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Commutators of lepton mass matrices associated with seesaw and leptogenesis

The origin of tiny neutrino masses and the baryon number asymmetry of the Universe are naturally interpreted by the canonical seesaw and leptogenesis mechanisms, in which there are the heavy Majorana neutrino mass matrix $M_{\rm R}$, the Dirac neutrino mass matrix $M_{\rm D}$, the charged-lepton mass matrix $M_\ell$ and the effective (light) neutrino mass matrix $M_ν$. We find that ${\rm Im}\left(\det\left[ M^\dagger_{\rm D} M_{\rm D}, M^\dagger_{\rm R} M_{\rm R} \right]\right)$, ${\rm Im}\left(\det\left[ M_\ell M^\dagger_\ell, M_νM^\dagger_ν\right]\right)$ and ${\rm Im}\left(\det\left[ M_\ell M^\dagger_\ell, M_{\rm D} M^\dagger_{\rm D} \right]\right)$ can serve for a basis-independent measure of CP violation associated with lepton-number-violating decays of heavy neutrinos, flavor oscillations of light neutrinos and lepton-flavor-violating decays of charged leptons, respectively. We first calculate these quantities with the help of a standard parametrization of the $6\times 6$ flavor mixing matrix, and then discuss their implications on both leptogenesis and CP violation at low energy scales. A comparison with the weak-basis invariants of leptogenesis as proposed by Branco {\it et al} is also made.