Exclusive Production of Vector Mesons

The experimental signals are the exclusive production of the vector mesons in the following decay modes

$$\rho^0\rightarrow \pi^+\pi^-~~~~~~ \phi\rightarrow K^+K^-~~~~... ...rightarrow\mu^+\mu^-,e^+e^-~~~~~ \Upsilon\rightarrow\mu^+\mu^-.$$

First results on $\omega\rightarrow \pi^+\pi^-\pi^o$ and higher vector mesons ( $\rho' \rightarrow \pi^+\pi^-\pi^o\pi^o$, $\rho' \rightarrow \pi^+\pi^-\pi^+\pi^-$ and $\psi'\rightarrow\mu^+\mu^-,e^+e^-$) have also been presented. A recent review of the HERA data can be found in Ref [9].

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 $J/\psi$ and $\Upsilon$. 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 $40 \raisebox{-.6ex}{${\textstyle\stackrel{<}{\sim}}$ }W \raisebox{-.6ex}{${\textstyle\stackrel{<}{\sim}}$ }180$ 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 $\rho ^0$ 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

$$b = b_0 + 2\mbox{$\alpha'$}\ln(W^2/M_V^2)$$

expected from Regge theory. The fitted value of $\mbox{$\alpha'$ }= 0.23 \pm 0.15 ^{+ 0.10}_{-0.07}$ GeV^-2 is consistent with a shrinkage of the t slope with increasing W² expected from soft diffractive processes.

  

Figure: ZEUS exclusive $\rho ^0$ t distributions characterised by $b_\rho$ as function of W together with a fit to the data discussed in the text and compared to a compilation of lower energy data [10].

The interaction radius, R_I, can be approximately related to the radii of the interacting proton and vector meson, $R_I \simeq \sqrt{R_P^2 + R_V^2}$. The variation of these b values is shown in Fig. 6(a) as a function of vector meson mass M_V². In Fig. 6(b), these slopes are presented as a function of increasing virtuality of the photon for $\rho ^0$ production data. In each case, the range of measured b-slopes varies from around 10 GeV^-2 ( $R_I \simeq 1.3$ fm) at low M_V² or Q² to 4 GeV^-2 ( $R_I \simeq 0.8$ fm) at the highest M_V² or Q² so far measured. Given $R_P \simeq 0.7$ fm, this variation in b-slopes corresponds to a significant decrease in the effective radius of the interacting vector meson from $R_V \simeq 1.1$ fm to $R_V \simeq 0.4$ fm as M_V² (at fixed $Q^2 \simeq 0$ GeV²) or Q² (at fixed $M_V^2 = M_\rho^2$) increase.

  

Figure: Exclusive vector meson production b-slopes as a function of (a) mass of the vector meson M_V² and (b) mean virtuality of the photon <Q²>.

  

Figure 7: W dependence of the exclusive vector meson and total photoproduction cross-sections compared to various power law dependences discussed in the text.

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 $\gamma p$ total cross-section is also shown in Fig. 7, rising with increasing energy as in hadron-hadron collisions and consistent with a value of $\epsilon \simeq 0.08$ i.e. the total cross-section increases as W^0.16.

Given the dominance of the pomeron trajectory at high W and a |t| distribution whose mean value $\vert\bar{t}\vert$ is given by 1/b, the diffractive cross-section rise is moderated from $W^{4\epsilon}= W^{0.32}$ to

$$W^{4(\epsilon - \mbox{$\alpha'$}\cdot\vert\bar{t}\vert)} \equiv W^{4\bar{\epsilon}} = W^{0.22}.$$

