Feynman Path Integral Formulation

8.9 Renormalization Group and Lattice Continuum Limit 2970.50.40.3kc0.20.10 2 4 6 8 10dFig. 8.10 Critical point k c = 1/8πG c in units of the ultraviolet cutoff as a function of dimensiond. Thecirclesatd = 3andd = 4 are the lattice results, suitably interpolated (dashed curve) usingthe additional lattice result 1/k c = 0ind = 2. The lower continuous curve is the analytical large-dlattice result of Eq. (7.150).which shows that the continuum limit is reached in the vicinity of the ultravioletfixed point (see Fig. 8.11). Phrased equivalently, one takes the limit in which thelattice spacing a ≈ 1/Λ is sent to zero at fixed ξ = 1/m, which requires an approachto the non-trivial UV fixed point k → k c . The quantity m is supposed to be arenormalization group invariant, a physical scale independent of the scale at whichthe theory is probed. In practice, since the cutoff ultimately determines the physicalvalue of Newton’s constant G, Λ cannot be taken to ∞. Instead a very large valuewill suffice, Λ −1 ∼ 10 −33 cm, for which it will still be true that ξ ≫ Λ which is allthat is required for the continuum limit.For discussing the renormalization group behavior of the coupling it will be moreconvenient to write the result of Eq. (8.77) directly in terms of Newton’s constant Gas( ) 1 ν [ ] G(Λ) νm = Λ− 1 , (8.79)a 0 G cwith the dimensionless constant a 0 related to A m by A m = 1/(a 0 k c ) ν . Note thatthe above expression only involves the dimensionless ratio G(Λ)/G c , which is theonly relevant quantity here. The lattice theory in principle completely determinesboth the exponent ν and the amplitude a 0 for the quantum correction. Thus fromthe knowledge of the dimensionless constant A m in Eq. (8.77) one can estimatefrom first principles the value of a 0 in Eq. (8.84). Lattice results for the correlationfunctions at fixed geodesic distance give a value for A m ≈ 0.72 with a significantuncertainty, which, when combined with the values k c ≃ 0.0636 and ν ≃ 0.335given above, gives a 0 = 1/(k c A 1/νm ) ≃ 42. The rather surprisingly large value for a 0