328
11. Loops and Renormalization II: QED
FIGURE 11.1
Counter terms in QED: (a) electron mass and wavefunction; (b) photon wavefunction; (c) vertex part.
according to chapter 7. We shall adopt the ‘renormalized perturbation theory’
approach and begin by introducing field strength renormalizations via
−1/2
ˆ
ˆ
ψ = Z
ψ 0
(11.2)
2
−1/2 μ
A ˆ μ
ˆ
= Z
A
(11.3)
3
0
where the ‘physical’ fields and parameters will now simply have no ‘0’ subscript. This will lead to a rewriting of the free and gauge-fixing part of (11.1):
1
1
ψ
¯ ˆ
0 (i∂ / − m 0 )ψ ˆ 0 − F ˆ 0μν F ˆ μν −
(∂ · A ˆ 0 )
2
4
0
2ξ 0
¯ ˆ
1 ˆ F ˆ μν −
1
= ψ(i∂ / − m)ψ ˆ − F μν
(∂ · A ˆ )
2
4
2ξ
¯
¯
F
μν
+ [(Z 2 − 1)ψ ˆ i∂ /ψ ˆ − δmψ ˆ ψ ˆ ] −
1 (Z 3 − 1)F ˆ μν ˆ
(11.4)
4
where ξ = ξ 0 /Z 3 and δm = m 0 Z 2 − m (compare (10.64)). We see the emer¯ ˆ
ˆ
ˆ · ˆ
gence of the expected ‘ψ . . . ψ’ and ‘F F ’ counter terms in (11.4), affecting
both the fermion and the gauge-field propagators. Next, we write the interaction in terms of a physical e, and the physical fields, together with a
compensating third counter term:
¯
¯
¯
ψ ˆ
0 γ
μ ˆ ˆ
ˆ ψ ˆ
ˆ ψ ˆ
−e 0
ψ 0 A 0μ = −eψγ
μ ˆ A μ − (Z 1 − 1)eψγ
μ ˆ A μ
(11.5)
where, with the aid of (11.2) and (11.3),
1/2
Z 1 e = e 0 Z 2 Z 3 .
(11.6)
The three counter terms are represented diagrammatically as shown in figures 11.1(a), (b) and (c), for which the Feynman rules are, respectively,
(a): i[k /(Z 2 − 1) − δm]

(b): − i(g
μν k
2
− k
μ k
ν )(Z 3 − 1)
(11.7)

(c): − ieγ
μ (Z 1 − 1).