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As is well-known, there exist totally three CP-violating phases in theleptonic sector if three ordinary neutrinos are massive Majorana particles. Inthis short note, we raise the question whether the number of sufficient andnecessary conditions for CP conservation in the leptonic sector with massiveMajorana neutrinos is three or four. An intuitive answer to this question wouldbe three, which is also the total number of independent CP-violating phases.However, we give a counter example, in which three conditions are in generalnot sufficient for CP conservation. Only for all the lepton masses and mixingangles within their experimentally allowed ranges can we demonstrate that it ispossible to find out three weak-basis invariants, which should be vanishing toguarantee leptonic CP conservation.

The number of sufficient and necessary conditions for CP conservation with Majorana neutrinos: three or four?