Color gluons are embodied ideas: symmetry incarnate.

THE CENTRAL IDEA OF QCD IS SYMMETRY. Now symmetry is a word that’s in common use, and like many such words its meaning is fuzzy at the edges. Symmetry can mean balance, pleasing proportions, regularity. In mathematics and physics, its meaning is consistent with all those ideas, but sharper.

The definition I like is that symmetry means you have a distinction without a difference.

The legal profession uses that phrase, distinction without a difference, as well. In that context, it typically means saying the same thing in different ways, or—to put it less politely—quibbling. Here’s an example, from the comedian Alan King:

My lawyer warned me that if I died without a will I’d die intestate.

To understand the mathematical concept of symmetry, it’s best to think about an example. We can build a nice little tower of examples, one that contains the most important ideas in easily digestible form, in the world of triangles. (See Figure 7.1.) You can’t move most triangles around without changing them (Figure 7.1a). Equilateral triangles, however, are special. You can rotate an equilateral through 120º or 240º (twice as much) and still get the same shape (Figure 7.1b). The equilateral triangle has nontrivial symmetry, because it permits distinctions (between a triangle and its rotated versions) that don’t, after all, make any difference (the rotated versions give the same shape). Conversely, if someone tells you that a triangle looks the same after rotation by 120º, you can deduce that it’s an equilateral triangle (or that the person’s a liar).

Figure 7.1 A simple example of symmetry. a. You can’t move a lopsided triangle without changing it. b. If you rotate an equilateral triangle through 120º around its center, it does not change.

The next level of complexity comes when we consider a set of triangles with different kinds of sides. (See Figure 7.2.) Now, of course, if we rotate one of the triangles through 120º, we don’t get the same thing—the sides don’t match. In Figure 7.2, the first triangle (RBG) rotates into the second triangle (BGR), the second triangle rotates into the third (GRB), and the third rotates into the first. But the complete set, containing all three triangles, is not changed.¹⁵

Conversely, if someone tells you that a triangle with three different kinds of sides, together with some other stuff, looks the same after rotation through 120º, you can deduce both that the triangle is equilateral and that there are two more equilateral triangles with different arrangements of the sides (or that the person’s a liar).

Figure 7.2 A more complicated example of symmetry. Equilateral triangles with different “colored” sides (here the colors are suggested as R(ed), B(lue), G(reen)) change under rotations through 120º; but the entire set of three goes over into itself.

Let’s add one last layer of of complexity. Instead of triangles with differently colored sides, let’s consider laws involving those triangles. For example, a simple law could be that if you squeeze the triangle it collapses in a neat way, so that its sides become bowed. Now suppose that we had investigated only RBG triangles, so that we really established the squeezing law only for those triangles. If we knew that rotating by 120º was a distinction without a difference—that is, that rotating by 120º defined a symmetry, in the mathematical sense—we’d be able to deduce not only that there had to be the other kinds of triangles, but also that they too collapse neatly when you squeeze them.

This series of examples exhibits, in simple forms, the powers of symmetry. If we know an object has symmetry, we can deduce some of its properties. If we know a set of objects has symmetry, we can infer from our knowledge of one object the existence and properties of others. And if we know that the laws of the world have symmetry, we can infer from one object the existence, properties, and behavior of new objects.

In modern physics, symmetry has proved a fruitful guide to predicting new forms of matter and formulating new, more comprehensive laws. For example, the theory of special relativity can be considered a postulate of symmetry. It says that the equations of physics should look the same if we transform all the entities in those equations by adding a common, constant “boost” to their velocities. That boost takes one world into another, distinct world moving with a constant velocity relative to the first. Special relativity says that that distinction makes no difference—the same equations describe behavior in both worlds.

Although the details are more complicated, the procedures for using symmetry to understand our world are basically the same as the ones we used in our simple examples from triangle world. We consider that our equations can be transformed in ways that could, in principle, change them—and then we demand that they don’t in fact change. The possible distinction makes no difference. Just as in the triangle world examples, so in general, for a symmetry to hold several things have to be true. The entities that appear in the equations will need to have special properties, to come in related sets, and to obey tightly related laws.

