Can we quantitatively understand quark and gluon confinement in quantum chromodynamics and the existence of a mass gap? Quantum chromodynamics, or QCD, is the theory describing the strong nuclear force. Carried by gluons, it binds quarks into particles like protons and neutrons. According to the theory, the tiny subparticles are permanently confined. A quark or a gluon cannot be taken from a proton because the strong force gets stronger with distance. How much do we understand of this phenomenon? and what do we hope to find about it by studying M-theory?
 A: Confinement cannot be rigorously shown in QCD with current techniques, because all analytic results in QCD are perturbative and the perturbative expansion breaks down at low energies where the coupling becomes strong.
QCD has a negative $\beta$-function, i.e. the Yang-Mills coupling grows at lower energies and becomes weaker at high energies. But the $\beta$-function itself is only known perturbatively, that is as a power series in the coupling strength. This expansion makes sense when the coupling is weak not when it is strong. This perturbative calculation of the $\beta$-function therefore teaches us two lessons:


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*The theory is asymptotically free, the coupling becomes weaker and weaker at high energies. Since the perturbative computation is justified in this regime, we can conclude that asymptotic freedom has been shown rigorously.

*At low energies the coupling becomes strong due to the $\beta$-function, but the calculation of the $\beta$-function itself relies on weak coupling. So we cannot say anything definite about this regime. However, the growing coupling is seen as a hint of confinement. One cannot, for example, compute meson masses in perturbative QCD.


Lattice computations however have been able to compute some meson masses, but they are of course numerical. Maybe somebody with more knowledge of lattice QCD can add some details.
There is the hope to study the strongly coupled regime of QCD using the holographic duality which maps QCD to a weakly coupled gravity theory (more precisely a string theory, which is closely related to M theory). There are some qualitative results, see e.g. here and here, but the holographic dual of QCD, if it exists, is not known and the quest for is has been unsuccessful. 