Here $\bar{\epsilon} = 0.055$ characterises the effective energy dependence after integration over t for b = 10 GeV^-2 (which is appropriate for $\rho ^0$ exchange as observed in Fig. 5). The observed shrinkage of the diffractive peak therefore corresponds to a relative reduction of the diffractive cross-section with increasing energy. Such a dependence describes the general increase of the $\rho ^0$, $\omega$ and $\phi$ vector meson cross-sections with increasing W. However, the rise of the $J/\psi$ cross-section is clearly not described by such a W dependence, the increase being described by an effective W^0.8 dependence. Whilst these effective powers are for illustrative purposes only, it is clear that in exclusive $J/\psi$ production a new phenomenon is occurring. For example, fits to the ZEUS data yield results for $\bar{\epsilon}$ which are inconsistent with soft pomeron exchange

$$\bar{\epsilon} = 0.230 \pm 0.035 \pm 0.025.$$

  

Figure: Exclusive $J/\psi$ production cross-sections as a function of (a) W and (b) $Q^2+M_\psi^2$.

The $J/\psi$ (charm) mass scale, $M_\psi^2$, is larger than the QCD scale, $\Lambda^2$, 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)

$$\sigma (\gamma p \rightarrow J/\psi~p) \propto [(xG(x,\bar{Q^2})]^2 \simeq W^{4\bar{\epsilon}} \simeq W^{0.8}$$

where $\bar{Q^2}= (Q^2 + M_{J/\psi}^2)/4 (\simeq 2.4$ GeV² for the photoproduction data) is chosen as the scale in the Ryskin model [11]. This approach enables discrimination among recent parametrisations of the proton structure function as shown in Fig. 8(a). Here, the measured cross-sections are compared to the Ryskin model for various choices of parton densities which describe recent F₂ data. The approach is therefore very promising as an independent method to determine the gluon distribution at low-x from the HERA data. Currently, however, there are model uncertainties and the calculations are only possible in leading order. The normalisation is therefore uncertain by up to a factor of two.

We also know from measurements of the DIS $\gamma^* p$ total cross-section that application of formula (1) results in a value of $\epsilon$ which increases with increasing Q², with $\epsilon \simeq$ 0.2 to 0.25 at $Q^2\simeq10$ 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 $\epsilon = \epsilon(Q^2)$ for $Q^2 \raisebox{-.6ex}{${\textstyle\stackrel{>}{\sim}}$ }Q_o^2$. The current data indicate that this transition occurs for $Q^2 \simeq 1$ GeV² [14].

$J/\psi$ electroproduction results are also available; the $Q^2+M_\psi^2$ dependence of the data is shown in Fig. 8(b). The cross-sections are fitted to $\sigma \propto 1/(Q^2+M_\psi^2)^n$ where the fitted line corresponds to $n \simeq 2.1$ 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 $n~\simeq~3$. Also shown are the lower-W EMC data where $n \simeq 1.5$. The $J/\psi$ electroproduction cross-section is of the same order as the $\rho ^0$ data. This is in marked contrast to the significantly lower photoproduction cross-section for the $J/\psi$, 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.

  

Figure: Q² dependence of the decay angular distributions for E665 $\rho ^0$ electroproduction data.

  

Figure: Ratio $R= \sigma _L/\sigma _T$ of $\rho ^0$ electroproduction data.

