  In this work we present the effective field theory of primordial statistical
anisotropies generated during anisotropic inflation involving a background
$U(1)$ gauge field. Besides the usual Goldstone boson associated with the
breaking of time diffeomorphism we have two additional Goldstone bosons
associated with the breaking of spatial diffeomorphisms. We further identify
these two new Goldstone bosons with the expected two transverse degrees of the
$U(1)$ gauge field fluctuations. Upon defining the appropriate unitary gauge,
we present the most general quadratic action which respects the remnant
symmetry in the unitary gauge. The interactions between various Goldstone
bosons leads to statistical anisotropy in curvature perturbation power
spectrum. Calculating the general results for power spectrum anisotropy, we
recover the previously known results in specific models of anisotropic
inflation. In addition, we present novel results for statistical anisotropy in
models with non-trivial sound speed for inflaton fluctuations. Also we identify
the interaction which leads to birefringence-like effects in anisotropic power
spectrum in which the speed of gauge field fluctuations depends on the
direction of the mode propagation and the two polarization of gauge field
fluctuations contribute differently in statistical anisotropy. As another
interesting application, our EFT approach naturally captures interactions
generating parity violating statistical anisotropies.
