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My understanding is that massive and massless QED share some key physical features including (see this PSE post and 8-4 of Ref. 1):

  • renormalizability
  • charge conservation

The key differences of massive QED being:

  1. A massive photon mediates a short range force.
  2. A massive photon has a 3rd, longitudinal polarization.
  3. Massive QED is not gauge invariant.

It seems to me that while renormalizability and charge conservation are features we might naturally like to impose on a QED, the consequences of a massive photon seem to be experimental questions, rather than theoretical constraints.

We can contrast this with non-Abelian gauge theory, where gauge invariance is required for renormalizability; so a massless gauge boson expected in the absence of a Higgs mechanism (see this PSE post and section 12-5-2 of Ref. 1).

One "benefit" of local gauge invariance is that it implies a global symmetry and thus a conserved Noether current. But I don't see the benefit to imposing a local gauge invariance instead of simply imposing the global symmetry directly, since either either way you get a conserved current.

Also, since gauge symmetry is a custodial symmetry for the photon mass, the photon mass is technically natural (see Ref. 2 section 22.6). So it seems that there is no fundamental problem with QED having a very small photon mass.

So my two related questions are:

  • Is there a theoretical reason to "prefer" that QED is massless?
  • Is there a theoretical reason to "prefer" a gauge invariant interaction for a matter field with an abelian global symmetry?

Please correct any misunderstandings I have in my discussion above!

References:

  1. Itzykson & Zuber's Quantum Field Theory
  2. Schwartz's Quantum Field Theory & the Standard Model
H. T. Tom
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  • GOing even more extreme, is it possible that the whole story of gauge is just a historical accident because we discovered $E$ and $B$ first and only after $A$? In the end the gauge story comes electromagnetism when you introduce the four potential $A$, but of course the Maxwell Equations may be a bit wrong and include the mass terms. – Quillo Nov 16 '21 at 15:29
  • This answer has some interesting insight: https://physics.stackexchange.com/a/4870/226902 – Quillo Nov 16 '21 at 15:47
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    Thanks for linking Rafael’s answer above—it seems to be consistent with the idea that the masslessness of the photon is simply an experimental fact, rather that something that is required or “preferred” by any theoretical concepts. – H. T. Tom Dec 08 '21 at 17:31

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