  The four-point correlation function of primordial scalar perturbations has
parity-even and parity-odd contributions and the parity-odd signal in
cosmological observations is opening a novel window to look for new physics in
the inflationary epoch. We study the distinct parity-odd and even prediction
from the axion inflation model, in which the inflaton couples to a vector field
via a Chern-Simons interaction, and the vector field is considered to be either
approximately massless ($m_A \ll $ Hubble scale $H$) or very massive ($m_A \sim
H $). The parity-odd signal arises due to one transverse mode of the vector
field being predominantly produced during inflation. We adopt the in-in
formalism to evaluate the correlation functions. Considering the vector field
mode function to be dominated by its real part up to a constant phase, we
simplify the formulas for numerical computations. The numerical studies show
that the massive and massless vector fields give significant parity-even
signals, while the parity-odd contribution is about one to two orders of
magnitude smaller.