Thus symmetry can be a powerful idea, rich in consequences. It’s also an idea that Nature is very fond of. Prepare for a public display of affection.

Nuts and Bolts, Hubs and Sticks

The theory of quarks and gluons is called quantum chromodynamics, or QCD. The equations of QCD are displayed in Figure 7.3.¹⁶

Figure 7.3 The QCD Lagrangian L written out here provides, in principle, a completedescription of the strong interaction. Here m_j and q_j are the mass and quantumfield of the quark of jth flavor, and A is the gluon field, with space-time indices μ, ν and color indices a, b, c. The values of the numerical coefficients f and t are completely determined by color symmetry. Aside from the quark masses, the couplingconstant g is the single free parameter of the theory. In practice, it takes ingenuityand hard work to calculate anything using L.

Pretty compact, no? Nuclear physics, new particles, weird behaviors, the origin of mass—it’s all there!

Actually, you shouldn’t be too quick to be impressed by the fact that we can write equations in compact form. Our clever friend Feynman demonstrated how to write down the Equation of the Universe in a single line. Here it is:

(1)

U is a definite mathematical function, the total unworldliness. It’s the sum of contributions from all the piddling partial laws of physics. To be precise, U = U_Newton + U_Einstein + ···. Here, for instance, the Newtonian mechanical Unworldiness U_Newton is defined by U_Newton = (F - ma) ²; the Einstein mass-energy Unworldliness is defined by U _Einstein = (E - mc ²) ² ; and so forth. Because every contribution is positive or zero, the only way the total U can vanish is for every contribution to vanish, so U = 0 implies F = ma, E = mc², and any other past or future law you care to include!

Thus we can capture all the laws of physics we know, and accommodate all the laws yet to be discovered, in one unified equation. The Theory of Everything!!! But it’s a complete cheat, of course, because there is no way to use (or even define) U, other than to deconstruct it into its separate pieces and then use those.

The equations displayed in Figure 7.3 are quite unlike Feynman’s satirical unification. Like U = 0, the master equations of QCD encode a lot of separate smaller equations. (For experts: the master equations involve matrices of tensors and spinors; the smaller equations, for their components, involve ordinary numbers.) There’s a big difference, though. When we unpack U = 0, we get a bunch of unrelated stuff. When we unpack the master equations of QCD, we get equations that are related by symmetry—symmetry among the colors, symmetry among different directions in space, and the symmetry of special relativity between systems moving at constant velocity. Their complete content is out front, and the algorithms that unpack them flow from the unambiguous mathematics of symmetry. So let me assure you that you really should be impressed. It’s a genuinely elegant theory.

One reflection of that elegance is that the essence of QCD can be portrayed, without severe distortion, in a few simple pictures. They’re displayed in Figure 7.5. We’ll discuss them presently.

But first, for comparison and as a warm-up, I’d like to display the essence of quantum electrodynamics (QED) in a similar format. QED is, as its name suggests, the quantum-mechanical account of electrodynamics. QED is a slightly older theory than QCD. The basic equations of QED were in place by 1931, but for quite a while people made mistakes in trying to solve them, and got nonsensical (infinite) answers, so the equations got a bad reputation. Around 1950 several brilliant theorists (Hans Bethe, Sin-Itiro Tomonaga, Julian Schwinger, Richard Feynman, and Freeman Dyson) straightened things out.

The essence of QED can be portrayed in the single picture of Figure 7.4a. It shows that a photon responds to the presence or motion of electric charge. Though it looks cartoonish, that little picture is much more than a metaphor. It is the core process in a rigorous representation of a systematic method of solving the equations of QED, which we owe to Feynman. (Yes, him again. Sorry Murray.) Feynman diagrams depict processes in space and time whereby particles travel from given places at one time to other places at some later time. In between, they can influence each other. The possible processes and influences in quantum electrodynamics are built up by connecting world-lines (that is, paths through space and time) of electrons and photons in arbitrary ways using the core process. It’s easier done than said, and you’ll get the general idea after pondering Figures 7.4b-f.

