In QCD, there is an SU(3) color symmetry for each flavor of quark as well as an SU(3) flavor symmetry for $u, d, s$ (although the latter is approximate). Why is there a gauge field for the SU(3) color symmetry but no gauge field for the SU(3) flavor symmetry? Put it more generally: What kind of symmetry has a gauge field and what kind of symmetry does not have a gauge field?
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There are two types of symmetries: global symmetries and gauge symmetries.${}^1$ Gauge symmetries act locally, in the sense that the parameters of the transformations are functions of spacetime. Instead global symmetries have parameters which are constants. These are two different things.
Gauge symmetries are associated to a gauge field, or connection (namely $A_\mu$). This field has to be a dynamical field and thus it appears in the Lagrangian. Global symmetries, on the other hand, have no gauge field associated to them.
In QCD the flavor symmetry is global and the color symmetry is gauge.
${}^1\;$ Actually it's improper to call the gauge symmetries "symmetries." Rather, it's better to say that they are a redundancy in the description of the degrees of freedom of the theory. But that's not relevant right now.
