  Measuring spectral distortions of the cosmic microwave background (CMB) is
attracting considerable attention as a probe of high energy particle physics in
the cosmological context, since PIXIE and PRISM have recently been proposed. In
this paper, CMB distortions due to resonant conversions between CMB photons and
light axion like particles (ALPs) are investigated, motivated by the string
axiverse scenario which suggests the presence of a plenitude of light axion
particles. Since these resonant conversions depend on the strength of
primordial magnetic fields, constraints on CMB distortions can provide an upper
limit on the product of the photon-ALP coupling constant g and the comoving
strength of primordial magnetic fields B. Potentially feasible constraints from
PIXIE/PRISM can set a limit g B < 10^{-16} GeV^{-1} nG for ALP mass, m_\phi <
10^{-14} eV. Although this result is not a direct constraint on g and B, it is
significantly tighter than the product of the current upper limits on g and B.
