Of Particular Significance

Celebrating the Standard Model: Atoms, Quarks and the Strong Nuclear Force

POSTED BY Matt Strassler

POSTED BY Matt Strassler

ON 07/14/2022

You may have heard the simplistic remark that "a proton is made of three quarks" (two up quarks and a down quark), which would suggest these quarks have mass of about 1/3 of a proton's. But something's clearly amiss. If you look at websites and other sources about particle physics, they all agree that up and down quark masses are tiny. If the proton were simply made of three quarks, it would naively have a mass of less than 1% of its actual mass. What can we all learn from this discrepancy?

For the general reader:

Last week I showed you, without any technicalities, how to recognize the elementary forces of nature in the pattern of particle masses and lifetimes. This week we’ll start seeing what we can extract just from the particles’ masses alone… and what we cannot. Today we’ll focus on quarks and the strong nuclear force.

A key factor in nature, which plays an enormous role in everyday life, is the mass of a typical atom. [Note: on this website, “mass” always means “rest mass”, which does not increase with a particle’s speed.] This in turn arises mainly from the masses of protons and neutrons, which are about equal, and tiny: about 0.00000000000000000000000000167 kg (or 0.00000000000000000000000000368 pounds). Since those numbers are crazy-small, physicists use a different measure; we say the mass is about 1 GeV/c2, and more precisely, 0.938 GeV/c2. In any case, it’s tiny on human scales, but it’s some definite quantity, the same for every proton in nature. Where does this mass come from; what natural processes determine it?

You may have heard the simplistic remark that “a proton is made of three quarks” (two up quarks and a down quark), which would suggest these quarks have mass of about 1/3 of a proton, or about 0.313 GeV/c2. But something’s clearly amiss. If you look at websites and other sources about particle physics, they all agree that up and down quark masses are less than 0.01 GeV/c2; these days they usually say the up quark has mass of 0.002 GeV/c2 and the down quark has 0.005 GeV/c2. So if the proton were simply made of three quarks, it would naively have a mass of less than 1% of its actual mass.

What’s going on? A first little clue is that different sources sometimes quote different numbers for the quark masses. There are six types of quarks; from smallest mass to largest, they are up, down, strange (u,d,s, the three light quarks), charm, bottom (c,b, the two somewhat heavy quarks) and top (t, the super-heavy quark.) [Their names, by the way, are historical accidents and don’t mean anything.] But some websites say the up quark mass is 0.003 instead of 0.002 GeV/c2, a 50% discrepancy; the bottom quark’s mass is variously listed as 4.1 GeV/c2, 4.5 GeV/c2, and so forth. This is in contrast to, say, the electron’s mass; you’ll never see websites that disagree about that.

The origin of all these discrepancies is that quarks (and anti-quarks and gluons) are affected by the strong nuclear force, unlike electrons, Higgs bosons, and all the other known elementary particles. The strong forces that quarks undergo make everything about them less clear and certain. Among numerous manifestations, the most dramatic is that quarks (and anti-quarks and gluons) are never observed in isolation. Instead they’re always found in special combinations, called “hadrons“. A proton is an example, but there are many more. And the strong nuclear force can have a big effect on their masses.

The Modern Proton and the Masses of Quarks

A proton, in fact, is not simply made from three quarks, the way a hydrogen atom is simply made from a proton and an electron. As I described in this article, it’s vastly more complex; it’s made from three quarks plus lots of gluons plus lots of pairs of other quarks and anti-quarks. So the simple intuition we get from atoms does not apply to hadrons like the proton.

Figure 1: (Left) A typical cartoon of the proton based on the 1960s concept that a proton is simply made from two up quarks and a down quark, each with a thrd of a proton’s mass. (Right) A modern image (from my own article) showing the 1970s conception of a proton as formed from two up quarks, a down quark, many gluons and many quark/anti-quark pairs, all with very small (or zero) masses and moving at extremely high speeds.

Since we never find quarks outside of hadrons, and hadrons are generally complicated, this poses real problems for measuring or even uniquely defining what quark masses are. Whereas one can isolate an electron and easily measure its rest mass (perhaps by wiggling it back and forth, or seeing how quickly it accelerates in a known electric field), one can’t do anything similar for a quark, since one can’t isolate it. In the end, this makes the very definition of a quark’s mass a bit ambiguous, and certainly subtle!

