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It is very suggestive then to accept that photons have in them strongly interacting components. Again, we can use a perturbation expansion, where the interaction Hamiltonian gives the photonhadron interaction. We can write the physical photon state as follows:
where 'n' labels all possible hadronic states which can be coupled to a photon. We can also write, as before, the expansion of a hadronic state n
Note that here the state | 0, n) is "almost physical." More precisely: as far as strong interactions go, it is physical. The situation is very similar to that in the K-K, regeneration problem: we are expanding our physical states into states which are "physical in their strong interactions."
Since we know strongly interacting vector mesons with the same quantum numbers as the photon, the simplest possible assumption one can make is to accept that the | 0, n) states are dominated by a vector meson (or mesons). In fact, there is a well-known model of photo-hadronic interactions (Vector Meson Dominance model) which assumes just that.
Some important consequences can be inferred from the above expansions even without specifying the nature of the hadronic components. Let us call m, the invariant mass of the hadronic component n). Then
The time during which the hadronic vacuum fluctuation lives is At≈ | (E,-En) |-1 and this also gives the distance it travels (l=cAt). Hence, when l≈2w/m2>>R (R= nuclear radius) we shall have shadowing fully developed. We can see, therefore, that full shadowing occurs at high enough energies. At low photon energies | E-E, | is large and the corresponding fluctuation cannot interact with the whole nucleus.
How can one test the VMD hypothesis? Let us denote the vector mesons by the letter V. Then (note that Z= 1 in our approximation)
Then the photoproduction of V (on a hadronic target) has the forward amplitude
Similarly, the elastic scattering of a high energy photon (Compton scattering) from a hadronic target has the forward amplitude
Note that restricting the high energy photon-nucleus interaction to only vector meson interactions (VMD model) is a very drastic step. There are other possible interactions of the same order of magnitude (~e) whose role one should discuss: for instance all kinds of such Compton-like processes which do not belong to the VMD model, where the photon is absorbed by the target (or part of it) and re-emitted. E.g., when the total Compton amplitude is a sum of individual (photon-nucleon) Compton amplitudes
where q is the three momentum transfer and r; are position vectors of the nucleons in the target nucleus, it gives the following contribution to the total photon-nucleus cross section (which is proportional to A):
where σr(p) and σr(yn) are the total photon-proton and photon-neutron cross sections, respectively. When the VMD model does reproduce the correct total photon-nucleon cross section the contribution given above is just a single scattering contribution of the multiple scattering of vector mesons and is properly taken care of by the VMD model description of photon-nucleus interactions. If, however, the VMD model fails to reproduce the total photon-nucleon cross section, the balance between the single and multiple scattering contributions given by the VMD model is disturbed and an additional contribution to the total cross section appears which is not screened (~A) (compare eq (4.10)). In fact, it is very likely that something like that may indeed take place. For instance in ref.  the authors working with the parton model find a not screened (~A) contribution amounting to ~20 percent of σr(yPb).
where ny is the ratio of the real to the imaginary part of M (y→V). So, finally, the relation which can be tested is as follows:
By measuring independently the forward vector meson production cross sections and the total photo cross section one can check the internal consistency of the VMD. There are some other tests but the above equation was used in the analysis published recently by D. O. Caldwell et al. . Another possible test is, e.g., the equation
Assuming M (V→V) purely imaginary (in the high energy limit) this becomes
First, let us discuss the formula (4.9) which is cruder and contains some nondirectly measurable parameters. The table below gives an idea of accuracy with which it is possible to test it (see K. Gottfried report in [S5]). At 7 GeV incident energy the left hand side is σr (y, nucleon) = 118±4 μb.
The sum 99±21 μb checks reasonably well with the value given above, 118±4 μb.
All these parameters are obtained from a host of various experiments: yv2/4 from ete-storage rings where the following process is observed:
συν and v (not shown above) from the A dependence of V photoproduction, ŋy was also extracted from Compton scattering and leptonic decays. From the table above one can also see that the p meson contribution is more important than the contribution of the other two vector mesons.
Let us go back to formula (4.8). It had been checked both for nucleon and nuclear targets (nuclear targets: Pb, Cu, C) .
Nucleon targets: The authors assumed the w and contributions to or (y, nucleon) to be 20±2 μb (remember: this is just a small contribution). Then, by measuring σr (y, nucleon) one can determine (do/d) (yp→pp) |。 for which one gets much too high a value. One may get agreement if one reduces Y2/4 from 0.62 to 0.37! But y,2/4π is well known from colliding beam experiments. So, it is unlikely that one should reduce it by a factor of almost 2! (Unless there is a strong dependence on the invariant mass of a virtual photon).
One can also get the correct answer when one accepts that there is a contribution from a Compton-like process (not given by the VMD model!) which does not show any screening and is proportional to A (see footnote 4).
Nuclear target: One could test (4.8) directly against experimental data for nuclear targets if the energy were high enough. Remember, however, that in order to have VMD active in its full strength one has to have l≈2w/my2>>R, where R is the nuclear radius, which condition is not well satisfied at existing photon energies: e.g., l, = 2.1 fm at 6 GeV. Hence one has to use a more sophisticated description which in fact allows for the hadronic fluctuation to fold back into a photon inside of the nucleus:
We shall come back to this point later. For the moment let us simply state the results of such a "sophisticated" description (the low energy version, in which A||#0, is given below in eq (4.10)) which was presented in the paper by Caldwell et al. . By investigating the A dependence of the OT (YA) cross section they found less screening than demanded by VMD but more than demanded by purely electromagnetic interactions. Hence the discrepancy is related to a partial breakdown of VMD, rather than to smaller y, (which would not change the screening). Indeed it seems that a Compton-like (non-VMD) contribution ~A which would give ~20 percent of or could make the theory and experiment agree. In fact one does not need a purely electromagnetic interaction to obtain a contribution ~A. It could come from a heavy vector boson whose
Conclusion: The VMD model is only approximately correct. There is no commonly accepted explanation of the discrepancies described above. Perhaps the Compton-like contributions to M (→) (as suggested by Brodsky, Close and Gunion ) should be added to VMD to explain the recent photoabsorption data.
4.2. Photoproduction of Vector Bosons
The breakdown of VMD which one sees from the results of Caldwell et al.  does not eliminate the possibility that the previously worked out relations between M (y-p) and M (p→p) are good approximations. It is in fact commonly accepted that they form a sound basis for analysis of production of vector mesons on various nuclear targets and, since it is comparatively well documented experimentally, it is instructive to outline it here.
The production amplitude of a vector boson is (in the limit w→→ ∞)
M (Y→V) =
(0, kv | tv | 0, kv' = k1)
where (tv) is just the elastic scattering amplitude of the vector meson V from the target nucleus (which we may take over from our previous discussion of hadronic elastic scattering from nuclei). So, in the high energy limit
In the high energy limit (w→∞) one can also describe this process as follows: the photon penetrates the nucleus up to a certain point where it converts into a V meson which scatters elastically from the other nucleons and then leaves the nucleus. Graphically