# Attract different quarks

## Quantum Chromodynamics on the Lattice

**Quantum chromodynamics describes how the building blocks of atoms interact with one another. But despite all modern aids, physicists are overwhelmed with exactly solving the mathematical equations behind them. So they use a trick.**

When people talk about the Higgs particle these days and that it explains the origin of mass, it is easy to forget that the majority of our own mass, the mass of our earth and the entire universe that is visible to us has a completely different origin. It is not the Higgs field that makes ordinary matter as heavy as it is - but the kinetic and binding energy of the smallest particles known to us that make up atomic nuclei. Because of the equivalence of mass and energy, expressed by Einstein in the formula E = mc², we notice this energy as mass. This leads to the somewhat strange situation that the inertia and heaviness of all things around us have their origin in the extremely fast movement of their many, very light components.

Elementary particles and basic forces

But what are these light building blocks of atomic nuclei? For more than thirty years now, physicists have agreed that protons and neutrons, which make up the atomic nucleus, themselves consist of three so-called quarks. The suggestion that this is the case, however, is much older, namely from 1964. At that time, the physicist Murray Gell-Mann recognized that this assumption could bring order to the already confusing zoo of "elementary" particles observed at the time.

If one assumes that a certain family of particles, the so-called baryons - which include protons and neutrons, for example - consists of three and another family, the so-called mesons, consists of only two of these quarks, then only three quarks are needed, in order to build up all baryons and mesons known at the time. Gell-Mann received the Nobel Prize in 1969 for this fundamental discovery, but the model initially had one major catch: Up until then, no one had succeeded in observing a single quark, and that has not changed to this day.

### A strong force

A theory therefore had to be found that takes this so-called quark inclusion into account - which contains quarks as fundamental objects, but which at the same time forbids observing a single, free quark. Such a theory should also provide something essential: It should explain why the atomic nucleus, which consists of electrically neutral neutrons and positively charged protons, can stick together at all. In other words, the theory must contain a force that holds the nucleus together. Since this force has to be stronger than the electrical repulsion between two positively charged protons in the atomic nucleus, it is also called the “strong nuclear force” or simply “strong force”.

Concept of color charge

In 1973 Gell-Mann, together with the two theoretical physicists Harald Fritzsch and Heinrich Leutwyler, proposed such a theory - quantum chromodynamics, or QCD for short. According to this, all quarks have an additional property, the so-called color charge, which can take on three different values. Analogous to the basic colors red, green and blue, these color charges can be combined: A red, a green and a blue quark together result, for example, in a color-neutral “white” particle. In addition, for every particle there is an antiparticle with exactly the opposite charge - in the case of blue this would be anti-blue, or the complementary color yellow. A color-neutral particle can therefore also be composed of a quark and a corresponding antiquark.

Interestingly, the color-neutral combinations of three quarks correspond exactly to the above-mentioned baryons, for example protons or neutrons, and the color-neutral combinations of one quark and one antiquark correspond to the above-mentioned mesons. If there were a mechanism in theory that explains why only color-neutral objects can be observed, many of the puzzles of the quark model would be solved.

### Matter and exchange particles

In fact, it is believed that this mechanism is known today. It is based on a property of the second type of particle that occurs in QCD alongside quarks, the so-called gluons. In contrast to the quarks, which represent the "matter particles" of the QCD, the gluons are the "exchange particles" - the particles that mediate the strong nuclear force between the matter particles. So they are the counterpart of the electric and magnetic fields that mediate the electromagnetic force between charges. More precisely, they are the exact counterpart of the particles that make up electrical and magnetic fields, the light particles or photons. And just as two electrical charges repel or attract each other by exchanging photons, so do quarks by exchanging gluons.

Difference between electric field and color field

There is, however, a major difference: While photons as exchange particles of the electromagnetic force only occur in one variant, in QCD a total of eight different types of gluons mediate the force between the quarks. Since the sum of the color components must not change, a gluon that transmits the force between two different color components must have a color itself: if it holds a blue and a red quark together, for example, it must be orange - composed of anti-blue or Yellow and red - and convert a blue into a red curd. As a result, there are six charged “colored” gluons in the QCD, which are joined by two uncharged “white” gluons.

