The Standard Model, and in particular, the quantum electrodynamics (QED) component of the Standard Model, assumes that the photon does not have mass (although photons do, of course, have energy, and hence are subject to gravity in general relativity in which gravity acts upon both matter and energy).
Now, almost nobody seriously thinks that the assumption of QED that the photon is massless is wrong, because the predictions of QED are more precise, and are indeed more precisely experimentally tested than any other part of the Standard Model, or for that matter almost anything in experimental physics whatsoever. There is no meaningful experimental or theoretical impetus to make the assumption that the photon is massless.
But, generalizations of Standard Model physics that parameterize deviations from the Standard Model expectation, provide a useful tool for devising experimental tests to confirm or contradict the Standard Model and to quantify how much deviation from the Standard Model has been experimentally excluded.
Also, any time one can demonstrate that it is possible to have a new kind of force with a massive carrier boson that behaves in a manner very much like QED, but not exactly like QED, that is theoretically rigorous, these theories, once their implications are understood, can be considered as possible explanations for explaining unsolved problems in physics.
For example, many investigators have considered a massive "dark photon" as a means by which dark matter fermions could be self-interacting very similar to the models discussed below, because self-interacting dark matter models seem to be better at reproducing the dark matter phenomena that astronomers observe, than dark matter models in which dark matter fermions interact solely via gravity and Fermi contact forces (i.e. the physical effects of the fact that two fermions can't be in the same place at the same time) with other particles and with each other.
The Proca Model and Podolsky Generalized Electrodynamics
A paper published in 2011 and just posted to arVix today for some reason evaluates the experimental limitations on this assumption.
The Proca model of Romanian physicist Alexandru Proca (who became a naturalized French citizen later in life when he married a French woman) was developed on the eve of World War II, mostly from 1936-1938 considers a modification of QED in which the photon has a tiny, but non-zero mass. Proca's equations still have utility because they describe the motion of vector mesons and the weak force bosons, both of which are massive spin-1 particles that operate only at short ranges as a result of their mass and short mean lifetimes.
Experimental evidence excludes the possibility that photons have Proca mass (according to the 2011 paper linked above and cited below) down to masses of about 10-39 grams (which is roughly equivalent to 10-6 eV/c2). This is on the order of 10,000 lighter than the average of the three neutrino masses (an average which varies by about a factor of ten between normal, inverted and degenerate mass hierarchies). The exclusion (assuming that no mass is discovered) would be about 100 times more stringent if an experiment proposed in 2007 is carried out. This exclusion for Proca mass is roughly equal to the energy of a photon with a 3 GHz frequency (the frequency of UHF electromagnetic waves used to broadcast television transmissions); visible light has more energy and a roughly 300 THz frequency (10,000 times more energetic).
The Particle Data Group's best estimate of the maximum mass of the photon is much smaller than the limit cited in the 2011 article, with a mass of less than 10-18 eV/c2 from a 2007 paper (twelve orders of magnitude more strict). A 2006 study cited by not relied upon by PDG claimed a limit ten times as strong. A footnote at the PDG entry based on some other 2007 papers notes that a much stronger limit can be imposed if a photon acquires mass at all scales by a means other than the Higgs mechanism (with formatting conventions for small numbers adjusted to be consistent with this post):
When trying to measure m one must distinguish between measurements performed on large and small scales. If the photon acquires mass by the Higgs mechanism, the large-scale behavior of the photon might be effectively Maxwellian. If, on the other hand, one postulates the Proca regime for all scales, the very existence of the galactic field implies m < 10-26 eV/c2, as correctly calculated by YAMAGUCHI 1959 and CHIBISOV 1976.Ordinarily a massive photon would break the gauge symmetry of QED, which would be inconsistent with all sorts of experimentally confirmed theoretical predictions that rely upon the fact that the gauge symmetry of QED is unbroken.
But, it is possible to find a loophole in the assumption that a massless photon would break the gauge symmetry of QED. Specifically, the Podolsky Generalized Electrodynamics model, proposed in the 1940s by Podolsky, incorporates a massive photon in a manner that does not break gauge symmetry. In this model, the photon has both a massless and massive mode, with the former interpreted as a photon and the later tentatively associated with the neutrino by the Podolsky when the model was formulated (an interpretation that has since been abandoned for a variety of reasons). In the Podolsky Generalized Electrodynamics model, Coulomb's inverse square law describing the electric force of a point charge is slightly modified. Podolosky Generalized Electrodynamics is equivalent to QED in the limit as Podolsky's constant "a" approaches zero.
Podolosky Generalized Electrodynamics is also notable because it can be derived as an alternative solution to one that uses a set of very basic assumptions to derive Maxwell's Equations from first principles, and does so in a manner that prevents the infinities found in QED because it has a point source (which Feynman and others solved for practical purposes with the technique of renormalization) from arising.
In the Podolsky Generalized Electrodynamics model, there is a constant "a" with units of length associated with the massive mode of the photon must have a value that is experimentally required to be smaller than the current sensitivity of any current experiments. But, "a" is required as a consequence of the "value of the ground state energy of the Hydrogen atom . . . to be smaller than 5.6 fm, or in energy scales larger than 35.51 MeV."
In practice (for reasons that are not obvious without reading the full paper), this means that deviations from QED due to a non-zero value of "a" could be observed only at high energy particle accelerators.
It isn't inconceivable, however, to imagine that Podolsky's constant had a value on the order of the Planck length (i.e. 1.6 * 10-35 meters), which would be manifest only at energies approaching the Planck energy which is far beyond the capacity of any man made experiment to create, a value which could be correct without violating any current experimental constraints that have been rigorously analyzed to date.
* B. Podolsky, 62 Phys. Rev. 68 (1942).
* B. Podolsky, C. Kikuchi, 65 Phys. Rev. 228 (1944).
* B. Podolsky, P. Schwed, 20 Rev. Mod. Phys. 40 (1948).
* R. R. Cuzinatto, C. A. M. de Melo, L. G. Medeiros, P. J. Pompeia, "How can one probe Podolsky Electrodynamics?", 26 International Journal of Modern Physics A 3641-3651 (2011) (arVix preprint linked to in post).