Cartan's Supersymmetry and the violation of CP symmetry

Abstract

Cartan's supersymmetriy fixes couplings of two types of fermions $ψ, ϕ$ and two types of vector fields $x_i$ and $x_{i'}$, ($i=1,2,3,4$). The electromagnetic interaction of leptons and quarks is expressed as ${^tψ}{\mathcal C}x_iψ$. In the case of coupling of leptons and quarks with $W$, we extend the coupling ${^tϕ}{\mathcal C}Xψ$ to ${^tϕ}{\mathcal C}X(1-γ_5)ψ$, where $X=x_i$ or ${x_i}'$, and unify the interactions in the form $ {^tϕ}{\mathcal C}\bar{x_i}ψ+{^tϕ}{\mathcal C} \bar{{x_i}'} {\mathcal C}ψ$, where $\bar{x_i}$ implies appropriate $x_i$ or $(-γ_5 x_i)$ is chosen. The $γ_5$ term induces in time component of $B^0\to K^0\, J/Ψ$, $K^0$ with the $\bar s$ quark dominated by the small component, while in space components of $\bar B^0\to \bar K^0\, J/Ψ$, $\bar K^0$ with the $s$ quark dominated by the small component. This asymmetry could be the origin of the $CP$ asymmetry in the $B^0\to K^0\, J/Ψ$. We apply the model to $B_{s(d)}^0\to D_{s(d)}^- D_{s(d)}^+$ decays and study violation of $CP$ symmetry via interference of tree and penguin contributions, and observe that the asymmetry is weak consistent to the experiment.