GLAS-PPE/97-1319 ^th December 1997

Diffraction: QCD Effects in Colour Singlet Exchange

Anthony T. Doyle¹
Department of Physics and Astronomy,
University of Glasgow.

Abstract

1.  Introduction

HERA Kinematics: The diffractive processes studied at HERA are of the form:

Signatures of diffraction: The processes studied build upon the basic elastic scattering process

In Fig. 2, the t distributions of pp elastic scattering data are illustrated as a function of the longitudinal momentum (p_L) of the outgoing proton transformed to a fixed-target rest frame. The patterns are similar to the diffraction patterns observed when light is scattered from an aperture and exhibit an exponential fall-off for values of t below @  1 GeV². This characteristic fall-off increases with increasing energy, a property known as shrinkage, and differs for different incident and outgoing systems. In order to characterise the t-dependence, a fit to the diffractive peak is performed. In the most straightforward approach, a single exponential fit to the t distribution, ds/d|t| µ e^-b|t| for |t|  ~  0.5 GeV² is adopted. Physically, the slope of the t dependence in diffractive interactions tells us about the effective radius of that interaction, R_I: if ds/dt µ e^-b|t|, then b @ R_I²/4.

Figure 2: Signatures of diffraction: ds/dt for ISR pp data as a function of p_L, the longitudinal momentum of the outgoing proton transformed to a fixed-target rest frame, from 24 GeV (uppermost plot) to 1496 GeV (lowest plot).

Regge trajectories: The subject of diffraction is far from new: diffractive processes have been measured and studied for more than thirty years [3]. Their relationship to the corresponding total cross-sections at high energies has been successfully interpreted via the optical theorem and Regge theory. At lower energies the colour singlet exchange of virtual mesons, called reggeons, contribute to the fall of the cross-section with increasing energy. At higher energies, the introduction of an additional trajectory, known as the pomeron trajectory, with a characteristic W² and t dependence is necessary [4]. The energy behaviour of the total cross-sections can then be described by the sum of two power-law dependences on the centre-of-mass energy, s º W²

In a Regge analysis, the diffractive data are interpreted via exchanges with spin J = a(t) = a(0) + a¢t and ds/dt µ ([(W²)/(W₀²)])^2(a(0)-1) e^-b|t|, with b = b₀ + 2a¢ln(W²/W₀²). At lower energies, these correspond to reggeon (i.e. approximately degenerate r, w, f and a) exchanges. At the highest energies, where the pomeron contribution dominates, the optical theorem relates the total cross-sections to the elastic, and hence diffractive, scattering amplitude at the same W². In Fig. 3, the trajectories, J = a(t), are shown as a function of M². The diffractive data probe the region of negative t. Given the dominance of the pomeron contribution at large W and an approximately exponential behaviour of the |t| distribution with slope b, whose mean |[`t]| value is given by 1/b at the mean [`W] of a given data sample, the diffractive cross-section rise is moderated from (W²)^2e to

Maps of the Pomeron: Whilst these Regge-based models gave a unified description of pre-HERA diffractive data, this approach is not fundamentally linked to the underlying theory of QCD. It was anticipated that at HERA and Tevatron energies if any of the scales Q², M² or t become larger than the QCD scale L², then it may be possible to apply perturbative QCD (pQCD) techniques, which predict changes to this power law behaviour. Qualitatively, the W dependence could be ascribed to the rise of the gluon density with decreasing x determined from the large scaling violations of F₂(x,Q²), where x is the Bjorken scaling variable, x = Q²/2P ·q @ Q²/(Q²+W²). QCD factorisation into a long-timescale and short-timescale process, where this timescale is characterised by 1/Q or 1/M or 1/Öt, leads to the following approaches.

• [·] For exclusive final states, e.g. vector meson production, with a hard scale the approach is very simple

  sdiff ~ Gp2 Ä ^ s

  i.e. two-gluon exchange where G_p² is the square of the gluon density of the proton at a representative value of x and [^(s)] represents the hard scattering process. The rise of F₂ with decreasing x, which constrains the gluon density, corresponds to an increase in the effective value of e. This brings us from the regime of dominance of the slowly-rising ``soft" pomeron to the newly emergent ``hard" behaviour and the question of how a transition may occur between the two. The QCD expectation is that the cross-sections should approximately scale as a function of t, corresponding to a weak dependence of e as a function of t and therefore a decrease of a¢ for the perturbative pomeron.
• [·] For inclusive diffraction with a hard scale

  sdiff ~ Gp Ä ^ s   ÄH

  i.e. leading-gluon exchange where G_p is the gluon density of the proton, [^(s)] represents the hard scattering process and H represents the hadronisation process. Here, the final state with a leading proton is seen as a particular hadronisation process [8]. For processes involving one incoming hadron, the above approaches can be tested and compared with experimental data.
• [·] Finally, one can break the process down further and invoke Regge factorisation where a flux of pomerons, f_{I P/p} lead to partons from the pomeron, f_{i/I P}, which interact and hadronise

  sdiff ~ fI P/p Äfi/I P Ä ^ s   ÄH

For processes involving two incoming hadrons, this approach has been generalised to

