A Tiny Slice of Quantum Chromodynamics

Yesterday, I attended the last summer Sunday at Brookhaven National Labs and in particular, did a brief tour of the engineering department that worked on building the RHIC (relativistic hadron ion collider) as well as hear a lecture by Gene van Buren on a solenoidal tracker attached to RHIC called the STAR detector. The RHIC is able to accelerate gold nuclei to 99.995% the speed of light; the track has about a 2.5 mile circumference and the particles travel around it about 80,000 per second. At speeds like this, the particles are very massive and experience very little time. This statement is not something we usually say in our everyday experiences but if you have some experience with special relativity, you know that in order to accelerate something, you need to apply force. At speeds near the speed of light, there is a diminishing return. In order to gain a fraction of speed, one needs to apply more and more force and asymptotically, one needs an infinite amount of force to get to the speed of light. If we view mass as a property which resists being moved (faster), then one way to view relativistic particles is that they’re very massive; cf. moment of inertia.

Some Engineering

To accelerate the particles to such speeds, they use electromagnets to propel two beams going in opposite directions around the track; the beams cross at 6 locations around the track. But because they need a high current, around 5000 amps, regular materials such as copper wires can’t bear that intensity. To put 200 amps through copper, you need a copper wire almost half and inch in diameter and lots of insulation to protect yourself from the heat radiating out due to resistance. However, if we use superconducting material, below their critical temperatures, resistance drops to 0. So they use niobium-titanium alloys, formed into fibers, wrapped around copper, and then they stretch it out (I assume it’s rather ductile). In order to cool the wires, they use liquid helium which has a boiling point of 4 Kelvin or -452.47 Fahrenheit. Thus, their track needs to have a pipe for the particle beam, a larger tube of the superconductor electromagnets, pipes for liquid helium supply, and also vacuum-insulation to keep out the heat.

I asked one of the engineers if they produce their own liquid helium. What I meant was whether they get a supply of gaseous helium and then pressurize it, let it cool, then pressurize again, let it cool, that sort of thing, until its liquid. I don’t know what the mechanics would actually be. His answer didn’t address that but rather pointed out that we can’t really chemically create helium (it’s really chemically inert); our supply of it is from radioactive decay and lots of helium is lost as it floats out of our atmosphere. The only process I know of to create helium is 93 million miles away in the core of the sun. I wonder if it’s possible to use liquid hydrogen. There’s the obvious danger of it combusting but if they’re in a vacuum with all the oxygen pumped out, that seems like a potential alternative. Of course, I’m just wildly postulating. I don’t know how low of temperatures the superconducting material requires; there are better superconductors now in the sense that their critical temperature is easier to achieve. But they would need to completely redo all 1,400 electromagnets which would be costly. RHIC has been running for about 25 years without any incident so their system is working. And of course, a wise person would never assume that everything works perfectly, such as ruling out the possibility of combusting liquid hydrogen.

Remark: For a theoretical mathematician like myself, this is when I would mention the vortex equations; for a survey on these equations and a bit about superconductors from a mathematical viewpoint, see the first lecture by Tim Perutz.

Experimental Nuclear Physics

Now, what is it that experimental nuclear physicists like Gene want to understand? He gave an excellent analogy. In liquid phase, water has the interesting property of being “sticky,” due to hydrogen bonds. This is what allows water to form droplets and for insects to walk on the surface of ponds. If we apply enough energy, then it becomes vapor and the water molecules have enough energy to be “free” of other water molecules; i.e. the hydrogen bonds are no longer all that relevant in comparison to the high energy of the water molecules. Similarly, if we have highly energized nuclei, the components inside, we might think, should behave like gases. But surprisingly, they do not! They form something known as quark-gluon plasma which has properties that we can associate to liquids, such as viscosity and pressure. But there are some unusual things about this “liquid.” Firstly, quark-gluon plasma is only sustained for a tiny duration, smaller than femtoseconds. Also, to reach that state, they need to get to 4 trillion degrees Celsius. This may seem like a lot but on their tiny scale, this has about as much energy as two mosquitos colliding. But one needs to remember that mosquitos are at least a billion billion times more massive than gold nuclei. This is one reason why using gold nuclei is actually inexpensive; they’re using so little of it. Gene said they’ve probably used less than the gold in his wedding ring in the last 25 years.

