Renormalization in QCD

Abstract

As we saw in the example of Section 2.3i, some amplitudes are divergent. This is due to the fact that field operators are singular objects: it is easy to trace the divergence of the dk integral in (2.3.4b) at large values of k to the occurrence, in position space, of products of field operators at the same space-time point. Because of this, we must, in order to discuss QCD (or indeed any local relativistic field theory), give a meaning to the integrals that appear when we evaluate Feynmann diagrams. This is called regularization and it amounts to altering the Lagrangian ℒ to ℒɛ in such a way that ℒ ɛ produces finite answers and, in some sense, as ɛ→0, ℒ ɛ→ℒ. Due to Bohr and Rosenfeld’s (1933, 1950) classical work, we know that field operators are intrinsically singular; therefore, any regularization must destroy some physical feature of the theory. Thus, Pauli-Villars regularization destroys hermiticity and gauge invariance for non-Abelian theories; lattice regularization destroys Poincare invariance, etc. Of course, in the limit as ɛ→0, these properties are recovered (if one was careful enough!). Because gauge and relativistic invariance are essential for QCD, we will use dimensional regularization that only destroys scale invariance. The method is related to so-called analytical regularization (Speer, 1968; Bollini, Giambiagi, and Gonzalez-Dominguez, 1964) and has been thoroughly deve-loped by ’t Hooft and Veltman (1972; see also Bollini and Giambiagi, 1972). It amounts to working in an arbitrary dimension, D = 4 — ɛ the physical limit is, ɛ→0 . Divergences appear as poles in 1/ɛ. As far as the author knows, there is no mathematically sound notion of dimension, in linear spaces, for D not a positive integer; but this should not worry us unduly: all we require are inter-polation formulas consistent with gauge and Poincare invariance, applicable to the evaluation of Feynman integrals. This we will accomplish in steps. First, consider a convergent integral of the (2π)D∫dDkf(k2), where typically f(k 2)=(k 2) r(k 2 −m ,r,m integers, and d D k=dk 0 dk 1...dk D−1; k 2 =(k 0)2−(k 1)2−...−(k D−1)2 Because f is analytic in the k 0 plane,1 we can rotate the integration from (− ∞, + ∞) to (− i∞, + i∞), a so-called Wick rotation We can recover an integration over (− ∞, + ∞) by then defining the new variable fix all the Cj (n) in (3.2.7). To have a unique theory, we have to give arbitrarily as many independent amplitudes as there are renormalization constants, Z.