The Standard Model More Deeply: Gluons and the Math of Quark “Color”

For readers who want to dig deeper; this is the second post of two, so you should read the previous one if you haven’t already. (Readers who would rather avoid the math may prefer this post.)

In a recent post I described, for the general reader and without using anything more than elementary fractions, how we know that each type of quark comes in three “colors” — a name which refers not to something that you can see by eye, but rather to the three “versions” of strong nuclear charge. In the post previous to today’s, I went into more detail about how the math of “color” works; you’ll need to read that post first, and since I will sometimes refer to its figures, you may want to keep in handy in another tab.

The Modern Math of Hadrons

Real protons are far more complicated than just three quarks. More generally, baryons are more than just three quarks, and mesons are more than a quark and an anti-quark. Fortunately, the addition of gluons is not so difficult.

Last time we encountered a language for discussing quarks (as lines or “vectors” in a space with three complex dimensions, one for each “color”.) Anti-quarks appeared as anti-lines, or “conjugate vectors”. We also saw that mesons are dot products (conjugate vector times vector) that are “colorless” — rotationally invariant under rotations of the three colors — while the determinant of a matrix made from three quarks is a similarly colorless baryon. (A similar determinant using anti-quarks gives an anti-baryon.)

Figure 0: For discussion, details and notation, see the post previous to this one.

The problem is that these concepts of mesons and baryons, as made only from a small number of quarks and anti-quarks, are far too naive to describe the real objects found in nature. At a minimum, we need to include gluons. How do we bring them into this story?

Within SU(3), a gluon, with color and anti-color, can be thought of as one line and one anti-line. Now, what can happen when a gluon encounters a quark? There are two possibilities. The combination may end up more complicated than before, with two free-standing lines and one anti-line. But it may also end up less complicated, with the color anti-line of the gluon canceling the color of the line of the quark, thus leaving only the color line of the gluon. The strong nuclear force tends to prefer the latter; it costs less energy. And so, if you find a gluon near a quark, their combined color will typically be the same as that of a single quark. This is all illustrated in Figure 8. In symbols, we might say that (g q ) ~ q.

Figure 8: (Left) While a quark can be represented as a line (above), or as a three-dimensional vector (below), a gluon, which has color and anti-color, can be represented as a line and an anti-line, or more accurately as a (3-color)-by-(3-anti-color) matrix. (Right) The combination of a gluon with a quark is energetically favored, by the strong nuclear force, to arrange for the gluon’s anti-color to cancel the quark’s color, leaving the combination with the same color-properties as a quark; we can view this as the anti-line canceling the quark’s line, or as the matrix product of a matrix and a vector yielding another quark-like vector.

More precisely, whereas a quark can be thought of as a line’s three coordinates (a “vector”), a gluon can be thought of as a three-by-three grid of coordinates (a “matrix”). [This would naively give nine independent numbers, but for SU(3) the sum of the three diagonal elements must be zero, leaving only eight; this is why there are eight gluons rather than nine, despite there being three colors and three anti-colors.] The product of a matrix and a vector is again a vector, as shown in Figure 8.

What happens when we add a second gluon? The same argument applies again, and we get the same answer (g g q) ~ q. We can repeat this over and over: if we add six gluons to a quark, the combination still is most likely to behave, as far as its strong nuclear charge, like a quark: (g g g g g g q) ~ q.

Figure 9: A chain of gluons with a quark can still behave, as far as its color, like a quark. As the figure illustrates schematically, this is characteristic of matrix multiplication, where the product of many matrices with a vector is still a vector.

What this means is if an anti-quark combined with a quark is color-invariant, as in Figure 0 at the top of this post, then an anti-quark combined with a chain of gluons and then a quark is also color-invariant, as in Figure 10.

Figure 10: We can attach an anti-quark at the left end of Figure 9 and obtain a colorless meson, less naive than the one shown in Figure 0 (and those in Figures 6 and 7 of the previous post.)

This chain of gluons with a quark at one end and an anti-quark at the other end is an improved mathematical view of a meson, compared to one with no gluons at all. Hmm… doesn’t it look a little bit like a string? Yes it does… and this is precisely why string theory and the strong nuclear force are related, and why mesons often behave like strings. (Baryons involve three strings joined to a more complicated object called a D-brane, see the end of this post.)

But what this shows you is why the math of a naive meson or baryon is essentially the same as the math of a real-world meson (such as a pion) or baryon (such as a proton): the dot-product and triple-product still have crucial roles to play, but now not merely between bare quarks and anti-quarks but rather between combinations of quarks and anti-quarks surrounded by nearby gluons with which they form color chains.

What about the extra quark/anti-quark pairs inside a colorless object? Naively these would break the object up into multiple colorless pieces, at least temporarily. But to make complete sense of what these pairs do and don’t do takes us somewhat beyond the math of color into the full physics of the strong nuclear force. That’s a level of complexity beyond what I can cover here.

Beyond Mesons and Baryons

Are mesons and baryons the only possibilities for colorless hadrons? No. Other combinations include (but are not limited to)

  • Double dot products of two gluons (called “glueballs”);
  • Dot products of two cross-products of two quarks and two antiquarks (called “tetraquarks”);
  • Triple products of a antiquark with two cross-products of two pairs of quarks (called “pentaquarks”).

