  Parity violating physics beyond the standard model of particle physics
induces a rotation of the linear polarization of photons. This effect, also
known as cosmological birefringence (CB), can be tested with the observations
of the cosmic microwave background (CMB) anisotropies which are linearly
polarized at the level of $5-10\%$. In particular CB produces non-null CMB
cross correlations between temperature and B mode-polarization, and between E-
and B-mode polarization. Here we study the properties of the so called
D-estimators, often used to constrain such an effect. After deriving the
framework of both frequentist and Bayesian analysis, we discuss the interplay
between birefringence and weak-lensing, which, albeit parity conserving,
modifies pre-existing TB and EB cross correlation.
