Production of Mesons at HERA
Abstract
Inelastic and diffractive production of mesons at HERA is reviewed. The data on inelastic photoproduction are described well within errors by the Colour Singlet Model in nexttoleading order. A search for colour octet processes predicted within the NRQCD/factorisation approach is conducted in many regions of phase space. No unambiguous evidence has been found to date. Diffractive elastic production of mesons has been measured in the limit of photoproduction ( up to the highest photon proton center of mass energies. The increase of the cross section is described by pQCD models. At larger , the dependence is found to be similar to that observed in photoproduction. First analyses of data at high , yield a powerlike dependence on . A LO BFKL calculation gives a good description of the data.
1 Introduction
The main interest in leptoproduction of mesons is a clarification of the production process. In the inelastic regime, many models have been proposed, and predictions within the theoretically favoured NRQCD/factorisation approach could not yet be identified nor could they be ruled out. In the diffractive regime, models based on pQCD have proven to work in principle and more detailed studies are underway experimentally and theoretically.
This review will cover production at HERA in inelastic and diffractive processes. Inelastic production at HERA has been studied in the limit of photoproduction ( and for 2 GeV. Data at high were published by the H1 collaboration [1] and new theoretical calculations are discussed in L. Zwirner’s contribution [2]. Here I concentrate on photoproduction.
Diffractive elastic production is studied with large statistics in the limit of photoproduction and also at 2 GeV, where the and dependences are analysed. Proton dissociative data at high values of , the momentum transfer squared at the proton vertex, are also analysed.
2 Inelastic Production of Mesons
The data from the H1[3] and ZEUS collaborations [4] on inelastic photoproduction are based on integrated luminosities of and , respectively, collected in the years 1996/97. The kinematic regions are characterized by for the ZEUS data and for H1. The usual kinematic quantities and are used, where denote the fourmomenta of the incoming and outgoing electron, of the exchanged photon and the incoming proton, respectively. An important variable is
where is the fourmomentum of the meson. In the proton rest frame is the relative energy of the with respect to the photon, . The variable is used to distinguish diffractive processes, where , from inelastic processes, where .
The inelastic production process at HERA is dominated by boson gluon fusion, see Figure 1b,c. At medium , direct processes dominate, where the photon couples directly to the charm quark, while at small resolved photon processes are expected to contribute, where the photon interacts via its hadronic component.
The theoretical description of meson production has undergone a change in the last few years. The Colour Singlet Model (CSM) [6] has been replaced by an effective field theory approach: non–relativistic QCD (NRQCD) and factorisation [7]. This approach contains the CSM but predicts additional contributions. In the CSM, the pair is produced in the hard scattering with the observed quantum numbers of the meson. The colour singlet state is achieved by emission of an additional gluon, i.e. the process is ^{1}^{1}1The ‘1’ or ‘8’ denote the colour state of the pair and spectroscopic notation is used for the angular momentum, .. In the NRQCD/factorisation approach the pair can also be in a colour octet state where different angular momentum states are also allowed.
At HERA we are in a unique position since for photoproduction full nexttoleading order calculations are available in the colour singlet model, which reduces the uncertainties in the normalisation of the predictions. So we will first show a comparison of the data with these calculations before we address possible contributions of colour octet (CO) intermediate states.
2.1 Comparison of data to Colour Singlet calculations
In Figure 2a,b the and the distribution are shown for data from ZEUS [4] and H1 [3] which agree well with each other. The prediction of the CSM in nexttoleading order [8] is overlaid. The band reflects the major theoretical uncertainties as detailed in the figure caption. A good description of the data is achieved. The necessity of including NLO contributions is demonstrated in the distribution of in Figure 2c. The theoretical band including NLO contributions describes the data well and is, at a , roughly a factor of 10 above the LO prediction. The discrepancy found in at values of between the data and the LO CSM calculation which led to the introduction of CO contributions within the NRQCD/factorisation approach was approximately a factor 40 [9].
A different approach to describe the data within the CSM is shown in Figure 3. Here the H1 data are shown with the results of two Monte Carlo implementations of the Colour Singlet Model in leading order. The dashed histograms represent the prediction from the EPJPSI [10] program which uses “standard” parton distributions integrated over the transverse momentum and evolved with DGLAP equations. The program CASCADE [11] uses the CCFM evolution equation and “unintegrated” parton distributions. These have a distribution in transverse momentum, , and are obtained from a fit to H1 data on . The data in Figure 3(note that now a cut has been applied) are in general well described by the CASCADE model except at very large values, where relativistic corrections are missing. The distribution of CASCADE agrees better with the data than EPJPSI, but there is still a systematic discrepancy towards high .
