hep–ph/9907315
HigherOrder Effects on Inelastic Photoproduction^{1}^{1}1To appear in the proceedings of the workshop on Monte Carlo Generators for HERA Physics, DESY, Hamburg, 21 April 1998 – 5 February 1999, edited by T. Doyle, G. Grindhammer, G. Ingelman, and H. Jung.
B.A. Kniehl
II. Institut für Theoretische Physik, Universität Hamburg,
Luruper Chaussee 149, Germany
Abstract: Approximately taking into account the higherorder effects due to multiplegluon initialstate radiation, we extract from the latest Tevatron data of prompt hadroproduction the leading colouroctet matrix elements within the nonrelativisticQCD (NRQCD) factorization formalism. We find that the matrix elements which describe the formation of mesons from colouroctet pairs in the angularmomentum states , with , which are responsible for the excess of the predicted cross section of inelastic photoproduction over the existing HERA data at high values of the inelasticity variable , are significantly reduced. As a consequence, the Tevatron and HERA measurements of inclusive production are reconciled in the NRQCD framework. and
1 Introduction
The cross section of inclusive hadroproduction measured in collisions at the Fermilab Tevatron [1, 2] turned out to be more than one order of magnitude in excess of what used to be the best theoretical prediction, based on the coloursinglet model (CSM). As a solution to this puzzle, Bodwin, Braaten, and Lepage [3] proposed the existence of socalled colouroctet processes to fill the gap. The central idea is that pairs are produced at short distances in colouroctet states and subsequently evolve into physical (coloursinglet) charmonia by the nonperturbative emission of soft gluons. The underlying theoretical framework is provided by nonrelativistic QCD (NRQCD) endowed with a particular factorization hypothesis, which implies a separation of shortdistance coefficients, which are amenable to perturbative QCD, from longdistance matrix elements, which must be extracted from experiment. This formalism involves a double expansion in the strong coupling constant and the relative velocity of the bound charm quarks, and takes the complete structure of the charmonium Fock space into account.
In the case of inelastic photoproduction, NRQCD with colouroctet matrix elements tuned [4] to fit the Tevatron data [1] predicts [5] at leading order (LO) a distinct rise in cross section as , where is the fraction of the photon energy transferred to the meson in the proton rest frame, which is not observed by the H1 [6] and ZEUS [7] collaborations at HERA. This colouroctet charmonium anomaly has cast doubts on the validity of the NRQCD factorization hypothesis [3], which seems so indispensible to interpret the Tevatron data in a meaningful way.
Here, we report on an attempt [8] to rescue the NRQCD approach by approximately taking into account dominant higherorder (HO) QCD effects. The basic idea is as follows. The predicted excess over the HERA data at close to unity is chiefly generated by colouroctet pairs in the states , , and [5], where we use the spectroscopic notation to indicate the spin , the orbital angular momentum , and the total angular momentum . On the other hand, in hadroproduction at the Tevatron, the contributions from the colouroctet and states fall off much more strongly with increasing transverse momentum () than the one due to the colouroctet state [4], which is greatly suppressed in the quasielastic limit of photoproduction [5]. Consequently, the nonperturbative matrix elements which are responsible for the colouroctet charmonium crisis are essentially fixed by the Tevatron data in the low regime. This is precisely where the LO approximation used in Ref. [4] is expected to become unreliable due to multiplegluon radiation from the initial and final states. In Ref. [9], this phenomenon was carefully analyzed in a Monte Carlo framework and found to significantly increase the LO cross section. In Ref. [8], fits to the latest prompt data taken by the CDF collaboration [2] at the Tevatron were performed incorporating this information [9] on the dominant HO QCD effects. The resulting HOimproved NRQCD predictions for inelastic photoproduction at HERA do not overshoot the H1 [6] and ZEUS [7] data any more.
2 Theoretical input
The underlying theoretical framework is explained in Ref. [8]. If is of order or below, we adopt the fusion picture, where the bound state is formed within the primary hardscattering process. In the high regime, we work in the fragmentation picture, where the bound state is created from a single highenergy gluon, charm quark or antiquark which is close to its mass shell. We take the renormalization scale and the common factorization scale to be , where is the transverse mass. We define the starting scale of the fragmentation functions (FF’s) as . For our LO analysis, we choose CTEQ4L [10] and GRVLO [11] as the proton and photon PDF’s, respectively, and evaluate from the oneloop formula with MeV [10]. Whenever we include higher orders, we adopt the renormalization and factorization scheme and employ CTEQ4M [10], GRVHO [11], and the twoloop formula for with MeV [10].
