Lepton flavour violating decays of $\mu$ and $\tau$ lepton in a gauge group $SU_L(2)\times SU_R(2)\times SU_l(2)$

Abstract

The electroweak unification group $ SU(2)_L\times SU(2)_R\times SU(2)_Y $ is proposed for the charged lepton flavor violating decays of the muon ($\mu$) and tau ($\tau$) leptons. The group $SU(2)_Y$ is in the lepton space. The left-handed leptons and anti-leptons are assigned to the fundamental representation $(2,2,\bar{2})$ of the semi-simple group. The gauge group $SU(2)_Y$ is spontaneously broken to $U(1)_{Y_1}$, where $Y_1=-L=\pm1$ is the hypercharge, by introducing a scalar multiplet $\Sigma$ which belongs to the triplet representation 3 of the $SU(2)_Y$ and is singlet under $SU(2)_L\times SU(2)_R$. At this stage charged vector bosons $Y^\pm$ of $SU(2)_Y$ which mediate the lepton flavor violating decays acquire masses and are decoupled with one Higgs scalar $H_\Sigma^0$. The residual group $SU(2)_L\times SU(2)_R\times U(1)_{Y_1}$ has all the features of the left-right electroweak unification group extensively studied in the literature. The probability for lepton flavor violating decays is $\left(\frac{\sin^2\theta_W}{1-2\sin^2\theta_W}\right)^2\left(\frac{m_{W_L}}{m_Y}\right)^4$.