257
8.8. Electron–proton elastic scattering and nucleon form factors
Comment (c)
+
The crossed version of this process, namely e e
−
→ μ
+ μ
− , is a very important
monitoring reaction for electron–positron colliding beam machines. It is also
+
basic to a discussion of the predictions of the quark parton model for e e
−
→
hadrons, which will be discussed in section 9.5. An instructive calculation
similar to this one leads to the result (see problem 8.18)
dσ α
2
=
(1 + cos
2 θ)
(8.194)
dΩ 4q 2
+
where all variables are defined in the e e
− CM frame, q
2 is now the square of
the CM energy, and the electron and muon masses have been neglected. The
total cross section, in the one-photon exchange approximation, is then
σ = 4πα
2 /3q
2 = 86.8 nb/q
2 (GeV
2 ),
(8.195)
where we have made use of equation (B.18) of appendix B.
The energy dependence of this cross section (∝ 1/q
2 ) is important, and
can be understood by a simple dimensional argument. A cross section has dimensions of a squared length, or in natural units (appendix B) inverse squared
mass or energy. Here both colliding particles are taken to be pointlike, with
no form factors involving a length parameter, and the mediating quantum is
massless. At energies much larger than the lepton masses, the only available
dimensional quantity is the CM energy. It follows that the cross section must
be inversely proportional to the square of the CM energy, in this ‘pointlike,
high energy’ limit. By the same token, deviations from this behaviour would
be evidence for non-pointlike leptonic structure.
8.8 Electron–proton elastic scattering and nucleon form
factors
In the one-photon exchange approximation, the Feynman diagram for elastic
electron–proton scattering may be drawn as in figure 8.18, where the ‘blob’ at
the ppγ vertex signifies the expected modification of the point coupling due to
strong interactions. The structure of the proton vertex can be analysed using
symmetry principles in the same way as for the pion vertex. The presence
of Dirac spinors and γ-matrices makes this a somewhat involved procedure:
problem 8.20 is an example of the type of complication that arises. Full details of such an analysis can be found in Bernstein (1968), for example. Here,
however, we shall proceed in a different way, in order to generalize more easily
to inelastic scattering in the following chapter. We focus directly on the ‘proton tensor’ B
μν , which is the product of two proton current matrix elements,
8.8. Electron–proton elastic scattering and nucleon form factors
Comment (c)
+
The crossed version of this process, namely e e
−
→ μ
+ μ
− , is a very important
monitoring reaction for electron–positron colliding beam machines. It is also
+
basic to a discussion of the predictions of the quark parton model for e e
−
→
hadrons, which will be discussed in section 9.5. An instructive calculation
similar to this one leads to the result (see problem 8.18)
dσ α
2
=
(1 + cos
2 θ)
(8.194)
dΩ 4q 2
+
where all variables are defined in the e e
− CM frame, q
2 is now the square of
the CM energy, and the electron and muon masses have been neglected. The
total cross section, in the one-photon exchange approximation, is then
σ = 4πα
2 /3q
2 = 86.8 nb/q
2 (GeV
2 ),
(8.195)
where we have made use of equation (B.18) of appendix B.
The energy dependence of this cross section (∝ 1/q
2 ) is important, and
can be understood by a simple dimensional argument. A cross section has dimensions of a squared length, or in natural units (appendix B) inverse squared
mass or energy. Here both colliding particles are taken to be pointlike, with
no form factors involving a length parameter, and the mediating quantum is
massless. At energies much larger than the lepton masses, the only available
dimensional quantity is the CM energy. It follows that the cross section must
be inversely proportional to the square of the CM energy, in this ‘pointlike,
high energy’ limit. By the same token, deviations from this behaviour would
be evidence for non-pointlike leptonic structure.
8.8 Electron–proton elastic scattering and nucleon form
factors
In the one-photon exchange approximation, the Feynman diagram for elastic
electron–proton scattering may be drawn as in figure 8.18, where the ‘blob’ at
the ppγ vertex signifies the expected modification of the point coupling due to
strong interactions. The structure of the proton vertex can be analysed using
symmetry principles in the same way as for the pion vertex. The presence
of Dirac spinors and γ-matrices makes this a somewhat involved procedure:
problem 8.20 is an example of the type of complication that arises. Full details of such an analysis can be found in Bernstein (1968), for example. Here,
however, we shall proceed in a different way, in order to generalize more easily
to inelastic scattering in the following chapter. We focus directly on the ‘proton tensor’ B
μν , which is the product of two proton current matrix elements,