At the center of every atom lies its nucleus, a tiny collection of particles called protons and neutrons. In this article we’ll explore the nature of those protons and neutrons, which are made from yet smaller particles, called quarks, gluons, and anti-quarks (the anti-particles of quarks.)  (Gluons, like photons, are their own anti-particles). Quarks and gluons, for all we know today, may be truly elementary (i.e. indivisible and not made from anything smaller).  But we’ll return to them later.

Strikingly, protons and neutrons have almost the same mass — to within a fraction of a percent:

• 0.93827 GeV/c² for a proton,
• 0.93957 GeV/c² for a neutron.

This is a clue to their nature: for they are, indeed, very similar. Yes, there’s one obvious difference between them — the proton has positive electric charge, while the neutron has no electric charge (i.e., is `neutral’, hence its name). Consequently the former is affected by electric forces while the latter is not. At first glance this difference seems like a very big deal! But it’s actually rather minor.  In all other ways, a proton and neutron are almost twins. Not only their masses but also their internal structures are almost identical.

Because they are so similar, and because they are the particles out of which nuclei are made, protons and neutrons are often collectively called “nucleons”.

Protons were identified and characterized around 1920 (though they were discovered earlier; the nucleus of a hydrogen atom is simply a single proton) and neutrons were discovered around 1933.  The fact that protons and neutrons are very similar was understood almost immediately.   But the fact that protons and neutrons have a measurable size, comparable in size to a nucleus (about 100,000 times smaller in radius than a typical atom), wasn’t learned til 1954.  That they are made from quarks, anti-quarks and gluons was gradually understood in a period lasting from the mid-1960s to the mid-1970s.  By the late 1970s and early 1980s, our understanding of protons and neutrons and what they are made of had largely stabilized, and has remained essentially unchanged since then.

It is much more difficult to describe nucleons than it is to describe atoms or nuclei. That’s not to say that atoms are altogether simple (you can read about my attempts here, which are made complicated by the subtleties of quantum mechanics) but at least one can say, without too much hesitation, that a helium atom is made from two electrons orbiting a tiny helium nucleus; and a helium nucleus is a relatively simple cluster of two neutrons and two protons. But a nucleon?  Here things are not so easy.  As I wrote elsewhere in my article “What’s a proton, anyway?” — useful reading for anyone who wants to understand the Large Hadron Collider [LHC] — an atom is like an elegant minuet, whereas a nucleon is like a wild dance party.

The complexity of the proton and of the neutron seems to be real, and not due to a lack of knowledge on the part of physicists. We have equations that we use for describing quarks, anti-quarks and gluons, and the strong nuclear forces that they exert on one another. [These equations are called “QCD”, short for “quantum chromodynamics”.]We can check the accuracy of those equations through many different measurements, including the rates for producing various types of particles at the LHC.  And when we put the QCD equations into a big computer, and make the computer calculate the properties of protons and neutrons, and other similar particles (collectively called “hadrons”), the computer’s predictions for the properties of these particles closely resemble what we see in the real world. So we do have good reason to believe that the QCD equations are right, and that our knowledge of the proton and neutron is based on the right equations. Yet having the right equations isn’t enough by itself, because

• simple equations can have very complicated solutions, and
• sometimes it is impossible to describe complicated solutions in a simple way.

As far as we can tell, that is the situation with nucleons: they are complicated solutions to the relatively simple equations of QCD, and there seems to be no way to describe them in a few words or pictures.

Because of the inherent complexity of nucleons, you, the reader, will have to make a choice at this point: how much of this complexity would you like to learn about? No matter how far you go, you will probably not be entirely satisfied; for although the answers to your questions may well become more enlightening as you learn more, the ultimate answer remains that the proton and neutron are complicated. So all I can offer you now is three layers of understanding, in increasing detail; you can choose to stop after any layer and move on to other subjects, or you can keep going to the last layer. Each layer begs questions that I can partially answer in the next layer, but the answers provided beg further questions. In the end — just as I do in professional conversations with my colleagues and advanced students — I can only appeal to data from real experiments, various powerful theoretical arguments, and the computer simulations I mentioned.

