Quark mixing sum rules and the right unitarity triangle

Abstract

In analogy with the recently proposed lepton mixing sum rules, we derive quark mixing sum rules for the case of hierarchical quark mass matrices with 1-3 texture zeros, in which the separate up and down type 1-3 mixing angles are approximately zero, and V ub is generated from V cb as a result of 1-2 up type quark mixing. Using the sum rules, we discuss the phenomenological viability of such textures, including up to four texture zeros, and show how the right-angled unitarity triangle, i.e., α ≈ 90• , can be accounted for by a remarkably simple scheme involving real mass matrices apart from a single element being purely imaginary. In the framework of grand unified theories, we show how the quark and lepton mixing sum rules may combine to yield an accurate prediction for the reactor angle.

Right unitarity triangles and tri-bimaximal mixing from discrete symmetries and unification

“…For hierarchical mass matrices, this corresponds to the 1-2 and the 2-2 elements of the mass matrix being either purely real or imaginary. Such textures have also been considered previously in [5][6][7], and in [5] a SUSY Grand Unified Theory (GUT) with a continuous family symmetry SU(3) × SU(3) responsible for such textures has been proposed, however the vacuum alignment responsible for the spontaneous breaking of the family symmetry was not studied.…”

“…Such a right unitarity triangle was suggested long ago, when the error bar on α was much larger, by Fritzsch and collaborators as a natural consequence of having quark mass matrices with zeros in the 1-3 element [4]. In the light of recent data, this observation has gained increased momentum, and there have been several papers that attempt to predict α ≈ 90 • by postulating up-type and down-type quark mass matrices with the elements M u,d ij being either purely real or purely imaginary, with texture zeros in the 1-3 elements, M u,d 13 = 0 [5][6][7] (see also [8]). Under these assumptions, it has been shown that the prediction α ≈ 90 • can be understood from a simple analytic "phase sum rule" relation [7] which relates the angle α to phases arising from the quark mass matrices.…”

“…In the light of recent data, this observation has gained increased momentum, and there have been several papers that attempt to predict α ≈ 90 • by postulating up-type and down-type quark mass matrices with the elements M u,d ij being either purely real or purely imaginary, with texture zeros in the 1-3 elements, M u,d 13 = 0 [5][6][7] (see also [8]). Under these assumptions, it has been shown that the prediction α ≈ 90 • can be understood from a simple analytic "phase sum rule" relation [7] which relates the angle α to phases arising from the quark mass matrices. To be precise, the "phase sum rule" can be expressed as α ≈ δ d 12 − δ u 12 [7], where the phases δ d 12 and δ u 12 are the arguments of the complex 1-2 rotation angles in the up-type and down-type quark mass matrices, as defined in [7].…”

“…Under these assumptions, it has been shown that the prediction α ≈ 90 • can be understood from a simple analytic "phase sum rule" relation [7] which relates the angle α to phases arising from the quark mass matrices. To be precise, the "phase sum rule" can be expressed as α ≈ δ d 12 − δ u 12 [7], where the phases δ d 12 and δ u 12 are the arguments of the complex 1-2 rotation angles in the up-type and down-type quark mass matrices, as defined in [7]. To explain α ≈ 90 • one might therefore simply try to realise δ d 12 = 90 • , δ u 12 = 0 or alternatively δ d 12 = 0, δ u 12 = −90 • in a model of flavour.…”

“…Both models illustrate that the right-angled CKM unitarity triangle can indeed be understood from an underlying flavour model with discrete symmetries which were introduced previously for the purpose of providing an explanation of tri-bimaximal lepton mixing. In both models the quark mass matrices have 1-3 texture zeros and satisfy the "phase sum rule" derived in [7], while the lepton mass matrices lead to the "lepton mixing sum rule" [14,15] together with a new prediction that the leptonic CP violating oscillation phase is close to either 0 • , 90 • , 180 • , or 270 • depending on the model, with neutrino masses being purely real (no complex Majorana phases). We summarise and conclude the paper in Section 5, providing appendices on the more technical aspects of the models including their ultraviolet completion.…”