Precisely where the Regge-based approach breaks down or where pQCD may be applicable is open to experimental question. In addition, once we observe ``hard" diffractive phenomena, we can ask whether the pQCD techniques applied to inclusive processes also apply to these exclusive colour singlet exchange reactions. The emphasis is therefore on the internal (in)consistency of a wide range of measurements of diffractive and total cross-sections. As an experimentalist navigating around the various theoretical concepts of the pomeron, it is sometimes difficult to see which direction to take and what transitions occur where (Fig. 4(a)). However, from an experimental perspective, the directions are clear, even if the map is not yet complete (Fig. 4(b)).

Outline: The HERA collider allows us to observe a broad range of diffractive phenomena at high W². What is new is that we have the ability to observe the variation of these cross-sections at specific points on the M² scale, from the r⁰ up to the U system as discussed in section 2. Similarly, the production cross-section can be explored as a function of Q², using a virtual photon probe. The observation of a significant fraction of events ( @ 10%) with a large rapidity gap between the outgoing proton and the rest of the final state in deep inelastic scattering (DIS) has led to measurements of the internal structure of the pomeron. In addition, the leading proton spectrometer data, where the diffracted proton is directly measured, enable the t distribution as well as the structure function to be determined simultaneously. These results are discussed in section 3. Studies of the hadronic final state in events with a large rapidity gap, including transverse energy flows, event shape distributions and high-p_T jets, have been used to provide complementary information on this structure. Also, the observation of rapidity gaps between jets, corresponding to large-t diffraction, are presented in section 4. Many of these hadronic final state investigations were initiated at p[`p] colliders. In section 5, the latest results from the Tevatron on diffractive dijet and W^± production, rapidity gaps between jets and first observations of hard double-pomeron exchange are presented. A comparison of hard diffractive event rates at HERA and the Tevatron is given and the interpretation of the observed differences is discussed.

2.  Exclusive Production of Vector Mesons

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 J/y and U. 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 ~ W ~ 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 r⁰ 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

Figure 5: ZEUS exclusive r⁰ t distributions characterised by b_r 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 @ [Ö(R_P² + R_V²)]. 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 r⁰ production data. In each case, the range of measured b-slopes varies from around 10 GeV^-2 (R_I @ 1.3 fm) at low M_V² or Q² to 4 GeV^-2 (R_I @ 0.8 fm) at the highest M_V² or Q² so far measured. Given R_P @ 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 @ 1.1 fm to R_V @ 0.4 fm as M_V² (at fixed Q² @ 0 GeV²) or Q² (at fixed M_V² = M_r²) increase.

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 gp total cross-section is also shown in Fig. 7, rising with increasing energy as in hadron-hadron collisions and consistent with a value of e @ 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 |[`t]| is given by 1/b, the diffractive cross-section rise is moderated from W^4e = W^0.32 to

The J/y (charm) mass scale, M_y², is larger than the QCD scale, L², 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 g^* p total cross-section that application of formula (1) results in a value of e which increases with increasing Q², with e @ 0.2 to 0.25 at Q² @ 10 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 e = e(Q²) for Q² ~ Q_o². The current data indicate that this transition occurs for Q² @ 1 GeV² [14].

J/y electroproduction results are also available; the Q²+M_y² dependence of the data is shown in Fig. 8(b). The cross-sections are fitted to s µ 1/(Q²+M_y²)ⁿ where the fitted line corresponds to n @ 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  @  3. Also shown are the lower-W EMC data where n @ 1.5. The J/y electroproduction cross-section is of the same order as the r⁰ data. This is in marked contrast to the significantly lower photoproduction cross-section for the J/y, 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 9: Q² dependence of the decay angular distributions for E665 r⁰ electroproduction data.

Figure 10: Ratio R = s_L/s_T of r⁰ electroproduction data.

