Consider the following Lagrangian:
$$ \tag 1 \mathcal{L} = \frac{\partial_{\mu}a(x)}{f}\sum_{q}\bar{q}\gamma^{\mu}\gamma_{5}q $$ This is a Lagrangian of the axion-like particles (ALPs) $a$ interacting with quarks $q$; $f$ is some dimensional constant.
At 1-loop level, the interaction $(1)$ generates the coupling
$$ \tag 2 \mathcal{L}_{G} = \frac{\alpha_{s}}{4\pi} c_{G}\frac{a}{f} G^{b}_{\mu\nu}\tilde{G}^{b,\mu\nu}, $$ where $c_{G} = c_{q}\sum_{q} C_{q}$. In the limit $2m_{q}\lesssim m_{a}$, $C_{q} \to 1$. In the opposite limit, it tends to zero as $m_{a}^{2}/12m_{q}^{2}$.
I would like to study the impact of the interaction $(2)$ on the low-energy hadronic interactions of the ALPs. The typical approach is to match the Lagrangians $(1)$, $(2)$ to the ChPT Lagrangian. So there, one considers the interaction of the ALP with the light quarks $u,d,s$ and gluons $G$, performs a chiral rotation of light quarks in order to convert $c_{G}$ to the pure quark sector (it would modify the quark mass term and the derivative term $(1)$), and writes down the ChPT terms.
My question is the following: should I include the contributions from $u,d,s$ quarks in $(2)$ when studying the ChPT matching? There are two problems I am facing when trying to answer this question:
the quark fields are ``active'' degrees of freedom, while their contribution to Eq. (2) looks like integrating them out.
Whether I should include all these contributions when considering decays of the ALPs into gluons. If including all of them to the width into gluons and not including the contributions of light quarks to $c_{G}$ for ChPT, I would not be able to match the rates of hadronic interactions from ChPT and perturbative QCD.
