# Quantum Chromodynamics (QCD)

Quantum Chromodynamics (QCD) describes the dynamics of the quarks, which interact via the gluon field. QCD is based on the notion that quarks possess a charge-like property, the "color". Color charges and color currents serve as sources of electric and magnetic color fields, the gluon fields. These ideas are described by the Maxwell equations of QCD. Contrary to electrodynamics, they are nonlinear equations since the gluons themselves also carry a color charge and hence interact with each other.

QCD is new theory based on the mentioned assumptions and cannot be derived from an underlying theory. One can only construct it in such that it is compatible with phsical experiments. QCD was first formulated in a paper of Fritzsch, Gell-Mann and Leutwyler in 1973.

In QCD the quarks combine to form protons and neutrons. Quarks come in 3 different types of charge, which in QCD is called ``color'': red, green and blue. The name alludes to the fact that, in nature, quarks appear always in ``white'' combinations: either by joining in baryons 3 quarks of the colors red, green and blue or by pairing 2 quarks in mesons, one with a color and the other with its anticolor (mesons). Mathematically the strong interaction is described by the group SU(3). It is mediated by massless bosons, which are called gluons and which are the counterparts of the photons in electromagnetism. The crucial difference is that the gluons themselves also carry a charge and thus interact with each other.

**Figure:** A baryon consists of 3, a meson of 2 quarks.

This leads to the phenomenon of **asymptotic freedom**: the nearer the quarks move together, the weaker they are coupled. A frequent analogy in this context pictures the quarks as being attached to the end of an elastic string. No force is exercised unless you try to pull the ends apart. In contrast, the electromagnetic force is supposed to increase in strength at short distances. One can imagine that the vacuum is actually filled with virtual electron-positron pairs, which form dipoles that screen the electric charge around a particle. In QCD, this effect produced by virtual quark-antiquark pairs is present, as well. Additionally, the vacuum contains gluons and photons. The photons are irrelevant because they are charge-neutral. However, the gluons do have color. Simply speaking, every gluon possesses one charge and one anticharge unit. The net effect of the gluons is that they **antiscreen**, i.e. increase, the charge that appears outwards. As one approaches a quark, the antiscreening decreases. Whether screening or antiscreening prevails depends on the number of different quark types, called ``flavours''. As long as there are less than 17 quark flavours, the strong force will be asymptotically free. Currently, we know of 6 flavours: u, d, c, s, t, b.

**Figure:** The strenght of the fundamental forces depends on the energy or, equivalently, length scale.

On the other end of the distance scale, at large separations, we encounter a different behaviour, known as **confinement**. As opposed to the electromagnetism, the strong force does not fall off quadratically with the distance, but becomes constant. Lattice simulations show that the field lines originating from the charges do not spread out radially in all directions, but are collimated in a flux tube - a string - which goes straight from one quark to the other. Since the field strength is proportional to the number of lines per area it thus continues unabated along the flux tube. To completely separate two quarks of opposite charge would take an infinite amount of energy. Actually, this would only be strictly true, if the quarks had infinite mass. In reality, when the energy expended suffices to create a quark-antiquark pair, the string breaks up into two quark-antiquark pairs. For this reason one does not experimentally observe free quarks.

**Figure:**The color field (left) is collimated, the electric field (right) spreads out.

**Figure:** String breaking by quark-antiquark pair production.*(J. Greensite, hep-lat/0301023)*

Phenomena of QCD physics which occur at strong coupling, such as confinement, are called non-perturbative. Since this occurs at large distances which is equivalent to low energies, one also uses the term infrared QCD. Lattice QCD plays a major rôle in the investigation of such aspects since they are in general not accessible by other techniques. Other non-perturbative effects include spontaneous chiral symmetry breaking and the anomaly of the chiral current.

**Figure:** Right- and left-handed quarks

**Chirality** is a property of fermions, to which the quarks belong. A quark can be considered to rotate about itself. It possesses a spin, i.e. an angular momentum, with a value of , measured in units of . Quantum mechanically, if the quark has zero mass, the axis of rotation can only be aligned parallel to its movement in space. (Actually, the quark masses are not zero, but depending on the phenomena we are interested in this is a rather good approximation.) The particle can spin either like a right-handed or a left-handed screw, and accordingly has right- or left-handed chirality. If the chiral symmetry is broken, particles can change their handedness. Physicists aim at understanding the mechanism of this symmetry breaking.

In quantum field theory, to calculate physical quantities one must average over all possible field configurations, weighted with some sort of Boltzmann factor, which depends on the action. The smaller the action of a configuration, the more important it is. Following this reasoning, one attempts to trace back the characteristic features of the strong interaction to the contribution of only a few (or even one) types of field structures, which dominate the averaging process, while all others are viewed as fluctuations, causing only small corrections.

One model addressing the issues related to chiral symmetry, involves **instantons**, special gauge field configurations that are concentrated about a spacetime point. They are deeply connected with the concept of topological charge. In the continuum, classical field configurations are ``smooth'' mappings from the gauge group to spacetime. By continuous deformations one can produce a range of other fields. Interestingly, for SU(N) gauge groups, not all of the configurations can be deformed into each other. Rather they fall into distinct classes, characterized by an integer, the so-called topological charge (or Pontryagin index or winding number). All fields within a class are connected by a continuous deformation, while fields from different classes are not. The branch of mathematics dealing with such questions is homotopy theory and the classes are called homotopy classes. The term winding number evokes the picture of a rubber band which can be wound around a tube only an integer number of times. If the band is not to lose contact with the tube surface, the winding number cannot be changed. To span the rubber band around the tube you have to stretch it and thereby you pump energy into it. Energetically it would be preferable if the rubber would contract but this is prevented by the tube. This is a sort of ``topological'' obstruction. Similarly, the instanton has non-zero energy, yet it is stable. The difference to the instanton in QCD is that the gauge field is wrapped around a sphere in spacetime.

To be precise, an instanton carries a topological charge of +1; equally, there are anti-instantons with . When putting several instantons together, their charges add up. Instanton ensembles are the fields with the least action within a given homotopy class.

Although instantons indeed provide an explanation for the spontaneous breaking and the anomaly of chiral symmetry, the hopes that they would also account for confinement have not materialized. A theory which is more successful on this battlefield is the vortex model.