# Where is the proton mass coming from?

I always thought that the mass of a proton would simply be the mass of the three separated quarks plus their binding energies. Nevertheless, I was once discussing with one of the developers of RICH detectors at LHC and he told me that this addition (quarks+binding) could account only for 30% of this energy (cannot remember what was the experiment performed to determine that), so where is the other 70% coming from?

• "plus their binding energies" That should be minus. 2 protons and 2 neutrons weigh more than a Helium-4 nucleus, for example. Commented May 11, 2018 at 3:40
• @SeanE.Lake The nuclear force (AKA residual strong force) has negative binding energies. The underlying strong force has positive binding energies—part and parcel of confinement. Commented May 11, 2018 at 5:17
• @dmckee This doesn't even make sense to me "negative binding energies," because it only makes sense to refer to negative interaction energies as binding energies. Do you mean positive interaction energies (i.e. a repulsive interaction)? Commented May 11, 2018 at 8:19
• This is more subtle than it appears. Because of confinement, there is no such thing as a 'separated quark'. We can only consider quarks in mesons or baryons or the QGP, where their behaviour depends on the colour-force field they're experiencing. That's why the 'mass of the quark' is not a generally defined concept and the tables of properties always show masses as 'of the order of', especially for light quarks. The analogy with nuclear masses (where there are no problems with considering free protons and free neutrons) is not helpful. Commented May 11, 2018 at 9:08
• @Sean The resolution of the difficulty that Roger discusses is "asymptotic freedom": the observation that as you increase interaction energies, the kinematics of hadronic final states become increasingly similar to those expected if the quarks were free, and because increasing energy implies shorter distance we assign the zero of strong interaction energy to zero separation instead of infinite separation as with other forces. But the system is still bound which means energy grows with separation so the potential energy must be positive. Commented May 11, 2018 at 15:33

As a quantum mechanical entity, the proton has to be described by the summed four vectors of its constituents. What is binding the constituents within a nucleon (hadron) is the strong force which exchanges innumerable gluons and generates quark antiquark pairs among them. The binding of nucleons in contrast is a spill over force from the strong force, called the strong nuclear force in contrast, and it is similar to the van der Waals forces that bind atoms with the electromagnetic force.

The strong nuclear force can be modeled similar to the electromagnetic in the atoms, and energy levels can be defined for the protons and neutrons within the nucleus. This is not true for the proton and its innumerable constituents in addition to the valence quarks.

A theory has been developed, QCD on the lattice, which does aim at computing fairly well the masses of hadrons, i.e. strong interacting resonances . Here is a review.

• so where is the mass coming from, all the virtual quarks/gluons that are moving inside the proton? Commented May 11, 2018 at 4:51
• It is the sum of all the fourvectors within the bag of proton which gives the invariant mass of the proton . Lattice QCD tries to apporximate this instantaneous picture and has a fairly good success, if you read the link, of establishing resonance masses . This is similar success to the shell model establishing nuclear masses in the periodic table. for the relativistic mathematics look up hyperphysics.phy-astr.gsu.edu/hbase/Relativ/vec4.html Commented May 11, 2018 at 5:03
• "The strong nuclear force can be modeled similar to the electromagnetic in the atoms, and energy levels can be defined for the protons and neutrons within the nucleus. This is not true for the proton and its innumerable constituents in addition to the valence quarks." Are you saying we cannot predict them or that they do not exist? If the latter, what disqualifies the p and n resonances listed by the PDG? Commented May 11, 2018 at 8:02
• Is there any experimental proof that virtual quarks and gluons moving inside the proton are the ones providing the proton with mass or is it all just theoretical? Commented May 11, 2018 at 9:51
• @Juanjo In the big picture this is an experimental fact—we measure the mass fraction and spin fraction attributable to sea partons. (I was even briefly and very loosely involved in one of these experiments NuSea/E866.) Getting the measured values and theoretical values to agree in detail, however, is still an open issue. Commented May 11, 2018 at 15:33

this addition (quarks+binding) could account only for 30% of this energy ..., so where is the other 70% coming from?

Kinetic energy of quarks and gluons.