The First Layer of Understanding

What are protons and neutrons made of?

To try to make things easy, many books, articles and websites will tell you that protons are made from three quarks (two up quarks and a down quark) and draw a picture like the one shown in Figure 1. The neutron is the same, but with one up quark and two down quarks, as also shown in Figure 1. This simple picture represents what some (but not all) scientists first believed protons and neutrons were, mainly in the 1960s. But this view was soon realized to be a significant oversimplification… to the point that it really is not correct.

More sophisticated sources of information will tell you that protons are made from three quarks (two up quarks and a down quark) that are held together with gluons — and they might draw the picture something like that shown in Figure 2, with gluons drawn like springs or strings holding the quarks together. Neutrons are again the same but with one up quark and two down quarks.

This is not quite as bad a way to describe nucleons, because it emphasizes the important role of the strong nuclear force, whose associated particle is the gluon (in the same way that the particle associated with the electromagnetic force is the photon, the particle from which light is made.)  But it is also intrinsically confusing, partly because it doesn’t really reflect what gluons are or what they do.

So there are reasons to go further and describe things as I have elsewhere on this website: a proton is made from three quarks (two up quarks and a down quark), lots of gluons, and lots of quark-antiquark pairs (mostly up quarks and down quarks, but also even a few strange quarks); they are all flying around at very high speed (approaching or at the speed of light); and the whole collection is held together by the strong nuclear force. I’ve illustrated this with Figure 3. Again, neutrons are the same but with one up quark and two down quarks; the quark whose identity has been changed is marked with a violet arrow.

Not only are these quarks, anti-quarks and gluons whizzing around, but they are constantly colliding with each other and converting one to another, via processes such as particle-antiparticle annihilation (in which a quark plus an anti-quark of the same type converts to two gluons, or vice versa) and gluon absorption or emission (in which a quark and gluon may collide and a quark and two gluons may emerge, or vice versa).

[If you don’t trust my Figure 3, you may want to readthis article, where I describe how processes at the LHC would be quite different if Figure 3 were wrong, and something like Figure 1 or Figure 2 were true instead.]

Let’s look at what all three descriptions have in common.

• two up quarks and the down quark (plus other stuff) for the proton
• one up quark and two down quarks (plus other stuff) for the neutron.
• the “other stuff” in neutrons is essentially the same as the “other stuff” in protons; i.e., all nucleons have the same “other stuff”
• the small difference in mass between the neutron and proton is due mainly to a difference between the down quark mass and the up quark mass

And because

• up quarks have electric charge ²/₃ e  (where “e” is the charge of the proton, -e the charge of the electron)
• down quarks have charge –¹/₃ e ,
• gluons have charge 0,
• any quark and its corresponding anti-quark have total charge 0 (for instance, an anti-down quark has charge +¹/₃ e, so a down quark and a down anti-quark have charge –¹/₃ e +¹/₃ e = 0),

each of the figures attributes the proton’s electric charge to the charge of two up quarks and one down quark, with the “other stuff” contributing zero charge in total; similarly the neutron’s electric charge is due to that of one up quark and two down quarks:

• the total electric charge of the proton is ²/₃ e + ²/₃ e – ¹/₃ e = e,
• the total electric charge of the neutron is ²/₃ e – ¹/₃ e – ¹/₃ e = zero.

Where the three descriptions differ is in

• how much “other stuff” is inside a nucleon,
• what that stuff is doing in there, and
• where a nucleon’s mass, and its mass-energy (i.e. the E=mc² energy that it has even when it is standing still), comes from. [Since the majority of the mass of an atom, and therefore of all ordinary matter, lies in the masses of protons and neutrons, this last point is rather important in understanding our own nature.]

