Feynman Path Integral Formulation

3.5 The Gravitational Case 89
Next one can make use of the freedom to rescale the metric, by setting
[ ( a1
1 −
ε + a )
2 √g √
G]
= g
ε 2 ′ , (3.99)
which restores the original unit coefficient for the cosmological constant term. The
rescaling is achieved by the following field redefinition
[ ( a1
g μν = 1 −
ε + a )
2 −2/d
G]
g
′
ε 2 μν . (3.100)
√
Hence the cosmological term is brought back into the standard form λ 0 g ′ , and one
obtains for the complete Lagrangian to first order in G
[
L →−
με
1 − 1 √g
16πG ε (b − 1 2 2)G] a √ ′ R ′ + λ 0 g ′ , (3.101)
where only terms singular in ε have been retained. From this last result one can
finally read off the renormalization of Newton’s constant
1
G → 1 [
1 − 1 ]
G ε (b − 1 2 a 2)G . (3.102)
From Eqs. (3.92) and (3.97) one notices that the a 2 contribution cancels out the
gauge-dependent part of b, giving for the remaining contribution b − 1 2 a 2 = 2 3 · 19.
Therefore the gauge dependence has, as one would have hoped on physical grounds,
disappeared from the final answer. It is easy to see that the same result would have
been obtained if the scaled cosmological constant Gλ 0 had been held constant, instead
of λ 0 as in Eq. (3.99). One important aspect of the result of Eq. (3.102) is
that the quantum correction is negative, meaning that the strength of G is effectively
increased by the lowest order radiative correction.
In the presence of an explicit renormalization scale parameter μ the β-function
for pure gravity is obtained by requiring the independence of the quantity G e (here
identified as an effective coupling constant, with lowest order radiative corrections
included) from the original renormalization scale μ,
μ d
dμ G e = 0
1
G e
≡
με
G(μ)
[
1 − 1 ]
ε (b − 1 2 a 2)G(μ)
. (3.103)
To zero-th order in G, the renormalization group β-function entering the renormalization
group equation
μ ∂ G = β(G) (3.104)
∂μ
is just given by
β(G)=ε G + ... (3.105)