Di raction and the Pomeron Halina Abramowicz School of Physics and Astronomy Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University, Israel 1 Intro duction In hadron-hadron scattering, interactions are classi ed by the characteristics of the nal states. In elastic scattering, b oth hadrons emerge unscathed and no other par- ticles are pro duced. In di ractive scattering, the energy transfer between the two interacting hadrons remains small, but one single disso ciation or b oth double dis- so ciation hadrons disso ciate into multi-particle nal states, preserving the quantum numb ers of the asso ciated initial hadron. The remaining con gurations corresp ond to inelastic interactions. The rst interpretation of di raction, develop ed byFeinb erg and Pomeranchuk [1] and Good and Walker [2], was that di erent comp onents of the pro jectile were di erently absorb ed by the target, leading to the creation of new physical states. This was one of the early indications of the comp osite nature of hadrons. In the Regge theory of strong interactions [3], di raction is the result of ex- changing a universal tra jectory with the quantum numb ers of the vacuum, the soft Pomeron, IP ,intro duced by Grib ov[4]. In the language of Quantum Chromo dynamics, the candidate for vacuum ex- change with prop erties similar to the soft Pomeron is two gluon exchange [5, 6]. As a result of interactions between the two gluons, a ladder structure develops. In p erturbative QCD, the prop erties of this ladder dep end on the energy and scales involved in the interaction, implying its non-universal character. In the high-energy limit, the prop erties of the ladder have b een derived for multi-Regge kinematics and the resulting exchange is called the hard BFKL p omeron [7, 8, 9]. A renewed interest in di ractive scattering followed the observation of a copious pro duction of di ractive-like events in deep inelastic scattering DIS at the HERA ep collider [10, 11] as well as the earlier observation of jet pro duction asso ciated with a leading proton in p p at CERN [12]. The presence of a large scale op ens the p ossibility of studying the partonic structure of the di ractive exchange as suggested by Ingelman and Schlein [13] and testing QCD dynamics. Moreover, the study of di ractive scattering o ers a unique opp ortunity to understand the relation b etween 495 Halina Abramowicz Di raction and the Pomeron the fundamental degrees of freedom prevailing in soft interactions { hadrons and Regge tra jectories, and those of QCD { quarks and gluons. One of the challenges is to establish theoretically and exp erimentally the reactions in which the soft comp onent is dominant and those in which the p erturbative QCD formalism is applicable. In this rep ort, the discussion will fo cus on single di ractive pro cesses in the presence of a large scale, mainly studied at HERA and at FNAL. 2 Kinematics of di ractive scattering The variables used to analyze di ractive scattering will be intro duced for ep DIS. Because DIS is p erceived as a two-step pro cess in which the incoming lepton emits a photon which then interacts with the proton target, the relevant variables can be readily generalized to p p interactions. , k ke e q quark β MX xIP,t LRG , pp pp Figure 1: Schematic diagram for di ractive DIS in ep interactions. A diagram for di ractive scattering in DIS, where the di racted state is separated from the scattered proton by a large rapidity gap LRG, is presented in Fig. 1. All the relevant four vectors are de ned therein. In addition to the usual DIS variables, 2 2 0 2 2 2 2 Q = q = k k , W = q + p , x = Q =2p q , and y = p q=p k, the variables used to describ ed the di ractive nal state are, 0 2 t = p p ; 1 2 2 0 M + Q q p p X ' ; 2 x = IP 2 2 q p W + Q 2 2 Q x Q = = ' : 3 2 0 2 2q P P x Q + M IP X x is the fractional loss of the proton longitudinal momentum. It is sometimes de- IP noted by . is the equivalent of Bjorken x, but relative to the exchanged ob ject. 496 Halina Abramowicz Di raction and the Pomeron M is the invariant mass of the hadronic nal state recoiling against the leading X 2 0 2 proton, M =q+pp . The approximate relations hold for small values of the X