The experimental signals are the exclusive production of the vector mesons
in the following decay modes
The relevant components of the H1 and ZEUS detectors are the inner tracking chambers which measure the momentum of the decay products; the calorimeters which allow identification of the scattered electron and are used in the triggering of photoproduced vector mesons; and the outer muon chambers used to identify muonic decays of the and . The clean topology of these events results in typical uncertainties on the measured quantities (t, M2, W2 and Q2), reconstructed in the tracking chambers, of order 5%. Containment within the tracking chambers corresponds to a W interval in the range GeV.
Photoproduction processes have been extensively studied in fixed-target experiments, providing a large range in W over which to study the cross-sections. The key features are the weak dependence of the cross-section on W, an exponential dependence on t with a b slope which shrinks with increasing W and the retention of the helicity of the photon by the vector meson.
In Fig. 5, the ZEUS results for exclusive
production as a function of t are shown.
An exponential fit to the ZEUS data in three W intervals yields b-slopes
which are fitted to the form
The interaction radius, RI, can be approximately related to the radii of the interacting proton and vector meson, . The variation of these b values is shown in Fig. 6(a) as a function of vector meson mass MV2. In Fig. 6(b), these slopes are presented as a function of increasing virtuality of the photon for production data. In each case, the range of measured b-slopes varies from around 10 GeV-2 ( fm) at low MV2 or Q2 to 4 GeV-2 ( fm) at the highest MV2 or Q2 so far measured. Given fm, this variation in b-slopes corresponds to a significant decrease in the effective radius of the interacting vector meson from fm to fm as MV2 (at fixed GeV2) or Q2 (at fixed ) increase.
Integrating over the measured t dependence, the W dependence of the results on exclusive vector meson photoproduction cross-sections are shown in Fig. 7. There is generally good agreement between the experiments on the measured cross-sections. The total cross-section is also shown in Fig. 7, rising with increasing energy as in hadron-hadron collisions and consistent with a value of i.e. the total cross-section increases as W0.16.
Given the dominance of the pomeron trajectory at high W and
a |t| distribution whose mean value
is given by 1/b, the diffractive cross-section
rise is moderated from
to
The
(charm) mass scale, ,
is larger than the QCD scale,
,
and it is therefore possible to apply pQCD
techniques.
The theoretical analysis predicts that
the rise of the cross-section is proportional to the square of the gluon
density at small-x (a pair of gluons with no net colour
is viewed as the perturbative pomeron)
We also know from measurements of the DIS total cross-section that application of formula (1) results in a value of which increases with increasing Q2, with 0.2 to 0.25 at GeV2 [12]. The fact that the corresponding relative rise of F2 with decreasing x can be described by pQCD evolution [13] points towards a predicted function for . The current data indicate that this transition occurs for GeV2 [14].
electroproduction results are also available; the dependence of the data is shown in Fig. 8(b). The cross-sections are fitted to where the fitted line corresponds to for the combined H1 and ZEUS data. This compares to the prediction of the Vector Dominance Model (VDM), applicable to soft photoproduction processes, where n = 1 (shown as the dashed line) and the Ryskin model where . Also shown are the lower-W EMC data where . The electroproduction cross-section is of the same order as the data. This is in marked contrast to the significantly lower photoproduction cross-section for the , even at HERA energies, also shown in Fig. 7. Further results in this area will allow tests of the underlying dynamics for both transverse and longitudinally polarised photons coupling to light and heavy quarks in the pQCD calculations.
One contribution to the DIS total cross-section is the electroproduction of low mass vector mesons, here typified by the data. The decay angle distributions of the pions in the rest frame with respect to the virtual photon proton axis from the E665 fixed-target experiment (7<W<28 GeV) are shown in Fig. 9 [15]. The measurements of this helicity angle of the vector meson decay determines for the (virtual) photon, assuming s-channel helicity conservation, i.e. that the vector meson preserves the helicity of the photon.
