Playing with Neutrino Masses

Most of what is known about neutrino masses and mixings results from studies of oscillation phenomena. We focus on those neutrino properties that are not amenable to such studies: $\Sigma$, the sum of the absolute values of the neutrino masses; $m_\beta$, the effective mass of the electron neutrino; and $m_{\beta\beta}$, the parameter governing neutrinoless double beta decay. Each of these is the subject of ongoing experimental or observational studies. Here we deduce constraints on these observables resulting from any one of six {\it ad hoc\} hypotheses that involve the three complex mass parameters $m_i$: (1) Their product or (2) sum vanishes; (3) Their absolute values, like those of charged leptons or quarks of either charge, do not form a triangle; (4) The $e$-$e$ entry of the neutrino mass matrix vanishes; (5) Both the $\mu$-$\mu$ and $\tau$-$\tau$ entries vanish; (6) All three diagonal entries are equal in magnitude. The title of this note reflects the lack of any theoretical basis for any of these simple assertions.