four-momentum transfer squared t and large W ,typical of high energy di raction. The need to preserve the identity of the target in di ractive scattering limits 2 the square of the momentum transfer, jtj < 1=R , where R is the radius of the T T target. The t distribution typically has an exp onential b ehavior, f t exp bjtj 2 with b ' R =6. The allowed M is also limited by the coherence requirement. The X T minimum value of t required to pro duce a given M from a target with mass m is X T 2 2 2 2 4 2 2 jt j ' m M + Q =W . For a typical hadronic radius of 1 fm, M < 0:2W min T X X and the hadronic nal state exhibits a large rapidity gap between the fragments of the di racted state and the unscathed target see Fig. 2. Therefore, in collider dN dy proton MX large rapidity gap yy p X Figure 2: Schematic representation of rapidity y distribution for single di raction events. exp eriments, di ractive events are identi ed either by the presence of a fast proton along the b eam direction or by the presence of a large rapidity gap in the central detectors. 3 Formalism of di ractive scattering To describ e di ractive DIS, it is customary to cho ose the variables x and t in addition IP 2 to the usual x and Q in the cross section formula. The di ractive contribution to F 2 D is denoted by F and the corresp onding di erential contributions are 2 D D dF dF D 3 D 4 2 2 F = ; F = : 4 2 2 dx dx dt IP IP Throughout this pap er, the contribution from the longitudinal structure function is omitted for simplicity. The four-fold di erential cross section for di ractive ep 497 Halina Abramowicz Di raction and the Pomeron scattering can b e written as 4 D 2 h i d 2 D4 ep 2 2 1+1y F x; Q ;x ;t: 5 = IP 2 2 4 dx dtdxdQ xQ IP The structure function F is related to the absorption cross section of a virtual 2 ? photon by the proton, . For di ractive scattering, in the limit of high W low x, p 2 D 2 d ? Q D 4 p 2 F x; Q ;x ;t= : 6 IP 2 2 4 dx dt IP This relation allows predictions for di ractive scattering in DIS based on Regge phe- ? nomenology applied to p scattering. In fact, many of the questions that are ad- dressed in analyzing di ractive scattering are inspired by Regge phenomenology as established in soft hadron-hadron interactions. 3.1 Regge phenomenology 2 For scattering of two hadrons a and b at squared center of mass energy s m ; t, a;b Regge phenomenology implies that, 01 IP ab s ; 7 tot 2 el 2 d ab el 0 tot b +2 ln st 0 IP ab ! ab = e ; 8 dt 16 ! 2 01 IP s 2 D D 0 b +2 ln t 1 d s 2 0 IP M X ab ! Xb e ; 9 2 2 2 dtdM M M X X X 0 where t = 0 + t is the parameterization of the IP tra jectory. The uni- IP IP IP versality of this parameterization has b een p ointed out by Donnachie and Landsho . 0 2 The value of 0 = 1:081 [14] and =0:25 GeV [15] were derived based on to- IP IP tal hadron-proton interaction cross sections and elastic proton-proton data. Recently, the IP intercept has b een reevaluated [16] leading to a value of 0 = 1:096 0:03. IP The IP intercept is sometimes presented as 0 = 1 + . IP Three implications are worth noting. 1. The slop e of the t distribution is increasing with ln s. This fact, b orne out by the data, is known as shrinkage of the t distribution. It is due to the fact that 0 6= 0 and has b een explained by Grib ov [4] as di usion of particles in the IP 0 2 exchange towards low transverse momenta, k , with 1=k see also [17]. T IP T 2. A steep and universal x dep endence of the di ractive cross section is exp ected, IP 1+2 D d =dx x . IP IP el D 3. The ratios = and = rise like s . Because >0, this is b ound to lead tot tot to unitarity violation. 498 Halina Abramowicz Di raction and the Pomeron 3.2 Perturbative QCD D QCD factorization for the di ractive structure function of the proton, F , is exp ected 2 D D to hold [18, 19, 20]. F is decomp osed into di ractive parton distributions, f , in a 2 i way similar to the inclusive F , 2 Z D 2 D x X IP dF x; Q ;x ;t x df z ; ; x ;t IP IP 2 i 2 ^ = F ;Q ;; 10 dz 2;i dx dt dx dt z 0 IP IP i ^ where F is the universal structure function for DIS on parton i, is the factorization 2;i D scale at which f are prob ed, and z is the fraction of momentum of the proton carried i by the di ractive parton i.