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We show that the large, direct CP-violation parameter A_ππ=-C_ππ, reported by the BELLE collaboration in the decays , implies an unusual situation in which the presence of a very large difference between two strong-interaction phases (~ -110°) plays an essential role. We make the demonstration within a model of strong, two-body quasi-elastic interactions between physical hadrons. The model can accommodate a large difference between two strong-interaction phases, for which it provides a natural enhancement.

We first briefly review tests on CPT invariance based on the consequences of the CPT theorem and then present some possible CPT tests due to exotic models in which some of the CPT conditions are lost, such as those without hermiticity.

Using the functional analysis method, we present the exact solutions of the double ring-shaped oscillator (DRSO) potential with certain parity in the cylindrical coordinates. Such a quantum system is separated to two differential equations, i.e. a one-dimensional harmonic oscillator plus an inverse square term and a two-dimensional harmonic oscillator plus an inverse square term. The key point is how to find the adapted symmetrical solutions of the one-dimensional harmonic oscillator plus an inverse square term at the singular point $z=0$. The obtained results are compared with those in the spherical coordinates. We also explore intimate connections $E_T(i\rho )=-E_T(\rho )$ and $E_z(iz)=-E_z(z)$ by substituting $\rho \to i\rho$ and $z\to iz$.

A right–right–left extension of the Standard Model is proposed. In this model, SM gauge group $\mathrm{\text{SU}}{(2)}_L\otimes \mathrm{\text{U}}{(1)}_Y$ is extended to $\mathrm{\text{SU}}{(2)}_L\otimes \mathrm{\text{SU}}{(2)}_R\otimes \mathrm{\text{SU}}{(2)}_R^{\prime }\otimes \mathrm{\text{SU}}{(2)}_L^{\prime }\otimes \mathrm{\text{U}}{(1)}_Y$. The gauge symmetries $\mathrm{\text{SU}}{(2)}_R^{\prime }$, $\mathrm{\text{SU}}{(2)}_L^{\prime }$ are the mirror counterparts of the $\mathrm{\text{SU}}{(2)}_L$ and $\mathrm{\text{SU}}{(2)}_R$, respectively. Parity is spontaneously broken when the scalar Higgs fields acquire vacuum expectation values (VEVs) in a certain pattern. Parity is restored at the scale of $\mathrm{\text{SU}}{(2)}_L^{\prime }$. The gauge sector has a unique pattern. The scalar sector of the model is optimum, elegant and unique.

In this paper, we consider the non-commutative quantum mechanics (NCQM) with parity (or space reflection) in two dimensions. Using the parity operators R_i, we construct the deformed Heisenberg algebra with parity in the non-commutative plane. We use this algebra to discuss the isotropic harmonic Hamiltonian with parity.

A left-right symmetric mirror model restoring parity at a high scale in a way such that the mirror fermions and mirror gauge sector simultaneously could exist at TeV scale is discussed. We also provide an ultraviolet completion of the model with vector-like fermions, and discuss some theoretical and phenomenological implications of this model.

We examine the basic matrix equation for a spin-2 field in spherical coordinates and a tetrad of Minkowski spacetime. The technique of Wigner D-functions is used to separate the variables. The operators of energy, the square and the third projection of the total angular momentum, and the space reflection operators are then diagonalized on the solutions. The separation of the variables is done, the problem reduces to 11 radial second-order differential equations for 11 variables related to the scalar and symmetric tensors. We then consider the case of a massless field. The four gauge solutions for the massless spin-2 field have been found in explicit form. These gauge solutions do not contribute to physically observable quantities such as the energy–momentum tensor. They are derived according to Pauli–Fierz theory from four independent spherical solutions for a spin-1 massless particle. We have also constructed two types of solutions, related to opposite parities, which are not the gauge solutions. These solutions describe physically observable states with spherical symmetry for the spin-2 massless field.