*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.)

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**.

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**.

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.**

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/c^{2} 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/c^{2} 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.

### 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.

## 9 Responses

Hello Matt,

thanks for these posts. I got lot of answers. And lot of new questions. I am an intrested underqualified. I do not understand the math of the QCD, but I searched something deeper than the simplified 3 color analogy. I would ask these:

– Is colorless means zero or rotation invariancy? I mean only zero dot product means colorless or any dot product? I thought 0 charge means colorless, but You wrote:

“We also saw that mesons are dot products (conjugate vector times vector) that are “colorless” — rotationally invariant under rotations of the three colors”

– every quark vector has the same length? But in different rotation? I think so.

– Every baryon consist of three perpendicular each other? If so, what about other 3-quark systems? These has also rotational-invariant volume (colorless?), but are not these baryons?

– if the anti-red is the minus of red, and I can choose the coordinate system arbitary, the same quark can become anti-quark? I think this is forbidden somehow by using line and anti-line in complex plane instead of plus and minus on real line

– But how line and anti-line differs from each other. Each has 2 coordinates (in one complex dimension). For example z = 1 – 2i. Is it a line or an anti line? Or are they rotating on the opposite direction for example?

– If I choose an other arbitary coord-system, why cannot change only one quark to anti-quark (x -> y, y -> x, z -> -z) of a barion? Maybe the volume can be plus and minus? Is it the difference between barion and anti-barion? Because it is not change if a rotated coord-system is choosen?

– Why is important, when You say that barion is the triple-product determined by 3 quark, and not the barion is these special connected 3 quark? I see that the volume is what rot-invariant. But it is strange for me, that the proton, what we see as an object, is not something, but a property of something. I heard from forms and geometric algebra. Maybe in the terminology of the geometric algebra the barions, mezons, gluons are k-forms?

– Why we need 3 complex dimension for QCD instead 6 real? How these differs from each other? Maybe rotating between (fixed) complex dimension happens only when gluon interaction? But rotating IN a complex dimension something what a quark can do without gluon interaction?

– For example how the red quark and the anti-red anti-quark of a free mezon moving in the red complex plane? Are they standing? Or are they rotating in a synchronized way? Or are they zigzag because of lots of gluons included? Can a lonely red-antired mezon leave the red complex plane without external effect?

– As You said “there are zillions of gluons, antiquarks, and quarks in a proton”. It is strange for me, that the mass of every proton is exactly same. Why this mass? Why not “zillion – 1”?

– Are free quarks theoreticaly impossible or we have not enough energy to create it? Or was it possible in the young universe only?

Maybe explaining the answers is impossible without very deep math. In this case a Yes/No/Quite different/Pointless question/… answer would help me.

Thanks: Lados.

Oops … I meant orientation in the 2nd question instead of rotation

– every quark vector has the same length? But in different orientation? I think so.

Lados

I’ll give this a shot, but there’s too much to go into detail. Remember the purpose of this post is to give you some intuition for why mesons and baryons are special when SU(3) is involved, but I’m not explaining in detail how quantum chromodynamics works.

– In this language, colorless means “unresponsive to any effect on color”, which would be a rotation of the 3-complex dimensional system. Since dot products are invariant under rotation, they are all colorless. (The corresponding language for electromagnetic charges is directly whether the charge adds up to zero. Instead, it is about whether a combination of objects is invariant under a corresponding rotation, in which each charged object is multiplied by a complex phase, e^{i q alpha}, where alpha is the angle of the phase rotation. The total phase rotation is a product of all these phases, and thus takes the form e^{i q_total alpha); only if q_total = 0 is the phase invariant.)

However,what’s far less obvious — I didn’t explain it, as it is a basic point in quantum physics and beyond the scope of the post — is that a colorless state has to be constructed as a dot product, without specifying the two vectors that have been dotted. That is, you don’t take a blue-green quark and dot it into a anti-blue-red anti-quark: instead the meson, viewed as a colorless quark-antiquark state, is always a correlated combination of quark and anti-quark states, red-antired + green-antigreen + blue antiblue.– Every quark vector has the same length, but it may be green, red, blue, or any combination thereof, indicating that it has a different orientation. But see above about constructing a meson state.

A color rotation is like a rotation of the entire system around a definite axis by a definite angle, so all the quarks rotate together (and same for antiquarks) in such a way that all dot products and cross-products and complex conjugated cross-products remain unchanged.

– A 3-quark system which is not combined together as a cross-product will not be invariant under such a rotation, and will not be colorless, so baryons are the only colorless option. Again, though, a baryon state is a particular correlated sum built from a cross-product, in analogy to what I said above about mesons.

The only simple rotationally invariant combinations in SU(3) are dot-products, cross-products, and the complex conjugate of cross-products.

– No, a quark cannot become an antiquark. To turn one into the other, I would need to be able to do rotations in 6 real coordinates, whereas in SU(3) I can only do rotations in 3 complex coordinates. This is crucial. With three complex coordinates, quarks and antiquarks always rotate differently. In the same sense, in electromagnetism, where I do complex phase rotations, no such rotation can turn something of charge +1 into something of charge -1.

Several of your following questions are on the same topic. A line (versus an anti-line) is determined by how it rotates when color is rotated; lines rotate one way, anti-lines rotate the other way. No coordinate system can change a line to an antiline, or a quark to an antiquark. (Same with electromagnetic charge +1 versus charge -1 and their different complex phase shifts.)