What’s a poor physicist to do? Well, since the culprit is the strong nuclear force being so darn strong and complicated, wouldn’t it be nice if we could make it weaker and see what happens? Sure would. And what do you know? The strong nuclear force does this for us, all on its own! For objects with mass well below the proton’s mass, the strong nuclear force is super-strong indeed; but for objects well above the proton’s mass, it becomes weaker (relative to other forces at that mass scale) and everything about it becomes a lot simpler. [Caution for future experts: there is an oversimplification in this statement.] The transition between super-strong and not-so-strong occurs in the region of 0.3 – 1.0 GeV/c2 in mass, or 0.3 – 1.0 GeV in energy… right around the proton mass. In fact, this is why protons and neutrons, and thus atoms, have the masses they do. The masses of atoms are set by the mass range in which the strong nuclear force transitions from kind-of-strong to really-really-strong.

Today I’m going to give you evidence for this incredibly important and surprising phenomenon, called “asymptotic freedom”, discovered in 1973 and awarded Nobel Prizes in 2004. We’ll do this through the patterns seen in some hadrons’ masses. Along the way we’ll see how to estimate the masses of the top, bottom and charm quarks (the three “heavy quarks”), gain some interesting insights into the masses of the up, down and strange quarks (the three “light quarks”), and see how the latter feed into the proton mass.

Our strategy will be to investigate some exotic “atoms” made, at least nominally, from a quark and an anti-quark. Hadrons of this class are called “mesons“, examples go by the names of pion, rho, kaon, omega, but we won’t care about the names. We’ll see that if the quark and anti-quark are both heavy (c or b), the atoms appear relatively simple, and in fact this is still roughly true even if only one of them is heavy. But if both the quark and anti-quark are light (u, d and/or s), then this is not true; the behavior of the corresponding meson is very different from an atom. Such mesons are more like protons, with an extremely complex interior.

We’ll extract these basic features of quarks, atom-like mesons and the strong nuclear force from just one figure! That’s the point of this post.

Four atoms: Hydrogen, anti-hydrogen, positronium and protonium

Since we’re going to focus on quark-antiquark “atoms”, let me first introduce you to four less complicated atoms, one familiar and three exotic. I want to show you how much we can learn from their masses. The technique, easy to understand in this context, can then be applied to mesons and quarks, with mixed results.

In ordinary life, hydrogen is the simplest atom; it consists of one electron (of charge -1) and one proton (of charge +1). But in high-tech experiments, physicists can make the anti-particles of electrons and protons (called positrons and anti-protons) and create three other simple electrically-neutral atoms, giving us four altogether; as shown in Figure 2, they are hydrogen (electron and proton), positronium (electron and positron), protonium (anti-proton and proton), and anti-hydrogen (anti-proton and positron.)

Figure 2: The contents of four simple atoms. (Nothing is to scale).

These four atoms have three different masses; that’s because hydrogen and anti-hydrogen have exactly the same mass (because two types of objects that are each other’s anti-particle always have exactly the same mass.) And also, the mass of an atom is roughly the sum of the masses of the objects it contains. Because of this, we can easily estimate the masses of the proton (and anti-proton) and of the electron (and positron) from the masses of these atoms.

  • Divide the protonium mass by 2 to get an estimate of the proton mass.
  • Divide the positronium mass by 2 to get an estimate of the electron mass.
  • As a check: their sum should give hydrogen’s mass.

For these atoms, it works great. We’ll have only partial success with mesons.

Simple Shifts To Atomic Masses

The reason it works so well with atoms, and less well with mesons, has to do with two types of shifts of an atom’s mass — changes that are extremely small for a relatively weak force like electromagnetism, but can be large for the strong nuclear force.