One contribution to the DIS $\gamma^* p$ total cross-section is the electroproduction of low mass vector mesons, here typified by the $\rho ^0$ data. The decay angle distributions of the pions in the $\rho ^0$ 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 $R= \sigma _L/\sigma _T$ 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 $\frac {1}{N}\frac{dN}{d\rm {cos} \it\theta_h} = \frac{3}{4}[1-r_{00}^{04}+(3 r_{00}^{04}-1)\rm {cos}^2\it\theta_{h}],$ where the density matrix element r₀₀⁰⁴ represents the probability that the $\rho ^0$ was produced longitudinally polarised by either transversely or longitudinally polarised virtual photons. The value of R is then obtained from $R = \frac{\sigma_L}{\sigma_T} = \frac{2(1-y)}{Y_+}\cdot \frac{r_{00}^{04}}{1-r_{00}^{04}}$, where $y = P \cdot q/P \cdot k$ 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 $\rho ^0$ (not shown) are consistent with the interaction of dominantly transversely polarised photons and hence $R \simeq 0$. 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 $\sigma_L$ is due to an increased flux of longitudinal photons, $\sigma_L \propto (Q^2/M_X^2) \sigma_T$. 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 $\rho ^0$ 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 $J/\psi$ 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 $\rho ^0$ cross-sections for finite values of Q² are shown in Fig. 11(a), compared to the corresponding photoproduction cross-sections (the $\phi$ 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 $Q^2 \simeq 6$ GeV². This is illustrated by comparison with the W^0.8 (dashed) and W^0.22 dotted lines for $Q^2 \simeq 6$ GeV² and $Q^2 \simeq 20$ 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, M_N, 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 $d\sigma/dM_N^2 \propto (1/M_N^2)^{1+\bar{\epsilon}}$ and integrating over the t-dependence CDF obtains $\epsilon$ = 0.100 $\pm$ 0.015 at $\sqrt{s} = 1800$ 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 $\bar{\epsilon}$ to vary from around 0.0 to 0.5, a choice which covers possible variations of $\bar{\epsilon}$ as a function of M_N and W. Combining all uncertainties, the overall systematic errors on the various cross-sections are typically $\simeq~20\%$ 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 $\rho ^0$ 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 $\bar{\epsilon}$ varying from $\bar{\epsilon} = 0.04 \pm 0.01 \pm 0.03~(Q^2 = 0.5~{\rm GeV}^2)$ to $\bar{\epsilon} = 0.19 \pm 0.05 \pm 0.05~(Q^2 = 20~{\rm GeV}^2)$ as indicated by the fitted lines in Fig. 11. These data are therefore consistent with a W^0.22 ( $\bar{\epsilon}$ = 0.05) dependence at the lowest Q² values and the W^0.8 ( $\bar{\epsilon}$ = 0.2) dependence at the highest Q² values.

  

Figure: Exclusive $\rho ^0$ virtual-photon proton cross-sections as a function of W for (a) all data and (b) ZEUS preliminary data, compared to the dependences discussed in the text.

An important point to emphasise here is that the relative production of $\phi$ to $\rho ^0$ 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 $\phi/\rho^0$ cross-sections approaches the SU(4) flavour prediction of $\rho:\omega:\phi:J/\psi = [\frac{1}{\sqrt{2}}(u\bar{u} - d\bar{d})]^2: [\frac{1}{\sqrt{2}}(u\bar{u} + d\bar{d})]^2: [s\bar{s}]^2: [c\bar{c}]^2 = 9:1:2:8$. 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 $J/\psi$ to $\rho ^0$ 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 $t\simeq 0$ to $t\simeq 2$ GeV².

  

Figure: t dependence of the ratio of exclusive production cross-sections $\sigma (\phi )/\sigma (\rho ^0)$ (upper plot) and $\sigma (J/\psi )/\sigma (\rho ^0)$ (lower plot) for the ZEUS photoproduction data.

  

Figure: Observation of exclusive $\Upsilon$ production: the invariant mass spectrum for $\mu ^+\mu ^-$ indicates a broad enhancement around 10 GeV above the fitted background.

Exclusive $\Upsilon$ photoproduction has also been observed [20]. The $\simeq$ 20 events observed in the $\mu ^+\mu ^-$ channel corresponds to a cross-section for predominantly (1S) production, as well as higher $\Upsilon$ states, of $0.9\pm0.3\pm0.3$ nb for $\sigma (\gamma p \rightarrow \Upsilon N)$ where M_N < 4 GeV and 80<W<280 GeV, Q² < 4 GeV². This is about 1% of the $J/\psi$ 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 $\simeq 20\%$, which exhibit two classes of W² behaviour: a slow rise consistent with that of previously measured diffractive data for low M_V² photoproduction data but a significant rise of these cross-sections above a finite value of M_V², t or Q². In general, the cross-sections at large W² can be compared to pQCD when either M_V², t or Q² become larger than the scale $\Lambda$. Precisely how the transition from the non-perturbative to the perturbative regime is made is currently being determined experimentally.