2.2 Search for Colour Octet Contributions
In Figure 4a the same differential cross sections, from the H1 and ZEUS data as before, are shown again as function of . The theoretical predictions within the NRQCD/factorisation approach in leading order [8] are shown in comparison. These include the direct photon contributions as well as resolved processes. At , direct photon contributions dominate and the CO contribution is dominant at 0.6. The uncertainties of the summed prediction, shown as a band, are mainly due to the uncertainties of the long range matrix elements, the parameters which, in the NRQCD/factorisation approach describe the transition of the state to a physical meson.
The long range matrix elements are denoted as e.g. to describe the transition of the state in a colour octet configuration with angular momentum state . They are not calculable but are believed to be universal. Numerical values have been determined in a number of theoretical analyses of data from the CDF collaboration using different approximations (see [8] for a summary and references). The most important matrix elements for photoproduction are and which are derived in the form of a linear combination , where is a parameter of order 3. In Figure 5, the values for (upper part) and the matrix element (lower part) extracted by a few groups using different approximations are shown following a compilation in [8]: the first group of 6 values is calculated in the LO collinear approximation. In the second group effects of higher orders have been taken into account approximately by using a parton shower Monte Carlo model (PYTHIA). The third group uses a distribution for the gluon and the last value in Figure 5 is derived in the factorisation approach.
The theoretical band in Figure 4a is calculated for and almost covering the full range of uncertainty. The strong rise predicted at large values () is not seen in the data which has led to several theoretical attempts for an explanation. In Figure 4b calculations of a different group [12] are compared to the same data as in 4a. The authors of [12] attempt to estimate effects of higher orders. Here the data are a factor 3 above the predicted sum of singlet and octet contributions for , while in Figure 4a the data are below the final calculation. The rise in due to CO contributions in Figure 4b takes place at . This large region is dominated by diffraction and no experimental results on inelastic processes are available. An intriguing question is of course a possible relation of such CO contributions with what is traditionally attributed to diffraction.
The NRQCD calculations shown in Figure 4a and b neglect the energy smearing of the transition of the state to the via emission of soft gluons. This effect is expected to be large at . In [13], an attempt was made to calculate this smearing using the technique of shape functions known from calculations in decays of . The calculations for the distributions are shown together with the data in Figure 6 ^{2}^{2}2The authors also extract the long range transition matrix elements fom B decays. The values, which were used for Figure 6, agree approximately with the values derived from the distribution of mesons in collisions taking into account higher orders via a PYTHIA Monte Carlo simulation [14].. In this comparison the standard cut 1 GeV is applied. The smeared calculations are given for two values of and show the expected decrease towards while the unsmeared curve (labelled “total partonic” in the figure) is seen to rise. However before decreasing the smeared curves are observed to increase above the unsmeared one (around ). The authors of [13] conclude that the theory is not well behaved and suggest applying a higher cut. Their prediction for is shown in the lower part of the figure. A comparison of this calculation with ZEUS results for a cut 2 GeV is shown in the right panel of Figure 6 and is seen to give an acceptable description of the data.
2.3 Resolved Contributions and Polarisation Analysis
The region of low values, , is explored in an analysis of H1 covering a region and . The result is shown in Figure 8a in comparison to predictions from [12]. At such low values, resolved photon contributions are expected and CO contributions are much larger than CS contributions. Here the smearing effects of soft gluon emission are not expected to play an important rôle. The number of parameters entering this comparison is larger and probably a mere cross section measurement will not suffice to distinguish between models. This can be judged from Figure 8b, where the predictions within the CSM are shown with the dominant uncertainties, which overall amount to a factor 5. The distribution of the data is shown in Figure 8c and looks – at least with present errors – very similar to the form found at medium values shown as a scaled curve.
Measurements of polarisation in were advocated to give independent proof of the presence of colour octet contributions. The polarisation is measured via the angular distribution of the decay leptons in the rest frame of the meson. Parametrising it as , corresponds to transverse (longitudinal) polarisation, while reflects no polarisation. The data for and from the CDF collaboration [9], though limited in statistics, do not show the expected increase of the polarisation parameter towards higher predicted within NRQCD[17].
