Neutrino mu-tau reflection symmetry and its breaking in the minimal seesaw

In this paper, we attempt to implement the neutrino $\mu$-$\tau$ reflection symmetry (which predicts $\theta^{}_{23} = \pi/4$ and $\delta = \pm \pi/2$ as well as trivial Majorana phases) in the minimal seesaw (which enables us to fix the neutrino masses). For some direct (the preliminary experimental hints towards $\theta^{}_{23} \neq \pi/4$ and $\delta \neq - \pi/2$) and indirect (inclusion of the renormalization group equation effect and implementation of the leptogenesis mechanism) reasons, we particularly study the breakings of this symmetry and their phenomenological consequences.Comment: 20 pages, 7 figures, accepted for publication in JHE

Probing the XYZXYZ states through radiative decays

In this work, we have adopted the spin rearrangement scheme in the heavy quark limit and extensively investigated three classes of the radiative decays: $\mathfrak{M}\to (b\bar{b})+\gamma$, $(b\bar{b})\to \mathfrak{M}+\gamma$, $\mathfrak{M} \to \mathfrak{M}^\prime+\gamma$, corresponding to the electromagnetic transitions between one molecular state and bottomonium, one bottomonium and molecular state, and two molecular states respectively. We also extend the same formalism to study the radiative decays of the molecular states with hidden charm. We have derived some model independent ratios when the initial or final states belong to the same spin flavor multiplet. Future experimental measurement of these ratios will test the molecular picture and explore the underlying structures of the $XYZ$ states.Comment: 21 pages, 10 tables Accepted by Phys.Rev.