Forward Rapidity Gaps  

The class of events which will be discussed in this section exhibits a rapidity gap extending to high values of $\eta$. An example is shown in Figure 9.

 

Figure:   A hard photoproduction event with a foward gap. The z-R display of the ZEUS detector is shown on the left hand side. In the upper right hand corner the $\eta$ and $\varphi$ coordinates of the calorimeter energy deposits are shown, weighted by their transverse energy. The lower right hand view is the x-y cross section.

There are two high transverse energy jets which are back to back in $\varphi$and no scattered e^- candidate. This is a hard photoproduction event. However there is no energy in the forward direction around the beam pipe which could be associated with the fragmentation products of the proton remnant. This is, therefore, a candidate diffractive hard scattering event.

This analysis proceeds in a similar way to that described in the previous section. First the uncorrected data are compared to Monte Carlo generated event samples which have been subjected to a full simulation of the ZEUS detector. In a second step the generated event samples are used to correct the data for the effects of the detector smearing and acceptance. Again, specific details of the analysis should be obtained from the publication [2].

The diffractive hard scattering process is understood to proceed as illustrated in Figure 10(a) where we have introduced two new momentum fraction variables. $x_{I\!\!P}$ represents the fraction of the proton's momentum which is carried by the pomeron and $\beta$ represents the fraction of the pomeron's momentum which is carried into the hard subprocess. Of course $x_{I\!\!P} \cdot \beta$ gives the familiar Bjorken-x_p variable.

 

Figure:   Kinematics of the diffractive hard photoproduction process. The meaning of the momentum fraction variables $x_{I\!\!P}$ and $\beta$ is illustrated in (a) where the fraction of the photon's momentum entering the hard subprocess, $x_{\gamma}$, the photon virtuality, P², and the squared momentum transfer which sets the energy scale of the hard subprocess, Q², are also indicated. The pseudorapidity of the particle with the highest pseudorapidity is denoted $\eta_{max}$ as illustrated in (b).

The other important variable for describing the diffractive hard photoproduction process is $\eta_{max}$. $\eta_{max}$ is defined to be the pseudorapidity of the most forward going particle (measured using the calorimeter) which has energy exceeding 400 MeV. The definition of $\eta_{max}$ is illustrated schematically in Figure 10(b).

The $\eta_{max}$ distribution for a sample of hard photoproduction events is shown in Figure 11. For this particular plot a subsample of events is shown for which the total hadronic invariant mass, M_X, (measured using all of the energy deposits in the calorimeter) satisfies M_X < 30 GeV. The data are shown by black dots and are not corrected for detector effects. The errors shown are statistical only. The data are peaked toward a value of $\eta_{max}$ which is close to the edge of the calorimeter acceptance. However there is a large contribution

 

Figure:   The $\eta_{max}$ distribution for a sample of hard photoproduction events. The data are shown by black dots with error bars representing statistical errors only. No corrections for detector effects have been made. The shaded histogram shows the prediction from the PYTHIA standard hard photoproduction events. The open histogram shows the prediction from the POMPYT simulation of $\gamma I\!\!P$ scattering where the $I\!\!P$ contains a hard gluon spectrum (see text). The Monte Carlo event samples have been subjected to a full simulation of the detector acceptance and smearing.

from events with very low values of $\eta_{max}$, indicating the presence of a forward rapidity gap. Also shown in this figure are two Monte Carlo predictions which include a full simulation of the ZEUS detector. The shaded histogram shows the PYTHIA prediction for standard hard photoproduction processes. It fails to describe the large rapidity gap events of the data which occur at low values of $\eta_{max}$. To describe these large rapidity gap events we must introduce the Monte Carlo program POMPYT [18], the prediction of which is shown by the open histogram.

POMPYT is a Monte Carlo implementation of the Ingelman-Schlein model [19] which assumes that the hard photoproduction cross section $\sigma^{jet}_{\gamma p}$ factorizes in the following way.  

$$\sigma^{jet}_{\gamma p} = f_{I\!\!P / p}(x_{I\!\!P},t) \otimes f_{a / I\!\!P}(\beta, Q^2) \otimes \hat{\sigma}(\hat{s},Q^2).$$ (1)

