Reviewing the State of QCD

A nice review article sums up the state of QCD (the part of the Standard Model pertaining to the strong force). Craig D. Roberts, "Hadron Physics and QCD: Just the Basic Facts" (January 26, 2015).  The abstract is as follows (typo in original corrected):

With discovery of the Higgs boson, the Standard Model of Particle Physics became complete. Its formulation is a remarkable story; and the process of verification is continuing, with the most important chapter being the least well understood. Quantum Chromodynamics (QCD) is that part of the Standard Model which is supposed to describe all of nuclear physics and yet, almost fifty years after the discovery of quarks, we are only just beginning to understand how QCD moulds the basic bricks for nuclei: pions, neutrons, protons. QCD is characterized by two emergent phenomena: confinement and dynamical chiral symmetry breaking (DCSB), whose implications are extraordinary. This contribution describes how DCSB, not the Higgs boson, generates more than 98% of the visible mass in the Universe, explains why confinement guarantees that condensates, those quantities that were commonly viewed as constant mass-scales that fill all spacetime, are instead wholly contained within hadrons, and elucidates a range of observable consequences of confinement and DCSB whose measurement is the focus of a vast international experimental programme.

In theory, QCD is a solved problem.

We believe that we know the exact form of all of the equations and we have decent experimental measurements of all of the relevant Standard Model fundamental physical constants that go into those equations (some of which are redundant degrees of freedom): the strong force coupling constant, the two electroweak coupling constants, the six quark masses, the three charged lepton masses, the W boson, Z boson, and Higgs boson masses, the Higgs field vacuum expectation value, the four parameters of the CKM matrix, and a few general purpose constants not particular to the Standard Model like the speed of light, Planck's constant, and pi.  We would like more precisely measurements of all of them, but that only limits the precision of the calculations we can do in the Standard Model.

(The other fundamental physical constants, the three absolute neutrino masses, the four parameters of the PMNS matrix, Newton's constant, and the cosmological constant, are not pertinent to QCD.)

But the promise of QCD to explain all of nuclear physics from first principles has yet to be realized in practice.  This is mostly because the mathematics is hard, largely because (1) higher order terms in the relevant infinite series approximations of the equation's predictions are more material, (2) the self-interactions of gluons greatly complicate the equations, and (3) some of the key physical constants, such as the light quark masses and the strong force coupling constant, aren't known with great precision.

These obstacles are interrelated.  Self-interactions are one of the reasons that higher order terms are more relevant, and the difficulty involved in doing the calculations is one of the reasons that the fundamental physical constants inferred from observable like hadron masses aren't very precisely known.

For example, the rest masses of the proton, the neutron, and the electron are all known with eight significant digit accuracy, and we know the masses of many of the baryons and mesons predicted by QCD and the quark model to four to six significant digits (often +/- 1 MeV in absolute terms and at least seven baryons and fifteen mesons, disproportionately made up of hadrons that include only up, down and strange quarks for which fundamental mass determinations are particularly imprecise, have absolute mass measurements of better than +/- 0.15 MeV precision).

But, the strong force coupling constant and the mass of the proton and neutron calculated from first principles using QCD are only known with an accuracy of about 1% (about 10 MeV).

Similarly, the least accurately known mass of a hadron made up only of up and down quarks (the spin-3/2 delta baryon), has an experimental measurement accuracy of about 0.2%  (about 2 MeV) and a theoretically predicted mass (calibrated using the proton and neutron masses) that differs from the experimental value by roughly 3% (about 30 MeV).  The linked paper in this paragraph explains at some length why these calculations are so difficult.