An atomic nucleus consists of protons and neutrons, held together by the strong nuclear force (i.e. gluons). The heavier a nucleus gets, the more neutrons have to be added to overcome the increasing repulsive electromagnetic force of the protons. In the end, the rest mass of the nucleus is less than the sum of the mass of its constituents. This difference is equivalent to the energy needed to break up (or free particles from) the nucleus.

So far so good.

A single proton, with a mass of about 1GeV, consists of three quarks (whose rest masses sum up to roughly 7MeV in total), and gluons (massless), which create virtual sea-quark pairs. The mass of the proton is thus significantly heavier than its real constituents, and the mediators of the strong force increase the proton mass (or more precisely, make up for most of it).

How are these two pictures naively compatible? Is this a consequence of asymptotic freedom? Or does the proton have an equivalent mass defect, i.e. a typical energy scale for inelastic scattering, so that the sum of the masses of constituent and sea quarks is larger than the mass of the proton (if one might for a second assume such a sum makes any sense)? And thus, are there sea quarks in a nucleus as a consequence of neutrons/protons exchanging gluons, which then make up for a (tiny) part of the mass of the atom?

  • $\begingroup$ The inter-nucleon force is a bit more complicated than you've given in that description of the Nuclear Force. $\endgroup$
    – Triatticus
    Commented Apr 25 at 19:21
  • $\begingroup$ You may enjoy Frank Wilczek's non-technical articles on this & related topics: frankwilczek.com/core.html $\endgroup$
    – PM 2Ring
    Commented Apr 25 at 21:36
  • $\begingroup$ Your "i.e., gluons" in the 2nd line is highly misleading: confining gluons enter only very indirectly in nuclear binding, which is due to an effective potential shaped by meson exchange. The nucleons are not confined in a nucleus, which can be split apart, and isolated constituents, or clumps thereof, fly out. $\endgroup$ Commented Apr 25 at 22:13
  • 1
    $\begingroup$ ...By sharp contrast, quarks and gluons are confined in the hadron, and cannot be pried out in isolation: the binding energy is basically infinite, which would generate a soup of new hadrons. Comparing a nucleon to a nucleus is a disastrously misleading framework of thinking about the respective systems. $\endgroup$ Commented Apr 25 at 22:13

1 Answer 1


The mass defect of a bound state is equal to the sum of the masses of isolated unbound constituents minus the mass of the bound state. The reason why these two quantities can differ is that we consider the mass of both the bound state and any isolated constituent to include the energy of the surrounding field. For example, the electromagnetic field surrounding a hydrogen atom has less energy than the sum of the energies of an electromagnetic field surrounding an isolated electron and an electromagnetic field surrounding an isolated proton. This is because superimposing the two individual electric fields on each other results in destructive interference in some regions.

So in order to determine the mass defect of a proton, you would need to know the mass of an isolated unbound quarks, with the field surrounding it. But you can't actually have isolated quarks in our universe. But what if you had a universe that was empty except for one quark; what would be the mass then? Well, basically, it would be infinite; the existence of a single uncancelled colour charge in the universe would force the entire universe to be a quark-gluon plasma (see also this answer).

A hadron is therefore, in fact, lighter by an infinite amount compared with the isolated quarks. Its mass defect has the expected sign, but is infinite in magnitude.

You may have heard that the mass of an up quark is something like 3 eV. That is not the mass that you would use to calculate mass defect (which, as stated above, is infinite). The typically quoted quark mass is the "current mass" of a quark, which means the apparent kinematic mass of the quark as a single particle inside a hadron, not including the surrounding gluon field and virtual quarks.


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