center-of-mass energy satisfie Wgg,35 GeV, whereWgg52EeAy1y2. In our calculation, we employed the Weiza¨cker-Williams approximation for the photon distribution, 09400 d l s n f g/e6~y!5 ae.m. 2p F11~12y!2 y logS Qmax 2 Qmin 2 D 12me 2yS 1 Qmax 2 2 1 Qmin 2 D G , ~5! with Qmin 2 5me 2y2/(12y), and Qmax 2 5(Eeu)2(12y) 1Qmin 2 . Here, the fraction of the parente6 energy (Ee) carried by the photons isy(5Eg /Ee), andu is the angular cut that guarantees that the photons are real. We useu 50.032 rad, as determined by the experiment. The inclusive subprocess cross sectionŝ(AB→cY) was calculated using the CEM; see Eqs.~1! and~3!. The partonic subprocesses contributing toJ/c production are depicted in Table I. Notice that both direct and resolved photons cont ute to charmonium production in the CEM. We evaluat numerically the tree level helicity amplitudes of the subp cesses displayed in Table I using theMADGRAPH @21# and HELAS @22# packages. The adaptive Monte Carlo progra VEGAS @23# was employed to perform the phase space in gration. In the framework of the CEM, the evaluation of th photon-photon production cross section contains only free parameters appearing in the perturbative QCD calc tion of the subprocesses presented in Table I, since the C free parameterrc can be fixed at the value extracted fro the photoproduction ofJ/c, i.e., rc50.5 @8#. We used the leading order GRV-G@24# and GRS-G@25# parton density functions as provided by the CERNPDFLIB package with the partonic subprocess center-of-mass energy as the facto TABLE I. Subprocesses contributing toJ/c production ingg collisions. Hereq stands for the light quark flavorsu,d,s. Direct Once resolved Twice resolved gg→cc̄g gq(q̄)→cc̄q(q̄) qq̄→cc̄g gg→cc̄g gq(q̄)→cc̄q(q̄) gg→cc̄g for n TABLE II. Cross sections for direct, once resolved, and twice resolved production processespT 2 .0.25 GeV2 using different sets of parton distribution functions~PDF!, charm masses, and renormalizatio scalesmR5jAmc 21bpT 2. Parameters Cross sections~pb! PDF mc j b Direct Once resolved Twice resolved Total GRV-G 1.3 1.0 0 1.72 13.3 1.00 16.1 GRS-G 1.3 1.0 0 1.72 13.0 0.94 15.7 GRS-G 1.2 1.0 0 2.75 21.8 1.73 26.3 GRS-G 1.4 1.0 0 1.02 7.5 0.51 9.07 GRS-G 1.3 0.5 0 3.26 46.8 6.42 56.4 GRS-G 1.3 0.5 1 2.42 28.4 3.15 34.0 GRS-G 1.3 1.0 1 1.44 9.65 0.61 11.7 GRS-G 1.3 2.0 0 1.17 5.99 0.29 7.45 GRS-G 1.3 2.0 1 1.03 4.84 0.22 6.08 9-2 TESTING COLOR EVAPORATION IN PHOTON-PHOTON . . . PHYSICAL REVIEW D68, 094009 ~2003! 10 -2 10 -1 1 10 1 2 3 4 5 6 7 8 9 10 pT 2 (GeV 2) dσ /d p T2 (p b/ Ge V 2 ) µR = ξ mc 1/2 < ξ < 2 (a) 10 -2 10 -1 1 10 1 2 3 4 5 6 7 8 9 10 pT 2 (GeV2) dσ /d p T2 (p b/ Ge V 2 ) µR = ξ (mc 2 + pT 2)1/2 1/2 < ξ < 2 (b) FIG. 1. Uncertainty in thepT 2 differential cross section originating from different choices of the renormalization scalemR . In ~a! we chose mR5jmc while in ~b! mR5jAmc 21pT 2. The shaded band was obtained by varying 1/2,j,2 and the solid line stands forj51. We fixed mc51.3 GeV, and used the GRS-G parton density function in both figures. t - nd ou ow th lts e e on e a e t s th s ie r M on o he the ell rder We II. tly - - g. D ood in re - ar- ec- ase red e- nd for tion scalemF5Aŝ. We verified that our predictions do no vary significantly (&10%) for other choices of the factoriza tion scales, e.g.,mF5 1 2 Aŝ, mF52Aŝ, or mF5mR . We also verified that the results are very similar for the GRV-G a GRS-G parton distributions~see Table II!. The strong cou- pling constant was evolved in leading order considering f active flavors andLQCD (4) 5300 MeV, while the charm quark mass was varied between 1.2 and 1.4 GeV. In order to access the theoretical uncertainties in the l est order CEM calculations, we analyzed the predictedJ/c transverse-momentum spectrum for different choices of renormalization scale (mR). We present in Fig. 1~a! the pre- dictedpT 2 spectrum obtained formR5jmc with 1 2 ,j,2 and mc51.3 GeV, as well as the DELPHI experimental resu @1,27#. We can see from this figure that the CEM describ well