I asked Gene whether, in this quark-gluon plasma, we observe quark confinement in any way (more on that soon). He said that we don’t really because the quarks are “free” within this soup of matter to move about. Similarly, a water molecule can move about in a glass of water, especially if we stir the water. The quarks aren’t free to leave the soup no more than a typical water molecule at room temperature inside the glass of water would fly up to the surface and then continue flying off. I mean, evaporation happens but in this analogy, let’s just ignore evaporation. I asked if this kind of freedom is what people in the field call asymptotic freedom; it’s not.

Some Theory of Quantum Chromodynamics

Above, I mentioned words such as “confinement” and “asymptotic freedom.” I will attempt to explain them as I understand but also give more background.

Let’s recall that one view of electromagnetism is to treat it as an abelian gauge theory. We use $\mathbb{R}$ or if quantized, $U(1)$, as the gauge group. Given any finite abelian group, its irreducible complex representations are 1-dimensional, by Schurs lemma. Though $U(1)$ is not finite, since the complex numbers are algebraically closed and that commuting operators are simultaneously diagonalizable, the irreducible complex representations of $U(1)$ are quite straightforward; this can be understood via Fourier analysis (strictly speaking, Fourier analysis tells us more than what purely representation theory considerations tell us). There is a “fundamental” representation of $U(1)$ which is how it acts on $\mathbb{C}$ naturally and it also has its adjoint representation coming from acting on the Lie algebra $\mathfrak{u}(1) \cong i\mathbb{R}$ (maybe complexify this?). This is rather important as we see there is one fermion and one boson in the theory: electrons and photons. By using principal $U(1)$-bundles over spacetime and understanding curvature, we’re able to obtain a very elegant description of Maxwell’s equations using a veyr minor amount of Hodge theory.

If we want to discuss the weak force, the structure group to use is $SU(2) \cong \text{Spin}(3) \cong Sp(1)$. How lucky we are that these three Lie groups are isomorphic; it gives us three viewpoints of them: as $2 \times 2$ complex matrices, as elements of a spin group (which we see double covers $SO(3)$ almost by definition), or as unit quaternions. The nonabelian properties make this an interesting group though because its maximal abelian torus is $U(1)$, the representation theory can be cleverly reduced to an understanding of $U(1)$. Now, $\dim_\mathbb{R} SU(2)=3$ and note that there are three bosons: $W^+,W^-,Z$ bosons.

For the strong force, the realm of quantum chromodynamics (QCD), we turn to $SU(3)$. The reason for this is that we’ve observed that quarks and gluons have an extra property that things like electrons do not have. In addition to electric charge, they also have something called color charge (hence, the name chromodynamics). While some people don’t think this is a useful term, I personally find it cute and somewhat helpful. As we’ll see, it’s useful for thinking about confinement. But I’m actually going a little backwards. Confinement was not discovered because of color. Rather, color was discovered because of confinement!

Having said that I’ll still go anachronistically. Let’s just accept that the first generation of quarks, the ones we see in “everyday” matter of protons and neutrons, can have properties that we’ll call red, green, blue. Antiquarks will have cyan, magenta, and yellow (or antired, antigreen, antiblue). Because there are these three colors, the Hilbert space for a single quark will be $\mathbb{C}^3$ and of course, the operators acting on it should be unitary so we need a subgroup of $U(3)$; we also want to preserve more so we’ll take $SU(3)$. Then, we let’s treat confinement as an axiom: the totality of the colors has to be white; so we need equal parts $r,g,b$. This is something that’s true of light: when we mix equal parts red, green, and blue light, we get white light. Of course, this is not what we learned in art class but that’s because we’re talking about paint pigments there and how they reflect light rather than how light waves interfere. Also, we impose that the total color must be invariant under $SU(3)$ action. As such, no individual quarks can exist because $SU(3)$ can act on a single quark and change its color! Similarly, no pairs can exist since there are no unit vectors in $\mathbb{C}^3 \otimes \mathbb{C}^3$ fixed by $SU(3)$. However, there are unit vectors in $\mathbb{C}^3 \otimes \mathbb{C}^3 \otimes \mathbb{C}^3$ fixed by $SU(3)$. They form a 1-dim family $\Lambda^3 \mathbb{C}^3 \subset \mathbb{C}^3 \otimes \mathbb{C}^3 \otimes \mathbb{C}^3$ and is spanned by $r \wedge g \wedge b$. It is also possible to take $\mathbb{C}^3 \otimes (\mathbb{C}^3)^*$ which represents a quark-antiquark pair. So then, for example, a green quark and a magenta antiquark have color coming out to be white.

