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