Photon Dissociation

Diffractive Event Selection: The study of the vector meson resonances enables specific points on the M² scale to be investigated. The inclusive dissociation into any low-mass state, X, from the (virtual) photon provides additional information: one of the major advances in the subject of diffraction has been the observation of large rapidity gap events in DIS and their subsequent analysis in terms of a diffractive structure function [21,22]. In addition, the relationship between these DIS measurements and those in the photoproduction regime provide insight into the transition of the diffractive structure function in the $Q^2\rightarrow 0$ limit. In these analyses, the typical signature of diffraction is a rapidity gap, defined by measuring the maximum pseudorapidity of the most-forward going particle with energy above 400 MeV, $\eta _{max}$, and requiring this to be well away from the outgoing proton direction. A typical requirement of $\eta_{max} < 1.5$ corresponds to a low mass state measured in the detectors of $\ln(M_X^2) \sim 4$ units and a large gap of $\ln(W^2) - \ln(M_X^2) \sim 8$ units with respect to the outgoing proton (nucleon system). In order to increase the lever arm in M_X², the H1 analysis has extended the $\eta _{max}$ cuts to 3.2. This is achieved by combining the calorimetry information with the forward muon system and proton remnant taggers. These extensions enable a cross-section beyond that due to simple diffractive processes to be determined at the expense of a significant non-diffractive contribution (up to $\simeq 50\%$). As illustrated in Fig. 14, the Monte Carlo description of the $\eta _{max}$ distributions shows a clear excess over the non-diffractive models (labelled MC Django) and a Monte Carlo to describe higher reggeon exchanges (labelled MC Pion). The data is well-described by a mixture (labelled MC mix) including an additional contribution due to the pomeron exchange, both in the $\eta _{max}$ distribution and the corresponding observed M_X distribution.

  

Figure: Diffractive event selection: H1 analysis of the $\eta _{max}$ distribution. The upper plot is the $\eta _{max}$ distribution, where a clear excess is seen over the non-diffractive Monte Carlo's discussed in the text. The lower plot shows the observed M_X distribution.

  

Figure: Diffractive event selection: ZEUS analysis of the $\ln M_X^2$ distributions for 134<W<164 GeV and Q² = 14 GeV². The solid line shows the extrapolation of the nondiffractive background determined from the fit to the data (dotted line) discussed in the text.

One of the major uncertainties comes from the estimation of the various contributions to the cross-section which depends on Monte Carlo techniques. This problem has been addressed in a different way in the ZEUS analysis [23]. Here, the mass spectrum, M_X², is measured as a function of W and Q², as shown in Fig. 15 for a representative interval, where the measured mass is reconstructed in the calorimeter and corrected for energy loss but not for detector acceptance, resulting in the turnover at large M_X². The diffractive data are observed as a low mass shoulder at low M_X². which becomes increasingly apparent at higher W. Also shown in the figure are the estimates of the non-diffractive contribution based on a direct fit to the data, discussed below.

The probability of producing a gap is exponentially suppressed as a function of the rapidity gap, and hence as a function of $\ln(M_X^2$), for non-diffractive interactions. The slope of this exponential is directly related to the height of the plateau distribution of multiplicity in the region of rapidity where the subtraction is made. The data can thus be fitted to functions of the form $dN/d \ln(M_X^2) = D + C {\rm exp}( b \cdot \ln(M_X^2))$, in the region where the detector acceptance is uniform, where b, C and D are determined from the fits. Here, D represents a first-order estimate of the diffractive contribution which is approximately flat in $\ln(M_X^2$). The parameter which determines the background is b. In general the measured value of b is incompatible with that of the ARIADNE Monte Carlo. This result in itself is interesting, since the fact that ARIADNE approximately reproduces the observed forward E_T ($\sim$ multiplicity) flow but does not reproduce the measured value of b suggests that significantly different correlations of the multiplicities are present in non-diffractive DIS compared to the Monte Carlo expectations. This method enables a diffractive cross-section to be determined directly from the data at the expense of being limited in the range of large masses that can analysed.

Finally, the advent of the leading proton spectrometers (LPS) at HERA is especially important in these diffractive measurements, since internal cross-checks of the measurements as a function of t, M², W² and Q² can be performed and underlying assumptions can be studied experimentally. Only in these measurements can we positively identify the diffracted proton and hence substantially reduce uncertainties on the non-diffractive and double dissociation backgrounds. This is illustrated in Fig. 16, where the x_L (where x_L = p'/p) distribution includes a clear diffractive peak for $x_L \simeq 1$. It should be noted, however, that the contribution from other Reggeon exchanges cannot be neglected until $x_L\raisebox{-.6ex}{${\textstyle\stackrel{>}{\sim}}$ }0.99$ (in fact the result at lower x_L can be simply interpreted via reggeon (approximated by pion) exchange, as discussed below.) However, new experimental uncertainties are introduced due to the need for precise understanding of the beam optics and relative alignment of the detectors. Reduced statistical precision also results due to the limited geometrical acceptance of the detectors ($\simeq$ 6%).

  

Figure 16: Observed x_L spectrum of the ZEUS LPS DIS data. The data are described by a sum of single diffraction (significant at high x_L) and pion exchange (significant at low x_L) with a small contribution due to double diffraction.

  

Figure: t distribution and corresponding b-slope as a function of W² compared to those of Chapin et al. for inclusive diffractive photoproduction.