In words, the jet cross section, $\sigma^{jet}_{\gamma p}$, may be written as the convolution of a term representing the flux of pomerons in the proton, $f_{I\!\!P / p}(x_{I\!\!P},t)$, with a term describing the flux of partons in the pomeron, $f_{a / I\!\!P}(\beta, Q^2)$, and with the subprocess cross section, $\hat{\sigma}(\hat{s},Q^2)$.The direct photoproduction subprocess cross section includes only the hard subprocesses, $\gamma q \rightarrow q g$ and $\gamma g \rightarrow q q$.In resolved photoproduction it includes in addition to the hard subprocesses $qq \rightarrow qq$, $qg \rightarrow qg$, etcetera, the flux of partons in the photon, $f_{a / \gamma}(x_{\gamma},Q^2)$.The hard subprocess cross sections are calculable in perturbative QCD and some experimental information exists which constrains the $f_{a / \gamma}(x_{\gamma},Q^2)$. Therefore $\hat{\sigma}(\hat{s},Q^2)$ is a known input in Eqn. 1. The pomeron flux factor, $f_{I\!\!P / p}(x_{I\!\!P},t)$, may be determined using Regge inspired fits to hadron-hadron data. The remaining unknown ingredient is the pomeron structure. We neglect the energy scale dependence of $f_{a / I\!\!P}(\beta, Q^2)$ and consider two extreme possibilities for its $\beta$ dependence. The first, $\beta f_{a / I\!\!P}(\beta) = 6 \beta (1 - \beta)$, yields a mean $I\!\!P$ momentum fraction of $\langle \beta \rangle = 1 / 2$ and is therefore known as the hard parton density. The second, $\beta f_{a / I\!\!P}(\beta) = 6 (1 - \beta) ^ 5$, has $\langle \beta \rangle = 1 / 7$ and is called the soft parton density. Finally, it is not clear whether there should be a momentum sum rule for the pomeron, that is, whether $\Sigma_{I\!\!P} \equiv \int_0^1 d\beta \sum_a \beta f_{a / I\!\!P}(\beta)$must equal 1 or not.

The open histogram in Figure 11 shows the POMPYT prediction for a $I\!\!P$ consisting entirely of gluons with the hard momentum spectrum. A fairly satisfactory description of $\eta_{max}$ may be achieved. In addition the POMPYT prediction is able to describe the M_X distribution, and the distribution of the photon proton centre-of-mass energies, $W_{\gamma p}$, for rapidity gap events with $\eta_{max} < 1.8$.(The results are similar for a $I\!\!P$ composed entirely of hard quarks.) For this reason we say that the data are consistent with containing a contribution from diffractive hard photoproduction processes.

In the second step of the analysis we correct the data for all effects of detector acceptance and smearing. We present in Figure 12 the ep cross section for photoproduction of jets of E_T^jet > 8 GeV as a function of the jet pseudorapidity. This cross section is for events which have a rapidity gap characterized by $\eta_{max} < 1.8$.

 

Figure:   Cross section $d \sigma / d \eta^{jet}$ for photoproduction of jets with E_T^jet > 8 GeV in events with $\eta_{max} < 1.8$. The inner error bars show the statistical errors and the outer error bars the systematic uncertainty added in quadrature -- excluding the systematic uncertainty due to the calorimeter energy scale which is shown by the shaded band. The PYTHIA prediction for standard hard photoproduction processes is shown by the dashed line. The POMPYT predictions $d\sigma^{\mbox{\scriptsize\it hard gluon}}/d\eta^{jet}$, $d\sigma^{\mbox{\scriptsize\it hard quark}}/d\eta^{jet}$ and $d\sigma^{\mbox{\scriptsize\it soft gluon}}/d\eta^{jet}$ for diffractive hard processes with different parton distribution functions and $\Sigma_{I\!\!P} = 1$ are shown by the solid lines.

The PYTHIA prediction for this cross section for non-diffractive processes is shown by the dashed line. It is too low in overall magnitude to describe the data as well as being disfavoured in shape. The POMPYT predictions, $d\sigma^{\mbox{\scriptsize\it hard gluon}}/d\eta^{jet}$,$d\sigma^{\mbox{\scriptsize\it hard quark}}/d\eta^{jet}$ and $d\sigma^{\mbox{\scriptsize\it soft gluon}}/d\eta^{jet}$ for the hard gluon, the hard quark and the soft gluon pomeron parton densities respectively where $\Sigma_{I\!\!P} = 1$ are shown by the solid curves. The soft parton density very rarely gives rise to sufficient momentum transfer to produce two E_T^jet > 8 GeV jets and so $d\sigma^{\mbox{\scriptsize\it soft gluon}}/d\eta^{jet}$ lies far below the cross sections for the hard parton densities in overall normalization. $d\sigma^{\mbox{\scriptsize\it soft gluon}}/d\eta^{jet}$ is inconsistent with the data in overall magnitude as well as being disfavoured in shape. We do not consider soft parton densities further. $d\sigma^{\mbox{\scriptsize\it hard quark}}/d\eta^{jet}$is consistent with the data in shape but too small in magnitude. $d\sigma^{\mbox{\scriptsize\it hard gluon}}/d\eta^{jet}$is capable of describing both the shape and magnitude of the measured cross section. Note, however, that the non-diffractive contribution to the data has not been subtracted, nor has the double dissociation contribution.

