  Recent literature has shown that photon-photon forward scattering mediated by
Euler-Heisenberg interactions may generate some amount of the circular
polarization ($V$ modes) in the cosmic microwave background (CMB) photons.
However, there is an apparent contradiction among the different references
about the predicted level of the amplitude of this circular polarization. In
this work, we will resolve this discrepancy by showing that with a quantum
Boltzmann equation formalism we obtain the same amount of circular polarization
as using a geometrical approach that is based on the index of refraction of the
cosmological medium. We will show that the expected amplitude of $V$ modes is
expected to be $\approx$ 8 orders of magnitude smaller than the amplitude of
$E$-polarization modes that we actually observe in the CMB, thus confirming
that it is going to be challenging to observe such a signature. Throughout the
paper, we also develop a general method to study the generation of $V$ modes
from photon-photon and photon-spin-1-massive-particle forward scatterings
without relying on a specific interaction, which thus represent possible new
signatures of physics beyond the Standard Model.
