Inelastic Photoproduction of Charmonium

Figure: Differential cross sections for the production of direct $J/\psi$ at the Tevatron as a function of $p_t$. The data points are CDF measurements from Run I [191,192]. The dotted curves are the CSM contributions. The solid curves are the NRQCD factorization fits, and the other curves are individual color-octet contributions to the fits (taken from [165]).

Many models have been suggested to describe inelastic charmonium production in the framework of perturbative QCD, such as the color-singlet model (CSM) [167,169,168,170], the color-evaporation model [171,172] and soft color interactions [173]. Most recently the ansatz of non-relativistic quantum chromodynamics (NRQCD) factorization was introduced in which colour octet $c\bar{c}$ states contribute to the charmonium production cross section.

Theoretical calculations based on the NRQCD factorization approach [174,175,176] are available in leading order [177,178,179,180,181,182]. In the NRQCD factorization approach the size of the color octet contributions, which are described by long distance matrix elements (LDME), are additional free parameters and have been determined in fits to the Tevatron data [183]. The NRQCD factorization approach contains the color singlet model which is recovered in the limit in which the long distance matrix elements tend to zero.

For $J/\psi$ and $\psi(2S)$ photoproduction, the CSM calculations are available including next-to-leading order contributions [184,185]. Alternatively, using the CSM, inelastic $J/\psi$ production can be modeled in the $k_t$ factorization approach (see section 2.5) using an unintegrated ($k_t$ dependent) gluon density in the proton [187,186,190].

Figure 41 shows data from CDF [191,192] together with CSM calculations to leading order and fitted color octet contributions. It can be seen that the color octet contributions are large, leading to a good description of the data. Unfortunately those long distance matrix elements which are most important in $J/\psi$ and $\psi(2S)$ photoproduction at HERA, are not well constrained by the Tevatron data and thus contain large uncertainties [165]. The new charmonium results from the Tevatron Run-II (see e.g.fig.7) which provide much more statistics and extend to lower values of $p_{t,\psi }$ could help to reduce the uncertainties of the LDME significantly.

It should be noted that next-to-leading-order corrections might change the size of the color octet contributions substantially. Although the NLO terms have not been calculated in the NRQCD approach, effects that are similar to those in the CSM may be expected, in which the NLO terms lead to an increase in the cross section of typically a factor two, with a strong $p_{t,\psi }$ dependence.

Figure: The rate for inelastic $J/\psi$ photoproduction at HERA as a function of a) $z$ and b) $p_{t,\psi }$. The open band represents the LO NRQCD factorization prediction [165]. The shaded band represents the NLO color-singlet contribution [185,165]. The dotted line in b) denotes the LO color-singlet contribution. The data points are from the H1 [29] and ZEUS [32] measurements.

Figure 42 shows the measurements of the $J/\psi$ cross section by the H1 collaboration [29] and the ZEUS collaboration [32], compared with the theoretical predictions given in Ref. [165]. The variable $z$ denotes the fraction of the photon energy in the proton rest frame that is transferred to the $J/\psi$ and is defined as

$$z=\frac{(E-p_z)_{J/\psi}}{(E-p_z)_{\rm hadrons}},$$ (10)

where $E$ and $p_z$ in the numerator are the energy and $z$-component of the momentum of the $J/\psi$ and $E$ and $p_z$ in the denominator are the sums of the energies and $z$-components of the momenta of all the hadrons in the final state.

The $J/\psi$ data are not corrected for feeddown processes from diffractive and inelastic production of $\psi(2S)$ mesons ( $\approx 15\%$), the production of $b$ hadrons with subsequent decays to $J/\psi$ mesons, or feeddown from the production of $\chi_c$ states. The latter two contributions are estimated to contribute between 5% at medium $z$ and 30% at the lowest values of $z$.

