Diractive Structure Functions in Dis

A review of theoretical models of di ractive structure functions in deep inelastic scattering (DIS) is presented with a view to highlighting distinctive features, that may be distinguished experimentally. In particular, predictions for the behaviour of the di ractive structure functions F 2 ; F D L ; F D(charm) 2 are presented. The measurement of these functions at both small and high values of the variable and their evolution with Q is expected to reveal crucial information concerning the underlying dynamics. 1 Models of hard di ractive structure functions in DIS It is natural to start with a de nition of what we mean by the terms `hard' and `di ractive' when applied to scattering of electrons and protons. High energy scattering processes may be conveniently classi ed by the typical scales involved. By hard scattering we mean that there is a least one short distance, high momentum, scale (e.g. high pT -jet, boson virtuality, quark mass) in the problem that gives one the possibility of using factorization theorems and applying perturbative QCD. In case of di ractive DIS this is the photon virtuality, Q, however this hard scale is not necessarily enough and indeed QCD factorization may not even be applicable to all hard di ractive scattering in DIS (see [1, 2, 3] for discussions and refs). It has been shown to be applicable to di ractive production of vector mesons initiated by a longitudinally polarized photon [4]. For the time being we will use the de nition, due to Bjorken, that a di ractive event contains a non-exponentially suppressed rapidity gap. Rapidity is the usual experimental variable related to the trajectory of an outgoing particle relative to the interaction point: given approximately by ln(tan( =2)) (in a cylindrical system of co-ordinates centered on the interaction point, with the z-axis along the beam pipe and polar angle ). This rather obscure sounding de nition results from the fact that within perturbative QCD large rapidity gaps (LRG) are suppressed because a coloured particle undergoing a violent collision will emit radiation that would ll up the gap. The suppression factor increases with the interval of rapidity but it's absolute magnitude for di ractive processes in DIS is uncertain. An additional source of rapidity gap supression comes from an overall damping factor associated with multiple interactions. The amount of damping is found to be much smaller in DIS than that typical for soft processes (e.g. proton proton collisions see [5]) making LRG events more likely. Supported by MINERVA Theoretically, for `di ractive' electron proton scattering in DIS one must observe the proton in the nal state. In practice this is very di cult for HERA kinematics since the highly energetic scattered proton disappears down the beam pipe in most events. This means that the current measurements also contain contributions from interactions in which the scattered proton dissociates into higher mass states. This uncertainty is considerably alleviated by the advent of the Leading Proton Spectrometer (LPS) which will provide crucial information about di raction (for the rst data from the LPS see [6]). The signi cance of the di erence between the experimental working de nition of di raction and the theoretical one is an interesting but as yet unresolved problem (it is certainly possible to produce large gaps in rapidity in `nondi ractive' processes, e.g. via secondary trajectory exchanges). Such LRG events occur naturally in processes known to be governed by soft processes (e.g proton anti-proton scattering at high energies). This is explained naturally in the context of Regge theory : at high enough energies one reaches the so-called Regge limit (s t and s all external masses) and all hadronic total cross sections are expected to be mediated by Pomeron exchange and to exhibit the same energy behaviour. This expectation is born out by the data (see e.g. [7]), which shows that a wide variety of high energy total elastic cross sections have the same energy dependence which is attributed to the trajectory of the soft pomeron. The energy dependence for di raction in these processes is discussed in e.g. [8]. Scattering of virtual photons and protons at small enough x corresponds to the Regge limit of this subprocess (ŝ t̂, ŝ Q;M Proton ). It is natural to ask if the di ractive events observed in the DIS sample also exhibit the universal behaviour even though we are now considering o shell scattering for which, strictly speaking, Regge theory does not necessarily have to apply. One of the reasons why hard di raction at HERA at small x is so interesting is that as x decreases, for xed large Q there should be a transition between the hard short distance physics associated with moderate values of x and the physics of the soft pomeron which is widely believed to dominate at very small x. It is a theoretical and experimental challenge to establish whether LRG events in DIS in the HERA range are governed by hard or soft processes or whether they are actually a mixture of both. The purpose of this report is to discuss the current theoretical models for di ractive structure functions in an attempt to address this problem, and, in particular, to outline the benchmark characteristics of the various