Feynman Path Integral Formulation

98 3 Gravity in 2 + ε Dimensions
Note that this added momentum-dependent cutoff term violates both the weak field
general coordinate invariance [see for instance Eq. (1.11)], as well as the general
rescaling invariance of Eq. (3.79).
The solution of the resulting renormalization group equation for the two couplings
G(μ) and λ(μ) is then truncated to the Einstein and cosmological term,
a procedure which is equivalent to the derivative expansion discussed previously.
A nontrivial fixed point in both couplings (G ∗ ,λ ∗ ) is then found in four dimensions,
with complex eigenvalues ν −1 = 1.667±4.308i for a gauge parameter choice
α → ∞ [for general gauge parameter the exponents can vary by as much as seventy
percent (Lauscher and Reuter, 2002)]. In the special limit of vanishing cosmological
constant the equations simplify further and one finds a trivial Gaussian fixed point at
G = 0, as well as a non-trivial ultraviolet fixed point with ν −1 = 2d(d − 2)/(d + 2),
which in d = 4 gives now ν −1 = 2.667. So in spite of the apparent crudeness of the
lowest order approximation, an ultraviolet fixed point similar to the one found in the
2 + ε expansion is recovered.
3.7 Running of α(μ) in Gauge Theories
QED and QCD provide two invaluable illustrative cases where the running of the
gauge coupling with energy is not only theoretically well understood, but also verified
experimentally. This section is just intended to provide some analogies and
distinctions between the two theories, in a way later suitable for a comparison with
the gravitational case. Most of the results found in this section are well known (see,
for example, Frampton, 2000), but the purpose here is to provide some contrast (and
in some instances, a relationship) with the gravitational case.
In QED the non-relativistic static Coulomb potential is affected by the vacuum
polarization contribution due to electrons (and positrons) of mass m. To lowest order
in the fine structure constant, the contribution is from a single Feynman diagram
involving a fermion loop. One finds for the vacuum polarization contribution ω R (k 2 )
at small k 2 the well known result (Itzykson and Zuber, 1980)
e 2
k 2 → e 2
k 2 [1 + ω R (k 2 )]
[
∼ e2
k 2 1 + α k 2 ]
15π m 2 + O(α2 )
, (3.135)
which, for a Coulomb potential with a charge centered at the origin of strength −Ze
leads to well-known Uehling δ -function correction
V (r) =
(
1 − α
15π
)
Δ −Ze
2
m 2 4π r
= −Ze2
4π r − α −Ze 2
15π m 2 δ (3) (x) . (3.136)
It is not necessary though to resort to the small-k 2 approximation, and in general a
static charge of strength e at the origin will give rise to a modified potential