According to Einstein $E= mc^2$ so $m = E/ c^2$. So energy present in an atom contributes to its mass. So if a photon is a carrier of energy shouldn't it have mass since energy contributes to mass of an atom?
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$E=mc^2$ is the formula for an object with 0 momentum. If an object has momentum then the formula is $E^2/c^2=m^2 c^2+p^2$. A photon always has momentum and that momentum is $p=E/c$ so $m=0$
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