Introduction

Prologue: In 1992, H. Fritzsch summarised the status of ``QCD 20 Years On" [1] with the words ``A very large amount of data in strong-interaction physics is described by the pomeron singularity... The fact that the physics of the pomeron is very simple needs to be explained in equally simple terms in QCD... Using HERA the experimentalists will be able to study the region of very low x in deep inelastic scattering... Furthermore, both in $p\bar{p}$-scattering at high energies and at HERA the structure functions of the pomeron can be studied in more detail." Now, 25 years on from the first developments of QCD, at the 25th SLAC Summer Institute we can discuss a series of diffractive measurements that have been made in the intervening period at HERA and the Tevatron.

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

$$e~(k)~+~ p~(P) \rightarrow e'~(k')~+~p'(P')~+~X,$$

where the photon dissociates into the system X and the outgoing proton, p', remains intact, corresponding to single dissociation, as illustrated in Fig. 1. The measurements are made as a function of the photon virtuality, $Q^2 \equiv -q^2=-(k~-~k')^2,$ the centre-of-mass energy of the virtual-photon proton system, W²=(q + P)², the invariant mass of the dissociated system, X, denoted by M² and the four-momentum transfer at the proton vertex, given by t = (P - P')².

  

Figure 1: Kinematic variables of diffractive ep scattering at HERA

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

$$AB \rightarrow AB$$

which is constrained through the measurement of the corresponding total cross-section $AB\rightarrow$ anything, via the optical theorem. The diffractive processes measured are then

$$AB \rightarrow XB~~~~~~~~ {\rm single~dissociation}$$

$$AB \rightarrow XY~~~~~~~~ {\rm double~dissociation}$$

where the systems X and Y have a limited mass compared to the overall energy available, W. The process is thought of as being mediated by the exchange of an object with vacuum quantum numbers (in particular, no colour is exchanged in the process). A signature for diffraction is via the momentum fraction carried by the exchanged colour singlet state. When this fraction is less than 1% of the momentum of, say, particle B (in the infinite momentum frame of B) we can interpret this as being largely due to pomeron exchange. The kinematics of producing two low-mass outgoing states (X and B or X and Y) with a small momentum fraction exchanged between them therefore leads to a rapidity gap. The high energy available at HERA provides a large rapidity span, $\Delta(\eta^*)$, of $\sim$ 12 units (here, $\Delta(\eta^*) \equiv \eta^*_{\rm max} - \eta^*_{\rm min}$ where $\eta^*_{\rm max} \sim \ln(W/m_p) = 5$ and $\eta^*_{\rm min} \sim -\ln(W/m_\pi) = -7$ for a typical $W \simeq 150$ GeV). A colour singlet exchange of reggeons (dominating at lower W) and the pomeron (dominating at higher W) can be used to interpret the data on single dissociation and double dissociation reactions. Fluctuations from processes where colour is exchanged may also generate low-mass states. However, these will be exponentially suppressed: the (Poisson) probability of not producing a given particle in the rapidity gap $\Delta\eta$ when the two systems are colour connected is $exp(-\lambda\Delta\eta)$, where $\lambda$ is the mean particle density for a given $\Delta\eta$ interval. This exponential fall-off reflects the plateau in the corresponding (non-diffractive) multiplicity distribution as a function of $\eta$ which increases relatively slowly (logarithmically) with increasing W [2].

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 $\simeq$ 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, $d\sigma/d\vert t\vert \propto e^{-b\vert t\vert}$ for $\vert t\vert~\raisebox{-.6ex}{${\textstyle\stackrel{<}{\sim}}$ }~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 d $\sigma/dt \propto e^{-b\vert t\vert}$, then $b \simeq$ R_I²/4.

  

Figure: Signatures of diffraction: $d\sigma /dt$ 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, $s \equiv W^2$

$$\sigma_{\rm tot} = A\cdot(W^2)^\epsilon + B\cdot(W^2)^{-\eta}$$ (1)

where W is measured in GeV, $\epsilon = \mbox{$\alpha_{_{I\hspace{-0.2em}P}}$ }(0) - 1$ and $\eta$ is defined to be positive such that $\eta = -(\mbox{$\alpha_{_{I\hspace{-0.2em}R}}$ }(0)-1)$. Here, $\mbox{$\alpha_{_{I\hspace{-0.2em}P}}$ }(0)$ and $\mbox{$\alpha_{_{I\hspace{-0.2em}R}}$ }(0)$ are the pomeron and reggeon intercepts (i.e. the values of the parameters at t = 0 GeV²), respectively. A wide range of total cross-section data are used to determine the parameters $\epsilon$ and $\eta$. The fall-off at low energy due to reggeon exchange constrains the value of $\eta \simeq 0.45$. The slow rise of hadron-hadron total cross-sections with increasing energy indicates that the value of $\epsilon \simeq 0.08$ i.e. the total cross-sections increase as W^0.16, although the $p\bar{p}$ data from CDF at two $\sqrt{s}$ values indicate $\epsilon = 0.112 \pm 0.013$ [5]. Recent fits using all pp and $\bar{p}p$ data are consistent with a value of $\epsilon = 0.08 \pm 0.02$ which will be used here to characterise this behaviour [6].

