Electrically neutral particles such as neutrinos can have nonvanishing magnetic dipole moments. Spin-1 particles, e.g., deuterium nuclei, can also have dipole moments. Googling seems to show that the Z boson has a magnetic moment.
So is there an elementary argument that explains why the photon has a zero magnetic moment? By analogy with the Bohr magneton and nuclear magneton, we might actually expect that zero mass would produce infinite dipole moment. Altschul 2007 has some discussion of empirical bounds and difficulties in creating a theory in which the moment doesn't vanish, but I'm having trouble translating anything in the paper into an elementary argument for why it's so much easier to have the moment vanish.
Altschul, "Astrophysical Bounds on the Photon Charge and Magnetic Moment." Astropart. Phys. 29 no. 4, pp. 290–298 (2008), arXiv:0711.2038.
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