Figure 1 would have you believe that quarks are basically one third of a nucleon, somewhat the way a proton or a neutron represents one quarter of a helium nucleus or one twelfth of a carbon nucleus. Were this picture right, the quarks in a nucleon would move around relatively slowly (at speeds much slower than the speed of light) with relatively weak forces between them (though with some kind of powerful force keeping them from escaping). The mass of an up quark, and that of a down quark, would be about 0.3 GeV/c², about one third of the mass of a proton. But this simple picture, and the ideas that go with it, just isn’t correct.

Figure 3 (I’ll come back to Figure 2) gives an entirely different view of a proton, as a seething cauldron of particles rushing around at speeds approaching the speed of light. These particles are colliding with one another; in these collisions, some of those particles are annihilated, while others are created in their place. The gluons are massless particles, while up quarks have masses about 0.004 GeV/c² and down quarks have masses about 0.008 GeV/c² — hundreds of times smaller than the mass of the proton. Where the proton’s mass-energy comes from is complicated: some of it is from the mass-energy of the quarks and anti-quarks, some of it is from the motion-energy of the quarks, anti-quarks and gluons, and some of it (possibly positive, possibly negative) is from the energy stored in the strong nuclear forces that are needed to hold the quarks, anti-quarks and gluons together to form the proton.

In a sense, Figure 2 tries to split the difference between Figure 1 and Figure 3. It simplifies Figure 3 by removing the many quark-antiquark pairs, which one might argue are ephemeral, as they constantly appear and disappear, and are not essential. But it tends to give the impression that the gluons found in a nucleon are directly part of the strong nuclear force that holds the proton together. And it doesn’t really make very clear where the mass of the proton comes from.

Figure 1 has another flaw, when we look beyond the narrow confines of the proton and neutron. It is not so good for explaining some of the properties of other hadrons, like the pion and the rho meson. Figure 2 shares some of these problems.

These limitations of Figures 1 and 2 are the reason that I choose to convey, both to my students and here on this website, the image shown in Figure 3. But I must warn you already that there are many limitations to the picture, too, which I’ll get into in later layers.

Still, it is worth noting that the extreme internal complexity implied by Figure 3 is to be expected for an object held together with a force as strong as the strong nuclear force. If you want to know why, you can read on in the second layer of detail… after we discuss the mass of the proton and neutron.

One more comment: the three quarks (two ups and a down for a proton) that aren’t a part of a quark/anti-quark pair are often referred to as “valence quarks”, with the pairs of quarks and anti-quarks called “sea quarks”.  This language is technically very useful in many contexts.  But it gives the false impression that if you somehow could look inside a proton, and you looked at a particular quark, you could quickly identify whether it was a sea quark or a valence quark.  You can’t do that; there’s no way to tell.

The Mass of the Proton, and That of the Neutron

Since the proton and neutron masses are so similar, and since the proton and neutron differ only by the replacement of an up quark with a down quark, it seems likely that their masses arise in the same way, from the same source, with the difference in their masses due to a some minor difference between up quarks and down quarks. But the three figures above suggest three very different views of where the proton mass comes from.

Figure 1 would suggest that the up and down quarks are simply ¹/₃ of the mass of the proton and of the neutron: about 0.313 GeV/c², or maybe a bit more or less due to the energy needed to hold the quarks together in the proton. And since the difference between the proton and neutron masses is just a fraction of a percent, the difference between the up and down quark mass should also be a fraction of a percent.

Figure 2 is a bit less clear. What fraction of the proton’s mass comes from the gluons? But certainly one would gather from the picture that much of the proton’s mass comes from the quarks’ masses, as in Figure 1.

But Figure 3 reflects the more subtle way in which the proton’s mass actually comes about (as we can check directly using computer calculations of the proton, and indirectly using other mathematical methods). It is very different from what is suggested in Figure 1 and 2, and not so simple.

To understand how this works, one should think first not in terms of the mass m of the proton but in terms of its mass-energy E = mc², the energy associated to its mass. The right conceptual question to ask is not “where does the proton’s mass m come from”, after which you calculate its mass-energy E by multiplying m by c², but rather the reverse: ask “where does the proton’s mass-energy E come from”, and then calculate the mass m by dividing E by c².

