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10. Loops and Renormalization I: The ABC Theory
FIGURE 10.1
O(g
4 ) contribution to the process A + B → A + B, involving the modification
of the C propagator by the insertion of a loop.
present chapter we shall concentrate mainly on the particular loop appearing
in figure 6.8. This will lead us into the physics of renormalization for the ABC
theory, which – as a Yukawa-like theory – is a good theoretical laboratory for
studying ‘one-loop physics’, without the complications of spinor and gauge
fields. In the following chapter, we shall discuss one-loop diagrams in QED,
emphasizing some important physical consequences, such as corrections to
Coulomb’s law, anomalous magnetic moments and the running coupling constant.
10.1 The propagator correction in ABC theory
[2] 2 )
10.1.1 The O(g 2 ) self-energy Π (q
C
We consider figure 6.8, reproduced here again as figure 10.1. In section 6.3.5,
we gave the extra rule (‘(iii)’) needed to write down the invariant amplitude
for this process. We first show how this rule arises in the special case of
figure 10.1.
Clearly, figure 10.1 is a fourth-order process, so it must emerge from the
term
∫ ∫ ∫ ∫
(−ig)
4
′
′
d
4 x 1 d
4 x 2 d
4 x 3 d
4 x 4 <0|a ˆ A (p A )ˆ a B (p B )
4!
ˆ
× T {φ ˆ A (x 1 )φ ˆ B (x 1 )φ ˆ C (x 1 ) . . . φ A (x 4 )φ ˆ B (x 4 )φ ˆ C (x 4 )}
†
†
′
′
× a ˆ (p A )ˆ a (p B )|0>(16E A E B E A E B )
1/2
(10.1)
A
B