167
6.3. Applications to the ‘ABC’ theory
Now let us figure out what are all the surviving terms in the vev in (6.74).
′
As far as contractions involving ˆ
a A (p ) are concerned, we have only three
A
non-zero possibilities:
′
†
′
′
<0|a ˆ A (p A )ˆ a (p A )|0>
<0|a ˆ A (p A )φ ˆ A (x 1 )|0>
<0|a ˆ A (p A )φ ˆ A (x 2 )|0>. (6.84)
A
†
′
†
There are similar possibilities for ˆ
a A (p A ), ˆ
a B (p B ) and ˆ
a B (p B ). The upshot is
that we have only the following pairings to consider:
′
†
′
†
<0|a ˆ A (p A )ˆ a (p A )|0><0|a ˆ B (p B )ˆ a (p B )|0>
A
B
× <0|T (φ ˆ A (x 1 )φ ˆ A (x 2 ))|0><0|T (φ ˆ B (x 1 )φ ˆ B (x 2 ))|0><0|T (φ ˆ C (x 1 )φ ˆ C (x 2 ))|0>;
(6.85)
′
†
′
<0|a ˆ A (p A )ˆ a (p A )|0><0|a ˆ B (p B )φ ˆ B (x 1 )|0>
A
†
× <0|φ ˆ B (x 2 )ˆ a (p B )|0><0|T (φ ˆ C (x 1 )φ ˆ C (x 2 ))|0><0|T (φ ˆ A (x 1 )φ ˆ A (x 2 ))|0>
B
+ x 1 ↔ x 2 ;
(6.86)
′
†
′
<0|a ˆ B (p B )ˆ a (p B )|0><0|a ˆ A (p A )φ ˆ A (x 1 )|0>
B
†
× <0|φ ˆ A (x 2 )ˆ a (p A )|0><0|T (φ ˆ C (x 1 )φ ˆ C (x 2 ))|0><0|T (φ ˆ B (x 1 )φ ˆ B (x 2 ))|0>
A
+ x 1 ↔ x 2 ;
(6.87)
′
†
′
<0|a ˆ A (p A )φ ˆ A (x 1 )|0><0|φ ˆ A (x 2 )ˆ a (p A )|0><0|a ˆ B (p B )φ ˆ B (x 1 )|0>
A
†
× <0|φ ˆ B (x 2 )ˆ a (p B )|0><0|T (φ ˆ C (x 1 )φ ˆ C (x 2 ))|0>
B
+ x 1 ↔ x 2 ;
(6.88)
′
†
′
<0|a ˆ A (p A )φ ˆ A (x 1 )|0><0|φ ˆ A (x 2 )ˆ a (p A )|0><0|a ˆ B (p B )φ ˆ B (x 2 )|0>
A
†
× <0|φ ˆ B (x 1 )ˆ a (p B )|0><0|T (φ ˆ C (x 1 )φ ˆ C (x 2 ))|0>
B
+ x 1 ↔ x 2 .
(6.89)
′
†
We already know that quantities like <0|a ˆ(p )ˆ a (p A )|0> yield something
A A
′
proportional to δ
3 (p A − p ) and correspond to the initial A-particle going
A
‘straight through’. The other factors in (6.85) which are new are quantities
′
like <0|a ˆ A (p )φ ˆ A (x 1 )|0>, which has the value (problem 6.4)
A
1
'
′
ip A ·x1
<0|a ˆ A (p A )φ ˆ A (x 1 )|0> = √
e
(6.90)
2E ′
A
which is proportional (depending on the adopted normalization) to the wave′
function for an outgoing A-particle with 4-momentum p A .
We are now in a position to give a diagrammatic interpretation of all
of (6.85)–(6.89). In these diagrams, we shall not (as we did in figure 6.2)
draw two separately time-ordered pieces for each propagator. We shall not
indicate the time-ordering at all and we shall understand that both timeorderings are always included in each propagator line. Term (6.85) then has
the structure shown in figure 6.3(a); term (6.86) that shown in figure 6.3(b);
term (6.87) that in figure 6.3(c); term (6.88) that in figure 6.3(d ); and term
6.3. Applications to the ‘ABC’ theory
Now let us figure out what are all the surviving terms in the vev in (6.74).
