Feynman Path Integral Formulation

3.6 Phases of Gravity in 2+ε Dimensions 95
μ d
dμ m ≡ μ d [
] A m μ |G(μ) − G c | ν = 0 , (3.122)
dμ
provided G runs in accordance with Eq. (3.117). To one-loop order one has from
Eqs. (3.106) and (3.120) ν = 1/(d −2). When the bare (lattice) coupling G(Λ)=G c
one has achieved criticality, m = 0. How far the bare theory is from the critical point
is determined by the choice of G(Λ), the distance from criticality being measured
by the deviation ΔG = G(Λ) − G c .
Furthermore Eq. (3.121) shows how the (lattice) continuum limit is to be taken.
In order to reach the continuum limit a ≡ 1/Λ → 0 for fixed physical correlation
ξ = 1/m, the bare coupling G(Λ) needs to be tuned so as to approach the ultraviolet
fixed point at G c ,
Λ → ∞ , m fixed , G → G c . (3.123)
The fixed point at G c thus plays a central role in the cutoff theory: together with the
universal scaling exponent ν it determines the correct unique quantum continuum
limit in the presence of an ultraviolet cutoff Λ. Sometimes it can be convenient to
measure all quantities in units of the cutoff and set Λ = 1/a = 1. In this case the
quantity m measured in units of the cutoff (i.e. m/Λ) has to be tuned to zero in order
to construct the lattice continuum limit: for a fixed lattice cutoff, the continuum
limit is approached by tuning the bare lattice G(Λ) to G c . In other words, the lattice
continuum limit has to be taken in the vicinity of the non-trivial ultraviolet point.
The discussion given above is not altered significantly, at least in its qualitative
aspects, by the inclusion of the two-loop correction of Eq. (3.110). From the expression
for the two-loop β-function
μ ∂
∂μ G = β(G)=ε G − 2 3 (25 − c)G2 − 20 3 (25 − c)G3 + ... (3.124)
for c massless real scalar fields minimally coupled to gravity, one computes the roots
β(G c )=0 to obtain the location of the ultraviolet fixed point, and from it on can
then determine the universal exponent ν = −1/β ′ (G c ). One finds
G c =
3
2(25 − c) ε − 45
2(25 − c) 2 ε2 + ...
ν −1 = ε + 15
25 − c ε2 + ... (3.125)
which gives, for pure gravity without matter (c = 0) in four dimensions, to lowest
order ν −1 = 2, and ν −1 ≈ 4.4 at the next order. 2
Also, in general higher order corrections to the results of the linearized renormalization
group equations of Eq. (3.119) are present, which affect the scaling away
from the fixed point. Let us assume that close to the ultraviolet fixed point at G c one
can write for the β-function the following expansion
2 If one does not expand the solution in ε, one finds from the two-loop result ν −1 = 2d −(1/6)(19+
√
60d − 95) which gives a smaller value ≈ 2.8ind = 4, as well as rough estimate of the uncertainty.