Probability density function for neutrino masses and mixings

The anarchy principle leading to the seesaw ensemble is studied analytically with the usual tools of random matrix theory. The probability density function for the seesaw ensemble of $N\times N$ matrices is obtained in terms of a multidimensional integral. This integral involves all light neutrino masses, leading to a complicated probability density function. It is shown that the probability density function for the neutrino mixing angles and phases is the appropriate Haar measure. The decoupling of the light neutrino masses and neutrino mixings implies no correlation between the neutrino mass eigenstates and the neutrino mixing matrix and leads to a loss of predictive power when comparing with observations. This decoupling is in agreement with some of the claims found in the literature.

Figure 1

Probability density function for the seesaw ensemble with $N=1$. The red curve corresponds to the analytic result while the histogram corresponds to numerical results (with $5\times 10^4$ dimensionless light neutrino mass matrices generated). The left panel shows matrices with real elements ($\beta =1$) while the right panel shows matrices with complex elements ($\beta =2$).

Figure 2

Level density at large $N$ for the seesaw ensemble. The red curve corresponds to the approximated analytic result while the histogram corresponds to numerical results (with $10^3$ singular values generated from $50\times 50$ matrices). The left panel shows matrices with real elements ($\beta =1$) while the right panel shows matrices with complex elements ($\beta =2$). Although the fits seem good, the approximated analytic result for $\beta =2$ is without a doubt not correct.

Figure 3

Comparison between the probability density function at $N=1$ and the level density at large $N$ for the seesaw ensemble. The red curve corresponds to probability density function, the black curve corresponds to the approximated analytic result while the histogram corresponds to the same numerical results as in Fig. 2. The left panel shows matrices with real elements ($\beta =1$) while the right panel shows matrices with complex elements ($\beta =2$).