  If a primordial magnetic field (PMF) is present during photon decoupling and
afterward, a finite neutrino mass can affect all modes of the CMB. In this
work, we expand on earlier studies of the scalar mode effects by constructing
the vector and tensor mode equations in the presence of massive neutrinos and a
PMF. We compute the power spectrum of the various modes in an illustrative
example and find that the neutrino mass can significantly affect the vector and
tensor modes when a PMF exists, while the effects are negligible for no PMF.
The most prominent result of the present analysis is the behavior of the EE
component of the tensor mode at low multipoles. For massive neutrinos the EE
mode can become comparable to the observed primary anisotropy. Therefore, if
and when the EE mode power spectrum is measured at low multipoles the
possibility exists to place a strong constraint on the sum of the neutrino
masses.