One contribution to the DIS g^* p total cross-section is the electroproduction of low mass vector mesons, here typified by the r⁰ data. The decay angle distributions of the pions in the r⁰ 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 = s_L/s_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

[1/N][dN/(dcos q_h)] = [3/4][1-r₀₀⁰⁴+(3 r₀₀⁰⁴-1)cos²q_h], where the density matrix element r₀₀⁰⁴ represents the probability that the r⁰ was produced longitudinally polarised by either transversely or longitudinally polarised virtual photons. The value of R is then obtained from

R = [(s_L)/(s_T)] = [(2(1-y))/(Y₊)]·[(r₀₀⁰⁴)/(1-r₀₀⁰⁴)], where y = P ·q/P ·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 r⁰ (not shown) are consistent with the interaction of dominantly transversely polarised photons and hence R @ 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 s_L is due to an increased flux of longitudinal photons, s_L µ (Q²/M_X²) s_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 r⁰ 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/y 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 r⁰ cross-sections for finite values of Q² are shown in Fig. 11(a), compared to the corresponding photoproduction cross-sections (the f 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 Q² @ 6 GeV². This is illustrated by comparison with the W^0.8 (dashed) and W^0.22 dotted lines for Q² @ 6 GeV² and Q² @ 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 ds/dM_N² µ (1/M_N²)<sup>1+[`(e)]</sup> and integrating over the t-dependence CDF obtains e = 0.100 ± 0.015 at Ös = 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 [`(e)] to vary from around 0.0 to 0.5, a choice which covers possible variations of [`(e)] as a function of M_N and W. Combining all uncertainties, the overall systematic errors on the various cross-sections are typically @  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 r⁰ 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 [`(e)] varying from [`(e)] = 0.04 ±0.01 ±0.03 (Q² = 0.5 GeV²) to [`(e)] = 0.19 ±0.05 ±0.05 (Q<sup>2</sup> = 20 GeV<sup>2</sup>) as indicated by the fitted lines in Fig. 11. These data are therefore consistent with a W<sup>0.22</sup> ([`(e)] = 0.05) dependence at the lowest Q² values and the W^0.8 ([`(e)] = 0.2) dependence at the highest Q² values.

An important point to emphasise here is that the relative production of f to r⁰ 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 f/r⁰ cross-sections approaches the SU(4) flavour prediction of

r:w:f:J/y = [[1/(Ö2)](u[`u] - d[`d])]²:[[1/(Ö2)](u[`u] + d[`d])]²:[s[`s]]<sup>2</sup>:[c[`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/y to r⁰ 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 @ 0 to t @ 2 GeV².

Exclusive U photoproduction has also been observed [20]. The @ 20 events observed in the m⁺m^- channel corresponds to a cross-section for predominantly (1S) production, as well as higher U states, of 0.9±0.3±0.3 nb for s(gp ® UN) where M_N <  4 GeV and 80 < W < 280 GeV, Q² < 4 GeV². This is about 1% of the J/y 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 @ 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 L. Precisely how the transition from the non-perturbative to the perturbative regime is made is currently being determined experimentally.

3.  Photon Dissociation

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_X²), 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_X²) = D + C exp( b ·ln(M_X²)) , 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_X²). 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 ( ~ 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 @ 1. It should be noted, however, that the contribution from other Reggeon exchanges cannot be neglected until x_L ~ 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 ( @  6%).

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±0.9±1.0 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 r⁰ 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

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

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 x dependence in all intervals of b and Q² i.e. F₂^LP(3) µ xⁿ. This is consistent with a factorisable ansatz of

F₂^D(3)(b, Q², x) = f_{I R}(x) ·F₂^{I R}(b,Q²), where f_{I R}(x) measures the flux of reggeons in the proton and F₂^{I R}(b,Q²) is the probed structure of these reggeons. The exponent of x is identified as n = 1-2·[`(h)], where [`(h)] measures the effective x dependence ( º W² dependence at fixed M_X² and Q²) of the cross-section, integrated over t. The data are consistent with n @ 0, corresponding to [`(h)] @ 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 x ~ 0.01, where x is now identified as x_{I P}. Here, the data fall approximately as x_{I P}^-1 (equivalent to a flat cross-section with increasing W) and therefore the data are plotted as x_{I P}·F₂^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 20: H1 preliminary F₂^D(3) data as function of x_{I P}in various intervals of b and Q² ( 10^-4 < x_{I P} < 0.05; 0.01 < b < 0.9; 2.5 < Q² < 65 GeV²). The fitted line corresponds to the QCD fit discussed in the text.