Perfectly definite mathematical rules specify, for each Feynman diagram, how likely it is for the process it depicts to occur. The rules for complicated processes, perhaps involving many real and virtual charged particles and many real and virtual photons, are built up from the core process. It is like making constructions with TinkerToys. The particles are different kinds of sticks you can use, and the core process provides the hubs that join them. Given these elements, the rules for construction are completely determined. For example, Figure 7.4b shows a way in which the presence of one electron affects another. The rules of Feynman diagrams tell you how likely it is that exchange of one virtual photon, as depicted there, makes the electrons swerve by any specified amount. In other words, they tell you the force! This diagram encodes the classic theory of electric and magnetic forces that we teach to undergraduates. There are corrections to that theory when you take into account rarer processes, involving the exchange of two virtual photons, as shown in Figure 7.4c. Or a photon can break free, as in Figure 7.4d; that’s what we call electromagnetic radiation, one form of which is light. You can also have processes where all the particles are virtual, as in Figure 7.4e. None of the particles involved can be observed, so that “vacuum” process might seem academic or metaphysical, but we’ll see that processes of this kind are tremendously important.¹⁷

Figure 7.4 a. The essence of QED: photons respond to electric charge. b. A good approximation to the force between electrons, due to exchange of virtual photons. c. A better approximation includes contributions like this. d. Let there be light! An accelerated electron can emit a photon. e. A totally virtual process. f. Radiation of an electron-positron pair. The antielectron, or positron, is represented as an electronwith the arrows reversed.

Maxwell’s equations for radio waves and light, Schrödinger’s equation for atoms and chemistry, and Dirac’s more refined version, which includes spin and antimatter—all that and more is faithfully encoded in these squiggles.

In the same pictorial language, QCD appears as an expanded version of QED. Its more elaborate set of ingredients and core processes are displayed in Figure 7.5, which has a correspondingly more elaborate caption.

At this pictorial level QCD is a lot like QED, but bigger. The diagrams look similar, and the rules for evaluating them are similar, but there are more kinds of sticks and hubs. More precisely, while there is just one kind of charge in QED—namely, electric charge—QCD has three.

The three kinds of charge that appear in QCD are called, for no good reason, colors. These “colors” of course have nothing to do with color in the ordinary sense; rather, they are deeply similar to electric charge. In any case, we’ll label them red, white, and blue. Every quark has one unit of one of the color charges. In addition, quarks come in different species, or flavors. The only two flavors that play a role in normal matter are called u and d, for up and down.¹⁸ Quark “flavors” have no more to do with how anything tastes than quark colors have to do with how they look. Also, these mixed-metaphorical names for u and d (Zen koan: What is the taste of up?) do not imply that there is any real connection between flavors and directions in space. Don’t blame me; when I get the chance, I give particles dignified scientific-sounding names like axion and anyon.

Continuing the analogy between QED and QCD, there are photon-like particles, called color gluons, that respond in appropriate ways to the presence or motion of color charge, much as photons respond to electric charge.

So there are u quarks with a unit of red charge, d quarks with a unit of blue charge—six different possibilities altogether. And instead of one photon that responds to electric charge, QCD has eight color gluons that can either respond to different color charges or change one color charge into another. So there are quite a large variety of sticks, and many different kinds of hubs that connect them. With all these possibilities, it seems as though things could get terribly complicated and messy. And so they would, were it not for the overwhelming symmetry of the theory. For example, if you interchange red with blue everywhere, you must still get the same rules. The symmetry of QCD allows you to mix the colors continuously, forming blends, and the rules must come out the same for blends as for pure colors. This extended symmetry is extremely powerful. It fixes the relative strength of all the hubs.

