I am reading Section $3$ of this review titled The Strong CP Problem and Axions by R. D. Peccei and also the post here.
A famous solution of the Strong CP problem in QCD is offered by the proposal of the Peccei-Quinn symmetry and its spontaneous breakdown. In a nutshell, the idea is that a global anomalous chiral ${\rm U(1)}$ symmetry, called the Peccei-Quinn symmetry, exists at very high energies. It is also broken at very high energies to produce a low-energy effective Lagrangian $$\mathscr{L}_{\rm eff}=\mathscr{L}_{{\rm SM}}+\Big(\bar{\theta}+\xi\frac{a}{f_a}\Big)\frac{g^2}{32\pi^2}G^a_{\mu\nu}\tilde{G}^{\mu\nu}_a-\frac{1}{2}\partial_\mu a\partial^\mu a+\mathscr{L}_{\rm int}(\partial_\mu a,\psi)\tag{1}$$ where $\bar{\theta}$ is called the effective theta parameter of QCD, $G^a_{\mu\nu}$ are the gluon field strength tensor, $a$ is a Goldstone boson field (called axion), $f_a$ is the scale at which the symmetry is broken and $\xi$ is a dimensionless coupling.
I wonder how the Lagrangian of Eq.$(1)$ can be derived. I tried to sketch a possible way to motivate a derivation but it remains incomplete. Any suggestions about how to proceed further?
My attempt We start by defining the ${\rm U(1)}_{\rm PQ}$ transformation on the Standard Model (SM) fields and also extend the SM with an additional complex scalar field $\phi$ (with a potential $V(\phi)$ and some interaction terms with the SM fields).
Assuming $V(\phi)$ is minimized at $|\phi|=f_a\big/\sqrt{2}$, we can write $\phi$ as $$\phi=\frac{1}{\sqrt{2}}(f_a+\rho)\exp\Big[i\frac{a}{f_a}\Big].\tag{2}$$ When expanded, the gradient term for $\phi$ gives rise to $$(\partial_\mu\phi^*)(\partial^\mu\phi)=\frac{1}{2}(\partial_\mu\rho)(\partial^\mu\rho)+\frac{1}{2}(\partial_\mu a)(\partial^\mu a)+\frac{1}{2f_a^2}(\rho^2+2f_a\rho)(\partial_\mu a)(\partial^\mu a).\tag{3}$$ A typical potential becomes $$V(\phi)=\frac{\lambda}{4}\Big(|\phi|^2-\frac{f_a^2}{2}\Big)^2=\frac{\lambda}{16}\Big(\rho^2+2f_a\rho\Big)^2\tag{4}$$
Question Since $a$ be the axion field, we get the desired kinetic term $-\frac{1}{2}\partial_\mu a\partial^\mu a$ (apart from a sign). But we should also obtain another physical field $\rho$. Why is the field $\rho$ absent in the Lagrangian $(1)$?
Note I do not know of any reference that starts with the high energy Lagrangian and from that systematically derives the effective Lagrangian $(1)$. If there is any reference containing a detailed derivation of $(1)$, please suggest.