section_path: Quarks and the mass of a proton
section_name: Quarks and the mass of a proton
paragraph_text: Quarks and the mass of a proton
  
  In quantum chromodynamics, the modern theory of the nuclear force, most of the mass of protons and neutrons is explained by special relativity. The mass of a proton is about 80–100 times greater than the sum of the rest masses of its three valence quarks, while the gluons have zero rest mass. The extra energy of the quarks and gluons in a proton, as compared to the rest energy of the quarks alone in the QCD vacuum, accounts for almost 99% of the proton's mass. The rest mass of a proton is, thus, the invariant mass of the system of moving quarks and gluons that make up the particle, and, in such systems, even the energy of massless particles confined to a system is still measured as part of the rest mass of the system.
  
  Two terms are used in referring to the mass of the quarks that make up protons: current quark mass refers to the mass of a quark by itself, while constituent quark mass refers to the current quark mass plus the mass of the gluon particle field surrounding the quark. These masses typically have very different values. The kinetic energy of the quarks that is a consequence of confinement is a contribution (see Mass in special relativity). Using lattice QCD calculations, the contributions to the mass of the proton are the quark condensate (~9%, comprising the up and down quarks and a sea of virtual strange quarks), the quark kinetic energy (~32%), the gluon kinetic energy (~37%), and the anomalous gluonic contribution (~23%, comprising contributions from condensates of all quark flavors).
  
  The constituent quark model wavefunction for the proton is | p ↑ ⟩ = 1 18 (2 | u ↑ d ↓ u ↑ ⟩ + 2 | u ↑ u ↑ d ↓ ⟩ + 2 | d ↓ u ↑ u ↑ ⟩ − | u ↑ u ↓ d ↑ ⟩ − | u ↑ d ↑ u ↓ ⟩ − | u ↓ d ↑ u ↑ ⟩ − | d ↑ u ↓ u ↑ ⟩ − | d ↑ u ↑ u ↓ ⟩ − | u ↓ u ↑ d ↑ ⟩). {\displaystyle \mathrm {|p_{\uparrow }\rangle ={\tfrac {1}{\sqrt {18}}}\left(2|u_{\uparrow }d_{\downarrow }u_{\uparrow }\rangle +2|u_{\uparrow }u_{\uparrow }d_{\downarrow }\rangle +2|d_{\downarrow }u_{\uparrow }u_{\uparrow }\rangle -|u_{\uparrow }u_{\downarrow }d_{\uparrow }\rangle -|u_{\uparrow }d_{\uparrow }u_{\downarrow }\rangle -|u_{\downarrow }d_{\uparrow }u_{\uparrow }\rangle -|d_{\uparrow }u_{\downarrow }u_{\uparrow }\rangle -|d_{\uparrow }u_{\uparrow }u_{\downarrow }\rangle -|u_{\downarrow }u_{\uparrow }d_{\uparrow }\rangle \right)}.}
  
  The internal dynamics of protons are complicated, because they are determined by the quarks' exchanging gluons, and interacting with various vacuum condensates. Lattice QCD provides a way of calculating the mass of a proton directly from the theory to any accuracy, in principle. The most recent calculations claim that the mass is determined to better than 4% accuracy, even to 1% accuracy (see Figure S5 in Dürr et al.). These claims are still controversial, because the calculations cannot yet be done with quarks as light as they are in the real world. This means that the predictions are found by a process of extrapolation, which can introduce systematic errors. It is hard to tell whether these errors are controlled properly, because the quantities that are compared to experiment are the masses of the hadrons, which are known in advance.
  
  These recent calculations are performed by massive supercomputers, and, as noted by Boffi and Pasquini: "a detailed description of the nucleon structure is still missing because... long-distance behavior requires a nonperturbative and/or numerical treatment..." More conceptual approaches to the structure of protons are: the topological soliton approach originally due to Tony Skyrme and the more accurate AdS/QCD approach that extends it to include a string theory of gluons, various QCD-inspired models like the bag model and the constituent quark model, which were popular in the 1980s, and the SVZ sum rules, which allow for rough approximate mass calculations. These methods do not have the same accuracy as the more brute-force lattice QCD methods, at least not yet.