10.1. The propagator correction in ABC theory
309
The corrected propagator (10.14) then becomes
i
(10.21)
[2]
dΠ
2
[2]
2
2
C
2
−
2
−
q
m C − Π (m
m
+ · · ·
C
ph,C ) − (q
ph,C )
|
|
|
|
dq 2
q 2
2
=m ph,C
i
=
[
]
. (10.22)
[2]
dΠ
2
C
2
(q
2
− m ph,C ) 1 −
+ O(q
2
− m ph,C )
2
|
|
|
|
dq 2
q 2 =m 2
ph,C
The expression (10.22) has indeed the expected form for a ‘physical C’ propa2
2
gator, having the simple behaviour ∼1/(q
2
− m
) for q
2
≈ m
. However,
ph,C
ph,C
the normalization of this (corrected) propagator is different from that of the
2
‘free’ one, i/(q
2
− m ), because of the extra factor
C
[
] −1
[2]
dΠ C
1 −
.
|
|
|
|
dq 2 2 2
ph,C
q =m
To the order at which we are working (O(g
2 )), it is consistent to replace this
expression by
[2]
dΠ C
1 +
.
|
|
|
|
dq 2
2
q 2 =m ph,C
Let us see how this factor may be understood.
Our O(g
2 ) corrected propagator is an approximation to the exact propagator which we may write as <Ω|T (φ ˆ C (x 1 )φ ˆ C (x 2 ))|Ω>, in coordinate space, where
|Ω> is the exact vacuum. The free propagator, however, is <0|T (φ ˆ C (x 1 )φ ˆ C (x 2 ))|0>
as calculated in section 6.3.2. Consider one term in the latter, θ(t 1 − t 2 )×
<0|φ ˆ C (x 1 )φ ˆ C (x 2 )|0>, and insert a complete set of free-particle states ‘1 =
∑ |n>
∑
θ(t 1 − t 2 )
<0|φ ˆ C (x 1 )|n>.
(10.23)
n
The only free particle state |n> having a non-zero matrix element of the free
−ik·x
field φ ˆ C to the vacuum is the 1 − C state, for which <0|φ ˆ C (x)|C, k> = e
as
we learned in chapters 5 and 6. Thus (10.23) becomes (cf equation (6.92))
∫
d
3
k
−iω k (t1−t2)+ik·(x1 −x2 )
θ(t 1 − t 2 )
e
(10.24)
(2π) 3 2ω k
which is exactly the first term of equation (6.92). Consider now carrying out a
similar manipulation for the corresponding term of the interacting propagator,
obtaining
∑
θ(t 1 − t 2 )
<Ω|φ ˆ C (x 1 )|n>
(10.25)
n
309
The corrected propagator (10.14) then becomes
i
(10.21)
[2]
dΠ
2
[2]
2
2
C
2
−
2
−
q
m C − Π (m
m
+ · · ·
C
ph,C ) − (q
ph,C )
|
|
|
|
dq 2
q 2
2
=m ph,C
i
=
[
]
. (10.22)
[2]
dΠ
2
C
2
(q
2
− m ph,C ) 1 −
+ O(q
2
− m ph,C )
2
|
|
|
|
dq 2
q 2 =m 2
ph,C
The expression (10.22) has indeed the expected form for a ‘physical C’ propa2
2
gator, having the simple behaviour ∼1/(q
2
− m
) for q
2
≈ m
. However,
ph,C
ph,C
the normalization of this (corrected) propagator is different from that of the
2
‘free’ one, i/(q
2
− m ), because of the extra factor
C
[
] −1
[2]
dΠ C
1 −
.
|
|
|
|
dq 2 2 2
ph,C
q =m
To the order at which we are working (O(g
2 )), it is consistent to replace this
expression by
[2]
dΠ C
1 +
.
|
|
|
|
dq 2
2
q 2 =m ph,C
Let us see how this factor may be understood.
Our O(g
2 ) corrected propagator is an approximation to the exact propagator which we may write as <Ω|T (φ ˆ C (x 1 )φ ˆ C (x 2 ))|Ω>, in coordinate space, where
|Ω> is the exact vacuum. The free propagator, however, is <0|T (φ ˆ C (x 1 )φ ˆ C (x 2 ))|0>
as calculated in section 6.3.2. Consider one term in the latter, θ(t 1 − t 2 )×
<0|φ ˆ C (x 1 )φ ˆ C (x 2 )|0>, and insert a complete set of free-particle states ‘1 =
∑ |n>
∑
θ(t 1 − t 2 )
<0|φ ˆ C (x 1 )|n>
(10.23)
n
The only free particle state |n> having a non-zero matrix element of the free
−ik·x
field φ ˆ C to the vacuum is the 1 − C state, for which <0|φ ˆ C (x)|C, k> = e
as
we learned in chapters 5 and 6. Thus (10.23) becomes (cf equation (6.92))
∫
d
3
k
−iω k (t1−t2)+ik·(x1 −x2 )
θ(t 1 − t 2 )
e
(10.24)
(2π) 3 2ω k
which is exactly the first term of equation (6.92). Consider now carrying out a
similar manipulation for the corresponding term of the interacting propagator,
obtaining
∑
θ(t 1 − t 2 )
<Ω|φ ˆ C (x 1 )|n>
(10.25)
n