Proton Mass

Kate sees that the chart of the fundamental leptons and quarks shows that the up and down quark masses are ~ 5 MeV/c² each. Yet the proton, which is composed of two up quarks and one down quark as the combination uud, has an enormous mass of 938 MeV/c². She asks why there is such a large mass difference between constituents and the final product.

Answer

Quantum chromodynamics describes the interactions of the quarks. The up and down quark masses are listed as ~ 5 MeV/c² each. However, these “current” quarks are not what is meant by having them confined inside a proton by the color fields. Instead one must use the effective mass the “constituent” mass which accounts for this confinement and which can be estimated from the Heisenberg uncertainty principle. Since δxδp_x ≥ h/4π, and each quark is confined within the proton radius of about one Fermi, we estimate δp_x ~ 100 MeV. In three dimensions, the total  MeV/c². So at least 510 MeV/c² of the proton mass is to be associated with the “constituent” mass of the three quarks within the proton. The remainder is the energy contributions of the gluons holding the proton together.

Most of the properties of protons, except the spin, seem to be determined by these three “valence quarks,” much like the valence electrons determine the important chemical properties of atoms. However, when the proton’s innards are probed more energetically, more structure is found, up to four or five more particles, called “virtual quarks.” In addition, up to 30 gluons can be detected. The proton is revealing its inner sanctum to investigators, and the view is becoming quite interesting. Quarks, antiquarks, and gluons can be said to form a thick “soup” inside the proton, and theoretical and experimental physicists are working together to figure out the recipe.

Today we know that the three valence quarks cannot alone account for the proton’s spin. The whole “sea” of quarks, antiquarks, and gluons each possess spin, so one must first determine the contribution made by each individual member of this seething mass. The results so far suggest that the sea of quarks makes a minimal contribution to the overall spin of a nucleon!