Figure 7.5 a. Quarks (antiquarks) carry one positive (negative) unit of color charge. They play a role in QCD similar to that of electrons in QED. A complication is that there are several distinct kinds, or flavors, of quarks. The two that are importantfor normal matter are the lightest ones, called u and d. (To be honest, there are also different flavors of electrons, called muons and τ leptons, but I’ve been suppressingirrelevant complications.) b. There are 8 different color gluons. Each carriesaway a unit of color charge and brings in another color (possibly the same). The total of each color charge is conserved. There would seem to be 3 × 3 = 9 possibilitiesfor gluons. But one particular combination, the so-called color singlet, which responds equally to all charges, is different from the others. We must remove it if we are to have a perfectly symmetric theory. Thus we predict that exactly 8 gluons exist. Fortunately, that conclusion is vindicated by experiment. Gluons play a role in QCD similar to that of photons in QED. c. Two representative core processes, where gluonssimply respond to, or both respond to and transform, the color charge of quarks. d. A qualitatively new feature of QCD, compared to QED, is that there are processes whereby color gluons respond to one another. Photons do not.

For all their similarities, however, there are a few crucial differences between QCD and QED. First of all, the response of gluons to color charge, as measured by the QCD coupling constant, is much more vigorous than the response of photons to electric charge.

Second, as shown in Figure 7.5c, in addition to responding to color charge, gluons can change one color charge into another. All possible changes of this kind are allowed. Yet each color charge is conserved, because the gluons themselves can carry unbalanced color charges. For example, if absorption of a gluon changes a blue-charged quark into a red-charged quark, then the gluon that got absorbed must have carried one unit of red charge and minus one unit of blue charge. Conversely, a blue-charged quark can emit a gluon with one unit of blue charge and minus one unit of red charge; it becomes a red-charged quark.

A third difference between QCD and QED, the most profound, is a consequence of the second. Because gluons respond to the presence and motion of color charge, and gluons carry unbalanced color charge, it follows that gluons, quite unlike photons, respond directly to one another.

Photons, by contrast, are electrically neutral. They don’t respond to one another very much at all. We’re all familiar with this, even if we’ve never thought much about it. When we look around on a sunny day, there is reflected light bouncing every which way, but we see right through it. The laser sword fights you’ve seen in Star Wars films wouldn’t work. (Possible explanation: it’s a movie about a technologically advanced civilization in a galaxy far, far away, so maybe they’re using color gluon lasers.)

Each of these differences makes calculating the consequences of QCD more difficult than calculating the consequences of QED. Because the basic coupling is stronger in QCD than in QED, more complicated Feynman diagrams, with lots of hubs on the inside, make relatively larger contributions to any process. And because there are various possibilities for routing the flows of color, and more kinds of hubs, there are many more diagrams at each level of complexity.

Asymptotic freedom allows us to calculate some things, such as the overall flows of energy and momentum in jets. That’s because the many “soft” radiation events don’t much affect the overall flow, and we can ignore them in our calculations. Only the small number of hubs where “hard” radiation occurs demand attention. Without too much work, then, using pencil and paper, a human being can work out predictions for the relative probability of different numbers of jets coming out at different angles with different shares of energy. (It helps if you give the human being a laptop and send her through a few years of graduate school.) In other cases the equations have been solved—approximately—only after heroic efforts. We’ll discuss the heroic efforts that enable us to calculate the mass of the proton, starting from massless quarks and gluons—and thus to identify the origin of mass—in Chapter 9.

Quarks and Gluons 3.0: Symmetry Incarnate

In trying to do justice to the way in which postulating an enormous amount of symmetry—what we call local symmetry—forces us to include color gluons in our equations, and thus to predict their existence and all their properties, I’m reminded of one of my favorite Piet Hein Grooks:

lovers meander in prose and rhyme,
trying to say—
for the thousandth time—
what’s easier done than said.

Anyway, on to the prose and rhyme.

