Feynman Path Integral Formulation

3.2 Perturbatively Non-renormalizable Theories: The Sigma Model 77theories, including gravity, there is no local order parameter, so this quantity has noobvious generalization there.In general the ε-expansion is only expected to be asymptotic. This is already seenfrom the expansion for ν which has recently been computed to four loops (Hikamiand Brezin, 1978; Bernreuther and Wegner, 1986; Kleinert, 2000)ν −1 = ε +ε2N − 2 + ε 32(N − 2)− [30 − 14N + N 2 +(54 − 18N)ζ (3)]ε 4× + ... (3.39)4(N − 2) 3which needs to be summed by Borel-Padé methods to obtain reliable results in threedimensions. For example, for N = 3 one finds in three dimensions ν ≈ 0.799, whichcan be compared to the 4 − ε result for the λφ 4 theory to five loops ν ≃ 0.705, tothe seven-loop perturbative expansion for the λφ 4 theory directly in 3d which givesν ≃ 0.707, with the high temperature series result ν ≃ 0.717 and the Monte Carloestimates ν ≃ 0.718, as compiled for example in a recent comprehensive review(Guida and Zinn-Justin, 1998).There exist standard methods to deal with asymptotic series such as the one inEq. (3.39). To this purpose one considers a general seriesand defines its Borel transform asf (g)=F(b)=∞∑n=0∞∑n=0f n g n , (3.40)f nn! bn . (3.41)One can attempt to sum the series for F(b) using Padé methods and conformal transformations.The original function f (g) is then recovered by performing an integralover the Borel transform variable bwhere the familiar formulaf (g)= 1 g∫ ∞0∫ ∞0dbe −b/g F(b) , (3.42)dzz n e −z/g = n!g n+1 , (3.43)has been used. Bounds on the coefficients f n suggest that in most cases F(z) isanalytic in a circle of radius a around the origin, and that the integral will convergefor |z| small enough, within a sector |argz| < α/2 with typically α ≥ π (Le Guillouand Zinn-Justin, 1980).