Feynman Path Integral Formulation

3.5 The Gravitational Case 91
1993a,b) one now finds β 0 = 2 3
(25 − c) which is consistent with the above quoted
Thirring result. 1
Fig. 3.5 Two loop graviton
diagrams in 2 + ε gravity. (a) (b)
Fig. 3.6 Two loop gravitonghost
diagrams in 2 + ε gravity.
(a) (b) (c)
In the meantime the calculations have been laboriously extended to two loops
(see Figs. 3.5 and 3.6) (Aida and Kitazawa, 1997), with the result
μ ∂
∂μ G = β(G)=ε G − β 0 G 2 − β 1 G 3 + O(G 4 ,G 3 ε,G 2 ε 2 ) , (3.110)
with β 0 = 2 3 (25 − c) and β 1 = 20
3
(25 − c).
Of some interest is the fact that N = 1 supergravity in 2 + ε dimensions also
seems to give rise to a non-trivial ultraviolet fixed point (Kojima, Sakai and Tanii,
1994). These authors consider a model with a vielbein eμ a , a Majorana Rarita-
Schwinger field ψ μ and a real auxiliary scalar field S, with indices a,b,... and
μ,ν,... running from 0 to d − 1. The action taken to be of the form
I SG = 1 ∫ [
d d x dete R + i ¯ψ μ γ μνρ D ν ψ ρ − d − 2 ]
16πG
d − 1 S2 , (3.111)
with γ μνρ ≡ 1 3! (γ μ γ ν γ ρ ± permutations). There are some subtleties associated with
the dimensional reduction of supergravity that will not be discussed here. To lowest
1 For a while there was considerable uncertainty about the magnitude of the graviton contribution
to β 0 , which was quoted originally as 38/3 (Tsao, 1977), later as 2/3 (Gastmans et al, 1978;
Weinberg, 1977; Christensen and Duff, 1978), and more recently as 50/3 (Kawai, Kitazawa and
Ninomiya, 1993). As discussed in (Weinberg, 1979), the original expectation was that the graviton
contribution should be d(d − 3)/2 = −1 times the scalar contribution close to d = 2, which would
suggest for gravity the value 2/3. Direct numerical estimates of the scaling exponent ν in the lattice
theory for d = 3 (Hamber and Williams, 1993) give, using Eq. (3.125), a value β 0 ≈ 44/3 andare
therefore in much better agreement with the larger, more recent values.