A Question about Spin Induced Magnetic Moment

Does photon has spin induced magnetic moment? I think the answer is no. But why? You know, electron has. Does this have something to do with the fact that electron is massive while photon massless?

 

Some guy says it has something to do with photon's charge conjugation quantum number. If you have two photons' wave function [tex]\Psi_{p,\sigma}, \Psi_{p',\sigma'}[/tex], you can couple them to a electromagnetic current to measure the interaction and magnetic moment:

[tex](\Psi_{p',\sigma'},J^\mu\Psi_{p,\sigma})[/tex]

But RHS has charge conjugation number +1 while LHS has charge conjugation number -1. So it's zero!

I couldn't understand charge conjugation quantum number, though this theory sounds fancy.