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Polarization in (quasi-)two-body decays and new physics

We consider two-body and quasi-two-body decays of the type $f_1 \to f_2 B$, where $f_1$ and $f_2$ are spin-1/2 fermions and $B$ a spin-0 or spin-1 boson. After recalling the non-covariant formalism for decay amplitudes, we derive the expression of the differential decay width and of the polarizations of the final spinning particles, both on- and off-shell. We find an intriguing geometrical interpretation of the results about the polarization. We also illustrate some methods for measuring the polarizations of the resonances and for optimizing data analysis. Then we propose applications to semi-leptonic weak decays, with a major attention to the $T$-odd component of the polarization; this may help to find, simultaneously, possible time-reversal violations and hints to physics beyond the standard model. We suggest also a $CPT$ test. Last, we discuss some $T$-odd observables for the production process of $f_1$ and for the study of the strong final state interactions of non-leptonic decays.