Photoproduction Results: ZEUS has measured the photon dissociation t distribution using the LPS, as shown in Fig. 17. An exponential fit to the data yields a b-slope parameter, $b= 7.3\pm0.9\pm1.0$ GeV^-2. A comparison of the data with lower W data from Chapin et al. shows that the result is consistent with shrinkage, as previously discussed in relation to exclusive $\rho ^0$ production. H1 results on the photon dissociation cross-sections as a function of M_X² in two W intervals are shown in Fig. 18. Regge theory predicts the form of the cross-section as a function of M_X and W, as discussed with respect to proton dissociation. The cross-section is therefore fitted to the form

$$d\sigma/dM_X^2 \propto (1/M_X^2)^{1+\bar{\epsilon}} (W^2)^{\bar{\epsilon}}$$

The contributions due to reggeon exchange are fixed using the lower energy data. Integrating over the t dependence of $\bar{\epsilon}$ yields a value of $\epsilon = 0.07\pm0.02\pm0.02$(sys)$\pm0.04$(model), again, consistent with soft pomeron exchange.

  

Figure 18: Inclusive diffractive photoproduction cross-sections of Chapin et al. (lower W) and H1 (higher W) data compared to the fit discussed in the text.

Deep inelastic structure of diffraction: A new era for diffraction was opened with the study of the dissociation of virtual photons. Here, the photon can be considered as probing the structure of the exchanged colourless object mediating the interaction. The deep inelastic structure of colour singlet exchange is therefore being studied. In the presentation of the results, the formalism changes [24], reflecting an assumed underlying partonic description, and two orthogonal variables are determined

$$\xi \equiv \mbox{$x_{_{I\hspace{-0.2em}P}}$}= \frac{(P-P')\cd... ...ta = \frac{Q^2}{2(P-P')\cdot q} \simeq \frac{Q^2}{M_X^2 + Q^2},$$

where $\mbox{$x_{_{I\hspace{-0.2em}P}}$ }$ is the momentum fraction of the pomeron within the proton and $\beta$ is the momentum fraction of the struck quark within the pomeron. The structure function is then defined by analogy to that of the total ep cross-section

$$\frac{d^3\sigma_{diff}}{d\beta dQ^2 d\mbox{$x_{_{I\hspace{-0.... ...+ \; F_2^{D(3)}(\beta,Q^2,\mbox{$x_{_{I\hspace{-0.2em}P}}$}),$$

where the contribution of F_L and radiative corrections are neglected and an integration over the t variable has been performed.

  

Figure: H1 data for F₂^LP(3) as function of $\xi$ for the leading proton analysis.

In addition to the structure of the pomeron, corresponding to large x_L, it is also possible to study the structure of the reggeons that contribute at lower x_L. H1 has analysed the leading proton data at lower x_L ( 0.7<x_L<0.9) and employed the formalism noted above to measure the structure of the exchange for reasonably forward protons, as shown in Fig. 19 [25]. The data are consistent with a flat $\xi$ dependence in all intervals of $\beta$ and Q² i.e. $F_2^{LP(3)} \propto \xi^{n}$. This is consistent with a factorisable ansatz of $F_2^{D(3)}(\beta, Q^2, \xi) = f_{I\hspace{-0.2em}R}(\xi) \cdot F_2^{I\hspace{-0.2em}R}(\beta,Q^2),$ where $f_{I\hspace{-0.2em}R}(\xi)$ measures the flux of reggeons in the proton and $F_2^{I\hspace{-0.2em}R}(\beta,Q^2)$ is the probed structure of these reggeons. The exponent of $\xi$ is identified as $n = 1-2\cdot\bar{\eta}$, where $\bar{\eta}$ measures the effective $\xi$ dependence ( $\equiv W^2$ dependence at fixed M_X² and Q²) of the cross-section, integrated over t. The data are consistent with $n \simeq 0$, corresponding to $\bar{\eta} \simeq 0.5$. These data involve colour singlet exchange and need to be explained in terms of QCD, but they are clearly not of a diffractive nature.

The area of interest for diffraction is in the behaviour of small values of $\xi \raisebox{-.6ex}{${\textstyle\stackrel{<}{\sim}}$ }0.01$, where $\xi$ is now identified as $\mbox{$x_{_{I\hspace{-0.2em}P}}$ }$. Here, the data fall approximately as $\mbox{$x_{_{I\hspace{-0.2em}P}}$ }^{-1}$ (equivalent to a flat cross-section with increasing W) and therefore the data are plotted as $\mbox{$x_{_{I\hspace{-0.2em}P}}$ }\cdot F_2^{D(3)}$ in Fig. 20. In the H1 case, the measurement is presented with no explicit subtraction for the non-diffractive contribution and quoted for limited masses of the dissociated proton system (M_N < 1.6 GeV). The measurement relies upon a good understanding of the various contributions to the cross-section in and around the measured region: the control plots in Fig. 14 illustrate how well this is achieved by combining the different Monte Carlo contributions.

  

Figure: H1 preliminary F₂^D(3) data as function of $x_{_{I\hspace {-0.2em}P}}$

Fits of the form $F_2^{D(3)} = A(\beta,Q^2) \cdot \mbox{$x_{_{I\hspace{-0.2em}P}}$ }^{n(\beta)}$ are performed where the normalisation constants $A(\beta,Q^2)$ are allowed to differ in each $\beta , Q^2$ interval. The fits are motivated by the factorisable ansatz of $F_2^{D(3)}(\beta, Q^2, \mbox{$x_{_{I\hspace{-0.2em}P}}$ }) = f_{I\hspace{-0.2e... ...(\mbox{$x_{_{I\hspace{-0.2em}P}}$ }) \cdot F_2^{I\hspace{-0.2em}P}(\beta,Q^2),$ where $f_{I\hspace{-0.2em}P}(\mbox{$x_{_{I\hspace{-0.2em}P}}$ })$ measures the flux of pomerons in the proton and $F_2^{I\hspace{-0.2em}P}(\beta,Q^2)$ is the probed structure of the pomeron. The exponent of $x_{_{I\hspace {-0.2em}P}}$