Mellin transform approach to rephasing invariants

In the low-energy effective theory of neutrinos, the Haar measure for unitary matrices is very likely to give rise to something similar to the observed Pontecorvo-Maki-Nakagawa-Sakata matrix. Assuming the Haar measure, we determine the probability density functions for all quadratic, quartic Majorana, and quartic Dirac rephasing invariants for an arbitrary number of neutrino generations. We show that for a fixed number of neutrinos, all rephasing invariants of the same type have the same probability density function under the Haar measure. We then compute the moments of the rephasing invariants to determine, with the help of the Mellin transform, the three probability density functions. We finally investigate the physical implications of our results in function of the number of neutrinos.

Figure 1

Quadratic rephasing invariant PDFs for different values of $N$. The red curves correspond to the analytic results while the histograms correspond to the numerical results with a sample of $5\times {10}^4$ unitary matrices.

Figure 2

Quartic Majorana rephasing invariant PDFs for different values of $N$. The red curves correspond to the analytic results, while the histograms correspond to the numerical results with a sample of $5\times {10}^4$ unitary matrices.

Figure 3

Quartic Dirac rephasing invariant PDFs for different values of $N$. The red curves correspond to the analytic results, while the histograms correspond to the numerical results with a sample of $5\times {10}^4$ unitary matrices.

Figure 4

Quartic rephasing invariant average values (solid red lines and black dots) in function of the neutrino number for Majorana (left panel) and Dirac (right panel) rephasing invariants. In the left panel, the blue and green dots represent the maximum allowed values (for $N=3$) calculated from the experimental values for the mixing angles. In the right panel, the blue dot represents the best-fit observed value (for $N=3$) and the green dot represents, as in the left panel, the maximum allowed value (for $N=3$) also calculated from the experimental values for the mixing angles.