Recall E=mc2, which equates mass and energy.

The logic can be constructed in many ways, and the following is one such. Take an isolated system (called a "particle") and accelerate it to some velocity

**v** (a vector). Newton defined the "momentum"

**p** of this particle (also a vector), such that

**p** behaves in a simple way when the particle is accelerated, or when it's involved in a collision. For this simple behaviour to hold, it turns out that

**p** must be proportional to

**v**. The proportionality constant is called the particle's "mass"

*m*, so that

**p** =

*m***v**.

In special relativity, it turns out that we are still able to define a particle's momentum

**p** such that it behaves in well-defined ways that are an extension of the newtonian case. Although

**p** and

**v** still point in the same direction, it turns out that they are no longer proportional; the best we can do is relate them via the particle's "relativistic mass"

*m*rel. Thus

**p** = mrel

**v** .

When the particle is at rest, its relativistic mass has a minimum value called the "rest mass"

*m*rest. The rest mass is always the same for the same type of particle. For example, all protons have identical rest masses, and so do all electrons, and so do all neutrons; these masses can be looked up in a table. As the particle is accelerated to ever higher speeds, its relativistic mass increases without limit.

It also turns out that in special relativity, we are able to define the concept of "energy"

*E*, such that

*E* has simple and well-defined properties just like those it has in newtonian mechanics. When a particle has been accelerated so that it has some momentum

*p* (the length of the vector

**p**) and relativistic mass

*m*rel, then its energy

*E* turns out to be given by

*E* =

*m*relc2 , and also

*E*2 =

*p*2c2 +

*m*2restc4 . (1)

There are two interesting cases of this last equation:

- If the particle is at rest, then
*p = 0*, and *E = m*rest*c*2.
- If we set the rest mass equal to zero (regardless of whether or not that's a reasonable thing to do), then
*E = pc*.

In classical electromagnetic theory, light turns out to have energy

*E* and momentum

*p*, and these happen to be related by

*E = pc*.

Quantum mechanics introduces the idea that light can be viewed as a collection of "particles": photons. Even though these photons cannot be brought to rest, and so the idea of rest mass doesn't really apply to them, we can certainly bring these "particles" of light into the fold of equation (1) by just considering them to have no rest mass. That way, equation (1) gives the correct expression for light,

*E = pc*, and no harm has been done. Equation (1) is now able to be applied to particles of matter

*and* "particles" of light. It can now be used as a fully general equation, and that makes it very useful.

Alternative theories of the photon include a term that behaves like a mass, and this gives rise to the very advanced idea of a "massive photon". If the rest mass of the photon were non-zero, the theory of quantum electrodynamics would be "in trouble" primarily through loss of gauge invariance, which would make it non-renormalisable; also, charge conservation would no longer be absolutely guaranteed, as it is if photons have zero rest mass. But regardless of what any theory might predict, it is still necessary to check this prediction by doing an experiment.

It is almost certainly impossible to do any experiment that would establish the photon rest mass to be exactly zero. The best we can hope to do is place limits on it. A non-zero rest mass would introduce a small damping factor in the inverse square Coulomb law of electrostatic forces. That means the electrostatic force would be weaker over very large distances.

Likewise, the behavior of static magnetic fields would be modified. An upper limit to the photon mass can be inferred through satellite measurements of planetary magnetic fields. The Charge Composition Explorer spacecraft was used to derive an upper limit of 6 × 10−16 eV with high certainty. This was slightly improved in 1998 by Roderic Lakes in a laboratory experiment that looked for anomalous forces on a Cavendish balance. The new limit is 7 × 10−17 eV. Studies of galactic magnetic fields suggest a much better limit of less than 3 × 10−27 eV, but there is some doubt about the validity of this method.

See:

E. Fischbach et al., Physical Review Letters

**73**, 514–517 25 July 1994.

Chibisov et al., Sov. Ph. Usp.

**19**, 624 (1976).

See also the Review of Particle Properties at

http://pdg.lbl.gov
You can get a mass metric from the energy of a photon, everything else is a matter of definition of the English words. It is a universal statement to say that photons do not have rest mass.

Coulomb's inverse square law implies that the photon does not have a mass. So if you wish find the lower limit for the photon mass, you should look at deviations from the Coulomb's law at large distances. If photon does indeed have a mass then the force between charged particles will go as the

Yukawa potential*.

Renormalization of the mass of an electron is studied within the framework of the Extended Holstein model at strong coupling regime and nonadiabatic limit. In order to take into account an effect of screening of an electron-phonon interaction on a polaron it is assumed that the electron- phonon interaction potential has the Yukawa form and screening of the electron-phonon interaction is due to the presence of other electrons in a lattice. The forces are derived from the Yukawa type electron-phonon interaction potential. It is emphasized that the early considered screened force is a particular case of the force deduced from the Yukawa potential and is approximately valid at large screening radiuses compared to the distances under consid- eration. The Extended Holstein polaron with the Yukawa type potential is found to be a more mobile than polaron studied in early works at the same screening regime.

See:

https://arxiv.org/pdf/1308.2197.pdf
*Yukawa potential is the effective non-relativistic description of the interaction of two particles due to the exchange of a massive particle of mass . It was proposed by H. Yukawa as a low-energy description of the strong interactions between nucleons, due to the exchange of massive particles, now known as pions.

See: Dark Universe phenomenology from the Yukawa potential at

The particle of electromagnetic radiation is often assumed to be massless, but the laws of physics do not require that assumption. If the photon has a mass, however, it must be exceedingly small.

Hartmann352