Feynman Path Integral Formulation

8.9 Renormalization Group and Lattice Continuum Limit 297
0.5
0.4
0.3
kc
0.2
0.1
0 2 4 6 8 10
d
Fig. 8.10 Critical point k c = 1/8πG c in units of the ultraviolet cutoff as a function of dimension
d. Thecirclesatd = 3andd = 4 are the lattice results, suitably interpolated (dashed curve) using
the additional lattice result 1/k c = 0ind = 2. The lower continuous curve is the analytical large-d
lattice result of Eq. (7.150).
which shows that the continuum limit is reached in the vicinity of the ultraviolet
fixed point (see Fig. 8.11). Phrased equivalently, one takes the limit in which the
lattice spacing a ≈ 1/Λ is sent to zero at fixed ξ = 1/m, which requires an approach
to the non-trivial UV fixed point k → k c . The quantity m is supposed to be a
renormalization group invariant, a physical scale independent of the scale at which
the theory is probed. In practice, since the cutoff ultimately determines the physical
value of Newton’s constant G, Λ cannot be taken to ∞. Instead a very large value
will suffice, Λ −1 ∼ 10 −33 cm, for which it will still be true that ξ ≫ Λ which is all
that is required for the continuum limit.
For discussing the renormalization group behavior of the coupling it will be more
convenient to write the result of Eq. (8.77) directly in terms of Newton’s constant G
as
( ) 1 ν [ ] G(Λ) ν
m = Λ
− 1 , (8.79)
a 0 G c
with the dimensionless constant a 0 related to A m by A m = 1/(a 0 k c ) ν . Note that
the above expression only involves the dimensionless ratio G(Λ)/G c , which is the
only relevant quantity here. The lattice theory in principle completely determines
both the exponent ν and the amplitude a 0 for the quantum correction. Thus from
the knowledge of the dimensionless constant A m in Eq. (8.77) one can estimate
from first principles the value of a 0 in Eq. (8.84). Lattice results for the correlation
functions at fixed geodesic distance give a value for A m ≈ 0.72 with a significant
uncertainty, which, when combined with the values k c ≃ 0.0636 and ν ≃ 0.335
given above, gives a 0 = 1/(k c A 1/ν
m ) ≃ 42. The rather surprisingly large value for a 0