Seesaw type-1 and integrating out heavy fields

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 $\nu_{R}$)?

Or this interaction is the prediction?

$$G _{ ij} \sigma \bar{\nu} _L ^i \nu _R ^j \rightarrow G _{ij} G _{jk}\sigma ^2 \bar{\nu} _L ^i \nu _L ^k$$ This isn't an exact expression since you are doing the matching after diagonalization in which case you are integrating out the interaction basis right handed neutrinos as opposed to the mass basis right handed neutrinos.