It’s useful to classify the contributions to the proton’s mass-energy into three groupings

• A) the mass-energy (or “rest-energy”) of the quarks and anti-quarks that it contains (the gluons, being massless particles, contribute nothing)
• B) the motion-energy (or “kinetic energy”) of the quarks, anti-quarks and gluons as they move around
• C) the interaction-energy (or “binding energy” or “potential energy”) stored in the strong nuclear forces (more precisely, in the “gluon fields”) that are holding the proton together

What Figure 3 suggests is that the particles inside the proton are rushing around at high speed, and there are many massless gluons in the proton, so contribution (B) is bigger than contribution (A). Typically, in most physical systems, (B) and (C) turn out to be of comparable size, though often (C) is actually negative! So the proton mass-energy (and similarly the neutron mass-energy) is mostly coming from a combination of (B) and (C), with (A) a small contributor. And therefore this is also true of the proton and neutron mass; they are arising not so much from the masses of the particles they contain but from themotion-energies of the particles they contain and from interaction-energy, associated with the gluon fields that exert the forces holding the proton together.  (The balance of energies is very different in most other systems we’re familiar.  For instance, in atoms and in the solar system, (A) dominates, with (B) and (C) much smaller and comparable to each other.)

To summarize all this, consider that

1. Figure 1 suggests that contribution (A) is where the proton’s mass-energy comes from
2. Figure 2 suggests that contributions (A) and (C) are both important, with some impact from (B)
3. Figure 3 suggests that (B) and (C) are important, with limited impact from (A).

And we know that Figure 3 is essentially right. That’s because we can do computer simulations to check it, and because (most importantly!) we know, from various powerful arguments that theorists have developed, that if the up and down quark masses were zero (and everything else was left unchanged), the proton mass would barely change from what we observe it to be. So it would appear that the masses of the quarks can’t be important contributors to the mass of the proton.

If Figure 3 is right, the quark and anti-quark masses are rather small.  How small are they really? The up quark mass (same as the mass of the anti-quark) is at most 0.005 GeV/c², far, far smaller than the 0.313 GeV/c² suggested by Figure 1. (The up quark mass is hard to measure and its apparent value is shifted by subtle effects, so in fact it might be much smaller than 0.005 GeV/c².) And the down quark mass is about 0.004 GeV/c² larger than the up quark mass.  That means the mass of any quark or antiquark is less than a percent of the proton’s mass.

Notice also that (in contrast to what Figure 1 would imply) this means that the ratio of the down quark mass to the up quark mass is not close to one! In fact the down quark mass is roughly double the mass of the up quark, or more. The reason that the neutron and proton masses are so similar is not that the up and down quark masses are similar, but that the up and down quark masses are both very small — the difference between them is small relative to the proton and neutron masses.  And remember that to turn a proton into a neutron you merely have to replace one of its up quarks by a down quark (Figure 3); that replacement is enough to make the neutron slightly heavier than a proton, and shift its electric charge from +e to zero.

By the way, the fact that the various particles inside the proton are colliding with each other, appearing and disappearing in the process, doesn’t affect this discussion — because in every such collision, energy is conserved (i.e. the amount of it is unchanged).  The mass-energy and motion-energy of the quarks and gluons may change, and their overall interaction-energy may change, but the total energy of the proton doesn’t change, even though the stuff inside is continually rearranged.  So the proton’s mass is constant, despite the maelstrom within.

Ok.  It’s important to take a moment to drink this all in.  How remarkable is this!  Almost all mass found in the ordinary matter around us is that of the nucleons within atoms.  And most of that mass comes from the chaos intrinsic to a proton or neutron — from the motion-energy of a nucleon’s quarks, gluons and anti-quarks, and from the interaction-energy of the strong nuclear forces that hold a nucleon intact.  Yes: our planet, our bodies and our breath are what they are as a result of a silent, and until recently unimaginable, internal pandemonium.