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The measured speed of neutrinos can't be separated from $c$ within experimental limits. The standard model includes no mechanism to grant neutrinos mass. The mass of neutrinos can't be detected. Nevertheless, if you were to conclude that neutrinos are massless you would be wrong (as they experience subjective time and so can't travel at $c$).

Can we falsify the hypothesis that photons have a tiny mass and travel just a little bit slower than $c$? What's the largest mass photons could have?

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  • $\begingroup$ Did you read what Wikipedia has to say on this topic? $\endgroup$ – G. Smith Jan 21 '20 at 5:21
  • $\begingroup$ (headpalm) I read 7 or 8 wikipedia articles to try to figure this out this and never the one on photons! $\endgroup$ – Robert Cooper Jan 21 '20 at 5:22
  • $\begingroup$ Hi Robert. One difference between photons and neutrinos is that the Higgs mechanism predicts the mass of the photon is zero so we have good reason to think this is the case. By contrast the Standard Model says nothing about the neutrino mass - the zero neutrino mass originally assumed was just an assumption without any theoretical justification. So the fact that the neutrino turned out to have a mass is less of a surprise than it would be if the photon turned out to have a mass. $\endgroup$ – John Rennie Jan 21 '20 at 5:27