In trying to understand exactly what confinement means, I have been reading 't Hooft s original paper on 2D QCD at large $N$. In the paper he shows that the quark propagator pole is moved to infinity, signalling the breakdown of the notion of a free propagating quark. This is often called a proof of confinement in literature. If this is the case in 4D QCD as well, then what exactly do we mean by quark mass? On the the other hand we expect freely propagating quarks at high energies. My feeling is that two point functions are insensitive to properties of the RG flow, since they involve only one momentum scale which can be lorentz transformed away. So maybe higher point functions have a pole that is scale dependent?
So what is 'current quark mass'? If you can play God and snap your finger to turn the strong interaction off. The quarks would be surfing around freely, instead of being confined like a helpless black Friday shopper stuck in a crowded Target mall. If you measure the mass of this 'free' quark, that is the 'current quark mass', stemming from the Higgs mechanism and interactions with the electroweak $SU(2)*U(1)$ gauge gang.
Obviously there is no contradiction between finite 'current quark masses' and 'no freely propagating quarks' in QCD with confinement.
In real life you can not play God to switch off QCD, unless you are Morgan Freeman. Is there an alternative way to measure the 'current quark mass'? Yes there is. When QCD is turned on, quark and antiquark bind together via strong interaction to form pions. This binding (quark and antiquark condensation) breaks dynamical chiral symmetry and should have generated a massless pseudoscalar Goldstone boson according to the usual Goldstone folklore.
In reality, the above would-be 'massless pseudoscalar boson' is NOT massless. The underlying reason is that the global chiral/axial symmetry is actually broken by the non-zero 'current quark mass' even before the dynamical chiral symmetry breaking mechanism is kicked in. The would-be 'massless pseudoscalar boson' is thus turned into 'pseudo-Nambu-Goldstone boson' with mass of the order of the 'current quark mass'. Therefore, the measurement of the mass of the pseudo-Nammbu-Goldstone boson (i.e. the pseudoscalar meson) can be used to infer the 'current quark masses'.