10.1. The propagator correction in ABC theory
301
of the Dyson expansion. Since it is basically a u-channel exchange process
′
′
(u = (p A − p )
2 = (p − p B )
2 ), the vev’s involving the external creation and
B
A
annihilation operators must appear as they do in equation (6.89) (‘ingoing
A, outgoing B
′ at one point x 2 ; ingoing B, outgoing A
′ at another point x 1 ’)
rather than as in equation (6.88) (‘ingoing A and B at x 2 ; outgoing A
′ and
B
′ at x 1 ’). In (10.1), however, we unfortunately have four space–time points
to choose from, rather than merely the two in (6.74). Figuring out exactly
which choices are in fact equivalent and which are not is best left to private
struggle, especially since we are not seriously interested in the numerical value
of our fourth-order corrections in this case. Let us simply consider one choice,
analogous to (6.89). This yields the amplitude (cf (6.91))
∫ ∫ ∫ ∫
i(p −pB)·x1 i(p −pA)·x2
A
B
(−ig)
4
d
4 x 1 d
4 x 2 d
4 x 3 d
4 x 4 e
'
e
'
× <0|T {φ ˆ C (x 1 )φ ˆ C (x 2 )φ ˆ A (x 3 )φ ˆ B (x 3 )φ ˆ C (x 3 )φ ˆ A (x 4 )φ ˆ B (x 4 )φ ˆ C (x 4 )}|0>
(10.2)
and we have discarded the numerical factor 1/4!. Once again, there are many
terms in the expansion of the vev of the eight operators in (10.2). But, with
an eye on the structure of the Feynman amplitude at which we are aiming
(figure 10.1), let us consider again just a single contribution
∫ ∫ ∫ ∫
d
4
i(p −pB)·x1 i(p −pA)·x2
A
B
(−ig)
4
x 1 d
4 x 2 d
4 x 3 d
4 x 4 e
'
e
'
× <0|T (φ ˆ C (x 1 )φ ˆ C (x 3 ))|0><0|T (φ ˆ C (x 2 )φ ˆ C (x 4 ))|0>
× <0|T (φ ˆ A (x 3 )φ ˆ A (x 4 ))|0><0|T (φ ˆ B (x 3 )φ ˆ B (x 4 ))|0>
(10.3)
which contains four propagators connected as in figure 10.2.
As we saw in section 6.3.2, each of these propagators is a function only
of the difference of the two space–time points involved. Introducing relative
coordinates x = x 1 − x 3 , y = x 2 − x 4 , z = x 3 − x 4 and the CM coordinate
1
X = (x 1 + x 2 + x 3 + x 4 ), we find (problem 10.1) that (10.3) becomes
4
∫ ∫ ∫ ∫
'
'
'
d
4 X d
4 x d
4 y d
4
i(p +p −pA−pB)·X i(p −pB)·(3x−y+2z)/4
A
B
A
(−ig)
4
z e
e
i(p −pA)·(−x+3y−2z)/4 D C (x)D C (y)D A (z)D B (z)
× e B
'
(10.4)
where D i is the position–space propagator for type-i particles (i = A, B, C),
defined as in (6.98). The integral over X gives the expected overall 4-momentum
′
′
′
′
conservation factor, (2π)
4 δ
4 (p A +p −p A −p B ). Setting q = p A −p = p −p B
B
B
A
(where 4-momentum conservation has been used), (10.4) becomes
∫ ∫ ∫
′
′
(−ig)
4 (2π)
4 δ
4 (p A + p B − p A − p B )
d
4 x d
4 y d
4 z e
iq·x D C (x)
× e
−iq·y D C (y)e
iq·z D A (z)D B (z).
(10.5)
301
of the Dyson expansion. Since it is basically a u-channel exchange process
′
′
(u = (p A − p )
2 = (p − p B )
2 ), the vev’s involving the external creation and
B
A
annihilation operators must appear as they do in equation (6.89) (‘ingoing
A, outgoing B
′ at one point x 2 ; ingoing B, outgoing A
′ at another point x 1 ’)
rather than as in equation (6.88) (‘ingoing A and B at x 2 ; outgoing A
′ and
B
′ at x 1 ’). In (10.1), however, we unfortunately have four space–time points
to choose from, rather than merely the two in (6.74). Figuring out exactly
which choices are in fact equivalent and which are not is best left to private
struggle, especially since we are not seriously interested in the numerical value
of our fourth-order corrections in this case. Let us simply consider one choice,
analogous to (6.89). This yields the amplitude (cf (6.91))
∫ ∫ ∫ ∫
i(p −pB)·x1 i(p −pA)·x2
A
B
(−ig)
4
d
4 x 1 d
4 x 2 d
4 x 3 d
4 x 4 e
'
e
'
× <0|T {φ ˆ C (x 1 )φ ˆ C (x 2 )φ ˆ A (x 3 )φ ˆ B (x 3 )φ ˆ C (x 3 )φ ˆ A (x 4 )φ ˆ B (x 4 )φ ˆ C (x 4 )}|0>
(10.2)
and we have discarded the numerical factor 1/4!. Once again, there are many
terms in the expansion of the vev of the eight operators in (10.2). But, with
an eye on the structure of the Feynman amplitude at which we are aiming
(figure 10.1), let us consider again just a single contribution
∫ ∫ ∫ ∫
d
4
i(p −pB)·x1 i(p −pA)·x2
A
B
(−ig)
4
x 1 d
4 x 2 d
4 x 3 d
4 x 4 e
'
e
'
× <0|T (φ ˆ C (x 1 )φ ˆ C (x 3 ))|0><0|T (φ ˆ C (x 2 )φ ˆ C (x 4 ))|0>
× <0|T (φ ˆ A (x 3 )φ ˆ A (x 4 ))|0><0|T (φ ˆ B (x 3 )φ ˆ B (x 4 ))|0>
(10.3)
which contains four propagators connected as in figure 10.2.
As we saw in section 6.3.2, each of these propagators is a function only
of the difference of the two space–time points involved. Introducing relative
coordinates x = x 1 − x 3 , y = x 2 − x 4 , z = x 3 − x 4 and the CM coordinate
1
X = (x 1 + x 2 + x 3 + x 4 ), we find (problem 10.1) that (10.3) becomes
4
∫ ∫ ∫ ∫
'
'
'
d
4 X d
4 x d
4 y d
4
i(p +p −pA−pB)·X i(p −pB)·(3x−y+2z)/4
A
B
A
(−ig)
4
z e
e
i(p −pA)·(−x+3y−2z)/4 D C (x)D C (y)D A (z)D B (z)
× e B
'
(10.4)
where D i is the position–space propagator for type-i particles (i = A, B, C),
defined as in (6.98). The integral over X gives the expected overall 4-momentum
′
′
′
′
conservation factor, (2π)
4 δ
4 (p A +p −p A −p B ). Setting q = p A −p = p −p B
B
B
A
(where 4-momentum conservation has been used), (10.4) becomes
∫ ∫ ∫
′
′
(−ig)
4 (2π)
4 δ
4 (p A + p B − p A − p B )
d
4 x d
4 y d
4 z e
iq·x D C (x)
× e
−iq·y D C (y)e
iq·z D A (z)D B (z).
(10.5)