p142

No-renormalizable interactions

(Addendum to those who know RG well)

In the perturbative renormalization in field theory, the intereaction term with a parameter $g$ of negative energy dimension (i.e., $[g] = E^a$ with $a < 0$)is called an non-renormalizable term, which behaves as (a) in Fig. 3.3. Its effect does not diverge in the low energy limit $E \rightarrow 0$ (The relevant dimensionless quantity is $gE^{-a}$ which vanishes in the low energy limit). Therefore, to describe observable low energy phenomena (i.e., in the field theory) we may ignore all the non-renormalizable interactions. That is, in consequence, we may choose a model relying on renormalizability. In other words, the model need not reflect the reality of the high energy world (rather, non-renormalizable interactions do corresponds to definite reality at high energy scales, but they cannot be observed by the observation at our scales). Furthermore, even relativistic invariance need not be demanded at high energy scales (we have only to demand it at the low energy scales; notice that symmetry tends to be recovered at low energy scale. Look at the lattice gauge theory).