So those are the rules of confinement and I’ve presented a simple representation theory description for it. Indeed, confinement is observed experimentally. But despite this being a description, it’s not really explanatory in a satisfying way. It would be like describing to an alien visiting earth for the first time the traffic laws in the US. We can say, “When you see a red light and there are no cars in front of you, you need to slow down your vehicle until it comes to a stop before the line marking entry into the intersection.” That would be a description of the law. But if we want to explain why the law is in place, we might need to explain that the traffic light is meant to control traffic flow, that it’s for the safety of the drivers on the road, and that sort of thing. If the alien is somewhat indestructible, maybe the notion of being kept safe from blunt force trauma has never been relevant to it and so we would also need to explain that humans are fragile and not designed for moving at 45 mph; hence the purpose of these traffic laws is partially for safety (something New York drivers should remember more often). Why do I bring up this analogy? It’s to emphasize that confinement is still something of a mystery in QCD. We would like a theory for QCD in which confinement comes out as a consequence, not take it as an axiom. As my friend Dallas said: “Any current description of confinement (including the dyons) that isn’t just numerical results from lattice gauge theories requires some approximations and assumptions.” Moreover, though I’ve talked about quark confinement, it’s actually the case that in QCD theories without fermions (without quarks), we still see confinement phenomena. So it’s clear that confinement should have more to do with gluons, not quarks. How mysterious!

Okay, now that we’ve covered confinement, let’s discuss some implications. One implication is that we can never whack a quark with so much energy that it gets separated from its triplet of buddies. What would happen is that the energy would actually create a quark-antiquark pair so that the quark that is “leaving” gets a companion so that the total color is still white while the duo left behind gains a friend so that they stay a triplet. This may be shocking to those unfamiliar with QCD or quantum field theory in general, to hear of particles popping in and out of existence. But the truth can be stranger than fiction.

I’ve also been putting off half of the theory; I’ve only mentioned the fermions of QCD which are the quarks. There are also the bosons which are the force-mediating particles in particle physics. For QCD, they are called gluons because they “glue” the quarks together. The glue is strong enough to overcome the fact that protons, which are all positively charged, generally want to repel each other due to the electromagnetic forces. The QCD force has to be stronger than the electromagnetic forces and so physicists, with their extreme wisdom and creativity, have dubbed this force the strong force. Though I’m taking a bit of a light-hearted jab at them, I think it’s usually best to name something in a way that describes what it is/does.

So what are some facts about gluons? For one, since $\dim_\mathbb{R} SU(3) = 8$, there are eight types of gluons; Murray Gell-Mann’s so-called 8-Fold Way (an allusion to the 8-Fold Path of Buddhism, I suppose). For another, unlike photons which don’t self-interact, gluons have color charge and so they do interact. In fact, within the nucleus of an atom, it’s just an absolute zoo. We’ve got quarks exchanging gluons all the time, we have particles popping in and out of existence, we have gluons interacting with other gluons as well (via flux tubes). If we’re feeling confident from the success of Feynman diagrams and path integrals for QED (quantum electrodynamics), we might apply the same kinds of tools to QCD but except in a few cases, there are infinities appearing and my sense is that the usual renormalization scheme doesn’t help. Lastly, the mass of quarks is actually relatively small (and comes from interaction with the Higgs field, something true of all particles with mass). But the mass of a proton is much larger than the combined mass of the triplets; where is that mass coming from? The answer is that it’s in the binding energy which is due to the interaction of all the gluons. If you’re confused as to why I just said the word “energy” when talking about mass, recall the equation of Einstein that you’ve known for a long time relating energy and mass: $E=mc^2$. Indeed, physicists knew in the first half of the 20th century that the binding energy of atomic nuclei contained great power if we could unleash it. Of course, this was done in a spectacular and weaponized fashion at Los Alamos in the Trinity test and then done again twice against Japan which brought a devastating conclusion to WWII.

