How to Spin like a Proton

The universe that we knew about before dark energy and dark matter is described by the Standard Model of Particles and Interactions. All of these particles have various physical properties associated with them including the following.

  • mass
  • charge (electric, weak, strong/color)
  • spin

I’m not going to worry about the mass, but I will need to discuss the other two.

If you clicked on the link above, you may have learned that the proton ,which along with the neutron makes up the nuclear matter in all atoms, is not itself a fundamental particle. It’s composed of 3 ‘valence’ quarks. I’ll describe why I add the term valence later.

Now the Standard Model describes how quarks work, but it doesn’t describe how protons work. So how do we measure the properties of quarks? By using protons, e.g. the Higgs discovery at the LHC in 2012 used the collisions of protons with protons. Why don’t we measure quarks directly? Because we can’t.

Everyone is familiar with the gravity and electromagnetic forces. Let’s take gravity for an example. On the surface of the Earth, the force of gravity on a mass m has some value F. If we move to a distance twice as far from the center of the Earth, that force would drop to F/4, three times as far, F/9, ten times as far, F/100, … In other words the further you go away, the weaker the force. In the Standard Model, forces show up through the exchange of force carrying particles, e.g. photons for electromagnetism. In an oversimplified explanation, the reason that gravity (or electromagnetism) gets weaker at larger distances is because the force carriers in the field do not interact with each other and spread uniformly around the masses involved.

Quarks carry a mass charge and an electric charge, so the electromagnetic force between two quarks does indeed mimic the example give for gravity. It falls off with distance. But quarks also carry a strong/color charge, and this is where things get different. The force carrier for a strong field is the gluon, but unlike the photon, the gluon carries a strong charge itself and interacts with other gluons. This has a curious effect in that instead of the field getting weaker at larger distances, it actually gets stronger!

So two quarks would naturally want to stay next to one another. That’s not unlike what we have with electric charges. An electron and a proton have opposite electric charges and will attract each other and try to recombine into a hydrogen atom, i.e. a final state with 0 net electric charge. This is what we normally encounter in nature, atoms and molecules with 0 net electric charge. The same thing goes for strong, or colored, matter. We encounter objects which are ‘colorless’ in nature. Nevertheless, if I want to make electrically charged matter from electrically neutral matter, I can do so by separating the charges. I’m writing this on a machine that uses that ability. But if I try to separate two colored charges, the field just goes stronger. And here is where Einstein comes in. The field energy keeps growing until one exceeds the energy needed to produce a new quark (mc^2), so that you just end up with two particles, both colorless. Consequently, producing colored matter from colorless matter becomes the paradox of creating a string with only one end. These colorless combinations of quarks are called hadrons (That’s why it’s the Large Hadron Collider).

You’re probably wondering when I’ll get to spin, so without going into further details on how quarks combine into colorless hadrons, I just wanted to assert that colorless objects primarily come (99.999999999999%) in two varieties: baryons composed of three quarks and mesons composed of a quark and an antiquark. Protons and neutrons are composed of three of the lightest quarks (up,up,down) for the proton and (down,up,down) for the neutron, modulo the binding energy which I’ll get back to.

Now we finally get to spin. Each of the quarks has the properties of a type of particle called a fermion, which just means that it obeys something called Fermi-Dirac statistics (Did that help?). Fermions have spins which are integer increments from 1/2. This means that they can have spins of …, 3/2, 1/2, -1/2, … When you combine three fermion quarks into a proton, you necessarily get something which is also a fermion. You can understand this by taking 1/2 and adding it three times with any signs, e.g. 1/2-1/2+1/2. You’ll always end up with something that’s still a fermion.

Well now we know where the proton spin comes from. It comes from the three quarks. Well, not quite. Here’s where that binding energy comes into play. A proton has a mass of around 1 GeV (GeV is a unit of energy and if one sets c=1 and reverses Einstein’s equation E=mc^2, then it’s a unit of mass, i.e. ‘natural’ units). But the up and down quarks have masses which are much less than 1/3 GeV, in the range of 0.001 GeV. Where does the extra mass come from? The binding energy of the color charges! Also, this means that that binding energy can produce lots of up and down quarks by itself, so that a proton at any one time could be (up,up,antiup,up,down) or (up, down, antidown, up, down) or … In fact, the next heaviest quark at ~0.5 GeV, the strange quark, can also be made relatively easily, i.e. (up, up, strange, antistrange, down). This is why I used the term ‘valence’ quark earlier. They’re the quarks left over after removing the quark-antiquark pairs in the gluon field forming the binding energy. Also the gluons making up the field of this binding energy also have spin, but since they are bosons their spin is 1. Consequently, a proton is not simply a colorless 3 quark state. It is a colorless 3 quark state in a massive, colorless swarm of gluons.

Well this all sounds very complicated, but why should I expect this swarm of gluons or gluons with quark-antiquark pairs to have any net spin rather than simply sum to zero, i.e. (1-1-1+1-1+1-…=0)? Put in as many binding energy particles as you want, one can make simple symmetry arguments that the spin contributions should cancel. If fact, a simple test of this is to measure the magnetic moments of the proton and neutron. The magnetic moments of the proton and neutron are related to the spins of the electrically charged particles inside them. One can then compare the measurement to the value if one sums the magnetic moments of the three valence quarks. The result is that there is very good agreement for multiple decimal places.

So why did I bother you with this. Well, there’s the rub. Magnetic moments are good, but that’s still a global, summed type of measurement. An attempt to measure what the individual quarks were actually doing was finally attempted in the 1980’s at Stanford and at CERN. Both of these scattered electrons (SLAC) or muons (CERN) off of polarized protons which really meant scattering polarized photons off of supposedly polarized quarks within the proton since electrons and muons interact with quarks only electromagnetically. This allowed them to probe a separate parameter which was the fraction of the proton’s momentum carried by the quarks. I won’t bother you with that explanation, other than to say that when one integrates over this variable, the integral is what is related to the fraction of the proton spin carried by the quarks. As you might expect, since I’m explaining it, the result was not consistent with the simple model used for the magnetic moments. In fact, the result was consistent with none of the proton spin coming from the quarks.

All of a sudden, this became a sexy topic, and several new experiments (including the one I worked on, the SMC) were started and took data during the 1990’s. The result was that the zero contribution changed to a 50% contribution from the valence quarks with the other 50% still a mystery, even to this day. The problem comes from the complications at low momentum fraction where the simple valence model (up, up, down) becomes exceedingly complex (up, up, down, antiup, down, antidown, up, antidown, up, down, antiup, …) for reasons that I will not delve into in further detail here.

But if you’d really like to go into the gory details, here’s a good link.


2 thoughts on “How to Spin like a Proton

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