Diffractive Charmonium Production

Figure 47: Diagram for diffractive charmonium production via exchange of two gluons in a color-singlet state.

Figure: Total cross section and cross sections for production of various vector mesons in $\gamma p$ collisions as a function of $W_{\gamma p}$, as measured at HERA and in fixed-target experiments.

At HERA, the dominant production channel for quarkonia with quantum numbers of real photons (i.e. $J^{PC}=1^{--}$) is through diffractive processes. In perturbative QCD, the diffractive production of vector mesons can be modeled in the proton rest frame by a process in which the photon fluctuates into a $q\bar{q}$ pair at a long distance from the proton target. The $q\bar{q}$ subsequently interacts with the proton via a color-singlet exchange, i.e.in lowest order QCD via the exchange of a pair of gluons with opposite color (see fig.47)  [194,195,196,197,198,199,200]. At small $\vert t\vert$, where $t$ is the momentum-transfer-squared at the proton vertex, the elastic process dominates, in which the proton stays intact. Toward larger values of $\vert t\vert$, the dissociation of the proton into a small-invariant-mass baryonic state becomes dominant. Measurements of diffractive vector-meson production cross sections and helicity structure from the H1 [201,202,28,203,204,205,206,207,208] and ZEUS [209,210,211,212,213,214,215,216,217] collaborations are available for $\rho^0$, $\omega$, $\phi$, $J/\psi$, $\psi'$, and $\Upsilon$ production, spanning the ranges of $0 \simeq Q^2 < 100$ GeV$^2$, $0 \simeq \vert t\vert < 20$ GeV$^2$, and $20 < W_{\gamma p} < 290$ GeV. ( $W_{\gamma p}$ is the $\gamma p$ center-of-mass energy.) In Figure 48, the elastic photoproduction cross sections are shown. Perturbative calculations in QCD are available for the kinematic regions in which at least one of the energy scales $\mu^2$ (i.e. $Q^2$, $M_V^2$ or $\vert t\vert$) is large and the strong-coupling constant $\alpha_s(\mu^2)$ is small [218,,,,,,].

In the presence of such a `hard' scale, QCD predicts a steep rise of the cross section with $W_{\gamma p}$. At small $Q^2$, $\vert t\vert$ and meson masses $M_V$, vector-meson production is known to show a non-perturbative `soft' behavior that is described, for example, by Regge-type models [225,226,227,228,229]. Toward larger values of $\vert t\vert$, in the leading logarithmic approximation, diffractive $J/\psi$ production can be described by the effective exchange of a gluonic ladder. At sufficiently low values of Bjorken-$x$ (i.e. large values of $W_{\gamma p}$), the gluon ladder is expected to include contributions from BFKL evolution [61,62,63,64,], as well as from DGLAP evolution [59,231].

The elasticity variable $z$ defined in equation (10) is often used to demark the boundary between the elastic and inelastic regions, with a typical demarcation for $J/\psi$ production being $z>0.95$ for the diffractive region and $z<0.95$ for the inelastic region. However, at large $z$, there is actually no clear distinction between inelastic $J/\psi$ production and diffractive $J/\psi$ production in which the proton dissociates into a final state with large invariant mass, owing to the fact that the two processes can produce the same final-state particles. In the region of large $z$, both inelastic (section 6.4.1) and diffractive production processes are expected to contribute to the cross section. At the same time, calculations in perturbative QCD that assume a diffractive color-singlet exchange are capable of describing the production cross sections at large $z$ [207,214,215]. A unified description in QCD of the large $z$ region, taking into account both inelastic and diffractive contributions, is yet to be developed.