Type-1 and integrating out heavy fields

Question

Let's assume seesaw 1 type of generation of left neutrino Majorana mass:

$$ L_{m} = -G_{ij}\begin{pmatrix} \bar{\nu}_{L}& \bar{l}_{L}\end{pmatrix}^{i}i\sigma_{2}\begin{pmatrix}\varphi_{1}^{*} \\ \varphi^{*}_{2} \end{pmatrix}\nu_{R}^{j} - M_{ij}(\nu_{R}^{T})^{i}\hat{C}\nu_{R}^{j} + h.c. $$ After using the unitary gauge and shifting the vacuum we can get the mass terms $$ \tag 1 L_{m} = -\frac{1}{2}\begin{pmatrix} \nu_{L} & \nu_{R}^{c}\end{pmatrix}^{T}\hat{C}^{-1}\begin{pmatrix} 0 & \hat{m}_{D}^{*} \\ \hat{m}_{D}^{\dagger} & \hat{M}^{\dagger} \end{pmatrix}\begin{pmatrix} \nu_{L}\\ \nu_{R}^{c}\end{pmatrix}+h.c., $$ where $\nu^{c} = \hat{C}\bar{\nu}^{T}, \quad \hat{m}^{ij}_{D} = \eta G^{ij}$,

and interaction terms with the Higgs field $\sigma $: $$ \tag 2 L_{int} = -\sigma G_{ij} \bar{\nu}_{L}^{i}\nu_{R}^{j} + h.c. $$ We can "diagonalize" $(1)$ if we assume that $||m_{D}|| << ||M||$: $$ L_{m} \approx \nu_{L}^{T}\hat{M}_{1}\nu_{L} - \nu_{R}^{T}\hat{M}_{2}\nu_{R} + h.c., \quad \hat{M}_{1}= -\hat{m}_{D}^{T}\hat{M}^{-1}\hat{m}_{D}, \quad \hat{M}_{2} = \hat{M}. $$ But how to "delete" interaction term $(2)$ (i.e., to integrate out heavy field $\nu_{R}$)?

This post imported from StackExchange Physics at 2014-08-09 08:47 (UCT), posted by SE-user Andrew McAddams