Fits of the form F₂^D(3) = A(b,Q²) ·x_{I P}^n(b) are performed where the normalisation constants A(b,Q²) are allowed to differ in each b,Q² interval. The fits are motivated by the factorisable ansatz of

F₂^D(3)(b, Q², x_{I P}) = f_{I P}(x_{I P}) ·F₂^{I P}(b,Q²), where f_{I P}(x_{I P}) measures the flux of pomerons in the proton and F₂^{I P}(b,Q²) is the probed structure of the pomeron. The exponent of x_{I P} is identified as n = 1+2·[`(e)], where [`(e)] measures the effective x_{I P} dependence ( º W² dependence at fixed M_X² and Q²) of the cross-section, integrated over t, as discussed in relation to exclusive vector meson production. However, the data are now sufficently precise that a single value for n is not sufficient. In fact, the contribution from reggeon exchange cannot be excluded since the data extend to x_{I P} beyond approximately 0.01 and therefore a full fit to the pomeron and reggeon contributions is required. A fit to the pomeron and reggeon contributions has therefore been performed

Figure 21: Regge fit to sample bins of the H1 F₂^D(3) data.

The ZEUS virtual-photon proton cross-sections measured at fixed M_X² and W, measured using the M_X method (and therefore explicitely subtracting the non-diffractive contribution) can be converted to x_{I P}·F₂^D(3) at fixed b and x_{I P}. These results are shown in Fig. 22 as the ZEUS(M_X) [23] analysis, compared to the ZEUS LPS analysis of x_{I P}·F₂^D(4), integrated over the measured t range, in comparable intervals of b and Q² as a function of x_{I P}.

The double dissociation contribution is estimated with similar uncertainties to the vector meson case. Common systematic errors are similar to those for the F₂ analyses ( ~ 10%) with additional acceptance uncertainties due to variations of the input diffractive Monte Carlo distributions. The LPS data therefore provide an excellent calibration to cross-check the background subtraction methods. The overall cross-sections in overlapping b and Q² intervals are in good agreement. In the region of overlap, the H1 and ZEUS LPS data points also agree well. The ZEUS LPS data beyond x_{I P}of 0.01 again tend to turn over, as observed in the H1 data.

The corresponding ZEUS LPS measurement of the t distribution is shown in Fig. 23, measured in the range x_L > 0.97, 5 < Q² < 20 GeV², 0.015 < b,0.5 and 0.073 < |t| < 0.4 GeV². The slope can be characterised by a single exponential fit with b = 7.1 ±1.1^+0.7_-1.0 GeV^-2 [23]. This is very similar to the slope obtained in the photoproduction case and is somewhat high compared to the value of b @ 4.5 expected for a predominantly hard pomeron but lies within the range of expectations of 4 ~ b ~ 10.

The ZEUS M_X data are largely restricted to the small x_{I P} < 0.01 and b > 0.1 region and therefore purely pomeron contributions are significant. A direct fit to the data in fixed M_X intervals yields a similar value for a_{I P}(0). This is illustrated in Fig. 24 where the the Q² dependence of the extracted a_{I P}(0) is compared to the H1 result as well as the soft pomeron prediction. The results for a_{I P}(0), obtained using different experimental methods, are compatible. These values are however incompatible with the predicted soft pomeron behaviour of a_{I P}(0) = 1.08 ±0.02. As noted previously in relation to exclusive r⁰ production (corresponding to the large b region), the contribution of longitudinal photons is significant. An upper estimate of the effect of s_L made by assuming s_L = (Q²/M_X²) s_T rather than s_L = 0 increases the measured value of a_{I P}(0) by about 0.05. Similarly, if no shrinkage were assumed, the value of a_{I P}(0) would increase by a¢·1/b = 0.035.

Figure 24: a_{I P}(0) values extracted from H1 and ZEUS preliminary data. The ZEUS data in two M_X intervals as a function of Q² are compared to the ranges of the fit to the H1 data indicated by the dashed lines and the soft pomeron prediction indicated by the shaded band.

The values can be compared with [`(e)] @ 0.2 obtained from the exclusive photoproduction of J/y mesons and the ZEUS exclusive r⁰ electroproduction data at large Q². These values are also compatible with fits to the inclusive DIS F₂ (i.e. the dependence of the corresponding total cross-sections) which yield e @ 0.2 to 0.25 in the measured Q² range [12]. In the model of Buchmüller and Hebecker [8], the effective exchange is dominated by one of the two gluons. In terms of e, where the optical theorem is no longer relevant, the diffractive cross-section would therefore rise with an effective power which is halved to e @ 0.1 to 0.125. Given the uncertainties, the measured values are within the range of these estimates.

The overall cross-sections in each b, Q² interval are similar and one can integrate over the measured x_{I P} dependence in order to determine [F\tilde]₂^D(b, Q²), a quantity which measures the internal structure of the pomeron up to an arbitrary integration constant. In Fig. 25 the H1 data are compared to preliminary QCD fits [26]. The data are fitted in the range 0.0003 < x_{I P} < 0.05 using the functional form shown in Fig. 20.

