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db exp (i▲•b) √ đ2s dz p (8, 2) [ dô exp [−id• (b+1⁄28)]ƒ»‚‹") (8) exp (¿A¦¦1⁄41⁄22)
and the analogous expression for the proton (fig. 44). Hence this part of the cross section, where these contributions dominate, can only confirm what we know from po photoproduction on free nucleons.
At large momentum transfers, however, where the double scattering dominates, the amplitude is given approximately by (see also fig. 45)
d2ôF (8)ƒ„»'» (1⁄2A+8)ƒ»yTM (1⁄2▲−8) + [ d2ôF (8)ƒ„‚‹TM (1⁄2▲−8)ƒ»x» (1⁄2A+8).
So, by measuring the cross section at large momentum transfers we can extract fnucleon because we know fynucleon quite well. In fact, since F (8) is much steeper than ƒ, and ƒ, it acts as a Dirac (2) function, and the measured differential cross section is proportional (the proportionality factor being known) to the elastic p-nucleon cross section.
In this experiment , the recoiling deuteron was detected. This guaranteed the coherence of the process and excluded any excitations of the target nucleus.
There are a few points which should be stressed again at this stage:
(i) This is the most direct measurement of the elastic p-nucleon cross section in existence.
(ii) If one wants to extract on or y,2/4π, one can do it very safely because the analysis, at large enough q2, depends insensitively on 7. First one extracts (do/dt) PNN and then by taking the ratio
one determines the coupling y. (Here a is the fine structure constant.)
(iii) The results of this analysis check beautifully with the results obtained from po photoproduction on larger nuclei (whose cross sections, incidentally, do depend sensitively on 7, as we have already indicated before).
(iv) The extraction of (do/dt) \,N,N does not depend on the VMD hypothesis. However, it assumes the multiple scattering model of Glauber completely and literally. For more details we refer to ref. .
Let us now go briefly to photoproduction of po on various nuclei and take as an example the DESY-MIT experiment mentioned already . Here, the target was not detected and the measured cross section contained contributions from nonelastic processes. The reaction measured was Y+A→po+A
The following targets were used: H2, Be, C, Al, Ti, Cu, Ag, Cd, In, Ta, W, Au, Pb, U. The parameters obtained from fitting the differential cross sections to their very extensive numerical data are , PN=27.7±1.7 mb, Y,2/4T=0.59±0.08,
in good agreement with the more recent numbers already quoted. In analysis they assumed the ratio of the real to the imaginary part to be ŋ,=0.2 (in agreement with the dispersion relations calculations of the total photon cross sections.)
From the above description and the inspection of the data shown in ref  one sees that there are several points which bring about uncertainties in such an analysis. These uncertainties arise because one has to make some corrections in order to obtain the coherent p-production cross sections:
(i) nuclear excitations should be removed
(ii) processes which lead to π production (other than p-production) should be subtracted, (iii) one has to decide which invariant masses of are p°'s and which belong to some kind of background (compare the data). The problems of interference between and p productions are also relevant (see T. H. Bauer in ref. ).
(i) and (ii) can be reliably estimated. Take (i). The process can be computed in the same manner as the poor energy resolution cross-section. The table below shows some illustrative calculations by Yennie (see K. Gottfried report in [S5]).
So, as long as we measure small angle po photoproduction we can neglect nucleus excitations (this won't produce more than a few percent error). The contribution (ii) was also estimated not to exceed ~10 percent. The point (iii) makes the results of our analysis model dependent but again the extensive work of many experimental groups analysing their experiments seems to show that this model dependence produces ~10 percent uncertainty. This is how far one can trust such numbers as quoted above for σN and y,2/4π.
Photoproduction of and w vector mesons
The data on these two mesons are much poorer than on p. There are in existence, however, several experiments in which they were photoproduced, both on protons and nuclei. The parameters (o, y2/4π, n) were already given earlier in these notes. We shall not go into any details of these experiments. Let us stress only two points:
(i) is narrow, hence it is much safer to treat it as a well-defined particle.
