244
8. Elementary Processes in Scalar and Spinor Electrodynamics
Condition (8.132) simply means that the total charge is Ze. The left-hand side
of (8.131) can be transformed by two (three-dimensional) partial integrations
to give ∫
∫
2
(∇
2 e
−iq·x )A
0 (x) d
3
x = −q
e
−iq·x A
0 (x) d
3
x.
(8.133)
Using this result in (8.131), we find
F (q)
A ˜0 (q) =
Ze.
(8.134)
q 2
Thus referring to equation (8.44) for example, the net result of the non-point2
like charge distribution is to multiply the ‘point-like’ amplitude Ze
2 /q by
the form factor F (q) which in this simple static case has the interpretation of
the Fourier transform of the charge distribution. So, for this (infinitely heavy
π
+ case), the ‘blob’ in figure 8.8 would be represented by F (q).
To gain some idea of what F (q
2 ) might look like, consider a simple exponential shape for ρ(x) :
1
−|x|/a
ρ(x) =
e
(8.135)
(8πa 3 )
which has been normalized according to (8.132). Then F (q
2 ) is (problem 8.13)
F (q
2 ) =
1
.
(8.136)
(q 2 a 2 + 1) 2
We see that F (q
2 ) decreases smoothly away from unity at q
2 = 0. The characteristic scale of the fall-off in |q| is ∼ a
−1 from (8.136), which, as expected
from Fourier transform theory, is the reciprocal of the spatial fall-off, which is
approximately a from (8.135); the root mean square radius of the distribution
√
2
(8.135) is actually 12a (problem 8.13). Since q
2 = 4k
2 sin
2 θ/2, a larger q
means a larger θ: hence, in scattering from an extended charge distribution,
the cross section at larger angles will drop below the point-like value. This is,
of course, how Rutherford deduced that the nucleus had a spatial extension.
We now seek a Lorentz-invariant generalization of this static form factor.
In the absence of a fundamental understanding of the π
+ structure coming
from QCD, we shall rely on Lorentz invariance and electromagnetic current
conservation (one aspect of gauge invariance) to restrict the general form of
the ππγ vertex shown in figure 8.8. The use of invariance arguments to place
restrictions on the form of amplitudes is an extremely general and important
tool, in the absence of a complete theory.
8.4.2 Lorentz invariance
First, consider Lorentz invariance. We seek to generalize the point-like ssγ
vertex (cf (8.98) and comment (1) after (8.99))
μ
+
+
j (p, p
′ ) =
| ˆ j
μ
(0)|s , p> = e(p + p
′ )
μ
(8.137)
s +
em,s
8. Elementary Processes in Scalar and Spinor Electrodynamics
Condition (8.132) simply means that the total charge is Ze. The left-hand side
of (8.131) can be transformed by two (three-dimensional) partial integrations
to give ∫
∫
2
(∇
2 e
−iq·x )A
0 (x) d
3
x = −q
e
−iq·x A
0 (x) d
3
x.
(8.133)
Using this result in (8.131), we find
F (q)
A ˜0 (q) =
Ze.
(8.134)
q 2
Thus referring to equation (8.44) for example, the net result of the non-point2
like charge distribution is to multiply the ‘point-like’ amplitude Ze
2 /q by
the form factor F (q) which in this simple static case has the interpretation of
the Fourier transform of the charge distribution. So, for this (infinitely heavy
π
+ case), the ‘blob’ in figure 8.8 would be represented by F (q).
To gain some idea of what F (q
2 ) might look like, consider a simple exponential shape for ρ(x) :
1
−|x|/a
ρ(x) =
e
(8.135)
(8πa 3 )
which has been normalized according to (8.132). Then F (q
2 ) is (problem 8.13)
F (q
2 ) =
1
.
(8.136)
(q 2 a 2 + 1) 2
We see that F (q
2 ) decreases smoothly away from unity at q
2 = 0. The characteristic scale of the fall-off in |q| is ∼ a
−1 from (8.136), which, as expected
from Fourier transform theory, is the reciprocal of the spatial fall-off, which is
approximately a from (8.135); the root mean square radius of the distribution
√
2
(8.135) is actually 12a (problem 8.13). Since q
2 = 4k
2 sin
2 θ/2, a larger q
means a larger θ: hence, in scattering from an extended charge distribution,
the cross section at larger angles will drop below the point-like value. This is,
of course, how Rutherford deduced that the nucleus had a spatial extension.
We now seek a Lorentz-invariant generalization of this static form factor.
In the absence of a fundamental understanding of the π
+ structure coming
from QCD, we shall rely on Lorentz invariance and electromagnetic current
conservation (one aspect of gauge invariance) to restrict the general form of
the ππγ vertex shown in figure 8.8. The use of invariance arguments to place
restrictions on the form of amplitudes is an extremely general and important
tool, in the absence of a complete theory.
8.4.2 Lorentz invariance
First, consider Lorentz invariance. We seek to generalize the point-like ssγ
vertex (cf (8.98) and comment (1) after (8.99))
μ
+
+
j (p, p
′ ) =
| ˆ j
μ
(0)|s , p> = e(p + p
′ )
μ
(8.137)
s +
em,s