– When I say a “baryon is a cross-product”, I am using shorthand: the baryon state is actually much more complicated than I’m describing here, because it is made from three quarks, combined in a cross-product in their color, but spread out across a region of space the size of a proton, combined further with combinations of gluons and quark-antiquark pairs that preserve the cross-product structure and are also spread out across the proton. Again, I’m just telling you how the math of color works, not how things arrange themselves spatially, and I’m also disregarding the complexities of color when we account for the fact that a proton isn’t made from just three quarks alone but from something more elaborate involving three

extraquarks combined with lots of other things.– 3 complex vs 6 real dimensions: if the structure of SU(3) rotations is replaced with SO(6), as we would have with 6 real dimensions, then the math and correspondingly the physics is completely different. Quarks and anti-quarks would be the same and would have 6 colors, all of which could be transformed into one another by rotations. Data agrees with SU(3), not SO(6).

– “For example how the red quark and the anti-red anti-quark of a free mezon moving in the red complex plane?” Wrong question. That’s not how quantum physics works. It’s like asking how an electron moves around an atom. It doesn’t. It surrounds the atom and rotates its phase with a definite frequency.

– “Why not zillion – 1?” Again, this is not how quantum physics works. The word “zillions” was intended to avoid the question; the fact is that the number is not defined, because quantum mechanics allows superpositions. I’ve already used superposition in defining, for instance, the meson state, as red-antired + blue-antiblue + green-anti-green; if you ask “what color is the quark in the meson right now?”, the question has no answer, because all three colors appear in the state. Similarly, the proton state contains a superposition of many states of definite particle number, and so its particle number is not fixed/defined/certain.

The reason the mass is always the same is the same reason that the mass of a hydrogen atom is always the same; if you build multiple copies of a system from exactly identical ingredients [and all up quarks are identical, all gluons are identical, etc.] the ground state (the lowest energy level) of that system will always be the same. The ground state of three extra quarks plus any other colorless combination of gluons and quark-antiquark pairs is the proton, and so it always has the same mass.

– Free quarks are impossible in real-world QCD without high temperature. At low temperature, nature will always rearrange itself to create colorless objects because (no matter how much energy you put in) it is easy and energetically favorable for it to do so.

But this would not be true if real-world QCD were replaced with a world where we keep SU(3) as before but we increase the number of quark types (up, down, strange…) to a much larger number. Then the low-temperature behavior would be very different and free quarks would exist. So it is a matter of very subtle and complex details, taught only to advanced graduate students; I don’t think I’ve covered it on this website.

Hi Matt,

thanks for answers.

> “I’ll give this a shot, but there’s too much to go into detail.”

I tried to understand, but You are right. It is difficult to understand most of deeper properties without enough math. Thanks for analogies with electric charge, these also were useful.

> “In this language, colorless means “unresponsive to any effect on color”, which would be a rotation of the 3-complex dimensional system.”

> “a colorless state has to be constructed as a dot product, without specifying the two vectors that have been dotted.”

Is this language the geometric algebra? (Spacetime algebra with complex field?). For example bivectors have area without shape, as I know.

> “The only simple rotationally invariant combinations in SU(3) are dot-products, cross-products, and the complex conjugate of cross-products”

But cross-product is between two vector, is not it? Or do You mean triple product here? If so, does rot-invariant cross product mean an other observable object?

> “if you build multiple copies of a system from exactly identical ingredients [and all up quarks are identical, all gluons are identical, etc.] the ground state (the lowest energy level) of that system will always be the same. The ground state of three extra quarks plus any other colorless combination of gluons and quark-antiquark pairs is the proton, and so it always has the same mass.”

Clear answer, thanks. But the 3 extra quarks seems have distinguished role in this system.

Can I think 3 extra quark es the “coee” of the baryon, and the other objects as the field of the barion? But stronger field on smaller place so with lot of vibration? Maybe other objects counterbalancing the effect generated by the 3 extra quarks?

> “Several of your following questions are on the same topic.”

Yes, the core of my questions would be this:

Of course I cannot visualise the moving of a quark in SU(3). But I hoped to visualize a useful projection of it. As we can see a projection of the 3D moving in 2D, if we draw a “Mercedes” sign on the paper (positive parts of coordinates) as usual. We can draw a radiation hazard sign on a paper. Every black section symbolizes a quarter of a complex coordinate. (A picture would be better here. Sorry for the silly analogy) Can I see the moving of the coordinates of a quark in this projection? Of course lot points are of overlaping, but we can mark the 3 coordinates with different colour ( for example red, green, blue 🙂 ) I guess I would see ellipses? But what would be more interesting, can I see the synchronized moving coordinates of the 2 quarks of a mezon or the 3 quarks of the baryon on this projection? I think no, because as You wrote, 2 quarks became a different object (dot product), without specifying the two vectors (quarks). Maybe no such thing as 2 distinct quarks in mezon? (I try not to forget that this SU(3) is not our space. Moving in it is not moving in space)

Thanks for answers again: Lados.

Hi, can you explain us about, thanks: Experimental determination of the QCD effective charge https://www.mdpi.com/2571-712X/5/2/15

Yes and no. I am already planning to address this in part in a later post, but not in the kind of depth that would allow discussion of what these authors are claiming about “effective charge” at distances comparable to the proton’s radius. In the meantime, depending on your background and interests, you might find this article useful: https://profmattstrassler.com/articles-and-posts/particle-physics-basics/the-known-forces-of-nature/the-strength-of-the-known-forces/ .

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.

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.

>… “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?