246
8. Elementary Processes in Scalar and Spinor Electrodynamics
As we have seen in (8.100), the current conservation condition is equivalent
to the condition
j
μ
q μ <π
+ (p
′ )| ˆ em,π (0)|π
+ (p)> = 0
(8.148)
on the pion electromagnetic vertex.
In the case of the point-like s
+ this is clearly satisfied since
′
q · (p + p) = 0
(8.149)
with the aid of (8.142). In the general case we obtain the condition
′
q μ [F (q
2 )(p + p)
μ + G(q
2 )q
μ ] = 0.
(8.150)
2
The first term vanishes as before, but q / = 0 in general, and we therefore
conclude that current conservation implies that
G(q
2 ) = 0.
(8.151)
In other words, all the virtual strong interaction effects at the π
+ π
+ γ vertex are described by one scalar function of the virtual photon’s squared 4momentum:
′
′
e(p + p)
μ
eF (q
2 )(p + p)
μ .
→
(8.152)
‘point pion’
‘real pion’
F (q
2 ) is the electromagnetic form factor of the pion, which generalizes the
static form factor F (q
2 ) of section 8.4.1. The pion electromagnetic vertex is
then
μ
j (p, p
′ ) = eF (q
2 )(p + p
′ )
μ .
(8.153)
π+
The electric charge is defined to be the coupling at zero momentum transfer,
so the form factor is normalized by the condition (cf (8.132))
F (0) = 1.
(8.154)
To lowest order in α, the invariant amplitude for e
− π
+
→ e
− π
+ is therefore
μ
μ
given by replacing j s + (p, p
′ ) in (8.97) or (8.109) by j (p, p
′ ):
π +
(
)
′
−ig μν
iM e − π + = −ie(p + p
′ )
μ F ((p − p)
2 )
[+ieu ¯(k
′ , s
′ )γ ν u(k, s)].
′
(p − p) 2
(8.155)
It is clear that the effect of the pion structure is simply to multiply the ‘no2
structure’ cross section (8.122) by the square of the form factor, F (q =
′
(p − p)
2 ).
− π
+
′
For e
− π
+
→ e
in the CM frame we may take p = (E, p) and p =
2
(E, p
′ ) with |p| = |p
′
| and E = (m π + p
2 )
1/2 . Then
′
q
2 = (p − p)
2 = −4p
2 sin
2 θ/2
(8.156)
′
as in section 8.1, where θ is now the CM scattering angle between p and p .
8. Elementary Processes in Scalar and Spinor Electrodynamics
As we have seen in (8.100), the current conservation condition is equivalent
to the condition
j
μ
q μ <π
+ (p
′ )| ˆ em,π (0)|π
+ (p)> = 0
(8.148)
on the pion electromagnetic vertex.
In the case of the point-like s
+ this is clearly satisfied since
′
q · (p + p) = 0
(8.149)
with the aid of (8.142). In the general case we obtain the condition
′
q μ [F (q
2 )(p + p)
μ + G(q
2 )q
μ ] = 0.
(8.150)
2
The first term vanishes as before, but q / = 0 in general, and we therefore
conclude that current conservation implies that
G(q
2 ) = 0.
(8.151)
In other words, all the virtual strong interaction effects at the π
+ π
+ γ vertex are described by one scalar function of the virtual photon’s squared 4momentum:
′
′
e(p + p)
μ
eF (q
2 )(p + p)
μ .
→
(8.152)
‘point pion’
‘real pion’
F (q
2 ) is the electromagnetic form factor of the pion, which generalizes the
static form factor F (q
2 ) of section 8.4.1. The pion electromagnetic vertex is
then
μ
j (p, p
′ ) = eF (q
2 )(p + p
′ )
μ .
(8.153)
π+
The electric charge is defined to be the coupling at zero momentum transfer,
so the form factor is normalized by the condition (cf (8.132))
F (0) = 1.
(8.154)
To lowest order in α, the invariant amplitude for e
− π
+
→ e
− π
+ is therefore
μ
μ
given by replacing j s + (p, p
′ ) in (8.97) or (8.109) by j (p, p
′ ):
π +
(
)
′
−ig μν
iM e − π + = −ie(p + p
′ )
μ F ((p − p)
2 )
[+ieu ¯(k
′ , s
′ )γ ν u(k, s)].
′
(p − p) 2
(8.155)
It is clear that the effect of the pion structure is simply to multiply the ‘no2
structure’ cross section (8.122) by the square of the form factor, F (q =
′
(p − p)
2 ).
− π
+
′
For e
− π
+
→ e
in the CM frame we may take p = (E, p) and p =
2
(E, p
′ ) with |p| = |p
′
| and E = (m π + p
2 )
1/2 . Then
′
q
2 = (p − p)
2 = −4p
2 sin
2 θ/2
(8.156)
′
as in section 8.1, where θ is now the CM scattering angle between p and p .