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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)

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  • $\begingroup$ 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 $\endgroup$
    – JeffDror
    Commented Apr 29, 2014 at 10:18
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    $\begingroup$ I'd agree with Jeff's assessment, it seems badly phrased. A priori, a gauge symmetry by no means indicates renormalizability. $\endgroup$
    – JamalS
    Commented Apr 29, 2014 at 12:32
  • $\begingroup$ @JeffDror Pretending that the proof of "local gauge symmetry ensures renormalizability" be complicated (I'm not sure) even for the simple local gauge theories (e.g. QED) can we give heuristic or physical arguments to understand this feature? $\endgroup$
    – SRS
    Commented Nov 5, 2018 at 8:23

1 Answer 1

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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 terms of higher dimension.

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    $\begingroup$ 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. $\endgroup$
    – JeffDror
    Commented Apr 29, 2014 at 3:38
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    $\begingroup$ @JeffDror: I have edited my answer. $\endgroup$
    – Frederic Brünner
    Commented May 3, 2014 at 18:05
  • $\begingroup$ @FredericBrünner "Couplings with zero or positive mass dimensions lead to renormalizable theories." This statement is incorrect as it stands. The intermediate vector boson theory of weak interactions with explicitly put in mass term in the Lagrangian, is a theory with non-negative mass dimensional couplings. Yet the theory is non-renromalizable. Having only non-negative mass dimensional parameters is only a necessary condition for renormalizability, not sufficient. $\endgroup$
    – Solidification
    Commented Apr 13 at 6:05

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