The general conclusions from the b dependence are that the pomeron has a rather flat structure as a function of b. This is typically characterised by a symmetric b(1-b) dependence, but an additional significant contribution at low b is required which has been fitted in the ZEUS analysis [22]. The Q² behaviour is broadly scaling, consistent with a partonic structure of the pomeron. Probing more deeply, however, a characteristic logarithmic rise of [F\tilde]₂^D is observed in all b intervals. Most significantly, at large b a predominantly quark-like object would radiate gluons resulting in negative scaling violations as in the case of the large-x ( ~ 0.15) behaviour of the proton. The question of whether the pomeron is predominantly quarks or gluons, corresponding to a ``quarkball" or a ``gluemoron" [30], has been tested quantitatively by H1 using QCD fits to [F\tilde]₂^D [26]. A flavour singlet quark density input of the form zq(z) = A_q ·z^B_q(1-z)^C_q, where z is the momentum fraction carried by the quark, yields a high c²/DoF = 95/39, since the characteristic Q² behaviour is not reproduced. Adding a gluon contribution of similar form gives an excellent description of the data with c²/DoF = 36.8/37, as shown in Fig. 25. In Fig. 26, the corresponding parton distributions are shown. In general, the fits tend to favour inputs where the gluon carries a significant fraction, ~ 70 to 90%, of the pomeron's momentum.

Figure 25: H1 preliminary data on [F\tilde]₂^D(b, Q²) as a function of Q² (b) at fixed b (Q²). The data are compared to preliminary next-to-leading-order QCD fits where gluons also contribute at the starting scale Q₀² = 2.5 GeV², resulting in a fit where gluons carry ~ 80% of the momentum, indicated by the full line (c²/DoF = 36.8/37).

Figure 26: H1 preliminary parton distributions of the pomeron. The upper plots show the momentum distributions at low Q² = 5 GeV² and high Q² = 65 GeV² as a function of z º x_{i/I P}. The lower plot indicates the fraction of the total momentum carried by gluons and quarks as a function of Q².

4.  Hadronic final states and jet structure

Historically, the measurements of < p_T^*2 > , the mean transverse momentum-squared of the outgoing hadrons, as a function of x_F = p_L/p_L^max, the scaled longitudinal momentum distribution, provided insight into the structure of the proton. Here, the variables are measured in the hadronic centre of mass frame and with respect to the virtual photon-proton axis which is equivalent to the virtual photon-pomeron axis for small values of t. In Fig. 27(a), the H1 g^*I P data (full circles) are compared to the EMC g^*p data at similar M_X º W values. The data are also compared to the RAPGAP (RG) Monte Carlo predictions incorporating quarks and gluons (-QG) and quarks only (-Q) [31]. (MEPS) and (CDM) refer to the Matrix Elements plus Parton Showers and Colour Dipole Model fragmentation schemes, respectively. The H1 data are approximately symmetric about x_F = 0 with a relatively large < p_T^*2 > peaking around 0.6 GeV². The symmetry and relatively large p_T^* values reflect the underlying boson-gluon fusion process where a ``leading" gluon from the pomeron interacts with the virtual photon. This behaviour is in contrast to the EMC g^*p data where QCD radiation is suppresed in the negative-x_F (proton remnant) region. Quantitatively the RAPGAP Monte Carlo which incorporates the pomeron parton densities (-QG) gives a good description of the data, provided that quarks and gluons are incorporated whereas a model with only quark (-Q) fails to describe the data. These conclusions are relatively independent of the fragmentation scheme, but the colour dipole model tends to give a better description of the data.

Similarly, event shape variables have been used at e⁺e^- colliders in order to establish the existence of gluon Bremsstrahlung radiation. In this case, the measurement of e.g. mean thrust (the mean value of the scaled longitudinal momentum with respect to the axis which maximises this value) is sensitive to the gluon-induced diagrams. A comparison of < thrust > with e⁺e^- annihilation experiments as a function of the reciprocal of hadronic centre of mass is shown in Fig. 27(b). The diffractive data exhibit lower thrust values compared to e⁺e^- data for all values of M_X. This additional broadening is due to the boson gluon fusion process which has no analogue in e⁺e^- annihilation continuum region.

The general increase in thrust with increasing M_X (decreasing 1/M_X) is indicative of jet production. The question of the constituent content of the pomeron can also be addressed via measurements of diffractively produced jets in the photoproduction data [32]. Jets are reconstructed at large W (134 < W < 277 GeV) using the cone algorithm with unit cone radius and two jets with E_T^jet > 6 GeV. The diffractive contribution is identified as a tail in the h_max distribution of these events above the PYTHIA 5.7 [33] Monte Carlo expectation. In Fig. 28 the measured cross-section is compared to various model predictions as a function of b^OBS, an estimator of the fraction of the pomeron momentum transferred to the dijet system.

