Equivalent photon approximation. Consider the process in which electrons of very high energy scatter from a target. In leading order in α, the electron is connected to the target by one photon propagator. If the initial and final energies of the electron are E and E′, the photon will carry momentum q such that q2≈−2EE′(1−cosθ). In the limit of forward scattering, whatever the energy loss, the photon momentum approaches q2=0; thus the reaction is highly peaked in the forward direction. It is tempting to guess that, in this limit, the virtual photon becomes a real photon. Let us investigate in what sense that is true.
(a) The matrix element for the scattering process can be written as
M=(−ie)uˉ(p′)γμu(p)q2−igμνMν(q),
where Mν represents the (in general, complicated) coupling of the virtual photon to the target. Let us analyze the structure of the piece uˉ(p′)γμu(p). Let q=(q0,q), and define qˉ=(q0,−q). We can expand the spinor product as:
uˉ(p′)γμu(p)=A⋅qμ+B⋅qˉμ+C⋅ϵ1μ+D⋅ϵ2μ,
where A, B, C, D are functions of the scattering angle and energy loss and ϵi are two unit vectors transverse to q. By dotting this expression with qμ, show that the coefficient B is at most of order θ2. This will mean that we can ignore it in the rest of the analysis. The coefficient A is large, but it is also irrelevant, since, by the Ward identity, qμMμ=0.
(b) Working in the frame where p=(E,0,0,E), compute explicitly
uˉ(p′)γ⋅ϵiu(p)
using massless electrons, u(p) and u(p′) spinors of definite helicity, and ϵ1, ϵ2 unit vectors parallel and perpendicular to the plane of scattering. We need this quantity only for scattering near the forward direction, and we need only the term of order θ. Note, however, that for ϵ in the plane of scattering, the small 3^ component of ϵ also gives a term of order θ which must be taken into account.
(c) Now write the expression for the electron scattering cross section, in terms of ∣Mμ∣2 and the integral over phase space on the target side. This expression must be integrated over the final electron momentum p′. The integral over p′3 is an integral over the energy loss of the electron. Show that the integral over p⊥′ diverges logarithmically as p⊥′ or θ→0.
(d) The divergence as θ→0 appears because we have ignored the electron mass in too many places. Show that reintroducing the electron mass in the expression for q2,
q2=−2(EE′−pp′cosθ)+2m2,
cuts off the divergence and yields a factor of log(s/m2) in its place.
(e) Assembling all the factors, and assuming that the target cross sections are independent of the photon polarization, show that the largest part of the electron-target scattering cross section is given by considering the electron to be the source of a beam of real photons with energy distribution (x=Eγ/E):
Nγ(x)dx=xdx2πα[1+(1−x)2]log(m2s).
This is the Weizsacker-Williams *equivalent photon approximation*. This phenomenon allows us, for example, to study photon-photon scattering using e+e− collisions. Notice that the distribution we have found here is the same one that appeared in Problem 5.5 when we considered soft photon emission before electron scattering. It should be clear that a parallel general derivation can be constructed for that case.