Do gluons mediate the interactions between different flavors of quark?

Question

From some of the quite professional sources (Basics from QCD by CERN; QCD from PDG), the QCD lagrangian is written in the form of

$$ L = \Sigma_{f} \; \bar{\psi}^{(f)} i \gamma^\mu D_\mu \psi^{(f)} + F_{a \mu \nu}F^{a \mu \nu} $$

This lagrangian excludes the interaction between different flavors $f=1 .. 6$ of quarks mediated by the gluons.

The gluons are said to be "flavor blind", ignoring the flavor at all, which can either mean coupling only the quarks of the same flavor (as shown by this lagrangian) or coupling between different flavors in the same way.

If gluons couple only the same flavors, who is responsible for binding the different flavors? Say, in a proton? Or is this lagrangian a misleading yet very common mistake? Or is my interpretation wrong?

Note: There seem to be other sources that take care of the inter-flavor interactions. GSI slides

Answer 1

The answer is very simple. For example, let's consider QED with an electron and muon field, $$\mathcal{L} = \bar{\mu} (i \gamma^\mu D_\mu - m_\mu) \mu + \bar{e} (i\gamma^\mu D_\mu - m_e) e - \frac14 F_{\mu\nu} F^{\mu\nu}.$$ There is no term in the Lagrangian that directly couples a muon field and an electron field. So does this mean muons and electrons don't interact with each other electromagnetically? Of course not. A muon can emit a photon, which is then absorbed by an electron. The act of emitting a photon can't change a muon into an electron, but that doesn't mean they don't interact.

The answer to your question is exactly the same. Quarks can emit gluons, though this process does not change their flavor. Those gluons can then interact with quarks of different flavors, i.e. they mediate an interaction between different quark flavors.