466
O. Dimensional Regularization
introduced in section 15.3. From (15.54) the two corresponding values of Λ
are related by
(
)
∫
Λ
1
α
B
2
s (
B
|q |)
dx
ln
=
(O.18)
Λ A
2 α A ( q 2 ) β 0 x 2 (1 + . . .)
s | |
A 1
=
(O.19)
2β 0
where we have taken |q
2
| → ∞ in (O.18) since the left-hand side is independent of |q
2
|. Hence the relationship between the Λ’s in different schemes is
determined by the one-loop calculation which gives A 1 in (O.19). For example,
changing from MS to MS gives (problem 15.8)
Λ
2
= Λ
2
MS exp(ln 4π − γ),
(O.20)
MS
as the reader may check.
Finally, consider the integral
∫ d
d k
k
μ k
ν
μν
I d (Δ, n) ≡
.
(O.21)
(2π) d [k 2 − Δ + i∈] n
From Lorentz covariance this must be proportional to the only second-rank
tensor available, namely g
μν :
μν
I = Ag
μν
d
.
(O.22)
The constant ‘A’ can be determined by contracting both sides of (O.21) with
g μν , using g
μν g μν = d in d-dimensions. So
∫
1
d
d k
k
2
A = d
(2π) d (k 2 − Δ + i∈) n
(∫
d
∫
d
)
1
d k
1
d k
1
=
+ Δ
d
(2π) d (k 2 − Δ + i∈) n−1
(2π) d (k 2 − Δ + i∈) n
)
i(−1)
n
n+1
(
Δ
(d/2)−
−Γ(n
Γ(
=
− 1
(4π) d/2
− d/2)
n
+
− d/2)
d
Γ(n − 1)
Γ(n)
i( 1)
n Δ
(d/2)−n+1 Γ(n 1 d/2)
=
−
− −
{−n + (n
(4π) d/2
d
Γ(n)
− d/2)}
i(−1)
n−1 Δ
(d/2)−n+1 1 Γ(n 1 d/
=
(4 ) d/2
− − 2) .
(O.23)
π
2
Γ(n)
Using these results, one can show straightforwardly that the gauge-non-invariant
part of (11.18) – i.e. the piece in braces – vanishes. With the technique
of dimensional regularization, starting from a gauge-invariant formulation of
the theory the renormalization programme can be carried out while retaining
manifest gauge invariance.