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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