Asymptotic Behaviour in Quantum Field Theory

Abstract

Convergence is tested by applying power-counting arguments 10 • 11 ). Imagine that a subset S = {p~, ... , p~} of the loop momenta {p 1 , ••• , p£} is scaled to infinity [all p~ = O(n) with n ~ 00 ] with all other momenta held fixed*). The exponent c(r) for its asymptotic behaviour as p 1 , ••• , p£ become large is reduced by at least 1, while no increase occurs in the other c(S). Therefore, if a/aq.1.is applied a sufficient number of times tor, the result is completely convergent.Since rd. (the divergent part of r) is annihilated by these differentiations, it lV must be a polynomial of degree~ d(r) in q 1 , ••• , qL-l. In diagrammatic language ( Fig. 1), this means that rd. is a local vertex produced by shrinking the blob r lV to a point.An £-loop divergence of this type is easily cancelled by including a counterterm ~ff£ in the Lagrangian ;t with bare vertex equal to -rdiv' However, many lPI graphs do not possess this property of being "primitively divergent"~<*). An arbitrary £-loop lPI graph may contain divergent £ 1 -loop lPI subgraphs (£ 1 < £) which must first be made convergent by including suitable counterterms ~ff£' in ff; i.e., it will be necessary to include internal counterterm vertices in the blob in Fig. 1. So we try the following prescription 10 • 12 ): *) **)More precisely, write pj = nrj and let n tend to 00 , keeping all rj, Pi ('any pj), and qi fixed and not allowing any partial sum L' r of the rj to vanish.Tfle latter condition ensures that a p 1 -dependent propagator (which carries momentum n l' r + ~p1p' p + f 1 q) has the expected asymptotic dependence on n. The special directions l r = 0 are covered by looking at appropriate su b sets s' of s .A primitively divergent graph becomes convergent if any internal line is cut.-3 -i) Start with !t 0 , a Lagrangian from which propagators and vertices can be constructed.ii) Construct counterterms 62? 1 which remove all divergences in 1-loop lPI graphs generated by 21 0 • iii) Use a new Lagrangian 'l! 1 = :t + 6 'l! to generate 2-loop graphs and construct , a , 16 '/! 2 to get rid of the resulting lPI divergences, and so on, to any finite number of loops and vertices 13 ):J,1 = ii-I + ~£1.Obviously, we obtain finite results for diagrams in which, for each pair of divergent lPI subgraphs, one subgraph is entirely contained within the other ("nested" divergences) or the two subgraphs are disjoint. The difficult step in the proof 12 )of convergence is to disentangle overlapping divergences (sets of divergent lPI graphs which are neither disjoint nor nested). The result, valid for all polynomial interactions (renormalizable or otherwise), is that the above procedure renders all subintegrations convergent by power-counting*). In other words, the Given the constraint (1.6), the rule for renormalization counterterms ~Q generated by lPI diagrams with a single Q-insertion is simply dim(Q) (1. 8) For example, consider = I 2. d);i., 2m1(1. 9)as a trial Lagrangian. Only the two-and four-legged lPI amplitudes r( 2 ),r ( 4 ) are superficially divergent. Accor...