Suppose you have an action $S(\epsilon) = S_1 + S_2 + \epsilon\, S_\mathrm{int}$. Assume that $S_1$ is gauge invariant under the action of the group $G$ and $S_2$ is gauge invariant under the action of the group $H$, such that the action $S_1$ + $S_2$ is gauge invariant under the action of $G\times H$. Suppose that $S_\mathrm{int}$ breaks the gauge group down to $F \in G\times H$, that is, the action $S(\epsilon)$ is gauge invariant under the action of $F$ only.

This implies that $$ S(0)=S_1+S_2= \lim\limits_{\epsilon\, \rightarrow\, 0}\,S(\epsilon) $$ has a wider gauge group than $S(\epsilon)$, that is, the gauge symmetry of the action $S(\epsilon)$ is enhanced when sending $\epsilon$ to $0$. For clarity, by "gauge invariant" I mean that the theory has a redundancy of description.

Does this imply that the parameter $\epsilon$ is technically natural?

To clarify, I mean "natural" in the sense of 't Hooft, e.g. as discussed in

The question is motivated by the fact that I could only find the concept of technical naturalness associated with global symmetries in the literature. On the other hand, I did not find any statement saying that it does not hold in the case of gauge symmetries.

EDIT: I can provide a simpler example to clarify ever more what I mean. Consider the Proca lagrangian density for a real massive spin-1 field,

$$ \mathcal{L}=-\dfrac{1}{2}F^{\mu\nu}F_{\mu\nu}+m^2A_\mu A^\mu, \qquad F_{\mu\nu}= \partial_\mu A_\nu - \partial_\nu A_\mu. $$

The corresponding Proca action is not invariant under the gauge group $U(1)$, but taking the limit $m\rightarrow 0$ gives us the action for a free photon, which is gauge invariant under $U(1)$. Hence, sending $m\rightarrow 0$ enhances the gauge symmetry of the action.

In this particular case, my question becomes: is the Proca mass $m$ natural in the sense of 't Hooft? In other words, is a small Proca mass $m$ protected against large quantum corrections, the latter being proportional to the small mass itself?

  • $\begingroup$ By a “gauge symmetry” do you mean a redundancy of description or a symmetry whose parameters can be spacetime functions? $\endgroup$ – knzhou Jul 1 '19 at 10:47
  • $\begingroup$ I mean a redundancy of description in the theory. $\endgroup$ – Frank Jul 1 '19 at 11:32

Gauge symmetries are redundancies of the theory, i.e. they remove degrees of freedom -- consider QED, where the four components of the gauge field give rise to only two degrees of freedom. Hence, gauge symmetries cannot be approximate: two states related by a gauge symmetry are strictly equivalent (as BobKnighton points out, at least locally; globally, there might be subtleties, hwich do not change to argument here), and when the gauge group would change, the number of degrees of freedom would change as well.

(Note that that is still true for "spontaneously broken" theories such as electroweak theory in the Higs phase: The (original) action is always invariant under gauge transformations, Higgs vev or not.)

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  • $\begingroup$ In the considered system (which I now describe in more detail in the question) the action changes when changing the gauge group. Hence, as you say, the degrees of freedom also change. The thing is that, if all of these symmetries were global symmetries and not gauge symmetries, then, if I am not mistaken, the parameter $\epsilon$ would be considered to be natural in the sense of 't Hooft. Is that the case also when these symmetries are gauge symmetries? $\endgroup$ – Frank Jul 1 '19 at 12:15
  • $\begingroup$ "...two states related by a gauge symmetry are strictly equivalent..." This statement isn't strictly true. For instance, two nonabelian gauge fields which differ by a gauge transformation with nontrivial falloff at infinity can correspond to two different physical states (see, for instance, the classification of instantons). $\endgroup$ – Bob Knighton Jul 1 '19 at 12:32
  • $\begingroup$ @BobKnighton You're right, I've added a comment $\endgroup$ – Toffomat Jul 1 '19 at 13:26

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