With latest knowledge of QCD, is there any explanation for why the neutron is slightly heavier than the proton? Can it be boiled down to a simple formula?
There is an easy partial answer to the question. The proton is (uud) and the neutron is (udd). The up quark is 2.2MeV and the down quark is 4.7MeV. So there is a 2.5MeV mass increase with the neutron. The proton is 938.272MeV and the neutron is 939.565MeV which is then 1.293MeV heavier. That is odd! The neutron has a smaller mass difference than just based on quark masses.
The complete answer is a massively difficult problem. The QCD gauge bosons or gluons are self-trapped so that while they have no mass their self-interaction confines then and their energy in a form of mass. In fact this is majority of mass in hadrons and baryons. This forms the basis of the mass-gap problem. Lattice gauge theory on computers has shed some light on this and predicted the mass of baryons pretty well. An exact math-physics answer is waiting in the wings, and ClayMath as a million dollar prize for an answer. With the proton there is more of a mass-gap due to interactions with gluons. Also the proton being charged may have some renormalized mass correction just from QED that off sets the increase in mass just from quarks.
For years the expected explanation has been, that a strong-force contribution to the mass difference, that originates in the mass difference between down and up quarks, outweighs the electromagnetic contribution to the mass difference.
There were two theory papers on this theme, earlier this year 1 2. The mechanism I get from skimming them, is that the eta meson mixes with the neutral pion (see first paper, page 7), and then the pion interacts with the omega meson (see second paper, end of introduction). That is, I think these are the specific interactions which produce the strong-force contribution to the nucleon mass difference.