  

Figure: Regge trajectories: The degenerate regge trajectories are indicated by the solid line. The pomeron trajectory is indicated by the dashed line. Also indicated are the $\rho$, $\omega$, f and a resonances as well as the I(J^PC) = 0(2⁺⁺) glueball candidate state X(1900) observed by the WA91 collaboration [7].

In a Regge analysis, the diffractive data are interpreted via exchanges with spin $J = \alpha(t) = \alpha(0) + \alpha't$ and $d\sigma/dt \propto (\frac{W^2}{W_0^2})^{2(\alpha(0)-1)} e^{-b\vert t\vert}$, with $b = b_0 + 2\mbox{$\alpha'$ }\ln(W^2/W_0^2)$. At lower energies, these correspond to reggeon (i.e. approximately degenerate $\rho$, $\omega$, 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, $J = \alpha(t)$, 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 $\vert\bar{t}\vert$ value is given by 1/b at the mean $\bar{W}$ of a given data sample, the diffractive cross-section rise is moderated from $(W^2)^{2\epsilon}$ to

$$\sigma_{\rm {diff}} \simeq (W^2)^{2(\epsilon-\mbox{$\alpha'$ }\cdot \vert\bar{t}\vert)} \equiv W^{4\bar{\epsilon}}$$ (2)

where $\bar{\epsilon} = \epsilon - \mbox{$\alpha'$ }\cdot \vert\bar{t}\vert = \alpha(\bar{t}) - 1$ and $\mbox{$\alpha'$ }= 0.25~$GeV^-2 reflects the shrinkage of the diffractive peak as a function of t with increasing W². The observed shrinkage of the diffractive peak therefore corresponds to a relative reduction of the diffractive cross-section with increasing energy. This value may be compared with the corresponding parameter $\mbox{$\alpha'_{_{I\hspace{-0.2em}R}}$ }\simeq 0.9~$ GeV^-2 for reggeon exchanges.

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 $\Lambda^2$, 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, $x= Q^2/2P \cdot q \simeq Q^2/(Q^2+W^2)$. QCD factorisation into a long-timescale and short-timescale process, where this timescale is characterised by 1/Q or 1/M or $1/\sqrt{t}$, leads to the following approaches.

$\bullet$

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

$$\sigma_{\rm {diff}} \sim G_p^2 \otimes \hat\sigma$$

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 $\hat\sigma$ 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 $\epsilon$. 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 $\epsilon$ as a function of t and therefore a decrease of $\mbox{$\alpha'$ }$ for the perturbative pomeron.

$\bullet$

For inclusive diffraction with a hard scale

$$\sigma_{\rm {diff}} \sim G_p \otimes \hat\sigma \otimes H$$

i.e. leading-gluon exchange where G_p is the gluon density of the proton, $\hat\sigma$ 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.

$\bullet$

Finally, one can break the process down further and invoke Regge factorisation where a flux of pomerons, $f_{I\hspace{-0.2em}P/p}$ lead to partons from the pomeron, $f_{i/I\hspace{-0.2em}P}$, which interact and hadronise

$$\sigma_{\rm {diff}} \sim f_{I\hspace{-0.2em}P/p} \otimes f_{i/I\hspace{-0.2em}P} \otimes \hat\sigma \otimes H$$

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

$$\sigma_{\rm {diff}} \sim f_{I\hspace{-0.2em}P/p} \otimes f_{i/I\hspace{-0.2em}P} \otimes \hat\sigma \otimes f_{j/h} \otimes H$$

where f_j/h represents the partons from the hadron which has not diffractively dissociated. This Regge factorisation approach can therefore be experimentally tested when diffractive data from HERA and the Tevatron are compared.

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)).

  

Figure 4: Maps of the pomeron: (a) theoretical and (b) experimental directions.

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 $\rho ^0$ up to the $\Upsilon$ 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 ( $\simeq 10\%$) 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 $p\bar{p}$ colliders. In section 5, the latest results from the Tevatron on diffractive dijet and $W^\pm$ 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.