In computing the amplitude we have to sum over all nucleons of the target nucleus because the conversion of y into V can occur on any one of them. for both the y→V and V→V processes. So, we get again the formula (4.9). In all the formulae above we have used the profiles of nucleons smeared over the interior of the nucleus with the single nucleon density functions p (r): The conclusion of the above discussion: The two seemingly different pictures give the same results in the high energy limit. As we saw in the example of our discussion of or (y, A), present experiments are not quite in this limit (at several GeV I<R). Hence if we want to analyze the existing experimental data we have to keep the longitudinal momentum transfer different from zero: mv2 Then the graphical description of figure 38 is not valid any more. One has to allow the amplitude Mnucleon (Y→V) to oscillate with the function This factor partly destroys the coherence of the process of producing V's over the whole nuclear volume. We construct the production amplitude as the (previously introduced) picture of multiple scattering tells us to. For the independent particle ground state wave function we have Note that this expression satisfies the correct "weak interaction" limit: when the V interaction is For large A and the profiles y much narrower than the density p, we have M→fv, (0) [ db exp (i▲·b) [° dz p (b, 2) e1a¡¡• where fv, (0) is the forward production amplitude on one nucleon corresponding to the profile YvŸ. The above formulae can also be obtained from the expression . . . M (v→→V) = f ď3r ... ... Pra | ¥ (x1, ... ... r1) |2 Σ M,π-V (q) exp (iq⋅r;) where q= (4, 4), and M, i are the amplitudes for the production of a V meson on the jth nucleon with the screening of the other nucleons taken into account. For a given configuration of the nucleons we have 5 When we want to shift the scatterer by the distance r, the amplitude acquires the phase ear. In other words, we have to transform the amplitudes as follows: M (q) →eia.rM (q), where q is the three-momentum transfer vector. As long as there is only one scattering center this phase factor is irrelevant, but when there are more scattering centers we obtain M(q) = Σeiar¡Mj (q), which is the formula used here. 6 The argument in yy, which gives the elastic scattering vector meson-nucleon profiles, should be the transverse distance between a target nucleon and the incident particle. The geometry below shows that the argument of yy in the expression for M(y→V) should be: b'+si-sj. PROJECTION ON THE SCATTERING PLANE INCIDENT PARTICLE SYMMETRY AXIS PROJECTION ON A PLANE PERPENDICULAR TO THE INCIDENT becomes the formula given above. Note that the same reasoning gives, in the limit of large A, d2b dz exp (i▲ b+i▲||2)p(b, z)ƒvy (0) exp [−11⁄20 vn (1— invn)T,(b, z)] T', (b, 2) = A [ ** dz′p (b, 2') . This is again the same formula as before. Note that for nucleons at different positions z, the attenuation of the outgoing vector meson beam is different. So, in the last two expressions for M (y→V) our Mr→V (q) depend on the position of all the other nucleons. These formulae are now adapted to take care of nonnegligible A. They are being used in all standard analyses of photoproduction of vector bosons on nuclei [S3]. Let us also quote, for the sake of completeness (without giving derivation), the amplitude for elastic scattering of photons from a nucleus derived from the multiple scattering model, with A nonnegligible, in the limit of large A and VMD assumed: In this formula the single scattering is separated out and one should set = Σv (πa/yv2) vvv if we apply VMD for the single scattering too. These two contributions in M (y→y) can be sketched as follows: Note that the single scattering contribution always goes as A (just as any multiple scattering process) but it becomes progressively less important as A0 with increasing energy. Recall that the experiments discussed above [37] had shown that when or (YA) is computed from (4.10), the single scattering contribution is not the one given by the VMD model (compare also footnote 1). Now let us go back to photoproduction of vector mesons. The formulae discussed above were applied to analyze a multitude of experimental data. The first suggestion that one can obtain some important information on the properties of V-nucleon interactions came from S. D. Drell and J. S. Trefil [38]. Then a flood of papers followed. The references can be found in the review articles we referred to at the very beginning of these notes [S1, S3, S5]. A few general comments can be made by inspecting the formulae for photoproduction of vector mesons: (i) They depend very strongly on A (in the optical limit the dependence on A is exponential). (ii) There may be a very important interplay between the "phase factors" eis||2 and exp [iσvNvN1⁄2T (b, z)]. So, the differential cross section for photoproduction of vector mesons should, in general, be sensitive to both σ and N. This is very important because these quantities cannot be obtained directly from any other experiments because there are no vector meson beams available due to their short lifetime. Here such indirect "measurement" is possible because the vector mesons interact with nucleons before they decay. The differential photoproduction cross sections look very much like elastic hadron-nucleus cross sections. They exhibit a steep slope at small q2 and then a flat part at large q2 with, possibly, some diffractive minima. Just to give some idea of how the results look, let us describe briefly the results of po photoproduction on deuterium (R. L. Anderson et al. [39]) and a DESY-MIT experiment (H. Alvensleben et al. [40]) of po photoproduction on light, medium and heavy nuclei. First, the deuterium target. We can use our formulae derived above after specifying them for A=2. We get MDeut. (po) = = ik 2π đb exp (i▲·b) ƒ dr p(r) {[1 − Ypp (b−1⁄28) 0 (2) ]Ypy) (b+1⁄2s) exp (¡A¡¡1⁄22) +[1−y, (n) (b+11⁄2s) 0 ( − z) ]Ypy (P) (b−11⁄2s) exp (—¿A¡¡1⁄41⁄22)} The cross section experimentally measured looks as follows: double scattering contributions From the above formulae we can see that at small momentum transfers we are essentially measuring the amplitudes for photoproduction of po on neutrons and protons, modulated by the deuteron form factor F. For example, |