  1. The mass of the atom is slightly shifted by the energy needed to hold the atom together; that extra energy affects the atom’s mass because E=mc2. It’s a tiny effect for these atoms, largest for protonium where it is a (negative) 1 in 10,000 effect, but for mesons it will matter much more.
  2. Two-particle atoms like this secretly have two configurations; if you pull the atom off the shelf you may find it in either one. Electrons and protons are intrinsically spinning, in a limited sense; but because quantum physics is weird, you’ll always find [Figure 3] their spin orientations in a hydrogen atom are either (a) anti-aligned (with “total spin 0“), or (b) aligned (with “total spin 1“). Because aligning the spins requires extra energy, these two “spin states” of the atom have very slightly different masses, a few parts per billion for protonium and positronium, and less for hydrogen.

[For electron-positron atoms, these two configurations are referred to as para-positronium and ortho-positronium.]

Figure 3: Anti-aligned spins (left) make a simple atom’s mass very slightly smaller than aligned spins (right).

These mass shifts are so very small because the electromagnetic force is not very powerful, relatively speaking. [For those who read my slightly more technical posts, it is because the strength of the force is small: α = 1/137.04… << 1.] For the strong nuclear force, though — even where it is only kind-of-strong, but especially where it is really-really-strong, these shifts are a big deal!

Many Mesons From A Few Quarks

It’s certainly naive to take a simple strategy that works for atoms and apply it to mesons. But it will partially work, and where it fails, it will teach us something.

First, though, we need to deal with the top quark. The super-heavy quark’s mass is so big, and its lifetime so small, that the strong nuclear force has neither enough time nor enough strength to mess with it very much. Consequently, despite what I said above, experiments do observe the top quark in (extremely brief) isolation, during which time it decays to other particles that leave evidence in experiments (namely jets, which you can read about here.) From this evidence, the top quark can be identified as a sharp peak in data; Figure 4 shows an example from the CMS experiment operating at the Large Hadron Collider [LHC]. The location of the peak measures its mass — close to 172 GeV/c2.

Figure 4: Data (black dots) from the CMS experiment at the LHC, showing attempts to reconstruct a particle that decays to three quarks. The red and pink areas show what a top quark of 172 GeV/c2 is expected to look like. The grey area shows what the data would be expected to look like in the absence of a top quark.

But back now to the other five quarks. Roughly, just as the electromagnetic force can bind an electron and a positron into positronium, the strong nuclear force can bind a quark and anti-quark into atom-like “mesons”. With five types of quarks and anti-quarks, that’s 5×5=25 types of mesons, each with two spin states, for 50 total. However, some of these are each others’ anti-particles and have the same mass, so the number of mesons with independent masses is only 30 (=5*6/2 mesons x 2 spin states.) Of these, 27 have so far been observed in experiments; that’s enough for our purposes.

When I plot the masses of all of these mesons in Figure 6, putting their spin 0 states (green) and their spin 1 states (red) next to each other, patterns become obvious, and indeed I’ve organized the dots horizontally to make some of them clear.

Figure 5: The masses of the lowest-mass spin-0 and spin-1 mesons of each type. The proton mass, which roughly delineates where the strong nuclear force begins to decrease in strength (relative to other forces at the same mass and energy scale), is shown for comparison.

The masses come in clusters: there are 6 red dots around 0.8-1.0 GeV/c2, whose corresponding green dots lie well below that; then there are three green-red pairs around 2 GeV/c2 and three around 5, and isolated dots at 3, 6 and 10 GeV/c2. This clustering is already a sign of the quark-antiquark structures within.

We can see a sign of the strong nuclear force’s asymptotic freedom and its transition around 1 GeV/c2 from super-strong to not-so-strong. For all mesons above 1 GeV/c2 where both spin-states have been observed, the separation between them is small; this gives us some reason to hope that these cases are not so different from ordinary atoms. By contrast, for the mesons at or below 1 GeV/c2, the separations are very significant in all but one case, so these mesons are clearly subject to very strong internal forces. They are likely much more complicated than atoms — and more like protons. (By the way, the proton has a higher spin state too, called the Delta, with a mass 30% higher than the proton’s.)

