Get your protons right!

The atomic nucleus consists of protons and neutrons. The protons and neutrons are themselves made of three quarks each, held together by gluons. That much is clear. But just how do the gluons hold the quarks together?

The quarks and gluons interact through the strong nuclear force. The strong nuclear force does not have only one charge – like electromagnetism – but three charges. The charges are called “colors” and often assigned the values red, blue, and green, but this is just a way to give names to mathematical properties. These colors have nothing to do with the colors that we can see.

Colors are a handy terminology because the charges blue, red, and green can combine to neutral (“white”) and so can a color and its anti-color (blue and anti-blue, green and anti-green, and so on). The strong nuclear force is mediated by gluons which each carry two types of colors. That the gluons themselves carry a charge means that, unlike the photon, they also interact among each other.

The strong nuclear force has the peculiar property that it gets stronger the larger the distance between two quarks, while it gets weaker on short distances. A handy metaphor for this is a rubber string – the more you stretch it, the stronger the restoring force. Indeed, this string-like behavior of the strong nuclear force is where string-theory originally came from.

The strings of the strong nuclear force are gluon flux-tubes, that are connections between two color-charged particles where the gluons preferably travel along. The energy of the flux-tubes is proportional to their length. If you have a particle (called a “meson”) made of a quark and an anti-quark, then the flux tube is focused on a straight line connecting the quarks. But what if you have three quarks, like inside a neutron or a proton?

According to the BBC, gluon flux-tubes (often depicted as springs, presumably because rubber is hard to illustrate) form a triangle.

This is almost identical to the illustration you find on Wikipedia:

Here is the proton on Science News:

Here is Alan Stonebreaker for the APS:

This is the proton according to Carole Kliger from the University of Maryland:

And then there is Christine Davies from the University of Glasgow who pictured the proton for Science Daily as an artwork resembling a late Kandinsky:

So which one is right?

At first sight it seems plausible that the gluons form a triangle because that requires the least stretching of strings that each connect two quarks. However, this triangular – “Δ-shaped” – configuration cannot connect three quarks and still maintain gauge-invariance. This means it violates the key principle of the strong force, which is bad and probably means this configuration is not physically possible. The Y-shaped flux-tubes on the other hand don’t suffer from that problem.

But we don’t have to guess around because this is physics and one can calculate it. This calculation cannot be done analytically but it is tractable by computer simulations. Bissey et al reported the results in a 2006 paper: “We do not find any evidence for the formation of a Delta-shaped flux-tube (empty triangle) distribution.” The conclusion is clear: The Y-shape is the preferred configuration.

And there’s more to learn! The quarks and gluons in the proton don’t sit still, and when they move then the center of the Y moves around. If you average over all possible positions you approximate a filled Δ-shape. (Though the temperature dependence is somewhat murky and subject of ongoing research.)

The flux-tubes also do not always exactly lie in the plane spanned by the three quarks but can move up and down into the perpendicular direction. So you get a filled Δ that’s inflated to the middle.

This distribution of flux tubes has nothing to do with the flavor of the quarks, meaning it’s the same for the proton and the neutron and all other particles composed of three quarks, such as the one containing two charm-quarks that was recently discovered at CERN. How did CERN picture the flux tubes? As a Δ:

Now you can claim you know quarks better than CERN! It’s either a Y or a filled triangle, but not an empty triangle.

I am not a fan of depicting gluons as springs because it makes me think of charged particles in a magnetic field. But I am willing to let this pass as creative freedom. I hope, however, that it is possible to get the flux-tubes right, and so I have summed up the situation in the image below :