If we accept our "diffractive elastic scattering" operator T to be constructed à la Glauber (although, in principle, one can take something else for it, we prefer to use the Glauber model prescription because it works so well for elastic diffractive scattering) we have T=tp(pı-kı) +tn (pn 1 − kn 1) +t, (pr1-k, 1) — tn (Pn 1-kn 1) t,(p,1—k,1). These four pieces of T produce the following contributions: One can summarize the situation as follows: the coefficients d2” (or d2TM”) give the amplitude for the neutron-pion fluctuation of the incoming proton. The total production amplitude is the difference of the two main contributions. This represents both single and double scattering of the n- system in which the diffractive elastic scattering occurs either before or after the fluctuation takes place. Note that in these considerations we can have any target we want! The target is specified through the scattering operators t, and tn. Hence one can use the same technique to describe the processes on simple and composite (e.g., nuclei) targets [S4]. Relation of the above model to some well established techniques of describing diffractive dissociation. First of all, our description is quite similar to that applied by Cheng and Wu [21], Bjorken, Kogut, and Soper [27], and Jaroszewicz [30], for high energy bremsstrahlung and pair production processes in QED (see also [S4]). It is also very closely related to some one particle exchange models of diffractive production processes first suggested by Drell and Hiida [31] and continued by Deck [32], and M. Ross and Y. Y. Yam [33]. Our description also contains, as a special case, the so-called vector meson dominance models which are employed to describe interactions with hadronic targets at high energy, a process which will be discussed later. go over to AND where the dot means that complete (to all orders) elastic scattering amplitudes are to be inserted and the vertices are given by di; coefficients analogous to the ones in eqs (4.5), (4.6), and (4.7). (b) Pair production: the Feynman diagrams go over to AND AND m and in our formulation a double scattering process should be added: which makes our description different from the above two Feynman diagrams (in fact, more complete) but in total agreement with the Bethe and Maximon formulae [34] for the high energy limit of the pair production cross section in a strong Coulomb field [35]. In such a model, the exchanged virtual pion scatters elastically (diffractively) off the target proton. Later (see e.g., Ross and Yam [33]), two more diagrams were added: (b) These diagrams (except for the "diffractive" vertex, they are just Feynman diagrams) are intimately related to our description. In order to see it, one should do some kinematics. Let us work out some kinematical expressions associated with the vertex d2" of the processes (i)–(iv): where the four-vectors are denoted (p1, P2, Po) and we employed conservation of the three momentum at the vertex. (Note that since we are using noncovariant perturbation theory the energy is not conserved at the vertex.) Let us evaluate all expressions in the limit w∞ (longitudinal component of the incident. momentum very large). There are two independent variables which may be chosen as pi, B. (Here ẞ is an arbitrary parameter, 0<ß<1). One can compute similarly the invariant mass of the n- system: Mnr2 = (√ẞ2w2+p12+mn2 + √(1−ẞ)2w2+p12+m,2)2 - (p1-p1)2 - (Bw+(1−ẞ) w)2 Since the Feynman diagrams (a)-(c) give the following contributions where A's are elastic scattering amplitudes and V's are the vertex functions, we can see, using the relation Αγνωστοί and above proven equalities, that (assuming the vertices identical, which is the case for forward amplitudes) M(b)—— M(c), and that they cancel to a large extent (they cancel exactly in the forward direction if σpp=σpn). We can also see the correspondence between our diagrams and the "one pion exchange" diagrams: However, our diagram (iv) has no analogue in the Drell-Hiida-Ross-Yam-Deck model. The other difference is the lack of four-momentum conservation in (i)-(iv) (only three-momentum conservation). This last difference may sometimes be relevant (see e.g. A. Bialas, W. Czyż, and A. Kotański [29]). Example 3: Photoprocesses and vector meson dominance: It is now a well established fact that, at high energy, photons exhibit shadowing. The total photo cross sections for complex nuclei vary approximately as A°.9 in the few GeV energy range. |