Subatomic Physics

322 The Electromagnetic Interaction
If the photon transforms to the hadron state inside the nucleus, the hadron will
interact near the position of production. Since the production can occur anywhere,
the contribution to the total cross section is proportional to A, just like that of bare
photons. On the other hand, virtual hadrons created before striking the nucleus
interact with nucleons in the nuclear surface layer because of their short mean free
path. The corresponding contribution to the total cross section consequently is
proportional to the nuclear area, or to A 2/3 . At a given photon energy, the total
cross section is the sum of the three contributions, and it should be of the form
σ(γA)=aA + bA 2/3 . (10.99)
As stated above, the second term is due to photons that transform into hadrons
before striking the nucleus. Such hadrons have a chance to interact if they are produced
within a distance L, which at high photon energies is, according to Eq. (10.95,
bottom), large compared to nuclear diameters and proportional to Eγ. Other things
being equal, the coefficient b should thus be proportional to Eγ, and the surface
term should become dominant at energies large compared to mhc 2 . The behavior
of the cross section as expressed by Eq. (10.99) and Fig. 10.27 therefore can be
understood in terms of virtual hadrons.
The expression for the hadron cloud of the photon, ch|h〉, canbewritteninan
informative form by using perturbation theory. We assume the states of the various
hadrons and of the photon, in the absence of the electromagnetic interaction, to be
given by the Schrödinger equations
Hh|γ0〉 =0, Hh|n〉 = En|n〉. (10.100)
Hh is the strong Hamiltonian, |γ0〉 the state function of the bare photon, and
|n〉 represents a hadronic state. If the electromagnetic interaction is switched on,
hadronic states are superimposed onto the bare photon state:
|γ〉 = c0|γ0〉 + �
cn|n〉, |c0| 2 + �
|cn| 2 =1. (10.101)
n
Since Hem is weaker than Hh, the expansion coefficients cn are small and c0 ≈ 1.
The state of the physical photon is a solution of the complete Schrödinger equation,
n
(Hh + Hem)|γ〉 = Eγ|γ〉. (10.102)
Inserting the expansion (10.101) into Eq. (10.102) gives, with Eq. (10.100) and with
〈n|γ0〉 =0,cn ≪ 1,
cn = 〈n|Hem|γ0〉
. (10.103)
Eγ − En
The energy difference between the photon energy Eγ and the hadron energy En is
given by Eq. (10.93); for large photon energies, the expansion coefficient becomes,
with Eq. (10.94, bottom),
cn = 〈n|Hem|γ0〉 2Eγ
m2 . (10.104)
hc4