6.3. Applications to the ‘ABC’ theory
Equation (6.67) shows that, even if ˜
g
2 /16π is small (∼ 1/137 say) Γ can still
be surprisingly large if m C is, as in W
−
→ e
− + ¯
ν e for example.
6.3.2 A + B → A + B scattering: the amplitudes
We now consider the two-particle → two-particle process
A + B → A + B
(6.68)
′
in which the initial 4-momenta are p A , p B and the final 4-momenta are p A ,
′
′
′
p so that p A + p B = p A + p B . Our main task is to calculate the matrix
B
′
′
element
to lowest non-trivial order in g. The result will
B
be the derivation of our first ‘Feynman rules’ for amplitudes in perturbative
quantum field theory.
The first term in the S ˆ -operator expansion (6.42) is ‘1’, which does not
involve g at all. Nevertheless, it is a useful exercise to evaluate and understand
this contribution (which in the present case does not vanish), namely
′
′
†
†
′
′
<0|a ˆ A (p A )ˆ a B (p B )ˆ a (p A )ˆ a (p B )|0>(16E A E B E A E B )
1/2 .
(6.69)
A
B
We shall have to evaluate many such vacuum expectation values (vev) of products of ˆ
a
† ’s and ˆ
a’s. The general strategy is to commute the ˆ
a
† ’s to the left,
and the ˆ
a’s to the right, and then make use of the facts
†
<0|a ˆ = ˆ
a i |0> = 0
(6.70)
i
for any i = A, B, C. Thus, remembering that all ‘A’ operators commute with
all ‘B’ ones, the vev in (6.69) is equal to
′
†
′
†
′
<0|a ˆ A (p A )ˆ a (p A ){(2π)
3 δ
3 (p B − p B ) + ˆ
a (p B )ˆ a B (p B )}|0>
A
B
′
†
′
′
= <0|{(2π)
3 δ
3 (p A − p A ) + ˆ
a (p A )ˆ a A (p A )}(2π)
3 δ
3 (p B − p B )|0>
A
′
′
= (2π)
3 δ
3 (p A − p A )(2π)
3 δ
3 (p B − p B ).
(6.71)
′
′
The δ-functions enforce E A = E A and E B = E B so that (6.69) becomes
′
′
2E A (2π)
3 δ
3 (p A − p A )2E B (2π)
3 δ
3 (p B − p B ),
(6.72)
a result which just expresses the normalization of the states, and the fact
that, with no ‘g’ entering, the particles have not interacted at all, but have
′
′
continued on their separate ways, quite unperturbed (p A = p
= p ).
A , p B
B
This contribution can be represented diagrammatically as figure 6.1.
Next, consider the term of order g, which we used in C → A + B. This is
∫
′
′
−ig d
4 x
.
(6.73)
We have to remember, now, that all the φ ˆ i operators are in the interaction