ZEUS performed a first measurement of the polarisation in scattering. Using the flight direction of the in the laboratory system as a reference axis the results in Figure 8a were obtained. They are compared with the predictions of the “semihard” model [18] using unintegrated gluon density distributions based on BFKL evolution equations. The trend of the data is described within this model. There are no calculations within NRQCD/factorisation which one can compare directly with the data. In Figure 8b, calculated in the “recoil” system which uses the direction in the system as a reference axis, one can however see that the parameter increases with for the Colour Singlet Model while it remains small for the octet contributions.
Summarising, there is still no evidence of colour octet contributions in photoproduction at HERA. Due to the experimental and theoretical uncertainties they can however not be excluded.
3 Diffractive Production of Mesons
In the last few years it has been demonstrated that elastic production can be described by perturbative QCD in photoproduction and also at finite . Data for elastic diffractive processes are now available in a large kinematic range. A summary of the published H1 data (21 ) [1, 19] and the new preliminary data from ZEUS [20] (corresponding to 40 ) are shown in Figure 10. The data cover the regime of photoproduction, , extending almost to the kinematic limit in the photon proton energy . In the data go up to .
The curves in Figure 10 show fits to the data of the form and is also plotted as a function of in the insert of Figure 10. The value of is on average well above expected from Reggetype fits with a soft pomeron. The distribution is fitted by a functional form , yielding for H1 and for the ZEUS data.
The photoproduction data in Figure 10 are overlaid with predictions from two theoretical groups using different approaches within pQCD [21, 22]. The basic picture is the exchange of two gluons between the proton and the pair. Apart from a number of technical differences the groups handle the conversion to a meson differently: MRT use partonhadron duality, while FKS use a wave function.
The important prediction is for the slope of the data which is described well by all calculations shown with the exception of GRV98 partons. H1 has previously also found good agreement within the FKS calculations using CTEQ4M or MRSR2 partons [19].
Both collaborations have extracted the effective trajectory from photoproduction data. The procedure is to fit the data at fixed values of with a form . The resulting values of are displayed in Figure 11 with separate fits to a simple form . The more accurate ZEUS data yield . The H1 data are compatible with this result. The soft pomeron trajectory is ruled out. A NLO BFKL calculation of the intercept [23] is in agreement with the data.
The data shown so far are dominated by low values of since they are found to drop with an exponential behaviour: with . Data collected at high are shown in Figure 12. At high values of , the proton dissociates into a low mass system . ZEUS and H1, at present, measure slightly different cross sections: the ZEUS data are corrected to a range while H1 corrects to the full kinematically allowed region of . A LO calculation implementing the BFKL evolution equation [24] reproduces the data sets well using a value for .
4 Summary and Outlook
Inelastic photoproduction of mesons at HERA continues to be well described by the COlour Singlet Model calculated in nexttoleading order. At a , the measured cross section is reproduced by the NLO calculation, which is a factor of 10 above the LO prediction. A reasonable alternative description of the data on a Monte Carlo basis can be obtained using parton distributions not integrated over the transverse momentum based on the CCFM evolution equations. As regards Colour Octet contributions, only LO calculations are available. Due to normalisation uncertainties, the comparison with data at high values is still ambiguous. NRQCD resummations at high seem to necessitate a higher cut and then describe the data reasonably well. A first analysis of data at low values shows sensitivity to resolved contributions. However since in this regime more parameters enter, a mere cross section measurement may not be sufficient to decide on Colour Octet contributions. A first attempt to measure the meson polarisation looks promising. This and all other analyses will profit from the increased statistics which will soon become available.
For elastic production, the range in photoproduction has been extended almost to the kinematic limit. The cross section continues to rise. Parametrising it as , is measured to be large, of order 0.8. The data at higher are in rough agreement with such a steep rise. Comparisons of theoretical calculations within pQCD show a sensitivity to the gluon distribution. A “trajectory” measurement excludes the soft pomeron trajectory. The measurement of diffractive processes has been extended to values of . The data is well described by calculations based on the BFKL equation.
Acknowledgement
I wish to thank the organizers for letting me participate in this very
fruitful and stimulating meeting in beautiful surroundings. Thanks to all
‘theoretical’ and ‘experimental’ colleagues for discussions about their
data or theories!
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