In the real world, glueballs are believed to decay so rapidly to mesons that they cannot be observed; in the language of this post, they are too short-lived to be thought of as particles. If the masses of all the quarks had been large enough, larger than 1 GeV/c2 or so, then the glueballs’ fates would have been different: they might have been long-lived and had the lowest masses of all hadrons! We can simulate such a world on a computer, and glueballs can indeed be studied there. But it’s not the world we live in, so we don’t see glueballs in experiments.

The same was once thought to be true of tetraquarks and pentaquarks: too short-lived to observe. But one of the big discoveries at the Large Hadron Collider is that this seems not to be the case, as long as some of the quarks involved have masses that are large compared to 1 GeV/c2 and thus are relatively slow-moving — i.e., as long as they contain charm and/or bottom quarks. (Top quarks decay so quickly that they do not have time to form hadrons.) In fact, all the new tetraquarks and pentaquarks, excepting one candidate, contain at least two of these large-mass quarks or anti-quarks. This story is still unfolding, so it’s too early to get into details. But the existence of these objects, while perhaps a bit surprising, is certainly consistent with the math of SU(3) color.

Figure 11: a list of hadrons newly discovered at the LHC by the LHCb, ATLAS and CMS experiments, with their masses listed in in MeV/c2 = 0.001 GeV/c2 versus the date of their discovery. The color-coding reflects the most favored quark/anti-quark content of the hadron; a number of them are tetraquarks and pentaquarks.

What if the number of colors had been bigger than 3?

What if quarks had come in four colors, or eight? What would have been different?

In N ordinary dimensions, with SO(N) rotations, dot-products are the same as for SO(2) or SO(3); the length of a line, or the product of the lengths of two lines with the angle between them, have analogous formulas for every N. But volumes are different; instead of the volume of a three-dimensional object being of most interest, we would now focus on the volume of an N-dimensional object. Our triple-product would then become a quadruple product in four dimensions, a quintuple product in five, and an N-tuple product in N dimensions.

Similarly, in SU(N), the dot product between a line and an anti-line is still rotationally invariant, but to make something invariant only out of lines, we need N such lines, combined using the N-tuple product. And thus, when it comes to physics, we can still make mesons out of one quark and one anti-quark (plus our usual crowds of gluons and quark/anti-quark pairs), but to make a baryon requires N quarks, not just three. (In the language of Figure 0, a baryon would involve a determinant of an NxN matrix.)

One interesting consequence of this is that while in the real world mesons and baryons have similar masses, with baryon masses just a bit larger, in a world with many colors baryons would have much larger masses, roughly N/2 times larger. This would potentially make them much more difficult to produce, and much less common than are protons and neutrons in our universe.

While larger N may seem irrelevant to the real world, that might not be the case. There may be forces, yet to be discovered, whose charges come in a large number of versions. Moreover, string theory is most closely related to theories with large N, and in that context, “D-branes” are related to baryons. While that’s a story for another day, it is of considerable importance both to our conceptual understanding of string theory and to our understanding of the strong nuclear force itself.

5 thoughts on “The Standard Model More Deeply: Gluons and the Math of Quark “Color””

  1. >… “for SU(3) the sum of the three diagonal elements must be zero, leaving only eight; this is why there are eight gluons rather than nine, despite there being three colors and three anti-colors.”

    Is the constraint that the sum of the three diagonal elements must be zero inherent to SU(3) or imposed because this gluon, which would have infinite range, is never observed experimentally?

  2. The answer is yes. 🙂 If you assume SU(3) is the right symmetry group, then the constraint follows [it is the “S” in “SU”]..

    Conversely, if you start from experiment and work forward, then you find evidence that quantum field theory with some gauge symmetry describes the experiments. Various data will tell you there are three colors, as we have seen in previous posts. But a quantum field theory of U(3) would have a ninth particle which would be have very differently from the other eight gluons — in particular it would not interact with them and its interaction strength would shrink at long distance instead of increasing. Since this is not observed, a field theory of SU(3) is used.

    As you have noted, one might wonder whether the photon can be viewed as the ninth particle. But the theory of the weak interactions requires this *not* be the case, and instead the photon is a mixture of particles built from an SU(2) x U(1) theory, known as the electroweak field theory of Glashow and then of Weinberg and of Salam.

    This is then what eventually led Georgi and Glashow and their followers to combine SU(3) x SU(2) x U(1) via grand unification into SU(5), or SO(10), or SU(3)xSU(3)xSU(3). The group U(3) was not generally useful. Of course the experimental evidence for such grand unification is entirely missing; there are only a couple of circumstantial clues.

  3. Great explanation, thank you! I think the emergence of the electroweak unification theory is one of the most amazing stories in physics.

    To me, it’s rather sad that the full power of this extremely data-driven theory story has proven difficult to explain to a broader audience over the decades. That lack of accessibility has, unfortunately, left a vacuum into which many other far less experimentally based approaches have moved.

    I liked your point about emergence of electroweak unification theory as the specific block to speculations about the fixed gluon-photon ratios in quarks and flux tubes. But, of course, you are exactly right. Any GUT via that route would require full “backwards compatibility” with existing electro-week theory.

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