The non-diffractive contribution estimated from PYTHIA (not shown) is significantly lower than the data. Here, standard photon and proton parton distributions are adopted and the overall scale, which agrees with the non-diffractive data normalisation, is set by E_T^jet. Also shown are the predicted diffractive cross-sections from the LO QCD calculation plus parton showers of POMPYT, using a hard (z(1-z)) quark combined with either a hard, soft ((1-z)⁵) or singular gluon where a Donnachie-Landshoff flux factor is adopted. Sampling low-energy (soft) gluons corresponds to a small cross-section and can be discounted, whereas high-energy (hard) gluons and/or quarks can account for the cross-section by changing the relative weights of each contribution. The shape of the b^OBS distribution is clearly sensitive to the shape of the input gluon distribution.

The x_g^OBS distribution for these events, where x_g^OBS is the corresponding estimator of the fraction of the photon momentum transferred to the dijet system, is peaked around 1, indicating that at these E_T^jet values a significant fraction of events is due to direct processes where the whole photon interacts with the pomeron constituents.

So far we have only considered the case of small-t diffraction with respect to the outgoing proton. Further insight into the diffractive exchange process can be obtained by measurements of the rapidity gap between jets. Here, a class of events is observed with little hadronic activity between the jets [34]. The jets have E_T^jet > 6 GeV and are separated by a pseudorapidity interval (Dh) of up to 4 units. The scale of the momentum transfer, t, is not precisely defined but is of order (E_T^jet)². A gap is defined as the absence of particles with transverse energy greater than 300 MeV between the jets. The fraction of events containing a gap is then measured as a function of Dh, as shown in Fig. 29. The fit indicates the sum of an exponential behaviour, as expected for non-diffractive processes and discussed in relation to the diffractive DIS data, and a flat distribution expected for diffractive processes. At values of Dh ~ 3, an excess is seen with a constant fraction over the expectation for non-diffractive exchange at @ 0.07±0.03. This can be interpreted as evidence for large-t diffractive scattering. In fact, secondary interactions of the photon and proton remnant jets could fill in the gap and therefore the underlying process could play a more significant rôle. The size of this fraction is relatively large when compared to a similar analysis by DØ and CDF where a constant fraction at @ 0.01 is observed [36,37], as discussed below.

Figure 29: ZEUS gap-fraction, f(Dh), as a function of the rapidity gap between the two jets compared with the result of a fit to an exponential plus a constant.

5.  Diffraction at the Tevatron

The studies of colour singlet exchange between jets at HERA was inspired by earlier studies at the Tevatron. These studies determined percentage gap-fractions of 1.07±0.10^+0.25_-0.13% for DØ [36] and 0.86±0.12% for CDF [37]. The behaviour of this colour singlet fraction as a function of average dijet energy is shown in Fig. 30(a) for jet E_T^jet thresholds of 15 GeV (low), 25 GeV (medium) and 30 GeV (high) [38]. In Fig. 30(b), the gap-fraction is examined as a function of Dh. A simple two-gluon exchange model with no additional QCD dynamics would tend to produce a flat gap-fraction, but the tendency towards an increasing gap-fraction with Dh indicates an additional dynamical mechanism is necessary to describe the data. The differences in the overall gap-fractions observed at HERA near 10%, compared to those at Fermilab of approximately 1%, may reflect the higher W values of the Tevatron compared to HERA. But the fact that this difference is so large indicates differences in the underlying high-x_g gp interactions compared to the relatively low-x p[`p] interactions where spectator interactions are more likely to fill in the gap.

Diffractive dijet production has been studied at the Tevatron. The presence of a diffracted proton may be identified either by a large rapidity gap on either the proton or antiproton side or by directly detecting the leading proton (CDF roman pots). In the rapidity gap analyses, DØ measure uncorrected dijet rates with E_T^jet > 12 GeV and |h_jet| > 1.6 in coincidence with the multiplicity in the electromagnetic calorimeter (2.0 < |h| < 4.1) opposite the dijet system, as shown in Fig. 31(a). Similarly, CDF measure the multiplicity in the forward part of the calorimeter (2.4 < |h| < 4.2) in coincidence with the number of hits in the BBC scintillator counter close to the beampipe (3.2 < |h| < 5.9), as shown in Fig. 31(b). Here the jets are measured for E_T^jet > 20 GeV and 1.8 < |h_jet| < 3.5 the diffractive events concentrate in the region 0.005 < x < 0.015. The ratio of diffractive dijet events is measured by CDF to be R_GJJ = 0.75 ±0.05 ±0.09% and by DØ to be R_GJJ = 0.67 ±0.05%. These preliminary figures are therefore in good agreement and can be used to constrain the gluon content of the pomeron. In addition, R_[`p]JJ = 0.109 ±0.003 ±0.016% has been measured using the CDF roman pots in region of large x (0.05 < x < 0.1, E_T^jet > 10 GeV and |t| < 1 GeV²). This is a region where reggeon (quark-like) contributions are presumably important.

