Feynman Path Integral Formulation

3.5 The Gravitational Case 911993a,b) one now finds β 0 = 2 3(25 − c) which is consistent with the above quotedThirring result. 1Fig. 3.5 Two loop gravitondiagrams in 2 + ε gravity. (a) (b)Fig. 3.6 Two loop gravitonghostdiagrams 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 resultμ ∂∂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 = 203(25 − c).Of some interest is the fact that N = 1 supergravity in 2 + ε dimensions alsoseems 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,... andμ,ν,... running from 0 to d − 1. The action taken to be of the formI SG = 1 ∫ [d d x dete R + i ¯ψ μ γ μνρ D ν ψ ρ − d − 2 ]16πGd − 1 S2 , (3.111)with γ μνρ ≡ 1 3! (γ μ γ ν γ ρ ± permutations). There are some subtleties associated withthe dimensional reduction of supergravity that will not be discussed here. To lowest1 For a while there was considerable uncertainty about the magnitude of the graviton contributionto 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 andNinomiya, 1993). As discussed in (Weinberg, 1979), the original expectation was that the gravitoncontribution should be d(d − 3)/2 = −1 times the scalar contribution close to d = 2, which wouldsuggest for gravity the value 2/3. Direct numerical estimates of the scaling exponent ν in the latticetheory for d = 3 (Hamber and Williams, 1993) give, using Eq. (3.125), a value β 0 ≈ 44/3 andaretherefore in much better agreement with the larger, more recent values.