the shape of the distribution, despite the large unc tainty in the absolute value of the differential cross secti Notice that we are only changing a global factor (aS) for this choice ofmR when we varyj. Figure 1~b! displays thepT 2 spectrum for mR5jAmc 21pT 2 with 1 2 ,j,2 and mc 51.3 GeV. For this choice ofmR the uncertainties in thepT 2 distribution are smaller than for the previous choice ofmR . However, the shape of thepT 2 spectrum changes and th CEM prediction seems to diminish faster than the data largepT 2 . In Fig. 2 we display the contributions to theJ/c pT 2 spec- trum arising from direct, once resolved, and twice resolv processes. These distributions were obtained using GRS-G photon parton densities,mR5mc , and mc 51.3 GeV. As we can see, the once resolved processe responsible for the majority of the events (.85%), while direct and twice resolved processes account for less 15% of the total cross section. The most important proces gg→cc̄g. We also present in this figure the uncertaint associated with the charm quark mass; the shaded band resents the sum of all contributions takingmc51.3 60.1 GeV. Notice that the largest uncertainties in the CE prediction originate from the choice of the renormalizati scale. This is quite as expected since we are performing calculation in lowest order perturbative QCD. In fact, t 09400 r - e s r- . t d he are an is s ep- ur charmonium production ingg collisions is dominated by once resolved photon processes; therefore it is similar to photoproduction of charm quark pairs. In this case, it is w known that theK factor is of the order of 2–3@26#, and consequently we can anticipate that the next-to-leading o corrections to our results should be of this magnitude. summarize our results for the total cross section in Table In order to further compare our results with the recen published DELPHI results@2,27#, we evaluated the depen dence of the totalJ/c yield on the minimum transverse mo mentum forAs5197 GeV. The result is presented in Fi 3~a!. As can be seen from this figure, the choice of QC parameters we used in this analysis provides a very g description of the existing data, reinforcing our confidence the predictive power of the color evaporation model. Figu 3~b! displays the CEM predictions for theJ/c production cross section as a function of thee1e2 center-of-mass en ergy (As). Here we assumed thatpT 2.0.25 GeV2, mR5mc with mc51.360.1 GeV, and we used the GRS-G set of p ton distribution functions. As expected, the total cross s tion grows with the center-of-mass energy due to the incre in 10 -2 10 -1 1 10 10 2 1 2 3 4 5 6 7 8 9 10 pT 2 (GeV2) dσ /d p T2 (p b/ Ge V 2 ) Total Once Resolved Direct Twice Resolved FIG. 2. Differential cross section as a function of the squa transverse momentum of theJ/c. The shaded band shows the th oretical prediction obtained by varying the charm mass (mc51.3 60.1 GeV). We explicitly show the direct, once resolved, a twice resolved contributions as well as the total cross section mc51.3 GeV. 9-3 ntity. We s ÉBOLI, GREGORES, AND MIZUKOSHI PHYSICAL REVIEW D68, 094009 ~2003! 1 10 10 2 0 0.5 1 1.5 2 2.5 3 3.5 4 Minimum pT 2 (GeV 2) Cr os s Se ct io n (p b) √s = 197 GeV 1.2 GeV < Mc < 1.4 GeV (a) 10 10 2 100 200 300 400 500 600 700 800 900 1000 Energy (GeV) Cr os s Se ct io n (p b) pT 2 > 0.25 GeV2 1.2 GeV < Mc < 1.4 GeV (b) FIG. 3. Total cross section as function of the minimum squared transverse momentum~a! and thee1e2 center-of-mass energy~b!. In ~a! the arrow stands for the DELPHI measured total cross section while the dotted lines indicate the experimental error of this qua varied the charm quark mass asmc51.360.1 GeV to estimate the theoretical uncertainties. We usedAs5197 GeV for the minimum transverse-momentum dependence~a! and imposedpT 2.0.25 GeV2 for the center-of-mass energy dependence~b!. The remaining parameter are the same as for Fig. 2. In both figures, the solid line representsmc51.3 GeV. ns re lu e th s. d ta e e sm r t ing de n of be es, the photon-photon