CDF have also tagged diffractive W production using high-p_T electrons/positrons and missing p_T to tag the W and then searching for a rapidity gap on the opposite side as in the dijet gap analysis [39]. The corrected ratio for diffractive/non-diffractive W-production is measured to be

An important question when relating the various diffractive measurements which involve a hard scale is whether Regge factorisation, in terms of a pomeron flux and parton densities within the pomeron, is applicable. In particular, if this approach is to be useful, a universal flux is required (or a QCD description of the non-universal correction to the flux). In this context, the rates for diffractive processes at HERA are compared to those observed at the Tevatron below.

We have two sets of CDF data probing the pomeron structure at similar momentum scales, E_T^jet and M_W. Each probes the large z structure of the pomeron with the dijet and W data predominantly sampling the (hard) gluon and quark distributions, respectively. In addition, we have the corresponding DIS [22] and jet HERA data sampling the (hard) gluon and quark distributions, respectively. In Fig. 32 the momentum fraction carried by the (hard) gluon, c_g, is plotted versus the momentum fraction of partons in the pomeron assuming a Donnachie-Landshoff flux. The CDF data are consistent with a momentum fraction carried by the gluons of c_g = 0.7±0.2, in agreement with the ZEUS measurements of c_g ~ 0.55 ±0.25, taking into account the systematic uncertainties due (mainly) to the estimation of the non-diffractive background. This in turn can be compared with the H1 NLO parton distributions (see Fig. 26) which indicate c_g @ 0.8 in the high Q² region. There is therefore reasonable agreement on the parton content of the pomeron. However, the overall diffractive rates are significantly higher at HERA compared to the Tevatron. This is reflected in the difference in the overall level of the ZEUS and CDF data in Fig. 32. This corresponds to a significantly different flux of pomerons (i.e. breaking of Regge factorisation) which has been predicted in terms of QCD [40,41] to reduce the diffractive cross-sections for processes which have two strongly-interacting initial state hadrons. These effects are not apparent in the HERA data, where a virtual photon or a (predominantly) direct photon participate in the hard scattering process.

Figure 32: Momentum fraction of hard partons in the pomeron assuming a Donnachie-Landshoff flux versus the momentum fraction carried by the gluons in the pomeron. The ZEUS band corresponds to the allowed region using the fits to the DIS and jet data (statistical errors only). The shaded band corresponds to the allowed region from the CDF dijet and W analysis. The constraints given by the earlier UA8 jet data are also indicated.

Double pomeron exchange, where the p and [`p] remain intact, has also been studied by CDF and DØ for their dijet samples. This process should be directly sensitive to Regge-factorisation breaking effects. Both experiments find a ratio of hard double pomeron exchange events to non-diffractive events of @ 10<sup>-6</sup>. This is consistent with independent dissociation of the p and [`p], with probabilities @ 10^-3, but further studies are required to establish these rates and determine whether these can be explained by the factorisation-breaking calculations [41].

6.  Conclusions

The soft pomeron does not describe all diffractive data measured at HERA. As the photon virtuality or the vector meson mass increase a new dependence on W² emerges. As we investigate the pomeron more closely, a new type of dynamical pomeron may begin to play a rôle; a dynamical pomeron whose structure is being measured in DIS. These data are consistent with a partonic description of the exchanged object which may be described by pQCD.

The cross-sections for hard diffractive processes at HERA can be compared to those observed at the Tevatron. All data are consistent with a significant gluon contribution of the partons within the pomeron i.e. QCD factorisation appears to be observed. However, a universal pomeron flux does not describe the observed rates i.e. Regge factorisation does not apply. This is qualitatively expected from QCD corrections to the rates, but the effects are large and non-perturbative.

The experimental work focuses on extending the lever arms and increasing the precision in t, M², W² and Q² in order to explore this new structure. Before more precise tests can be made, further theoretical and experimental input is required to reduce the uncertainties due to non-diffractive backgrounds and proton dissociation as well as the treatment of F_L and radiative corrections.