The last thing I want to say in this section is what asymptotic freedom is. It is a phenomenon where, in extremely close quarters at extremely high energies, the strong force actually weakens. In fact, asymptotically as energy increases, the force weakens to zero. This allows physicists to do perturbative calculations at high energies (which one can very roughly, view as “starting off without quantum effects and then slowly turning them on to see what happens”). The significance may be lost to non-physicists but I’m told by physicists that perturbative computations are something they’re happy about and also, quantum field theory was being doubted before this discovery. After this discovery, QFT was “rehabilitated.” Anyways, asymptotic freedom was discovered by Frank Wilczek and David Gross in 1973 and independently by David Politzer. For this, the three were awarded the Nobel Prize in 2004. For some personal anecdotes from Wilczek, there is a lovely interview with him and Steve Strogatz in the Joy of X podcast episode.

There’s a lot more to say about QCD, of course, but I’m getting to the edge of what I know. Maybe I’ll mention one very last thing in this section. QCD also has what we call electric and magnetic charges though these are misleading names as they are different from the electric and magnetic charges of EM. Rather, these are chromoelectric and chromomagnetic charges. But interestingly, there is something superconductivity in QCD, similar to EM. In EM, below critical temperatures, magnetic field lines are repelled out of the material. In QCD, something similar occurs for the analogous magnetic fields.

Final Remarks on QCD

To conclude, let’s discuss quark-gluon plasma. Above, I said that it has liquid properties such as viscosity. One of the neat things about it is that it has the lowest possible viscosity that a liquid can have! I asked Gene whether he was making a theoretical sort of statement or empirical statement; i.e. was this a statement proven by theory or has it also been experimentally verified? He said that it was sort of both. Theorists have done computations to see what the lowest visocity of a liquid could be (assuming some definition of liquid) and found that quark-gluon plasma realizes that minimum. I think Gene then said that the viscosity has then been measured experimentally. However, QGP might not be the only substance reaching this minimum; some superfluids like liquid helium may also realize the minimum.

Appendix: Metascience

Above, I mentioned that one of the problems still open in physics is to give a satisfactory understanding of confinement. It may be that in order to explain it, we need an overhaul of QCD as it is currently. I don’t know whether there is any consensus among experts as to what must happen in order to have a satisfactory theory. But there is something that Feynman said that I want to share, partly because it has been lingering in my mind for several days now. This section could stand apart from everything I said above.

Newtonian mechanics has been and continues to be tremendously successful. It really did require quite a leap to imagine that the same force governing a falling apple was also the force governing the orbit of the moon around the earth and the orbit of the earth around the sun and the movement of all the stars and galaxies. And Newtonian mechanics can keep our satellites in the sky, get us to Mars, help us understand everyday events like throwing a baseball or driving a car. But there was something it couldn’t do: explain the precession of Mercury, a tiny, almost seemingly inconsequential deviation from the prediction. It seemed like just a tiny imperfection in Newtonian mechanics that maybe one could try smoothing over. One might think to just tweak Newtonian mechanics a little bit in order to account for Mercury’s precession. But as Feynman said, “You can’t tweak a perfect theory like Newtonian mechanics so that it produces tweaked results. It’s already a perfect theory! You need a whole separate perfect theory like general relativity.”

I like Feynman’s point of view. It’s not that Newtonian mechanics was wrong; it’s perfectly right about the things it has any business being right about. But as always, context is important and there are arenas of physics that Newtonian mechanics does not apply to. So if you want to be judgemental, then fine, it was incomplete but I don’t think it’s wrong. For what it was intended to do, for what Newton had in mind, I think it’s a perfectly good theory that went as far as it could in the scope that it applies to. Thus, to account for something beyond it’s scope, you can’t just add on a few small building blocks but need something entirely different. It’s like trying to break a land speed record with conventional car engines. You just won’t get very far; you need to imagine attaching a rocket engine.

I think this is one of the ways in which scientific revolutions occur. It’s possible for small changes to push the limits of science but sometimes, when those techniques have been pushed to the limit, even if we just want to push a little further to obtain results that look only slightly different, it can require a tremendous shift in thinking and a lot of work. This can go unappreciated when people only look at the outcomes and don’t see much difference, not aware of the entirely new building blocks beneath.