# How do gluons interact with quarks

## Thirty-six quarks and eight gluons

Experimental Physicist Thoughts

Elementary particle physics can be confusing. In the Standard Model there are simply too many particles and too many free parameters that the theory does not predict and that are therefore only determined experimentally. But there are also numbers in the standard model that are mandatory. Which can be explained logically. For example, why at first I naively assumed that there must be nine gluons, and why there are actually eight.

But first to the number of quarks: Many of my readers should be familiar with the fact that there are three families of elementary particles. The first family is sufficient to explain all stable matter. It consists of up quark, down quark, electron and electron neutrino. So two quarks. But I would also like to count the antiparticles in my little zoology of elementary particles.1 The anti-down quark and the anti-up quark have the same properties as quarks and only differ in their charges. So in the first family we have four quarks and four light particles2.

To the four quarks of the first family we have to add the four of the second family3 and the four of the third family4. So we have a total of twelve quarks. These quarks differ in their charge. There is an overview on the Quark page of the quantum world.

So quarks occur as particles and antiparticles. They come from different families and come in the flavors Up, Down, Strange, Charm and Top. They carry different electrical charges and, last but not least, they have another type of charge, the color charges. Each of the twelve quarks mentioned so far is available in three different versions, which are named after the primary colors red, green5 and blue. Antiquarks have the corresponding negative charges anti-red, anti-green and anti-blue. So we have to multiply the number of quarks by three again. There are thirty-six quarks in total. Eighteen Ordinary Matter and Eighteen Antiquarks. 6

Color charges work in a similar way to electrical charges. In electromagnetism there are positive and negative charges and there is a particle that couples to that charge. The photon. The photon can create a pair of charged particles; it can be emitted by a charged particle when it changes its orbit. It can be exchanged between two charged particles, thereby exchanging energy and momentum.

All of this works with the color charge too. There are three “positive” charges red, green, blue and three negative charges anti-red, anti-green and anti-blue. Just as in the electromagnetic force a particle is neutral, i.e. uncharged to the outside, if it has the same number of positive and negative charges, a particle is color-neutral to the outside when every red is balanced by anti-red, every green by anti-green and every blue by anti-blue. 7 But there is another way to achieve neutrality. A particle is also neutral if it contains the same amount of red, green and blue. 8

### Gluons

Colored quarks can also be created in pairs, emit particles when they change their path, or attract each other by exchanging particles. These particles are called gluons. Gluons differ from photons in that they are themselves carriers of the type of charge of which they are the exchange particles. Each gluon carries a color charge and an anti-color charge. If one forms all possible pairs, one would naively assume that there must be nine types of gluons. Each of the three colors combined with each of the three anti-colors. However, that is not entirely true.

We have to consider that particles in elementary particle physics are something completely different from the classical point particles in mechanics. They represent computational rules with which the dynamics of quantum objects among one another is described. The colors of the quarks are metaphors for a certain symmetry that underlies the strong nuclear force. This symmetry is called a special unitary group and treats something like generalized rotations in spaces of complex numbers. That sounds complicated and it is, but the rotation gives us a new approach to visually understand what it is about.

In an earlier post I wrote how interesting complex numbers are for physics. We can use them to describe vibrations and waves quite simply. And because quantum objects like quarks are described using wave functions, it is not surprising that we can describe the forces between quarks with complex numbers.

A photon that transmits electrical force can also be described as a complex number. As a wave, it has an amplitude9 and a phase10. The transmission of photons changes the speed of electrons, it changes the wave function of the electrons in a mathematically precisely defined way. It is very similar with the color interaction, only that it is a triad of different charges that we call color.

The mathematical structure of the interaction of quarks cannot work with simple complex numbers, it has to use quantities that contain three complex numbers. These are these special unitary groups. We can think of them as twists. Any rotation in space can be broken down into individual components. An airplane can move its nose up and down, this is called pitch, it can swing sideways, which is called yaw, or it can turn around its longitudinal axis, which is roll .

What the exchange of gluons does with the wave functions of quarks is similar to rotations. There are the six color-changing rotations mentioned above, mediated by gluons like red-anti-green or blue-anti-red. But there are two other basic rotations that leave the colors as they are and only change the phases, i.e. the relative positions of the wave functions to one another. These two rotations are superimpositions of the conceivable states red-anti-red, green-anti-green and blue-anti-blue. There are only two degrees of freedom for this phase rotation, similar to how you only need two coordinates on a three-dimensional sphere like our earth to uniquely determine a location.

In total there are eight gluons that are the carriers of forces between quarks. Six of them change their color charge, two leave the colors untouched and only rotate their relative phases. Another combination of red-anti-red, green-anti-green and blue-anti-blue is conceivable, but does not belong to the permitted rotations of the special unitary group and therefore does not exist as a gluon.

Remarks:
1. It is generally not customary to count antiquarks as quarks and will lead to a certain ambiguity here (and in other of my texts). But I am sure that you will always recognize in this context whether I mean quarks in the sense of a special particle or as a generic term for all quark-like elementary particles.
2. The light particles are called leptons. In addition to the above, the positron and the anti-electron neutrino also belong to the first family.
3. Strange and Charm quarks with antiparticles
4. Bottom, top and antiparticle
5. Some textbooks use the basic color yellow instead of green, but that doesn't matter because colour is just a vivid picture. The quarks are of course not really painted. Printer specialists may want to use magenta, cyan, and yellow.
6. The three different charges justify seeing red, green and blue up-quarks as different particles. After all, we see particles that differ in their electrical charge, for example up and down quarks, also as different particles. In addition, measurements of the quark generation rate in particle accelerators clearly show that every quark exists three times.
7. That is the reason why mesons, which essentially consist of a quark and an antiquark, appear uncharged in terms of color to the outside.
8. The proton, for example, consists essentially of two up quarks and one down quark, one of which must be red, green and blue.
9. the deflection of the wave

Joachim Schulz is group leader for sample environment at the European XFEL GmbH in Schenefeld near Hamburg. His scientific career began in quantum optics, where he studied the interaction of individual atoms with laser fields. Among other things, she introduced him to atomic physics with synchrotron radiation and cluster physics with free-electron lasers. For four years he planned, set up and carried out experiments on coherent X-ray diffraction on biomolecules at the Center for Free-Electron Laser Science (CFEL) in Hamburg. In his free time, for example, he writes "Joachim's Quantenwelt" on the blog or on his homepage.