333
11.4. The O(e
2 ) renormalized photon self-energy
tensor structure, and the Λ-dependence can be compensated by the ‘Z 3 − 1’
counter term which has the same tensor structure (cf figure 11.2(b)). But
what about the first line of (11.18)? Various gauge-invariant regularizations
have been used, the effect of all of which is to cause the first line of (11.18) to
vanish. The most widely used, since the 1970s, is the dimensional regularization technique introduced by ’t Hooft and Veltman (1972), which involves the
‘continuation’ of the number of space–time dimensions from four to d (< 4).
As d is reduced, the integrals tend to diverge less, and the divergences can be
isolated via the terms which diverge as d → 4. Using gauge-invariant dimensional regularization, the two terms in the first line of (11.18) are found to
cancel each other exactly, leaving just the manifestly gauge invariant second
line (see appendix O of volume 2).
We proceed to the next step, renormalizing the gauge-invariant part of
[2]
iΠ μν (q
2 ).
11.4 The O(e
2 ) renormalized photon self-energy
[2]
The surviving (gauge-invariant) term of Π μν is
∫
∫
1
d
4 k
′
x(1 − x)
iΠ
[2]
2
μν (q
2 ) = 8e
2 (q μ q ν − q g μν )
dx
(11.23)
0
(2π) 4 (k ′ 2 − Δ γ + i∈) 2
2 g μν − q μ q ν )Π
[2]
≡ i(q
(q
2 ).
(11.24)
γ
The d
4 k
′ integral in (11.23) is exactly the same as the one in (10.42), with Δ
replaced by Δ γ . It contains a logarithmic divergence, which we regulate as
before by a simple cut-off Λ, so that we are dealing with the gauge-invariant
[2] 2
quantity Π γ (q , Λ
2 ). The calculation leading to (10.55) then tells us that, as
Λ → ∞,
∫
(
)
2
1
e
1
Π
[2] 2
γ (q , Λ
2 ) = − π 2
dx ln Λ + (ln 2 − 1) − ln Δ γ .
(11.25)
2
0
The analogue of (10.11) is then (in the gauge ξ = 1)
−ig μν
−ig μρ
2 ρσ
− q
ρ σ )Π
[2] 2 , Λ
2 )
−ig σν
+
· i(q g
q
(q
·
γ
q 2
q 2
q 2
−ig μρ
2 ρσ
− q
ρ σ )Π
[2] 2
−ig στ
+
· i(q g
q
(q , Λ
2 ) ·
γ
q 2
q 2
2 τ η
− q
τ
· Π
[2] 2
−ig ην
· i(q g
q
η )
(q , Λ
2 ) ·
+ · · ·
γ
q 2
−ig μν
−ig μρ
ν Π
[2] 2
−ig μρ
ν (Π
[2] 2
=
+
P
ρ
(q , Λ
2 ) +
P τ
ρ P
τ
(q , Λ
2 )
2 + · · ·
γ
γ
q 2
q 2
q 2
(11.26)
11.4. The O(e
2 ) renormalized photon self-energy
tensor structure, and the Λ-dependence can be compensated by the ‘Z 3 − 1’
counter term which has the same tensor structure (cf figure 11.2(b)). But
what about the first line of (11.18)? Various gauge-invariant regularizations
have been used, the effect of all of which is to cause the first line of (11.18) to
vanish. The most widely used, since the 1970s, is the dimensional regularization technique introduced by ’t Hooft and Veltman (1972), which involves the
‘continuation’ of the number of space–time dimensions from four to d (< 4).
As d is reduced, the integrals tend to diverge less, and the divergences can be
isolated via the terms which diverge as d → 4. Using gauge-invariant dimensional regularization, the two terms in the first line of (11.18) are found to
cancel each other exactly, leaving just the manifestly gauge invariant second
line (see appendix O of volume 2).
We proceed to the next step, renormalizing the gauge-invariant part of
[2]
iΠ μν (q
2 ).
11.4 The O(e
2 ) renormalized photon self-energy
[2]
The surviving (gauge-invariant) term of Π μν is
∫
∫
1
d
4 k
′
x(1 − x)
iΠ
[2]
2
μν (q
2 ) = 8e
2 (q μ q ν − q g μν )
dx
(11.23)
0
(2π) 4 (k ′ 2 − Δ γ + i∈) 2
2 g μν − q μ q ν )Π
[2]
≡ i(q
(q
2 ).
(11.24)
γ
The d
4 k
′ integral in (11.23) is exactly the same as the one in (10.42), with Δ
replaced by Δ γ . It contains a logarithmic divergence, which we regulate as
before by a simple cut-off Λ, so that we are dealing with the gauge-invariant
[2] 2
quantity Π γ (q , Λ
2 ). The calculation leading to (10.55) then tells us that, as
Λ → ∞,
∫
(
)
2
1
e
1
Π
[2] 2
γ (q , Λ
2 ) = − π 2
dx ln Λ + (ln 2 − 1) − ln Δ γ .
(11.25)
2
0
The analogue of (10.11) is then (in the gauge ξ = 1)
−ig μν
−ig μρ
2 ρσ
− q
ρ σ )Π
[2] 2 , Λ
2 )
−ig σν
+
· i(q g
q
(q
·
γ
q 2
q 2
q 2
−ig μρ
2 ρσ
− q
ρ σ )Π
[2] 2
−ig στ
+
· i(q g
q
(q , Λ
2 ) ·
γ
q 2
q 2
2 τ η
− q
τ
· Π
[2] 2
−ig ην
· i(q g
q
η )
(q , Λ
2 ) ·
+ · · ·
γ
q 2
−ig μν
−ig μρ
ν Π
[2] 2
−ig μρ
ν (Π
[2] 2
=
+
P
ρ
(q , Λ
2 ) +
P τ
ρ P
τ
(q , Λ
2 )
2 + · · ·
γ
γ
q 2
q 2
q 2
(11.26)