Acknowledgements

References

[1]

H. Fritzsch, Aachen Workshop on ``QCD - 20 Years Later", World Scientific, (1992) 827.

[2]

N. Cartiglia (SSI 96), hep-ph/970325.

[3]

K. Goulianos, Phys. Rep. 101 (1983) 169; Nucl. Phys. B (Proc. Suppl.) 12 (1990) 110.

[4]

A. Donnachie and P.V. Landshoff, Nucl. Phys. B244 (1984) 322.

[5]

CDF Collab., F. Abe et al., Phys. Rev. D50 (1994) 5550.

[6]

J.R. Cudell, K. Kang and S.K. Kim, hep-ph/9601336.

[7]

WA91 Collab., S. Albatzis et al., Phys. Lett. B321 (1994) 509.

[8]

W. Buchmüller and A. Hebecker, DESY 95-077, W. Buchmüller, DIS97-AIP Conference Proceedings 147 (1997) 542.

[9]

J.A. Crittendon, hep-ex/9704009 (and references therein).

[10]

D. Aston et al., Nucl. Phys. B209 (1982) 56.

[11]

M. Ryskin, Z. Phys. C57 (1993) 89.

[12]

A. Levy, DESY 95-204.

[13]

M. Glück, E. Reya and A. Vogt, Phys. Lett. B306 (1993) 391.

[14]

ZEUS Collab., J. Breitweg et al., Jerusalem EPS 97-646.

[15]

E665 Collab., M. Adams et al., MPI-PhE-97-03.

[16]

S. Brodsky et al., Phys. Rev D50 (1994) 3134.

[17]

ZEUS Collab., M. Derrick et al., Phys. Lett. B356 (1995) 601.

[18]

H. Holtmann et al., hep-ph/9503441.

[19]

CDF Collab., F. Abe et al., Phys. Rev. D50 (1994) 5535.

[20]

E. Gallo, hep-ex/9710013.

[21]

H1 Collab., T. Ahmed et al., Phys. Lett. B348 (1995) 681.

[22]

ZEUS Collab., M. Derrick et al., Z. Phys. C68 (1995) 569.

[23]

ZEUS Collab., J. Breitweg et al., Jerusalem EPS 97-638. (Final results for the measurement of F₂^D(4) are available in DESY 97-184.)

[24]

G. Ingelman and K. Jansen-Prytz, Z. Phys. C58 (1993) 285.

[25]

H1 Collab., C. Adloff et al., Jerusalem EPS 97-379.

[26]

H1 Collab., C. Adloff et al., Jerusalem EPS 97-337. (Final results are available in Z. Phys. C76 (1997) 613.)

[27]

L. Lönnblad, Comp. Phys. Comm. 71 (1992) 15.

[28]

P. Bruni and G. Ingelman, DESY 93-187; Proceedings of the Europhysics Conference on HEP, Marseille 1993, 595.

[29]

N.N. Nikolaev and B.G. Zakharov, Z. Phys. C53 (1992) 331; M. Bertini et al., hep-ph/9710547.

[30]

F. Close and J. Forshaw, hep-ph/9509251.

[31]

C. Cormack, DIS97-AIP Conference Proceedings 147 (1997) 585. (Final results for the measurement of < thrust > are available in DESY 97-210.)

[32]

ZEUS Collab., J. Breitweg et al., Jerusalem EPS 97-648; ZEUS Collab., M. Derrick et al., Phys. Lett. B356 (1995) 129.

[33]

H.-U. Bengtsson and T. Sjöstrand, Comp. Phys. Comm. 46 (1987) 43; T. Sjöstrand, CERN-TH.6488/92.

[34]

ZEUS Collab., M. Derrick et al., Phys. Lett. B369 (1996) 55.

[35]

SLD Collab., K. Abe et al., Phys. Rev. Lett. 76 (1996) 4886.

[36]

DØ Collab., S. Abachi et al., Phys. Rev. Lett. 72 (1994) 2332; DØ Collab., S. Abachi et al., FERMILAB-PUB-95-302-E (1995).

[37]

CDF Collab., F. Abe, et al., Phys. Rev. Lett. 74 (1995) 855.

[38]

J. Perkins, DIS97-AIP Conference Proceedings 147 (1997) 533.

[39]

CDF Collab., F. Abe, et al., Phys. Rev. Lett. 78 (1997) 2698.

[40]

D. Soper, DIS97-AIP Conference Proceedings 147 (1997) 147.

[41]

J. Collins et al., Phys. Lett. B307 (1993) 161; J. Collins, hep-ph/9709499.

Footnotes:

¹ Supported by PPARC and DESY.