Extracting Quark Masses

Despite this, let’s apply the naive procedure we used to extract the masses of protons and electrons. In Figure 7, I’ve done this graphically, with arrows representing the mass of the quarks that we would extract from the procedure; red, yellow, green, blue and purple arrows indicate the masses of the u, d, s, c and b quarks. Two arrows connected together, one for the quark and one for the anti-quark, then give our naive guess for the mass of a corresponding meson, whose quark/anti-quark content I’ve indicated. Three minor details:

  • Each meson has the same mass as its anti-particle, which has quark and anti-quark types interchange.
  • Mesons may contain many gluons and many quark/anti-quark pairs in addition to the quark and anti-quark shown in the label.
  • The quark/anti-quark content of three low-mass mesons is more complicated (they represent a “superposition” of quark/antiquark combinations.)
Figure 6: Naive estimates of the quark masses using the assumption that mesons are like simple atoms. The arrows’ lengths indicate a best estimate; two arrows end-to-end estimate a meson’s mass. The quark and anti-quark specifying the meson are indicated, except for three mesons that are mixtures. This method can’t work for the low-mass mesons because their spin states have big mass differences, but it works pretty well for mesons with mass well above 1 GeV/c2. Despite this, all quark masses turn out to be overestimated by a few tenths of a GeV/c2.

In many cases the arrows do point to where the dots are; in particular, this

  • works well for all spin-1 mesons
  • works well for both spin-1 and spin-0 mesons that have at least one b or c quark (or anti-quark)
  • completely fails for most spin-0 mesons with u, d and s quarks!

Despite some partial success, there’s something very funny about the quark masses that we get this way. They’re always too high. Approximately,

Quarksu   d   s   c   b   
Modern Methods0.0020.0050.0961.34.2
Our Method0.370.370.511.64.7
Estimates of masses of quarks using professional scientific methods and using our crude, naive approach, which always overshoots by a few tenths of a GeV/c2.

(Different ways of applying today’s method give somewhat different results, but always qualitatively similar.) This is striking for two reasons:

  • For each quark, our mass estimate comes out about 0.3 to 0.5 GeV/c2 higher than modern methods obtain.
  • The sum of the masses for two u quarks and one d quark (or two d quarks and one u quark) comes within 15% of the proton’s or neutron’s mass, much closer than one obtains from the modern methods.

The Central Role of the Strong Nuclear Force

What is this telling us? One way to think about it — but this is only a rule of thumb, not an exact or rigorous statement backed by clear theoretical argument or calculation — is that the strong nuclear force, through its addition to a hadron of all those gluons, quark/anti-quark pairs, and binding energy, essentially shifts the quarks’ masses, adding three to five tenths of a GeV/c2 to the elementary mass of a quark. This shift is a small effect for the heavy quarks c and b, but is a dramatic effect for the light quarks u, d and s. Our method picks out the quark masses after this shift. Scientists, however, have highly sophisticated (though still imperfect) methods for undoing this shift, and picking out the more fundamental values of the quark masses — and this is what you see on most websites.

The importance of this shift is enormous! As we’ve seen, without it a proton or neutron would have a mass of less than 0.01 GeV/c2. And so the strong nuclear force’s mass shift contributes 99% of a proton’s and neutron’s mass — which means it contributes 99% of your mass, too.

These issues reflect the fact that most mesons, in particular those with light quarks or antiquarks in them, are not at all like hydrogen atoms, and aren’t made from just two particles. The mass shift somehow captures the presence of all those other particles, and the energy required to keep them both trapped and in motion.

Why is the shift roughly 0.3-0.4 GeV/c2 per quark? Qualitatively, its size makes sense: it is right around the mass at which the strong nuclear force transitions from super-strong to not-so-strong. A shift of 10 GeV/c2 would be way too big, and a shift of only 0.001 GeV/c2, while possible, might be surprisingly small. But why is the shift 0.3 rather than 0.15 or 0.5 GeV/c2? and why is it basically the same in all spin-1 mesons and in “baryons” (the class of hadrons to which protons and neutrons belong)? No one has fully understood the answers, either conceptually or mathematically. One clue as to how this shift works is seen in the fact that the spin-0 meson masses are so small — that the mass shifts are smaller there. This is well-understood, but it is a long story for a future post.

