Feynman Path Integral Formulation

94 3 Gravity in 2 + ε Dimensionsnot expect gravity to be screened. On the other hand the infrared growth of the couplingin the strong coupling phase G > G c can be written equivalently as[ ( ) mG(k 2 2 (d−2)/2]) ≃ G c 1 + a 0k 2 + ... , (3.117)where the dots indicate higher order radiative corrections, and which exhibits anumber of interesting features. Firstly the fractional power suggests new non-trivialgravitational scaling dimensions, just as in the case of the non-linear σ-model. Furthermore,the quantum correction involves a new physical, renormalization groupinvariant scale ξ = 1/m which cannot be fixed perturbatively, and whose size determinesthe scale for the quantum effects. In terms of the bare coupling G(Λ), itisgiven by( ∫ G(Λ)) dG′m = A m ·Λ exp −β(G ′ , (3.118))which just follows from integrating μ ∂∂μG = β(G) and then setting as the arbitraryscale μ → Λ. Conversely, since m is an invariant, one has Λ d m = 0; the running ofG(μ) in accordance with the renormalization group equation of Eq. (3.104) ensuresthat the l.h.s. is indeed a renormalization group invariant. The constant A m on ther.h.s. of Eq. (3.118) cannot be determined perturbatively, it needs to be computed bynon-perturbative (lattice) methods, for example by evaluating invariant correlationsat fixed geodesic distances. It is related to the constant a 0 in Eq. (3.117) by a 0 =1/(Am 1/ν G c ).At the fixed point G = G c the theory is scale invariant by definition. In statisticalfield theory language the fixed point corresponds to a phase transition, where thecorrelation length ξ = 1/m diverges and the theory becomes scale (conformally)invariant. In general in the vicinity of the fixed point, for which β(G)=0, one canwriteβ(G)If one then defines the exponent ν by∼G→Gcβ ′ (G c )(G − G c )+O[(G − G c ) 2 ] . (3.119)β ′ (G c )=−1/ν , (3.120)then from Eq. (3.118) one has by integration in the vicinity of the fixed pointm∼G→GcΛ · A m |G(Λ) − G c | ν , (3.121)which is why ν is often referred to as the mass gap exponent. Solving the aboveequation (with Λ → k) forG(k) one obtains back Eq. (3.117), with the constant a 0there related to A m in Eq. (3.121) by a 0 = 1/(Am 1/ν G c ) and ν = 1/(d − 2).That m is a renormalization group invariant is seen fromdΛ