Why is neutron slightly heavier than the proton? [duplicate]

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?

• Are you asking for a complete and proven rigorous explanation? Or will "the different valence quark combinations interact with the QCD vacuum in different ways, which produces states with different energies, as has been demonstrated using lattice QCD" be sufficient? – probably_someone Dec 27 '18 at 23:05
• "Different valence quark makeups" - proton is uud, neutron is udd. "Interact with the QCD vacuum in different ways" - since the interactions are non-perturbative at these energies, there's not a whole lot more I can say here. – probably_someone Dec 27 '18 at 23:09
• The quark masses account for less than 1% of the nucleon mass. Most of the mass comes from the kinetic energy of the gluons. Calculating that energy is tricky because gluons have color charge, which makes the integrals / Feynmann diagrams messy, unlike in QED, where they converge nicely. So you need a lot of computer time even to calculate an approximate solution. – PM 2Ring Dec 27 '18 at 23:54
• If protons were heavier than neutrons then protons would decay. The universe would be very different if hydrogen were unstable. I guess the only kind of stars would be neutron stars, and complex atoms wouldn't be produced. – PM 2Ring Dec 28 '18 at 0:03
• Possible duplicates: physics.stackexchange.com/q/85/2451 , physics.stackexchange.com/q/264420/2451 and links therein. – Qmechanic Dec 28 '18 at 0:40

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.

• I've just thought if all quarks had the same average momentum, then they're energy differences would be smaller than their rest energy differences. I wonder if that has something to do with it. – zooby Dec 28 '18 at 1:47
• @zooby Here's a numerical analysis using lattice QCD that also accounts for the different electric charges of the quarks: arxiv.org/abs/1505.07057. I mention this to corroborate Lawrence's statement that " Lattice gauge theory on computers has shed some light on this..." This is one of the relatively few numerical lattice QCD analyses I've seen that focuses on baryon mass differences due to the asymmetries (mass, charge) between up and down quarks. – Chiral Anomaly Dec 28 '18 at 3:14
• @ Zooby QCd has a property called anti-screening, where the renormalization group flow for the coupling constant increases it with lower energy. We can work well enough with scattering states at high energy because the coupling constant decreases $\sim~g^2/4\pi log(\Lambda/E)$ as I recall. At very low energy coupling becomes large and renormalization schemes for computing these bound states does not work well. This is the reason for lattice QCD. So the oddity is that as the energy or momentum of quarks in a hadron becomes small things go haywire. – Lawrence B. Crowell Dec 28 '18 at 23:26

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.