335
11.4. The O(e
2 ) renormalized photon self-energy
FIGURE 11.3
One-loop corrected photon propagator connected to a charged particle vertex.
Inserting (11.31) into (11.30) we obtain the final important expression for
the γ-propagator including the one-loop renormalized self-energy (cf (10.71)):
−ig μν
(11.32)
[2]
q 2 (1 − Π ¯ γ (q 2 ))
where
Π
[2]
2 , Λ
2 ) − Π
[2]
¯ (q
2 ) = Π
[2] (q
(0, Λ
2 ).
(11.33)
γ
γ
γ
Equation (11.25) then leads to the result
[
]
∫ 1
2
Π
[2]
¯ (q
2 ) = −
2α
dx x(1 − x) ln
m
,
(11.34)
γ
π
m 2 − q 2 x(1 − x)
0
[2]
which was first given by Schwinger (1949a). This ‘once-subtracted’ Π ¯ γ is
finite as Λ → ∞, and tends to zero as q
2
→ 0.
The generalization of (11.32) to all orders will be given by
−ig μν
(11.35)
q 2 (1 − Π ¯ γ (q 2 ))
[2]
where Π ¯ γ (q
2 ) is the all-orders analogue of Π ¯ γ in (11.32), and is similarly related to the 1-γ irreducible photon self-energy Π ¯ μν via the analogue of (11.24):
2
iΠ ¯ μν (q
2 ) = i(q g μν − q μ q ν )Π ¯ γ (q
2 ).
(11.36)
Because Π ¯ μν , and hence Π ¯ γ , has no 1–γ intermediate states, it is expected to