I am following Peskin Section 13.3, where they solve the nonlinear sigma model using Polyakov method. This system has Lagrangian \begin{equation} \mathcal{L}=\frac{1}{2g^2}|\partial_\mu\vec{n}|^2,\tag{13.67} \end{equation} where $\vec{n}$ is a unit vector. To find the renormalization flow of this Lagrangian the authors use a derivation first done by Polyakov where the unit vector field is decomposed in slow and fast variables. The slow vector field $\tilde{n}^i$ is obtained by integrating the Fourier modes of $n^i$ in the range $0<|k|<b\Lambda$ and rescaling the result to get another unit vector. After that the relation between $n^i$ and $\tilde{n}^i$ is given by \begin{equation} n^i(x)=\tilde{n}^i(x)(1-\phi^2)^{1/2}+\sum_{a=1}^{N-1}\phi_a(x)e^i_a(x),\tag{13.88} \end{equation} where the $e^i_a$ are unit vectors orthogonal to $\tilde{n}^i$ and the $\phi_a$ are expansion coefficients. Until now this expansion is completely general, however the authors state that the coefficients $\phi_a$ are fast variables that only have Fourier components in the range $b\Lambda<|k|<\Lambda$. This make sense intuitively given that $\tilde{n}^i$ is constructed only from slow Fourier components. Nevertheless, I would like to have a more formal derivation that this construction gives $\phi_a$ with Fourier components only exactly in the given range. Can anyone help me with that? I tried to play with the equations without success.
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