Deviations of the Lepton Mapping Matrix from the Harrison-Perkins-Scott Form

TL;DR

This paper investigates how the leptonic mixing matrix deviates from the Harrison-Perkins-Scott form due to a specific imaginary perturbation in the charged lepton mass matrix, deriving constraints and bounds on these deviations.

Contribution

It introduces a simple hypothesis for deviations from the HPS form caused by a maximally T-violating perturbation, and derives parameter constraints independent of perturbation details.

Findings

Tau lepton matrix elements deviate by no more than ~0.0035 from HPS values.

Two parameters determine the form of the electron and muon parts of the matrix.

Constraints on mixing parameters are derived assuming negligible muon-to-tau mass ratio effects.

Abstract

We propose a simple set of hypotheses governing the deviations of the leptonic mapping matrix from the Harrison-Perkins-Scott (HPS) form. These deviations are supposed to arise entirely from a perturbation of the mass matrix in the charged lepton sector. The perturbing matrix is assumed to be purely imaginary (thus maximally $T$-violating) and to have a strength in energy scale no greater (but perhaps smaller) than the muon mass. As we shall show, it then follows that the absolute value of the mapping matrix elements pertaining to the tau lepton deviate by no more than $O((m_\mu/m_\tau)^2) \simeq 3.5 \times 10^{-3}$ from their HPS values. Assuming that $(m_\mu/m_\tau)^2 $ can be neglected, we derive two simple constraints on the four parameters $\theta_{12}$, $\theta_{23}$, $\theta_{31}$, and $\delta$ of the mapping matrix. These constraints are independent of the details of the imaginary $T$-violating perturbation of the charged lepton mass matrix. We also show that the $e$ and $\mu$ parts of the mapping matrix have a definite form governed by two parameters $\alpha$ and $\beta$; any deviation of order $m_\mu/m_\tau $ can be accommodated by adjusting these two parameters.