The open band in fig.42 represents the sum of the color-singlet and color-octet contributions, calculated in leading order NRQCD. The uncertainty is due to the uncertainty in the color-octet NRQCD matrix elements. The shaded band shows the calculation of the color-singlet contribution to next-to-leading order in $\alpha_s$ [184,185] which describes the data quite well without the inclusion of a color-octet contribution. The next-to-leading-order QCD corrections are crucial in describing the shape of the transverse-momentum distribution of the $J/\psi$.

Figure: Inelastic $J/\psi$ production in the region $60<W_{\gamma p}<240$ GeV, $0.3<z<0.9$, and $p_{t,\psi }^2>1$ GeV$^2$, in comparison with a $k_t$ factorization model implemented in the Monte Carlo generator CASCADE[188,189]. In a) the differential cross section ${\rm d}\sigma /{\rm d}z$ is shown and in b) $d\sigma /d\ensuremath {p_{t,\psi }^2}$ in the range $0.3<z<0.9$.

The $k_t$ factorization approach [187,188,189,190] has been applied for $J/\psi$ production [186]. Figure 43 shows a comparison of the data with the predictions from the $k_t$ factorization approach as implemented in the Monte Carlo generator CASCADE. Good agreement is observed between data and predictions for $z\,\lower.25ex\hbox{$\scriptstyle\sim$}\kern-1.30ex\raise 0.55ex\hbox{$\scriptstyle <$}\,0.8$. At high $z$ values, the CASCADE calculation underestimates the cross section. The CASCADE predictions for the the $p_{t,\psi}^2$ dependence of the cross section (fig.43c) fit the data considerably better than the collinear LO calculations (dotted curve in fig.42b). This improved fit is attributed to the transverse momentum of the gluons from the proton, which contribute to the transverse momentum of the $J/\psi$ meson.

The polarization of the $J/\psi$ meson is expected to differ in the various theoretical approaches discussed here and could in principle be used to distinguish between them, independently of normalization uncertainties. The general decay angular distribution can be parameterized as

$$\frac{d\Gamma(J/\psi\to \ell^+\ell^-)}{d\Omega} \propto 1 + \lamb... ...os^2\theta + \mu \sin 2\theta \cos\phi + \frac{\nu}{2} \sin^2\theta \cos 2\phi,$$ (11)

where $\theta$ and $\phi$ refer to the polar and azimuthal angle of the three-momentum of the positive lepton with respect to a coordinate system that is defined in the $J/\psi$ rest frame [178]. The parameters $\lambda, \mu, \nu$ can be calculated within NRQCD or the CSM as a function of the kinematic variables, such as $z$ and $p_{t,\psi }$.

Figure: Polarization parameters $\lambda$ (left panels) and $\nu$ (right panels) in the target rest frame as functions of $z$ (top panels) and $p_{t,\psi }$ (bottom panels). The error bars on the data points correspond to the total experimental error. The theoretical calculations shown are from the NRQCD approach [178] (shaded bands) with color-octet and color-singlet contributions, while the curves show the result from the color-singlet contribution separately.

In fig.44, the data are shown, together with the results from two LO calculations: the NRQCD prediction, including color-octet and color-singlet contributions [178], and the color-singlet contribution alone. In contrast to the predictions shown in fig.44, in which $\lambda$ is zero or positive, the prediction of the $k_t$ factorization approach is that $\lambda$ should become increasingly negative toward larger values of $p_{t,J/\psi}$, reaching $\lambda \sim -0.5$ at $p_{t,\psi}=6$ GeV. However, at present, the errors in the data preclude any firm conclusions. In order to distinguish between full NRQCD and the color-singlet contribution alone, measurements at larger $p_{t,\psi }$ are required. The measured values of $\nu$, for which no prediction is available from the $k_t$ factorization approach, slightly favor the full NRQCD prediction.

In conclusion, it should be noted that calculations to next-to-leading order, which are not yet available in the framework of NRQCD factorization, could be an essential ingredient in a full quantitative understanding of charmonium production at HERA, and also at other experiments, such as those at the Tevatron.