Unfortunately, not all ingredients which would be necessary for a fully consistent NLO analysis are yet available. In the case of fusion, the NLO corrections to the partonic cross sections are only known for direct photoproduction in the CSM [12]. Furthermore, in the case of fragmentation, the NLO corrections to the FF’s at the initial scale are still unknown. In the case of direct photoproduction under typical HERA conditions, the QCD correction factor to the inclusive cross section in the CSM was found [12] to be as low as 1.2 in the inelastic regime . It is plausible that the factors for the coloroctet and resolvedphoton processes should be modest, too. However, the situation should be very different for inclusive hadroproduction at the Tevatron, especially in the low range, where one expects substantial HO QCD effects due to multiplegluon radiation. Such effects were estimated for the fusion mechanism in Ref. [9] by means of the Monte Carlo event generator PYTHIA [13] after implementing therein the relevant colouroctet processes, and they were indeed found to be very sizeable. The impact of these effects on the fit to the latest CDF data [2] is demonstrated in Table 1.
LO  HO  

GeV  GeV  
GeV  GeV  
GeV  GeV  
3.47  3.54  
7.49/12  3.96/12 
3 Predictions for charmonium photoproduction
We now explore the phenomenological consequences of this HO improvement for inclusive photoproduction in collisions at HERA, with beam energies GeV and GeV in the laboratory frame, assuming the maximum photon virtuality to be GeV. As in the H1 [6] and ZEUS [7] publications, we convert the cross sections to averaged cross sections by dividing out the photonflux factor. The contribution due to mesons with subsequent decay into mesons is approximately taken into account by multiplying the theoretical predictions by an overall factor of 1.15. The data are mostly concentrated in the low range, where the fusion picture should be valid.
In Fig. 1, we compare our LO and HOimproved predictions for the , , and distributions with the H1 [6] and ZEUS [7] data. Here, is the rapidity in the laboratory frame, which is taken to be positive in the proton flight direction. The circumstance that and are not separately fixed by the fit to the CDF data [2] induces some uncertainty in the colouroctet contributions to the cross sections of direct and resolved photoproduction and thus also in the total cross section. This uncertainty is encompassed by the results for and and those for and , which are actually shown in Fig. 1. We observe that, at LO, the colouroctet contribution of direct photoproduction is dominant for . Thus, it also makes up the bulk of the and distributions, which are integrated over . This contribution is responsible for the significant excess of the LO predictions over the experimental results for and at low and high , respectively. On the other hand, the HOimproved predictions tend to undershoot the data leaving room for a substantial factor due to the missing NLO corrections to the partonic cross sections. Now, the coloursinglet contribution of direct photoproduction, which is well under theoretical control [12], is by far dominant, except in the corners of phase space, at and , where the colouroctet contributions of resolved and direct photoproduction, respectively, take over. Of course, we should also bear in mind that the predictions shown in Fig. 1 still suffer from considerable theoretical uncertainties related to the choice of the scales and , the PDF’s, and other input parameters such as and [12]. From these observations, we conclude that it is premature at this point to speak about a discrepancy between the Tevatron [1, 2] and HERA [6, 7] data of inclusive production within the framework of NRQCD [3].







4 Conclusions
We determined the colouroctet matrix elements which appear in the NRQCD expansion [3] at leading order in by fitting the latest Tevatron data of prompt hadroproduction [2]. We found that the result for the linear combination of and is substantially reduced if the HO QCD effects due to the multiple emission of gluons, which had been estimated by Monte Carlo techniques [9], are taken into account. As an important consequence, the intriguing excess of the LO NRQCD prediction for inelastic photoproduction at close to unity [5] over the HERA measurements [6, 7] disappears. We assess this finding as an indication that it is premature to proclaim an experimental falsification of the NRQCD framework on the basis of the HERA data. Although we believe that our analysis captures the main trend of the HO improvement, we stress that it is still at an exploratory level, since a number of ingredients which would be necessary for a fully consistent NLO treatment of inclusive hadroproduction and photoproduction are still missing.
Achnowledgements. The author thanks Gustav Kramer for his collaboration on this work.
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