′
As far as contractions involving ˆ
a A (p ) are concerned, we have only three
A
non-zero possibilities:
′
†
′
′
<0|a ˆ A (p A )ˆ a (p A )|0>
<0|a ˆ A (p A )φ ˆ A (x 1 )|0>
<0|a ˆ A (p A )φ ˆ A (x 2 )|0>. (6.84)
A
†
′
†
There are similar possibilities for ˆ
a A (p A ), ˆ
a B (p B ) and ˆ
a B (p B ). The upshot is
that we have only the following pairings to consider:
′
†
′
†
<0|a ˆ A (p A )ˆ a (p A )|0><0|a ˆ B (p B )ˆ a (p B )|0>
A
B
× <0|T (φ ˆ A (x 1 )φ ˆ A (x 2 ))|0><0|T (φ ˆ B (x 1 )φ ˆ B (x 2 ))|0><0|T (φ ˆ C (x 1 )φ ˆ C (x 2 ))|0>;
(6.85)
′
†
′
<0|a ˆ A (p A )ˆ a (p A )|0><0|a ˆ B (p B )φ ˆ B (x 1 )|0>
A
†
× <0|φ ˆ B (x 2 )ˆ a (p B )|0><0|T (φ ˆ C (x 1 )φ ˆ C (x 2 ))|0><0|T (φ ˆ A (x 1 )φ ˆ A (x 2 ))|0>
B
+ x 1 ↔ x 2 ;
(6.86)
′
†
′
<0|a ˆ B (p B )ˆ a (p B )|0><0|a ˆ A (p A )φ ˆ A (x 1 )|0>
B
†
× <0|φ ˆ A (x 2 )ˆ a (p A )|0><0|T (φ ˆ C (x 1 )φ ˆ C (x 2 ))|0><0|T (φ ˆ B (x 1 )φ ˆ B (x 2 ))|0>
A
+ x 1 ↔ x 2 ;
(6.87)
′
†
′
<0|a ˆ A (p A )φ ˆ A (x 1 )|0><0|φ ˆ A (x 2 )ˆ a (p A )|0><0|a ˆ B (p B )φ ˆ B (x 1 )|0>
A
†
× <0|φ ˆ B (x 2 )ˆ a (p B )|0><0|T (φ ˆ C (x 1 )φ ˆ C (x 2 ))|0>
B
+ x 1 ↔ x 2 ;
(6.88)
′
†
′
<0|a ˆ A (p A )φ ˆ A (x 1 )|0><0|φ ˆ A (x 2 )ˆ a (p A )|0><0|a ˆ B (p B )φ ˆ B (x 2 )|0>
A
†
× <0|φ ˆ B (x 1 )ˆ a (p B )|0><0|T (φ ˆ C (x 1 )φ ˆ C (x 2 ))|0>
B
+ x 1 ↔ x 2 .
(6.89)
′
†
We already know that quantities like <0|a ˆ(p )ˆ a (p A )|0> yield something
A A
′
proportional to δ
3 (p A − p ) and correspond to the initial A-particle going
A
‘straight through’. The other factors in (6.85) which are new are quantities
′
like <0|a ˆ A (p )φ ˆ A (x 1 )|0>, which has the value (problem 6.4)
A
1
'
′
ip A ·x1
<0|a ˆ A (p A )φ ˆ A (x 1 )|0> = √
e
(6.90)
2E ′
A
which is proportional (depending on the adopted normalization) to the wave′
function for an outgoing A-particle with 4-momentum p A .
We are now in a position to give a diagrammatic interpretation of all
of (6.85)–(6.89). In these diagrams, we shall not (as we did in figure 6.2)
draw two separately time-ordered pieces for each propagator. We shall not
indicate the time-ordering at all and we shall understand that both timeorderings are always included in each propagator line. Term (6.85) then has
the structure shown in figure 6.3(a); term (6.86) that shown in figure 6.3(b);
term (6.87) that in figure 6.3(c); term (6.88) that in figure 6.3(d ); and term