8. RENORMALIZABLE SCALAR FIELD THEORIES 21 This fixed point, if it exists, would be a scaling limit of the theory it would necessarily be a scale-invariant theory. For instance, it is expected that Yang-Mills theory is renormalizable in this sense, and that the scaling limit is a free theory. This ideal definition is difficult to make sense of perturbatively (when we treat as a formal parameter). For instance, suppose a coupling constant c changes to c l c = c + c log l + · · · Non-perturbatively, we might think that 0, so that this flow converges to a fixed point. Perturbatively, however, is a formal parameter, so it appears to have logarithmic growth. Our perturbative definition can be interpreted as saying that a pertur- bative theory is renormalizable if, at first sight, it looks like it might be non-perturbatively renormalizable in this sense. For instance, if it contains coupling constants which are of polynomial growth in l, these will proba- bly persist at the non-perturbative level, implying that the theory does not converge to a fixed point. One can make a more refined perturbative definition by requiring that the logarithmic growth which does appear is of the correct sign (thus distin- guishing between c l c and c l− c). This more refined definition leads to asymptotic freedom, which is the statement that a theory converges to a free theory as l 0. 8. Renormalizable scalar field theories Now that we have a definition of renormalizability, the next question to ask is: which theories are renormalizable? It turns out to be straightforward to classify all renormalizable scalar field theories. 8.1. Suppose we have a local action functional S of a scalar field on Rn, and suppose that S is translation invariant. We say that S is of dimension k if S(Rl(φ)) = lkS(φ). Recall that Rl(φ)(x) = l1−n/2φ(−1lx). Every translation invariant local action functional S is a finite sum of terms of some dimension. For instance: R4 φ D φ and R4 φ4 are of dimension 0 R4 φ3 ∂xi φ is of dimension 1 R4 φ2 is of dimension 2
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