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1. If this is truely the case, and if we believe in E = mc2, then the photons have no energy of any kind either;
2. In this case, if an electron and a positron annihilate producing, say, two photons having each neither mass nor energy of any kind, then where the mass and the energy of the electron-positron pair go to?
3. Note that in this case (that is, assuming the invariant mass as the only legitimate entity) even this invariant mass would disappear entirely and with no traces.