The photon magnetic moment has not a perpendicular component and is fully paramagnetic

TL;DR

This paper defends the original claim that photons in a magnetized vacuum are fully paramagnetic, clarifying misconceptions about a supposed perpendicular diamagnetic component caused by mathematical errors.

Contribution

It clarifies that the photon magnetic moment has no perpendicular component and corrects previous misconceptions caused by differentiation mistakes.

Findings

Photon magnetic moment is fully paramagnetic.

No perpendicular diamagnetic component exists.

Critique of previous claims is mathematically invalid.

Abstract

Our paper Phys. Rev. D \textbf{79}, 093002 (2009), in which it was shown the paramagnetic behavior of photons propagating in magnetized vacuum, is criticized in Phys. Rev. D \textbf{81}, 105019, (2010) and even claimed that the photon has a diamagnetic component. Here it is shown that such criticism is inadequate and that the alleged "perpendicular component" is due to a mistake in differentiating a vanishing term with regard to the magnetic field $B$, or either by mistaking the derivative of a scalar product as that of a dyadic product. A discussion on the physical side of the problem is also made.