I am reading the section 11.4 of Peskin and Schroeder's book (page 373), and there is a step I could not follow.
To calculate the effective action of linear sigma model, the determinant of $\frac{\delta^{2}\mathcal{L}}{\delta\phi^{i}\delta\phi^{j}}$, and in peskin's book, it shows that
$$\frac{\delta^2\mathcal{L}}{\delta\phi^i\delta\phi^{j}}=-\partial^2\delta^{ij}+\mu^2\delta^{ij} - \lambda[(\phi\_{\mathrm{cl}}^k)^2\delta^{ij} + 2\phi\_{\mathrm{cl}}^i\phi\_{\mathrm{cl}}^j].\tag{11.67}$$
Then, orient the coordinates to make the $\phi^{i}\_{cl}$ points in the *N*th direction,
$$\phi^{i}\_{cl}=(0,0,...,0,\phi\_{cl}),\tag{11.68}$$
and the operator in the RHS of the first equation is just a KG operator $(-\partial^{2}-m^{2}\_i),$ where
$$m^2\_i=\begin{cases} \lambda\phi^2\_{cl}-\mu^2\quad acting\; on\;\eta^1,...,\eta^{N-1};\\3\lambda\phi^2\_{cl}-\mu^2\quad acting\; on\;\eta^N.\end{cases}\tag{11.69}$$
I think for all $\eta^i$, the value of $m^2\_i$ should equal to $3\lambda\phi^2\_{cl}-\mu^2$, how the $\lambda\phi^2\_{cl}-\mu^2$ is obtained?