Why does local gauge invariance suggest renormalizability?

Question

I'm reading Gauge Field Theories: An Introduction with Applications by Mike Guidry and this particular remark is not obvious to me:

A tempting avenue is suggested by the QED paradigm, for if a local gauge invariance could be imposed on the weak interaction phenomenology we might expect the resulting theory to be renormalizable. [Guidry, section §6.5, p. 232]

Is there an obvious argument for this "local gauge invariance suggests renormalizability" remark? I should add that I still tend to get lost in the streets of renormalization when unsupervised, i.e. I'm not familiar enough with the entire concept to have any real intuition about it. (references on renormalizability that might help are of course also welcome)

This post imported from StackExchange Physics at 2014-05-04 11:28 (UCT), posted by SE-user Wouter

Comments

I suspect that even though its badly phrased. They just mean that they are working with a large cutoff, as in the SM, and so the nonrenormalizable terms are small

This post imported from StackExchange Physics at 2014-05-04 11:28 (UCT), posted by SE-user JeffDror

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I'd agree with Jeff's assessment, it seems badly phrased. A priori, a gauge symmetry by no means indicates renormalizability.

This post imported from StackExchange Physics at 2014-05-04 11:28 (UCT), posted by SE-user JamalS

Answer 1

This statement is related to the fact that renormalizability of a theory depends of the mass dimension of the coupling constants in the Lagrangian. Couplings with zero or positive mass dimensions lead to renormalizable theories. As a consequence, writing down only terms with appropriate mass dimensions is required in order to construct a theory that is renormalizable.

In quantum electrodynamics, all operators consistent with both (local) gauge and Poincaré symmetry that are at most of mass dimension 4 automatically satisfy the above criterion. One could understand the statement in the reference in this way. Of course, this does not hold for term of higher dimension.

This post imported from StackExchange Physics at 2014-05-04 11:28 (UCT), posted by SE-user Frederic Brünner

Comments

Imposing only gauge invariance and Poincare symmetry we can still have other nonrenormalizable terms such as $\bar{\psi}\psi \bar{\psi}\psi$,$\bar{\psi}\psi F_{\mu\nu} F^{\mu\nu}$, etc.

This post imported from StackExchange Physics at 2014-05-04 11:28 (UCT), posted by SE-user JeffDror

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@JeffDror: I have edited my answer.

This post imported from StackExchange Physics at 2014-05-04 11:28 (UCT), posted by SE-user Frederic Brünner