The decay angular distribution can be written as where the density matrix element r0004 represents the probability that the was produced longitudinally polarised by either transversely or longitudinally polarised virtual photons. The value of R is then obtained from , where is the fractional energy loss of the electron in the proton rest frame and Y+ = 1 + (1-y)2. The kinematic factor 2(1-y)/Y+ is typically close to unity. This variation of R with Q2 is summarised in Fig. 10. The photoproduction measurements for the (not shown) are consistent with the interaction of dominantly transversely polarised photons and hence . The electroproduction data are consistent with a universal dependence on Q2 independent of W and show a transition from predominantly transverse to predominantly longitudinal photons with increasing Q2. This increase of is due to an increased flux of longitudinal photons, . At higher Q2 values, the cross-section due to longitudinal exchange is determined in leading-log pQCD [16] where the underlying interaction of the virtual photon with the constituent quarks of the is calculated. As noted previously (see Fig. 6(b)), the measured b-slope decreases by about a factor of two from the photoproduction case to values comparable to that in the photoproduced case. The basic interaction is probing smaller distances, which allowed a first comparison of the observed cross-section with the predictions of pQCD [17].
The W dependence of the (virtual-)photon proton cross-sections for finite values of Q2 are shown in Fig. 11(a), compared to the corresponding photoproduction cross-sections (the data, not shown, exhibit similar trends). There is a significant discrepancy between the ZEUS and H1 measured cross-sections at Q2 = 20 GeV2 as well as a smaller discrepancy between the E665 and NMC measurements at GeV2. This is illustrated by comparison with the W0.8 (dashed) and W0.22 dotted lines for GeV2 and GeV2. At each Q2 value, a simple dependence cannot account for all the data.
One of the key problems in obtaining accurate measurements of these exclusive cross-sections and the t slopes is the uncertainty of the double dissociation component, where the proton has also dissociated into a low mass nucleon system [18]. At HERA, the forward calorimeters will see the dissociation products of the proton if the invariant mass of the nucleon system, MN, is above approximately 4 GeV. A significant fraction of double dissociation events produce a limited mass system which is typically not detected. One expects that the dissociated mass spectrum will fall as and integrating over the t-dependence CDF obtains = 0.100 0.015 at GeV [19]. Precisely how the proton dissociates and to what extent the proton can be regarded as dissociating independently of the photon system is not a priori known. Currently, this uncertainty is reflected in the cross-sections by allowing the value of to vary from around 0.0 to 0.5, a choice which covers possible variations of as a function of MN and W. Combining all uncertainties, the overall systematic errors on the various cross-sections are typically for both the photoproduction and electroproduction measurements. The estimation of the double dissociation contribution has, however, historically been one of the most significant experimental problems with these measurements. Whether this is the source of the H1 and ZEUS discrepancy is not yet known.
The combined W dependence of the electroproduction data are, therefore, currently inconclusive. However, taking the ZEUS data alone, shown in Fig. 11(b) there are indications of a transition from the soft to the hard intercept with varying from to as indicated by the fitted lines in Fig. 11. These data are therefore consistent with a W0.22 ( = 0.05) dependence at the lowest Q2 values and the W0.8 ( = 0.2) dependence at the highest Q2 values.
An important point to emphasise here is that the relative production of to mesons approaches the quark model prediction of 2/9 at large W as a function of Q2. Similar observations have been made on the t dependence of this ratio for photoproduction data, as shown in the upper plot of Fig. 12. Here, the ratio of the cross-sections approaches the SU(4) flavour prediction of . The restoration of this symmetry indicates that the photon is interacting via quarks, rather than as a vector meson with its own internal structure. This therefore indicates the relevance of a gluonic interpretation of the pomeron and the applicability of pQCD to these cross-sections. Similarly, the relative production of to mesons is shown with the asymptotic prediction of 8/9 from the quark model in the lower plot of Fig. 12. In this case, it is evident that threshold effects for the heavy charm quark are still significant in the measured t range, however the ratio climbs by almost two orders of magnitude from to GeV2.
Exclusive photoproduction has also been observed [20]. The 20 events observed in the channel corresponds to a cross-section for predominantly (1S) production, as well as higher states, of nb for where MN < 4 GeV and 80<W<280 GeV, Q2 < 4 GeV2. This is about 1% of the cross-section, as shown in Fig. 13, emphasing the importance of the mass of the heavy quark in the production of exclusive vector mesons.
In conclusion, there is an accumulating body of exclusive vector meson production data, measured with a systematic precision of , which exhibit two classes of W2 behaviour: a slow rise consistent with that of previously measured diffractive data for low MV2 photoproduction data but a significant rise of these cross-sections above a finite value of MV2, t or Q2. In general, the cross-sections at large W2 can be compared to pQCD when either MV2, t or Q2 become larger than the scale . Precisely how the transition from the non-perturbative to the perturbative regime is made is currently being determined experimentally.