= Solution
Set
$$
A=\frac S{a^2},
\qquad
C=\frac{4JS^2}{a^2},
\qquad
\mathbf q=\widetilde{\mathbf n}\times
\partial_t\widetilde{\mathbf n}-\mathbf B.
$$
After adding $\lambda\mathbf m\mathbin\cdot\widetilde{\mathbf n}$, the terms involving the massive canting field are
$$
\mathcal L_m=-C|\mathbf m|^2
+\mathbf m\mathbin\cdot(A\mathbf q+lambda\widetilde{\mathbf n}).
$$
Completing the square gives
$$
\mathcal L_m
=-C\left|
\mathbf m-\frac{A\mathbf q+lambda\widetilde{\mathbf n}}{2C}
\right|^2
+\frac{|A\mathbf q+lambda\widetilde{\mathbf n}|^2}{4C}.
$$
The <Gaussian functional integral> over $\mathbf m$ contributes only a field-independent determinant. Hence
$$
\boxed{
\mathcal L_{eff}(\widetilde{\mathbf n},\lambda)
=-\frac{JS^2}{2}|\nabla\widetilde{\mathbf n}|^2
+\frac1{16Ja^2}
\left|
\mathbf q+\frac{a^2\lambda}{S}widetilde{\mathbf n}
\right|^2.}
$$
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