Feynman Path Integral Formulation

16 1 Continuum Formulation
1.5 One-Loop Divergences
Once the propagators and vertices have been defined, one can then proceed as in
QED and Yang-Mills theories and evaluate the quantum mechanical one loop corrections.
In a renormalizable theory with a dimensionless coupling, such as QED
and Yang-Mills theories, one has that the radiative corrections lead to charge, mass
and field re-definitions. In particular, for the pure SU(N) gauge action one finds
I YM = − 1
4g 2 ∫
dx trFμν 2 →− 1 ∫
4g 2 R
dx trF 2 R μν , (1.87)
so that the form of the action is preserved by the renormalization procedure: no new
interaction terms such as (D μ F μν ) 2 need to be introduced in order to re-absorb the
divergences.
In gravity the coupling is dimensionful, G ∼ μ 2−d , and one expects trouble already
on purely dimensional grounds, with divergent one loop corrections proportional
to
GΛ d−2 where Λ is an ultraviolet cutoff. 2 Equivalently, one expects to lowest
order bad ultraviolet behavior for the running Newton’s constant at large momenta,
G(k 2 )
G ∼ 1 + const. Gkd−2 + O(G 2 ) . (1.88)
These considerations also suggest that perhaps ordinary Einstein gravity is perturbatively
renormalizable in the traditional sense in two dimensions, an issue to which
we will return later in Sect. 3.5.
A more general argument goes as follows. The gravitational action contains the
scalar curvature R which involves two derivatives of the metric. Thus the graviton
propagator in momentum space will go like 1/k 2 , and the vertex functions like k 2 .
In d dimensions each loop integral with involve a momentum integration d d k,so
that the superficial degree of divergence D of a Feynman diagram with V vertices,
I internal lines and L loops will be given by
The topological relation involving V , I and L
is true for any diagram, and yields
D = dL+ 2V − 2I . (1.89)
L = 1 + I −V , (1.90)
2 Indeed it was noticed very early on in the development of renormalization theory that perturbatively
non-renormalizible theories would involve couplings with negative mass dimensions, and for
which cross-sections would grow rapidly with energy (Sakata, Umezawa and Kamefuchi, 1952).
It had originally been suggested by Heisenberg (Heisenberg, 1938) that the relevant mass scale
appearing in such interactions with dimensionful coupling constants should be used to set an upper
energy limit on the physical applicability of such theories.