260
8. Elementary Processes in Scalar and Spinor Electrodynamics
with the normalization
F 1 (0) = 1
(8.209)
F 2 (0) = 1
(8.210)
and the magnetic moment of the proton is not one (nuclear) magneton, as for
an electron or muon (neglecting higher-order corrections), but rather μ p =
1 + κ with κ = 1.79. Problem 8.20 shows that the ¯
uγ
μ u piece in (8.208) can
uiσ
μν
be rewritten in terms of ¯
u(p + p
′ )
μ u/2M and ¯
q ν u/2M . The first of these
is analogous to the interaction of a charged spin-0 particle. As regards the
second, we note that σ
μν is just
σ
μν
1
= i[γ
μ , γ
ν ]
(8.211)
2
which reduces to the Pauli spin matrices for the space-like components
(
)
σ
ij
σ
k
0
=
(8.212)
0 σ
k
with our representation of γ-matrices (σ
ij is a 4 × 4 matrix, σ
k is 2 × 2, and i,
j and k are in cyclic order). The second term in this ‘Gordon decomposition’
of ¯
uγ
μ u thus corresponds to an interaction via the spin magnetic moment –
with, in fact, g = 2. Thus the addition of the κ term in (8.208) corresponds
to an ‘anomalous’ magnetic moment piece. In terms of F 1 and F 2 one can
show that
A = F 1
2 + τκ
2
F
2
(8.213)
2
B = 2τ (F 1 + κF 2 )
2
(8.214)
where
τ = −q
2 /4M
2 .
(8.215)
The point-like cross section (8.193) is recovered from (8.207) by setting F 1 = 1
and κ = 0 in (8.213) and (8.214).
The functions F 1 and F 2 are, in turn, usually expressed in terms of the electric and magnetic form factors G E and G M , defined by G E = F 1 −τκF 2 , G M =
F 1 + κF 2 . We then find A = (G
2
E + τG
2 )/(1 + τ ) and B = 2τG
2 . The cross
M
M
section formula (8.207), written in terms of G E and G M , is known as the
‘Rosenbluth’ cross section.
Experimental data indicate that the q
2 -dependences of G E and G M for
the proton, and of G M for the neutron, are all quite well represented by the
function F (q
2 ) of (8.136) with q
2 replaced by −q
2 and with a ∼ 0.84 GeV
−1 ,
2
at least for values of −q up to a few GeV
2 (see, for example, Perkins 1987,
section 6.5).
Before we leave elastic scattering it is helpful to look in some more detail
at the kinematics. It will be sufficient to consider the ‘point-like’ case, which
8. Elementary Processes in Scalar and Spinor Electrodynamics
with the normalization
F 1 (0) = 1
(8.209)
F 2 (0) = 1
(8.210)
and the magnetic moment of the proton is not one (nuclear) magneton, as for
an electron or muon (neglecting higher-order corrections), but rather μ p =
1 + κ with κ = 1.79. Problem 8.20 shows that the ¯
uγ
μ u piece in (8.208) can
uiσ
μν
be rewritten in terms of ¯
u(p + p
′ )
μ u/2M and ¯
q ν u/2M . The first of these
is analogous to the interaction of a charged spin-0 particle. As regards the
second, we note that σ
μν is just
σ
μν
1
= i[γ
μ , γ
ν ]
(8.211)
2
which reduces to the Pauli spin matrices for the space-like components
(
)
σ
ij
σ
k
0
=
(8.212)
0 σ
k
with our representation of γ-matrices (σ
ij is a 4 × 4 matrix, σ
k is 2 × 2, and i,
j and k are in cyclic order). The second term in this ‘Gordon decomposition’
of ¯
uγ
μ u thus corresponds to an interaction via the spin magnetic moment –
with, in fact, g = 2. Thus the addition of the κ term in (8.208) corresponds
to an ‘anomalous’ magnetic moment piece. In terms of F 1 and F 2 one can
show that
A = F 1
2 + τκ
2
F
2
(8.213)
2
B = 2τ (F 1 + κF 2 )
2
(8.214)
where
τ = −q
2 /4M
2 .
(8.215)
The point-like cross section (8.193) is recovered from (8.207) by setting F 1 = 1
and κ = 0 in (8.213) and (8.214).
The functions F 1 and F 2 are, in turn, usually expressed in terms of the electric and magnetic form factors G E and G M , defined by G E = F 1 −τκF 2 , G M =
F 1 + κF 2 . We then find A = (G
2
E + τG
2 )/(1 + τ ) and B = 2τG
2 . The cross
M
M
section formula (8.207), written in terms of G E and G M , is known as the
‘Rosenbluth’ cross section.
Experimental data indicate that the q
2 -dependences of G E and G M for
the proton, and of G M for the neutron, are all quite well represented by the
function F (q
2 ) of (8.136) with q
2 replaced by −q
2 and with a ∼ 0.84 GeV
−1 ,
2
at least for values of −q up to a few GeV
2 (see, for example, Perkins 1987,
section 6.5).
Before we leave elastic scattering it is helpful to look in some more detail
at the kinematics. It will be sufficient to consider the ‘point-like’ case, which