luminosity. We verified that contributio of direct, once resolved, and twice resolved, processes a the same proportion as for the results presented forAs 5197 GeV; see Fig. 2. Taking into account the planned minosity of the futuree1e2 colliders, we can easily forese that it will be possible to extract very precise data on photon-photon charmonium production in these machine III. CONCLUSION In this paper we showed that the color evaporation mo for quarkonium production correctly describes DELPHI da on J/c via photon-photon collisions. Because of the rath large uncertainties in the data, it is not possible to use th to discriminate between the different proposed mechani for charmonium production. As far as the DELPHI data a considered, the NRQCD@27# and CEM frameworks presen equivalent results. na - ar , , 09400 in - e el r m s e Considering that the CEM is also successful in describ the photo- and hadroproduction of charmonium, we conclu that this model gives a robust and simple parameterizatio all charmonium physics. Moreover,gg reactions provide a clear proof of the importance of coloredcc̄ pairs in the pro- duction of charmonium, since the data on this reaction can explained only by considering resolved photon process which lead to coloredcc̄ pairs. ACKNOWLEDGMENTS This research was supported in part by Fundac¸ão de Am- paro àPesquisa do Estado de Sa˜o Paulo~FAPESP!, by Con- selho Nacional de Desenvolvimento Cientı´fico e Tecnolo´gico ~CNPq!, and by Programa de Apoio a Nu´cleos de Exceleˆncia ~PRONEX!. cs us s. nd @1# S. Todorova-Nova, in Proceedings of the 31st Internatio Symposium on Multiparticle Dynamics~ISMD 2001!, Datong, China, 2001, edited by B. Yuting, Y. Meiling, and W. Yuan fang, eConf C010901, 2001. @2# DELPHI Collaboration, J. Abdallahet al., Phys. Lett. B565, 76 ~2003!. @3# CDF Collaboration, F. Abeet al., Phys. Rev. Lett.69, 3704 ~1992!; 79, 572 ~1997!; 79, 578 ~1997!. @4# DO” Collaboration, S. Abachiet al., Phys. Lett. B370, 239 ~1996!. @5# G.T. Bodwin, E. Braaten, and G.P. Lepage, Phys. Rev. D51, 1125 ~1995!; 55, 5853~E! ~1997!. @6# E. Braaten, S. Fleming, and T.C. Yuan, Annu. Rev. Nucl. P Sci. 46, 197 ~1996!. @7# J.F. Amundson, O.J.P. E´boli, E.M. Gregores, and F. Halzen Phys. Lett. B372, 127 ~1996!. @8# J.F. Amundson, O.J.P. E´boli, E.M. Gregores, and F. Halzen Phys. Lett. B390, 323 ~1997!. l t. @9# O.J.P. Éboli, E.M. Gregores, and F. Halzen, inProceedings of the 26th International Symposium on Multiparticle Dynami (ISMD 96), Faro, Portugal, 1996, edited by J. Dias de De et al. ~World Scientific, Singapore, 1997!. @10# A. Edin, G. Ingelman, and J. Rathsman, Phys. Rev. D56, 7317 ~1997!. @11# C.B. Mariotto, M.B. Gay Ducati, and G. Ingelman, Eur. Phy J. C23, 527 ~2002!. @12# M. Kramer, Prog. Part. Nucl. Phys.47, 141 ~2001!. @13# G.A. Schuler and R. Vogt, Phys. Lett. B387, 181 ~1996!. @14# R. Gavai, D. Kharzeev, H. Satz, G.A. Schuler, K. Sridhar, a R. Vogt, Int. J. Mod. Phys. A10, 3043~1995!. @15# G.A. Schuler, hep-ph/9403387. @16# O.J.P. Éboli, E.M. Gregores, and F. Halzen, Phys. Lett. B395, 113 ~1997!. @17# O.J.P. Éboli, E.M. Gregores, and F. Halzen, Phys. Rev. D60, 117501~1999!. 9-4 o cl. r, TESTING COLOR EVAPORATION IN PHOTON-PHOTON . . . PHYSICAL REVIEW D68, 094009 ~2003! @18# O.J.P. Éboli, E.M. Gregores, and F. Halzen, Phys. Lett. B451, 241 ~1999!. @19# O.J.P. Éboli, E.M. Gregores, and F. Halzen, Phys. Rev. D67, 054002~2003!. @20# O.J.P. Éboli, E.M. Gregores, and F. Halzen, Phys. Rev. D64, 093015~2001!. @21# T. Stelzer and W.F. Long, Comput. Phys. Commun.81, 357 ~1994!. @22# H. Murayama, I. Watanabe, and K. Hagiwara, Report N KEK-91-11. 09400 . @23# G.P. Lepage, Report No. CLNS-80/447. @24# M. Gluck, E. Reya, and A. Vogt, Phys. Rev. D46, 1973 ~1992!; 45, 3986~1992!. @25# M. Gluck, E. Reya, and M. Stratmann, Phys. Rev. D51, 3220 ~1995!. @26# S. Frixione, M.L. Mangano, P. Nason, and G. Ridolfi, Nu Phys.B412, 225 ~1994!. @27# M. Klasen, B.A. Kniehl, L.N. Mihaila, and M. Steinhause Phys. Rev. Lett.89, 032001~2002!. 9-5