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We discuss domain walls from spontaneous breaking of Abelian discrete symmetries $Z_N$. A series of different domain wall structures are predicted, depending on the symmetry and charge assignments of scalars leading to the spontaneous symmetry breaking (SSB). A widely-existing type of domain walls are those separating degenerate vacua which are adjacent in the field space. We denote these walls as adjacency walls. In the case that $Z_N$ terms are small compared with the $U(1)$ terms, the SSB of $U(1)$ generates strings first and then adjacency walls bounded by strings are generated after the SSB of $Z_N$. For symmetries larger than $Z_3$, non-adjacent vacua exist, we regard walls separating them as non-adjacency walls. These walls are unstable if $U(1)$ is a good approximation. If the discrete symmetry is broken via multiple steps, we arrive at a complex structure that one kind of walls wrapped by another type. On the other hand, if the symmetry is broken in different directions independently, walls generated from the different breaking chains are blind to each other., 15 pages, 6 figures, 2 tables, discussions and refs added, published in PRD

We discuss a minimal flavor model with twin modular symmetries, leading to trimaximal ${\mathrm{TM}}_{1}$ lepton mixing in which the first column of the tri-bimaximal lepton mixing matrix is preserved. The model involves two modular ${S}_{4}$ groups, one acting in the neutrino sector, associated with a modulus field value ${\ensuremath{\tau}}_{SU}$ with residual ${Z}_{2}^{SU}$ symmetry, and one acting in the charged lepton sector, associated with a modulus field value ${\ensuremath{\tau}}_{T}$ with residual ${Z}_{3}^{T}$ symmetry. Apart from the predictions of ${\mathrm{TM}}_{1}$ mixing, the model leads to a new neutrino mass sum rule which implies lower bounds on neutrino masses close to current limits from neutrinoless double beta decay experiments and cosmology.

We explore the concept of effective alignments: contractions of multiple flavour symmetry breaking flavon fields. These contractions give rise to directions that are hard or impossible to obtain directly by breaking the flavour symmetry. Within this context, and using $S_4$ as the flavour symmetry to exemplify, we perform a phenomenological check of lepton flavour models built from pairing any two effective alignments up to order 2 (in flavon contractions). The check is performed for each pair of effective alignments in a framework with models of constrained sequential dominance type, in a basis where the charged leptons are diagonal. We thus obtain an indication of which effective alignments are interesting for model building, within this so-called $S_4$ landscape. We find three types of viable topologies and provide examples of models realizing this strategy for each topology., Comment: 18 pages, 5 figures, 6 tables

A phenomenological analysis of the general two Higgs doublet model is presented. Possible constraints of the Yukawa couplings are obtained from the $K^0-\bar{K}^0, B^0-\bar{B}^0$ and $D^0-\bar{D}^0$ mixings. A much larger $D^0-\bar{D}^0$ mixing than the standard model prediction is possible. It is shown that the emerging of various new sources of CP violation in the model could strongly affect the determination of the unitarity triangle. It can be helpful to look for a signal of new physics by comparing the extracted angle $\beta$ from two different ways, such as from the process $B\to J/\psi K_{S}$ and from fitting the quantities $|V_{ub}|, \Delta m_{B}$ and $\epsilon$.