Feynman Path Integral Formulation

16 1 Continuum Formulation1.5 One-Loop DivergencesOnce the propagators and vertices have been defined, one can then proceed as inQED and Yang-Mills theories and evaluate the quantum mechanical one loop corrections.In a renormalizable theory with a dimensionless coupling, such as QEDand Yang-Mills theories, one has that the radiative corrections lead to charge, massand field re-definitions. In particular, for the pure SU(N) gauge action one findsI YM = − 14g 2 ∫dx trFμν 2 →− 1 ∫4g 2 Rdx trF 2 R μν , (1.87)so that the form of the action is preserved by the renormalization procedure: no newinteraction terms such as (D μ F μν ) 2 need to be introduced in order to re-absorb thedivergences.In gravity the coupling is dimensionful, G ∼ μ 2−d , and one expects trouble alreadyon purely dimensional grounds, with divergent one loop corrections proportionaltoGΛ d−2 where Λ is an ultraviolet cutoff. 2 Equivalently, one expects to lowestorder 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 perturbativelyrenormalizable in the traditional sense in two dimensions, an issue to whichwe will return later in Sect. 3.5.A more general argument goes as follows. The gravitational action contains thescalar curvature R which involves two derivatives of the metric. Thus the gravitonpropagator 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,sothat the superficial degree of divergence D of a Feynman diagram with V vertices,I internal lines and L loops will be given byThe topological relation involving V , I and Lis true for any diagram, and yieldsD = 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 perturbativelynon-renormalizible theories would involve couplings with negative mass dimensions, and forwhich 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 scaleappearing in such interactions with dimensionful coupling constants should be used to set an upperenergy limit on the physical applicability of such theories.