Here, we assume that the flavor structure of the right-handed neutrinos is fully determined by $M_N$, so $B_N=F_X/X$ is a constant.

Then, SUSY breaking is transferred to the visible sector through the neutrino Dirac Yukawa interaction. Wave function renormalization from the interaction with right-handed neutrinos is given by [$$\begin{split} \delta Z_L = \frac{Y_{\nu}^{R \dagger} }{16 \pi^2} \Big( 1- \ln \frac{M^{R \dagger}M^R}{\Lambda^2} \Big) Y_{\nu}^{R}, ~~~~~ \delta Z_{H_u}= \mathrm{Tr} \delta Z_L \label{eq:wavere}\end{split}$$]{} where [$$\begin{split} \lambda_N^R = [Z_N^{-1/2}]^T \lambda_N Z_L^{-1/2 } Z_{H_u}^{-1/2},~~~~~ M^R = [Z_N^{-1/2 }]^T M_N Z_N^{-1/2},\end{split}$$]{} then analytically continued Majorana masses give the soft masses. From field redefinitions [$$\begin{split}& L \rightarrow \big( 1- \frac{\delta Z_L \vert_0}{2} \big)(1- \theta^2 \delta Z_L \vert_{\theta^2})L
 \\
 & H_u \rightarrow \big( 1- \frac{\delta Z_{H_u} \vert_0}{2} \big)(1- \theta^2 \delta Z_{H_u} \vert_{\theta^2})H_u,\end{split}$$]{} supersymmetric kinetic terms can be written in the simple form, [$$\begin{split}\Phi^{\dagger}(1+ \delta Z_{\Phi}) \Phi \rightarrow \Phi^{\dagger}(1+ \theta^2 \bar{\theta}^2 \delta Z_{\Phi} \vert_{\theta^2 \bar{\theta}^2}) \Phi \end{split}$$]{} then we can read off the one-loop corrections to the soft masses [$$\begin{split} \delta m_L^2 = - \delta Z_L \vert _{\theta^2 \bar{\theta}^2}~~~ \mathrm{and}~~~ \delta m_{H_u}^2 = - \delta Z_{H_u} \vert _{\theta^2 \bar{\theta}^2} \label{eq:smass}.\end{split}$$]{} In the expression, $B_N$ is just a constant, not a matrix. So $\ln(M_N^\dagger M_N)$ in the wave function renormalization 