Back when we discussed the colored triangles and their symmetry, there was a fussy footnote about how you were supposed to ignore the fact that the different triangles were in different places. It makes perfectly good logical and mathematical sense to do that. In mathematics we often ignore irrelevant details in order to concentrate on the most interesting, essential features. For example, in geometry it’s standard operating procedure to deal, conceptually, with lines that have zero thickness and continue forever in both directions. But from the point of view of physics, it’s a little peculiar to suppose that symmetry requires you to take no account of where things are. Specifically, for example, it seems peculiar to suppose that the symmetry between red color charge and blue color charge requires that you change red charged quarks into blue charged quarks, and vice versa, everywhere in the universe. It might seem more natural to suppose that you could make the changes only locally, without having to worry about distant parts of the universe.

That physically natural version of symmetry is called local symmetry. Local symmetry is a much bigger assumption than the alternative, global symmetry. For local symmetry is an enormous collection of separate symmetries—roughly speaking, a separate symmetry for each point of space and time. In our example, we can make the red-blue charge switch anywhere and at any moment. Each place and moment defines, therefore, its own symmetry. With a global symmetry you have to make the same switch everywhere and everywhen, and instead of infinitely many independent symmetries you have just one lockstep version.

Because local symmetry is a much bigger assumption than global symmetry, it imposes more restrictions on the equations or, in other words, on the form of physical laws. The restrictions from local symmetry are so severe, in fact, that at first sight they appear impossible to reconcile with the ideas of quantum mechanics.

Before explaining the problem, a quick summary of the relevant quantum mechanics: In quantum mechanics, we have to allow for the possibility that a particle might be observed at different places, with different probabilities. There’s a wave function that describes all these possibilities. The wave function has large values where the probability is large, and small values where the probability is small (quantitatively, the probability is equal to the square of the wave function). Furthermore, wave functions that are nice and smooth—that vary gently in space and time—have lower energy than those that have abrupt changes.

Now to the heart of the problem: Let’s suppose we have a nice smooth wave function for a quark carrying red color charge. Then apply our example of local symmetry in a small region, changing red color charge into blue color charge. After that change, our wave function has abrupt changes. Inside the small region, it has only a blue color component; outside, it has only a red color component. So we’ve changed a low-energy wave function, without abrupt changes, into a wave function that has abrupt changes and therefore describes a state of high energy. That change of state is going to change the behavior of the quark we’re describing, unmistakably, because there are many detectable effects of energy. For example, according to Einstein’s second law, you could determine the energy of the quark by weighing it. But the whole point of symmetry is that it’s not supposed to change the behavior of the things it transforms.¹⁹ We want to have a distinction without a difference.

So to get equations that have local symmetry, we must alter the rule according to which abrupt changes in the wave function necessarily have large energy. We have to suppose that the energy is not governed simply by the steepness of change in the wave function; it must contain additional correction terms. That’s where the gluon fields come in. The correction term contains products of various gluon fields (eight of them, for QCD) with the different color components of the quark wave functions. If you do things just right, then when you make a local symmetry transformation, the quark wave function changes, and the gluon field changes, but the energy of the wave function—including the correction terms—stays the same. There’s no ambiguity about the procedure: local symmetry dictates what you must do, every step of the way.

The details of the construction are very hard to convey in words. It really is, as the Grook says, “easier done than said,” and if you want to see it done for real, with equations, you’ll have to look at technical articles or textbooks. I’ve mentioned a few of the more accessible ones in the endnotes. Fortunately, you don’t need to work through the detailed construction to understand the grand philosophical point, which is this:

In order to have local symmetry, we must introduce the gluon fields. And we must arrange the way those gluons fields interact with quarks, and with one another, just so. An idea—local symmetry—is so powerful and restrictive that it produces a definite set of equations. In other words, implementing an idea leads to a candidate reality.

The candidate reality containing color gluons succeeds in embodying the idea of local symmetry. New ingredients—color gluon fields—are part of the recipe for its candidate world. Are they present in our world? As we’ve discussed, and even seen in photographs, they are indeed. The candidate reality, hatched from ideas, is reality itself.