These discrepancies between what one means by quark masses are not new; they confused physicists for a long time. In fact, with all these mysterious shifts and ambiguities, you’d be reasonable to wonder if physicists even now actually understand quarks and the strong nuclear force. To see that we do, I’ve shown in Figure 7 the results of computer simulations of the light quarks interacting with the strong nuclear force. With just two inputs (the masses of the lightest two types of spin-0 mesons), a whole host of things can be successfully calculated:

  • the masses of the full set of spin-1 mesons
  • the masses of the remainder of the spin-0 mesons (the most difficult of all the calculations, for technical reasons)
  • the masses of spin 1/2 and spin 3/2 “baryons” (including those of protons and neutrons.)

So the equations for the strong nuclear force do show that light quarks with masses as low as 0.002 GeV/c2 can turn into mesons with masses ranging from 0.14 to 1.02 GeV/c2, as well as baryons from the proton’s 0.938 GeV/c2 on up. Our understanding isn’t complete, but our equations are right, and our ability to extract information from them is likely to improve as computers and math develop further.

Figure 7: The calculation of hadron masses from various computer simulations; modified figure from https://arxiv.org/abs/1310.6087. The π and K masses are taken from data and used to specify the quark masses. The remainder of the hadron masses are then computed via numerical simulation of the strong nuclear force with low-mass up, down and strange quarks. Agreement between calculation (colored dots) and measurement (horizontal lines) is excellent for baryons and spin-1 mesons. Even for the spin-0 η and η’, particularly subtle to calculate, data and calculation agree to about 10% or better.

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13 Responses

  1. Fascinating, thanks! Your last comment about the very large neutrino mixing angles is especially interesting, as well as your general discussion of the almost-matches of the mass-basis and interaction-basis for quarks. My first random thought was whether there might be a correlation between chiral partnering and mixing angles — that is, might the absence of appreciable levels of weak-blind neutrino partners somehow “free up” the mixing angles?

    It’s remarkable that there is so much detail on the nuances of generations these days. Yet, to the best of my poor knowledge, there’s still no persuasive answer to Rabi’s 86-years-old question “Who ordered _that_?”

  2. Nice article. One question: Why did you group quarks by mass rather than properties (generations)? I know this same mixup was of great historical significance, but that was before three families were recognized. Isn’t this particular mass similarity more a red herring than a true insight?

    1. Because the quarks don’t actually organize themselves into families as precisely as is often implied. The only clear organization is vertical, not horizontal: there are u, c, t quarks with one electric charge, and d, s, b quarks with another electric charge. Who is paired with who, and why, is actually a long story, going under the name of “quark mixing”; I don’t think I’ve ever written about it thoroughly. A generation is not actually a property; for instance, there’s nothing in the standard model that says that the electron is in the same generation as the up quark. In fact there’s nothing in the standard model that relates the generations of quarks to the generations of leptons. Generations are a useful way to discuss the particles of the Standard Model, but the generational viewpoint is not deeply ingrained into its structure.

      1. If I’m reading your reply rightly — and I freely admit that I may not be — I think you are saying:
        (a) even with mixing, there are six experimentally distinguishable quarks;
        (b) all six possess specific, well-defined combinations of color and electric charge;
        (c) three of them have the same -1/3 electric charge, at least if you go by valence counting;
        (d) the other three have the same +2/3 electric charge, and finally;
        (e) starting with the lightest or down quark, they increase in mass in an alternating charge-type pattern.

        If that’s a correct reading — and again, it may not be — you are hypothesizing that none of the above patterns are significant in comparison to rough grouping based on masses, which, of course, are a bit difficult to determine for quarks.

        I gather there must be more reasoning behind your hypothesis. Do you have papers, or could you elaborate?

        1. I agree with a-e, except for nit-picking: for (c), it’s not a matter of valence counting, you can verify this using deeper principles such as anomaly cancellation, e+e- –> hadrons, and decay modes, and (e) actually the up quark is lightest, with down slightly heavier.

          I’m not trying to over-emphasize the grouping by masses either. I was just using it in this post because it gives us a nice way to learn about the strong nuclear force.

