Calculate proton mass

Overgenomen van Quora.com/ “How do I use the masses of the 2 up quarks and the one down quark to calculate the rest mass of a proton?”

Een antwoord op deze vraag werd gegeven door Stephen Selipsky , ex-particle theorist (Stanford Ph.D., research at CERN, BU, Yale) als volgt gegeven:

1. Get a Ph.D. in theoretical physics, then do a post-doctoral fellowship with a lattice gauge theory research group.
2. Use this acquired knowledge of four or five decades of lattice gauge theory and computing lore, fermion determinants and degrees of freedom adjustment on lattices, renormalization group extrapolation from finite lattice spacing to continuous spacetime, etc. to write efficient, accurate, physically meaningful algorithms to evaluate the path integral for the field theory of quarks moving at highly relativistic speeds ( γ∼40 ) in a strongly interacting non-perturbative gluon field distorting non-perturbative quark bilinear vacuum condensates. For state of the art accuracy, remember to include electromagnetism as well as chromodynamics.
3. Implement your algorithm in highly efficient numerical code to run in parallel on a world-class supercomputer. Code needs to be tuned to the individual computing architecture to properly use its capacity.
4. Buy, or obtain a research grant to support time on, the chosen supercomputer.
5. Run your code for a few weeks, save the terabytes of configuration ensembles, propagators and correlation functions, and analyze on workstation computers with your collaborators (you did do all the social and professional networking necessary to obtain competent collaborators on all this work, right?-)
6. Write up and publish the results, making sure that the planning stages of your work above pushed the state of the art enough to improve on current 1% level accuracy.

Note that the proton or neutron (“nucleon”) masses are within a few percent of the isospin-conserving limit of zero light quark masses; the actual values of “current quark” masses account for only about 1% of the total nucleon mass. Actually, the dominant parameter is the energy scale of QCD, or equivalently the value of the dimensionless QCD coupling constant measured at a given interaction energy.

For a “flavor” of what all this involves, see reviews like

• arxiv_1 en arxiv_2 (somewhat introductory presentations)

• review lattice qcd

• review quark model, section 15.6 and figure below, which comes from https://arxiv.org/pdf/1203.1204.pdf ;

• There are even programming toolboxes for putting it all together, like washington.edu, week 3

Edit: it’s not that hard to “calculate” the proton mass analytically within a factor of two or so. Consider the measured strong coupling constant at the Z pole energy, αs(E=91.2 GeV)=0.118 . Theory tells us that the virtual gluon charges bubbling in the vacuum are pushed out a little by a color charge (while virtual quark charges are polarized in but less so), making them anti-screen the color charge of a bare quark, so the coupling constant will be stronger at lower energies = larger distances. The coupling will keep growing at those lower energies until it becomes strong and confines quarks at energy scales below around ΛQCD≈250 MeV (so distances around ℏc/ΛQCD=0.8−1.0 fm ), see https://pdg.lbl.gov/2020/reviews/rpp2020-rev-qcd.pdf , Figure 9.3 and running coupling coefficients for running below the charm mass.

So three quarks in a bound state making up a proton must be confined inside 1.0 fm radius. The uncertainty principle (or the Dirac wave equation in a bag) says each quark with position uncertainty below 1.0 fm must have momentum spread above ℏ/1 fm=200 MeV/c ; with p/c much bigger than 5−10 MeV/c2 current masses, each quark’s kinetic energy is relativistic E≈pc . The kinetic energy of three quarks adds up to about 600 MeV; the strong force field energy necessary to confine them must give 33% to 90% more energy, as you can confirm with an easy variational calculation E(R confine)=A/R+BR3(+CR2 if you want to get fancy with bag wall energy). So the proton mass must be around 1000 MeV/c^2 based only on measured strength of the QCD interaction plus simple theoretical arguments. If you want to be more accurate, you need to work a lot harder (non-perturbatively) as discussed in the main answer.