The Proton's Mass Budget

The proton weighs 938.27208816(29) MeV. Its three valence quarks, at the masses the Higgs coupling gives them, weigh 9.02 MeV between them. The other 929.25 MeV is not missing. Quantum chromodynamics owns it, and lattice QCD has split it into four entries: quark energy, glue field energy, a quark-mass term built from the scalar condensates, and one quarter of the trace anomaly. This paper spends no new physics on that split. It runs a measuring instrument on it instead: the factor census and the coherent noise floor, the discipline this program built at home for its own residue. Every entry gets asked the same questions. What defines it. What measures it. What its error is, and whether that error was measured or inherited. Then the account gets graded against the coherence of its own inputs. The budget under test is the chiQCD accounting of Yang and collaborators (2018): in the MS-bar scheme at mu = 2 GeV, quark energy 32(4)(4) percent, glue field energy 36(5)(4) percent, quark-mass term 9(2)(1) percent, and a quarter of the trace anomaly 23(1)(1) percent. The scheme and the scale are printed with the numbers in this paper every time the numbers appear, because two of the four entries move when either changes. The verdict on the balance is this. The budget balances exactly at its central values: 32 + 36 + 9 + 23 = 100 percent of the proton mass, in the MS-bar scheme at mu = 2 GeV. The balance is built in at two joints, not found: the momentum fractions are normalized to sum to one, and the trace-anomaly term is taken as the remainder the trace sum rule leaves after the quark-mass term. The account is closed. It is not independently closed. The checks that could have failed sit elsewhere, and Section 4 grades them: the lattice's own proton mass lands 1.7 of its own standard deviations above the measured mass, and the glue momentum fraction lands within one standard deviation of the global fit it was checked against. The softest entry is the smallest one. The quark-mass term carries the largest relative error of the four, near 25 percent, and a second accounting fed with different sigma terms moves it from 9 percent to 11.2 percent. The anomaly entry prints the smallest error of the four and earned none of it: a remainder inherits its parent terms' errors in a shape the printed digits do not show. A comparison with this program's home bookkeeping is graded in Section 6. It is a rhyme, and it is printed at that size. The softest measured entry is the quark-mass term, near 25 percent relative, with a documented hundred-MeV convention lever behind it. The softest entry outright is the anomaly quarter, whose error bar is bookkeeping, not measurement. The comparison with the home two-ledger habit is RHYME. This paper claims no new decomposition and solves nothing.