In the final stage of the analysis the non-diffractive contribution was subtracted from the data using the PYTHIA prediction (which has been shown to provide a good description of inclusive jet cross sections in photoproduction [20]). A contribution of $(15 \pm 10)$% due to double dissociation processes was also subtracted. Then the assumption that $\Sigma_{I\!\!P} = 1$ was relaxed. The $I\!\!P$was assumed to be composed of a fraction c_g of hard gluons and a fraction 1-c_g of hard quarks. Then for various values of c_g the expression $\Sigma_{I\!\!P} \cdot [ c_g \cdot d\sigma^{\mbox{\scriptsize\it hard gluon}}/... ...et} + (1-c_g) \cdot d\sigma^{\mbox{\scriptsize\it hard quark}}/d\eta^{jet} ]$ was fit to the measured $d \sigma / d \eta^{jet}$ distribution to obtain $\Sigma_{I\!\!P}$. (The POMPYT predictions for $d\sigma^{\mbox{\scriptsize\it hard gluon}}/d\eta^{jet}$ and $d\sigma^{\mbox{\scriptsize\it hard quark}}/d\eta^{jet}$ were used in the fit.) The result of this series of fits is shown in Figure 13 by the solid line where the statistical uncertainty of the fit is indicated by the shaded band.

 

Figure:   Allowed regions of the $\Sigma_{I\!\!P} - c_g$ plane. The solid line and its shaded band of uncertainty show the constraint from the measurement of $d \sigma / d \eta^{jet}$. The dash-dotted lines and their shaded band of uncertainty show the constraint imposed from the measurement of F₂^D(3). (The upper dash-dotted curve is for two quark flavours and the lower dash-dotted curve is for three quark flavours.)

We find, for instance, that the data are do not favour a $I\!\!P$ which consists exclusively of hard gluons and simultaneously satisfies the momentum sum rule, $\Sigma_{I\!\!P} = 1$. (See [2], however, for a discussion of additional theoretical systematic uncertainties.)

Results from studies of diffractive hard electroproduction have been expressed in terms of the diffractive structure function, $F_2^{D(3)}(\beta,Q^2,x_{I\!\!P})$ [21,22]. The expression of factorization is then, $F_2^{D(3)}(\beta,Q^2,x_{I\!\!P}) = f_{I\!\!P / p}(x_{I\!\!P}) \cdot F_2^{I\!\!P}(\beta,Q^2)$.Integrating this over $x_{I\!\!P}$ and $\beta$ and then subtracting the integral over the pomeron flux thus gives the sum of the momenta of all of the quarks in the pomeron, $\Sigma_{I\!\!P} \cdot (1 - c_g)$.The ZEUS measurement [22], $\Sigma_{I\!\!P} \cdot (1 - c_g) = 0.32 \pm 0.05$, is shown in Figure 13 by the lower dot-dashed line. This is the result for two flavours of quark in the $I\!\!P$. The result for three flavours of quark is $\Sigma_{I\!\!P} \cdot (1 - c_g) = 0.40 \pm 0.07$, the upper dot-dashed line in Figure 13. The dark-shaded band shows the additional measurement uncertainty.

Assuming that the pomeron flux is the same in the measurement of F₂^D(3) and of $d\sigma^{jet}/d\eta^{jet}(\eta_{max} < 1.8)$ one can combine the two analyses to determine the allowed ranges, $0.5 < \Sigma_{I\!\!P} < 1.1$ and 0.35 < c_g < 0.7. However the $\Sigma_{I\!\!P}$ range is affected by additional uncertainties in the normalization of the pomeron flux factor. Taking into account all remaining systematic uncertainties of the measurements we find 0.3 < c_g < 0.8. This measurement is independent of the pomeron flux and of the total momentum carried by partons in the pomeron.

In summary, the distributions of $\eta_{max}$, M_X and $W_{\gamma p}$indicate that the events with a forward rapidity gap are consistent with a diffractive hard scattering via exchange of a low-t pomeron. The ep cross section $d \sigma / d \eta^{jet}$ for the photoproduction of jets of E_T^jet > 8 GeV in large rapidity gap events ($\eta_{max} < 1.8$)has been measured and is significantly larger than the cross section due to non-diffractive processes. A comparison of the $d \sigma / d \eta^{jet}$ measurement in photoproduction with the measurement of F₂^D(3) in electroproduction indicates that 30% to 80% of the momentum of the pomeron which is due to partons is carried by hard gluons.