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Because a photon has a momentum, we can calculate the relativistic mass of it according to the special relativity. Also according to this theory, the relativistic mass is proportional to the rest mass when the speed is determined, which means only when the rest mass isn't zero can the relativistic mass not equals to zero. So we cannot consider the rest mass of photon equals to zero. However, once we consider the photon travels at the speed of light, the proportional factor between the relativistic mass and the static mass, $\gamma$, runs to infinity. So, the rest mass has to be zero when the momentum is determined.

This seems contradictory.

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    $\begingroup$ No equations of motion for photons contain a mass term. $\endgroup$ Dec 18, 2021 at 8:58

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The photon is an elementary particle of mass zero in the table of the Standard Model of particle physics. In particle physics the four vector algebra of special relativity is used in the SM to model all experimental observations.

invar mass

This is how the invariant mass of a particle is defined by the four vector $(E,p_x,p_y,p_z)$. Composite particles add up to one four vector and the composite has an invariant mass.

Note when the invariant mass is zero . When the momentum is zero, i.e. in the rest frame of the particle, one gets the $E=m_0c^2$.

The relativistic mass is not used in particle physics because of the confusions as in your question. One has to use the four vectors to be consistent with special relativity. The relativistic mass is an outdated concept .

The SM has been validated in most experiments up to now( the exceptions pointing to searches for new physics). The data are consistent with the mass of the photon being zero.

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