In Section 12.1 of Peskin & Schroeder, they introduce the renormalization group for $\phi^4$ theory.
Let $b < 1$ and $\Lambda$ some UV cutoff. Define
$$\hat{\phi} = \begin{cases}
\phi(k) , \quad b\Lambda \leq |k| < \Lambda\\
0
\end{cases}$$
and redefine $\phi(k)$ as
$$\phi = \begin{cases}
\phi(k) , \quad |k| < b\Lambda\\
0
\end{cases}$$
so that we replace the original $\phi$ by $\phi + \hat{\phi}$. This leads to the functional integral:
$$Z = \int \mathcal{D}\phi e^{-\int \mathcal{L}(\phi)} \int \mathcal{D} \hat{\phi} \exp\Big(-\int d^dx\big[\frac{1}{2}(\partial_\mu \hat{\phi})^2 + \frac{1}{2} m^2\hat{\phi}^2 + \lambda(\frac{1}{6} \phi^3 \hat{\phi} + \frac{1}{4}\phi^2 \hat{\phi}^2 + \frac{1}{6}\phi \hat{\phi}^3 + \frac{1}{4!}\hat{\phi}^4)\big]\Big). \tag{12.5}$$
They introduce a term $\mathcal{L}_\text{eff}$ and note that it involves only the Fourier components $\phi(k)$ with $|k| < b\Lambda$. To carry out the integrals over the high momentum modes $\hat{\phi}$ they say on p. 396-397
...we will see below that the new terms in $\mathcal{L}_\text{eff}$ can be written in diagrammatic form. In this analysis, we treat the quartic terms in (12.5), all proportional to $\lambda$, as perturbations. Since we are mainly interested in the situation $m^2 \ll \Lambda^2$, we will also treat the mass term $\frac{1}{2}m^2\hat{\phi}^2$ as a perturbation. Then the leading-order term in the portion of the Lagrangian involving $\hat{\phi}$ is $$\int \mathcal{L}_0 = \frac{1}{2} \int_{b\Lambda \leq |k| < \Lambda} \frac{d^d k}{(2\pi)^d} \hat{\phi}^*(k) k^2 \hat{\phi}(k). \tag{12.7}$$
I am having a hard time following their work.
Where did (12.7) come from?
By quartic terms do they mean all terms of the form $\phi^n \hat{\phi}^m$ such that $n + m = 4$?