While the electric field between two electric charges can expand unhindered throughout the room, the color field between two quarks with opposite color charges behaves completely differently. In contrast to the electrically neutral photons, the gluons that make up the color field carry a color charge themselves. They therefore attract each other and limit the color field to a certain area of the room. If you try to separate the two quarks like electrical charges, the gluon field between the quarks forms a kind of tube. The greater the distance between the quarks, the more energy there is in this tube. Finally, the accumulated energy is converted into mass: into two more, again oppositely charged quarks. The end result of the experiment is not two free quarks, but two color-neutral mesons.

### Do arithmetic on the grid

In 1974, Gell-Mann's student Kenneth Wilson attempted to substantiate this vivid picture mathematically. Since it seemed hopeless to him to describe the bond states of quarks and gluons with the usual means of the so-called perturbation theory, in which one starts from a state of free quarks and gluons and adds corrections step by step, it was Wilson's goal to make the equations complete and exact to solve. Of course it's not that easy.

In QCD, quarks and gluons are described by quantum fields that spread throughout space at any point in time. For an exact solution of the equations, one would have to be able to calculate the value of these fields at each of the infinitely many spaces and times, which is practically impossible. Wilson therefore suggested replacing continuous space and time with a grid. The quantum fields of the quarks and gluons should no longer exist in the entire space-time, but only at the grid points or their connecting lines. This construction eliminates the infinity of the problem and it is now in principle possible to solve the basic equations of QCD with the help of a computer.

Spacetime grid

Although this basic idea of lattice QCD is very simple, there are significant practical difficulties in implementing it. The very first question: How do you get the solution of the QCD in continuous spacetime from the solutions of the lattice QCD. This is where a property called “asymptotic freedom” comes into play, for the discovery of which David Politzer, David Gross and Frank Wilczek received the Nobel Prize in 2004. Asymptotic freedom is, so to speak, the counterpart to quark inclusion and means that the strength of the interaction between the quarks decreases further and further for small distances and finally disappears. The finer the spacetime grid, the closer the results of the grid calculations to the correct results. On the other hand, the grid must also be so large that none of the particles contained in it “feel” that they do not exist in the entire room, but only in a small section of it.

Another hurdle arises from the fact that no free quarks can be observed. Consequently, physicists do not know the mass of a free quark a priori and can therefore not put it into the QCD equations from the start. The same also applies to the strength of the force or coupling constant. Because how strongly the quarks attract each other cannot be measured directly due to the quark inclusion. To find the correct values, physicists first study the effects of these two parameters on the properties of observable objects in their models. They then adjust the parameters so that these properties match those observed in nature.

### Grid QCD Achievements

Constant progress in understanding mathematical problems, new algorithms and, last but not least, the rapid development of computer technology have contributed to the fact that grid calculations can already answer many questions today. The mass of the proton and many other particles composed of quarks can now be calculated and thus the origin of the majority of the mass of the visible universe can be understood quantitatively. Physicists were also able to determine the masses of the quarks and the coupling constant of the QCD using lattice calculations - as well as the structure of the particles made up of quarks, above all that of the proton.

Proton and neutron

The lattice QCD even enables a glimpse into the early universe: shortly after the Big Bang, the temperatures in space were so high that all bond states “melted” and even free quarks and gluons could exist. The cosmos gradually cooled down and when it was around a hundred thousand times hotter than the core of our sun, the first protons and neutrons formed from the soup of free quarks and gluons. Lattice calculations have shown that this was a very constant process, at what temperatures it took place and how high the pressure and energy were.

Despite these partial successes, there is still a lot of work to be done in the coming years and probably decades. Thus, QCD, as the fundamental theory of strong nuclear force, should be able to predict all phenomena in nuclear physics. However, describing the interaction between protons and neutrons and the atomic nuclei made up of them, especially the massive ones, is currently still a difficult task for physicists. But it attracts the hope of tracking down currently unknown, exotic forms of matter.

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