It is shown that one can in principle constrain the $CP$-violating parameter $\delta$ from measurements of four independent $|V_{ij}{|}^2$’s, or three $|V_{ij}{|}^2$’s and a ratio of two of them, in the leptonic sector. To quantify our approach, using unitarity, we derive simple expressions in terms of four independent $|V_{ij}{|}^2$’s for $\cos \delta$, and an expression for ${sin}^2\delta$ from $J^2$. Thus, depending on the values for $|V_{ij}|$ and their accuracy, we can set meaningful limits on $|\delta |$. To illustrate numerically, if $|V_{\mu 1}{|}^2$ is close to 0.1 with a 10% precision, and if $|V_{e3}{|}^2$ is larger than 0.005 and for values of $|V_{e2}{|}^2$ and $|V_{\mu 3}{|}^2$ that stay within $\pm 0.1$ of the current experimental data leads to a bound: $\pi /2\le |\delta |\le \pi$. Alternatively, a certain combination of parameters with values of $|V_{e3}{|}^2$ larger than 0.01 leads to a closed bound of $73\le |\delta |\le 103$. In general, we find that it is better to use $|V_{\mu 1}{|}^2$ or $|V_{\tau 1}{|}^2$ as the fourth independent $|V_{ij}{|}^2$ and that, over most of the parameter space, $\delta$ is least sensitive to $|V_{e3}{|}^2$. With just three independent measurements (solar, atmospheric, and reactor), it is impossible to set limits on the $CP$ phase. In this respect, we study the use of ultrahigh energy cosmic ray neutrino fluxes as the additional fourth information. We find that within the SM, neutrino fluxes of all three flavors will be very similar but that pushing current neutrino data to their extreme values still allowed, ratios of cosmic neutrino fluxes can differ by up to 20%; such large discrepancies could imply negligibly small $CP$ violation. We also study a nonradiative neutrino decay model and find that the neutrino fluxes can differ by a factor of up to 3 within this model and that an accuracy of 10% on the neutrino fluxes is sufficient to set interesting limits on $\delta$.

• Received 5 May 2006

DOI:https://doi.org/10.1103/PhysRevD.74.033013