approaches to facilitate the search for appropriate experimental tests. In analogy to the total DIS cross section, the di ractive cross section in DIS can be written, d D dxIPdtdxdQ = 4 2 e.m xQ4 " 1 y + y 2[1 +RD(x;Q2; xIP; t)] # F 2 (x;Q ; xIP; t) (1) where D denotes di raction, R = F L =(F D 2 F D L ) and y = Q =sx ; t = 0 is usually assumed since the cross section is strongly peaked here. Ingleman and Schlein [9] suggested on the basis of expectations from Regge theory that the di ractive structure functions could be factorized as follows: F 2 (x;Q ; xIP; t) = f(xIP; t)F IP 2 ( ;Q ; t); (2) where Q is the photon virtuality, xIP is the fraction of the proton's momentum carried by the di ractive exchange and t is the associated virtuality, = Q=(M x + Q ) = x=xIP, with M 2 x the mass of the di ractive system. The last relation for in terms of x, the Bjorken variable, is a good approximation but only holds for negligible t and proton mass [10]. Due to lack of information on the remnant proton both xIP and t can only be estimated indirectly or have to be integrated out. The 1993 HERA data [11, 12, 13] con rmed the presence of events with large rapidity gap between the proton direction and the nearest signi cant activity in the main detector, in the total DIS cross section at the leading twist level (i.e. this contribution persisted to high values of Q). These events constitute approximately 10 % of the total sample (compared to 40% in photo-production). As has been known for many years and as Bjorken has recently pointed out [14] the fact the di ractive cross section is present in the total sample as a leading twist e ect (i.e. it `scales') at large Q and small x does not necessarily imply that the mechanism that creates these events is point-like. For a careful discussion of the kinematics of hard and soft di raction in a variety of di erent reference frames see [15]. The observed events were also not inconsistent with the Regge factorization of eq.(2). Since the cross section had the same power-like xIP dependence (in f) over a the wide range of ( ;Q ) that were measured it was tempting to postulate that a single mechanism or `exchange' was responsible for these events. The presence of the gap tells us that this object is a colour singlet and since the centre of mass energy was very high, the exchanged object became known as the `Pomeron'. From this observation it is natural to ask, following [9], if the partonic content of this `particle' may be investigated by changing and Q, with interpreted as the momentum fraction of the pomeron carried by the struck parton; f in this picture is interpreted as `the ux of pomerons in the proton'. This approach has led to a plethora of theoretical papers in which the parton content of the Pomeron at some small starting scale, Q0, is treated in various physically motivated ways (relying strongly on Regge theory). The DGLAP [16] equations of perturbative QCD (to a given logarithmic accuracy) are then used to investigate the evolution with Q of this parton content. Formally the use of the DGLAP equations is inapplicable for the description of di raction because the presence of the gap makes it impossible to sum over all possible nal hadronic states. Their use in this context is at the level of a plausible assumption. In some papers an analogy is drawn with the proton [10, 17, 18] and a momentum sum rule may be imposed on the parton content. Others models [10, 19, 20] take the view that that the Pomeron may be more like the photon and so can have, in addition, a direct coupling to quarks within the virtual photon. Although it is no longer clear once a direct coupling has been introduced whether the concept of a Pomeron structure function has any meaning. Fits [10, 17, 21] to the 1993 data on di raction reveal a partonic structure that is harder (more partons at high ) than the proton and that gluons contain a large fraction of pomeron momentum (up to 90 %) with a large fraction of these at high . Clearly in a quantitative sense such statements will depend on the physical assumptions used to parameterize the input distribution. However qualitatively these statements are reasonable. The paper of Gehrmann and Stirling [10] is particularly useful in discussing Pomeron structure function models in that it discusses and compares two models: model 1 which has only resolved component and imposes a momentum sum rule on the parton content and model 2 which also allows a direct coupling of the Pomeron to quarks. This leads to rather di erent predictions for the Q evolution of these two models (see curves labelled `GS(I), GS(II)' in g.(1)). Model 1 evolves in a way familiar to the evolution of the proton structure function in QCD, i.e. as Q increases there is a migration of partons from high to low . In model 2, as a result of the direct coupling of the pomeron to quarks (at ` = 1'), the high distribution is supplemented and, provided the direct component is large enough, one expects an increase of parton densities with Q over the whole range, which is also an expectation of the boson-gluon fusion model of [22] (see g.(1)). Q(GeV) x Ι Ρ • F 2 D (3 ) β=0.1 xΙΡ=0.005 NZ GS(I) GS(II) BW Q(GeV) β=0.8 xΙΡ=0.005 LND GK RS BP β x Ι Ρ • F 2 D (3 ) Q=8 xΙΡ=0.005 β Q=50 xΙΡ=0.005 0 0.02 0.04 0.06 0.08 0.1