          I’m not really making a hypothesis here, just stating facts that are pretty much agreed by everyone in the field nowadays. Recognizing now the level of your sophistication, let me say a little more. The point is that the quarks (and the leptons too) can be characterized in the mass basis (in which each has a definite mass) or in an interaction basis (in which each one has a definite SU(2) partner), and quark mixing can be understood as the failure of these two types of bases to line up. The surprise is that they *almost* line up — the mixing matrix is not that far from the unit matrix (i.e. all mixing angles are small.) This need not have been the case. Had the mixing matrix been random, so that top quarks are just as likely to decay to down quarks as bottom quarks, and charm quarks decayed more often to down quarks than strange quarks, then the whole notion of generational structure would be far less attractive and would not be apparent in the data.

          To say it another way: the particle to which the top quark decays — the “interaction-bottom quark” — is almost entirely mass-bottom quark, with a small admixture of mass-strange and mass-down. But if “interaction-bottom quark” were mostly “mass-down” quark, then why would we have put the heaviest charge -1/3 quark, the mass-bottom quark, in the same generation as the top quark? We would have been just as motivated to put the mass-down quark in that generation. Theorists would argue about it endlessly.

          And indeed, this is *precisely* the problem in the neutrino sector! We used to refer to the neutrinos as “electron-neutrino, muon-neutrino, tau-neutrino.” Now we know the mixing angles are very large, so no one does this anymore; we just call the mass eigenstates nu_1, nu_2, nu_3. The generational structure in the neutrinos is simply not there.

  3. Mass estimates surely look funny. In my understanding that for particles that could exist in isolation mass of bound state is always less than sum of masses of particles. For hadrons it’s other way round (for nucleons extremely so). Is there good explanation for that?

    1. Yes, binding energy can be positive as well as negative; consider the harmonic oscillator. If particles can escape the bound state, then the energy of binding has to be negative. But if you have a potential such that it requires positive (finite or infinite) energy to pull a particle out of a bound state, then you can easily have positive binding energy.

  4. OT: Matt, do you read and sometimes reply to good comments made under older articles?
    If yes, I think it would add to the quality of your blog if you had a small window displaying recent comments publicizing these articles, similar to what Sabine has on her Back Reaction blog when she used to reply to comments there, or Terence Tao’s blog etc.

  5. Just a retired engineer’s guess, who has labored many, many hours to resolve vibration issues in space packages and many hours of analyzing test data, could this shift be caused by the slight differences in the wave functions of the contents, i.e. beating?

    That was a nightmare for me trying to find it in package, aluminum enclosures, before CAD stations. Many, many accelerometers and stroboscope inspections while on the vibration table. 🙂

    Sorry, I’m probably oversimplifying the issue.

    1. No, it’s not that. It has more to do with the uncertainty principle, even though that’s not the full story either. If I trap an object in a box of size S, then it can’t have kinetic energy much less than hbar/S , so even if its mass is tiny, its energy will be at least as big as hbar/S… so if you start trapping quarks, via the strong nuclear force, in a space the size of a proton, whose size is about 10^-15 meters, you will find their energy is inevitably tenths of a GeV.

      1. I’m afraid my engineering background biases me in conceptualizing in relativistic concepts/deterministic outcomes but has the physics so far eliminated the possibility that the quarks can orbit faster than the speed of light? Is c the limit or can trapping these particles below a certain radius can increase their angular speeds beyond and so increase their energies?

        Maybe achieving FTL is part of the strong force mechanism?

        Sorry, my physics is very limited. Fun and inspiring just to understand the latest achievements in your field. Congrats to all of the physicists, especially at CERN.

  6. New ideas for interactions and particles: This paper [Editor’s note: Link deleted; I do not permit commenters to advertise their personal theories of the universe on this site; that’s what academic journals are for. Repeat offenders will be blocked.] examines the possibility to origin the Spontaneously Broken Symmetries from the Planck Distribution Law. This way we get a Unification of the Strong, Electromagnetic, and Weak Interactions from the interference occurrences of oscillators. Understanding that the relativistic mass change is the result of the magnetic induction we arrive to the conclusion that the Gravitational Force is also based on the electromagnetic forces, getting a Unified Relativistic Quantum Theory of all 4 Interactions.

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