Feynman Path Integral Formulation

3.5 The Gravitational Case 87
× × ×
(a) (b) (c)
Fig. 3.4 One-loop diagrams giving rise to coupling renormalizations in gravity. From left to right,
graviton loop, ghost loop and scalar matter loop.
1
k 2 (δ μαδ νβ + δ μβ δ να ) − 2 1
d − 2 k 2 δ μνδ αβ
(
− 1− 1 ) 1
α k 4 (δ μαk ν k β + δ να k μ k β + δ μβ k ν k α + δ νβ k μ k α )
+ 1 4(β − 1) 1
d − 2 β − 2 k 4 (δ μνk α k β + δ αβ k μ k ν )
[
4(1 − β)
+
(β − 2) 2 2 − 3 − β
]
2(1 − β) 1
−
α d − 2 k 6 k νk ν k α k β .
(3.88)
Normally it would be convenient to choose a gauge α=β = 1, in which case only
the first two terms for the graviton propagator survive. But here it might be advantageous
to leave the two gauge parameters unspecified, so that one can later show
explicitly the gauge independence of the final result. In particular the gauge parameter
β is related to the gauge freedom associated with the possibility, described
above, of rescaling the metric g μν . Note also the presence of kinematical poles in
ε = d − 2 in the second, fourth and fifth term for the graviton propagator.
To illustrate explicitly the mechanism of coupling renormalization, the cosmological
term will be discussed first, since the procedure is a bit simpler. The cosmological
term √ g is first expanded by setting g μν = g μν +h μν with a flat background
g μν = δ μν . One has
√ g = 1 +
1
2
h − 1 4 h μνh μν + 1 8 h2 + O(h 3 ) , (3.89)
with h = h μ μ. Terms linear in the fluctuation h μν are dropped, since in a properly
chosen background such terms are expected to be absent. The one-loop correction
to the 1 term in the above expression is then given by the tadpole diagrams for the
two quadratic terms,
− 1 4 h μνh μν + 1 8 h2 →− 1 4 + 1 8 . (3.90)
These are easily evaluated using the graviton propagator of Eq. (3.88). For the one
loop divergences (see Fig. 3.4) associated with the √ g term one then obtains