Abstract: This paper reports the observation of the decays B-(s)(0) -> D-s1(2536)K--/+(+/-) using proton-proton collision data collected by the LHCb experiment, corresponding to an integrated luminosity of 9 fb(-1). The branching fractions of these decays are measured relative to the normalisation channel B-0 -> (D) over bar (K+K-)-K-0. The D-s1(2536)(-) meson is reconstructed in the (D) over bar*(2007)K-0(-) decay channel and the products of branching fractions are measured to be B(B-s(0) -> D-s1(2536)K--/+(+/-)) x B(D-s1(2536)(-) -> D ($) over bar*(2007)K-0(-)) = (2.49 +/- 0.11 +/- 0.12 +/- 0.25 +/- 0.06) x 10(-5), B(B-0 -> D-s1(2536)K--/+(+/-) ) x B(D-s1(2536)(-) -> (D) over bar*(2007)K-0(-)) = (0.510 +/- 0.021 +/- 0.036 +/- 0.050) x 10(-5). The first uncertainty is statistical, the second systematic, and the third arises from the uncertainty of the branching fraction of the B-0 -> (D) over bar (K+K-)-K-0 normalisation channel. The last uncertainty in the B-s(0) result is due to the limited knowledge of the fragmentation fraction ratio, f(s)/f(d). The significance for the B-s(0) and B-0 signals is larger than 10 sigma. The ratio of the helicity amplitudes which governs the angular distribution of the D-s1(2536)(-) -> (D) over bar*(2007)K-0(-) decay is determined from the data. The ratio of the S- and D-wave amplitudes is found to be 1.11 +/- 0.15 +/- 0.06 and the phase difference between them 0.70 +/- 0.09 +/- 0.04 rad, where the first uncertainty is statistical and the second systematic.

Abstract: The so-called Golden Mass Relation provides a testable correlation between charged-lepton and down-type quark masses, that arises in certain flavor models that do not rely on Grand Unification. Such models typically involve broken family symmetries. In this work, we demonstrate that realistic fermion mass relations can emerge naturally in modular invariant models, without relying on ad hoc flavon alignments. We provide a model-independent derivation of a class of mass relations that are experimentally testable. These relations are determined by both the Clebsch-Gordan coefficients of the specific finite modular group and the expansion coefficients of its modular forms, thus offering potential probes of modular invariant models. As a detailed example, we present a set of viable mass relations based on the Gamma 4 approximately equal to S4 symmetry, which have calculable deviations from the usual Golden Mass Relation.

Abstract: In this paper we discuss SU(5)3 with cyclic symmetry as a possible grand unified theory (GUT). The basic idea of such a tri-unification is that there is a separate SU(5) for each fermion family, with the light Higgs doublet(s) arising from the third family SU(5), providing a basis for charged fermion mass hierarchies. SU(5)3 tri-unification reconciles the idea of gauge non-universality with the idea of gauge coupling unification, opening the possibility to build consistent non-universal descriptions of Nature that are valid all the way up to the scale of grand unification. As a concrete example, we propose a grand unified embedding of the tri-hypercharge model \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\text{U}}{\left(1\right)}_{Y}<^>{3}$$\end{document} based on an SU(5)3 framework with cyclic symmetry. We discuss a minimal tri-hypercharge example which can account for all the quark and lepton (including neutrino) masses and mixing parameters. We show that it is possible to unify the many gauge couplings into a single gauge coupling associated with the cyclic SU(5)3 gauge group, by assuming minimal multiplet splitting, together with a set of relatively light colour octet scalars. We also study proton decay in this example, and present the predictions for the proton lifetime in the dominant e+pi 0 channel.