One can, in fact, suspect that in the "elastic" scattering after a vector meson is produced, some kind of superposition of the vector mesons propagates through the nuclear matter (C. Rogers and Colin Wilkin ). We shall not go into these problems now.
A summary of the picture of high energy photon interactions with hadronic targets (which include nuclei). 1. We considered the high energy limit of elastic scattering of photons by nucleons and nuclei. We discussed the evidence for the existence, in the physical high energy photon, of some strongly interacting components. The condition for the applicability of the high energy limit was that the characteristic length, l, defined by the incident energy and the lowest available hadronic invariant
be much larger than the target radius, R. At presently available energies this condition is satisfied for nucleon targets but it is not satisfied for nuclear targets.
2. The consequence of this fact is that one cannot use the high energy limit description in interpreting the present experiments of photon interactions with nuclear targets (nucleon targets are OK). We derived the formulae corrected for a non-negligible longitudinal momentum transfer for the case of vector meson production (the formula for elastic photon scattering was also given without derivation).
3. The Vector Meson Dominance model was briefly discussed and some recent experiments which seem to show its incompleteness described.
4. Photoproduction of vector mesons from deuterium, and from light, medium, and heavy nuclei was also analyzed.
5. Diffractive Production of Hadrons in Hadron-Nucleus Collisions
The Standard Analysis
We have already discussed some general features of diffractive production processes by hadrons (in the limit of very high energy). Let us now look into a few details of such processes with special emphasis on coherent production processes on nuclei. As in the case of vector meson photoproduction, in the existing experiments the incident energy of hadrons was too low to neglect the longitudinal momentum transfer and the high energy limit description is, at these energies, not applicable. However, with our experience in photoproduction of vector mesons on nuclear targets we can easily remedy this situation. So, the only change we should introduce into the formulae given for vector
meson photoproduction is an attenuation of the incident beam of hadrons (the incident photons were not attenuated due to the weakness of electromagnetic interactions). Let us make this extension explicitly and obtain the so-called "one step" model for diffractive production of hadrons in hadronnucleus collisions. In this "one step" process, the production takes place on a nucleon inside the nucleus.
(a) coherent diffractive photoproduction of p mesons:
M (v→p) =før (0) A f. d
(b) coherent diffractive production of hadrons by hadrons:
M (1→2) = f (0) A ƒ d d2b dz exp (1-iα1)
hence the oscillating factor under the integral sign, ei-1 as the energy increases. The only other energy dependence in our expression is possibly in f21. When a diffractive production mechanism is effective, this amplitude is experimentally observed to produce a nucleon cross section which is approximately energy independent. Hence one should expect an increase of the do21/dA2 cross section (obtained from the 3 (1-2) amplitude given above) as the incident energy increases and e→1. (Note the difference with vector meson photoproduction: there, due to an intricate interference with ViovNvNA J dz'p, it is hard to predict what the limit A-0 will produce.) The existing experimental data on 3 production by pions seem to support this conclusion (H. Leśniak and L. Leśniak ).
In computing the curve shown above, they assumed σ1=σ2=25 mb and a realistic density distributions for the nuclear targets. In addition, the cross section was weighted with an invariant mass distribution W (m) obtained from coherent production of 3π on a nucleon.
There are many uncertainties in this calculation but the result seems to indicate that one gets to the limit A=0 rather slowly and with presently available experimental data one has to take All into account. The point of taking the total 3-nucleon cross section to be 25 mb, equal to the one cross section, needs some explaining and that will be done below. Here, let us say only a few words about the high energy limit (A||=0) and connect it with our earlier discussion. When A||=0 we can perform the integration over z (o' = σ (1— ia)):
[** dz p (b, z) exp (− 1⁄2201 ́A [*_ dz'p (b, 2')) exp (−1⁄4‰o2'A [” d
Hence, in the high energy (A||=0) limit the expression for M (1→2) is
Note that this is precisely the same expression that one would get from our model of the high energy diffractive process which assumes a weak transition between the initial (one pion) and the final (3 states). Let us repeat the arguments again.