by the three-three component which gives mass to the tau lepton and bottom quark respectively, those matrices can be calculated to be $Y_e\sim \text{diag}(0,0,Y_\tau)$ and $Y_d \sim \text{diag}(0,0,Y_b)$. This means that the largest elements in the trilinear RPV Yukawas are $\lambda_{3i3} = -\lambda_{i33} = Y_\tau\epsilon_i/\mu $ and $\lambda_{3i3}' = Y_b \epsilon_i/\mu$.

As an application of this rotation, consider the lepton number violating decay $\mu \to e \gamma$. This places the following approximate bound on the trilinear $R$-parity violating couplings [@Barbier:2004ez]: $$| \lambda_{23k} \lambda_{13k}| \lesssim 2 \times 10^{-4} \left(\frac{m_{\tilde \nu_3}}{100 \text{ GeV}}\right)^{-2} \ .$$ Using Eq. (\[lambda\]) yields $$\left|\frac{\epsilon_1 \epsilon_2 }{\mu^2}\right| \lesssim 2.5\times 10^{-3} \left(\frac{m_{\tilde \nu_3}}{100 \text{ GeV}}\right)^{-2}$$ as the most stringent constraint. This corresponds to $\tan \beta = 55$, approximately the upper bound on $\tan \beta$ that keeps $Y_\tau$ perturbative up to the GUT scale. The dependence on $\tan \beta$ is due to the fact that the SUSY Yukawa coupling $Y_\tau = \sqrt 2 \, m_\tau/v_d$, where $m_\tau$ is the tau mass. This is negligible due to the suppression of the lepton Yukawa coupling and the $\mu$ term. One would expect $\epsilon_{i}$ values much lower than this bound due to constraints from neutrino masses, as we shall see later. It 