ar X iv : 0 90 7 . 53 18 v 1 [ he p - ph ] 3 0 Ju l 2 00 9 Confinement & Chiral Symmetry Breaking : The fundamental problems of hadron physics

The most intriguing phenomena of hadron physics are confinement as well as dynamical and anomalous chiral symmetry breaking. Despite the fact that the theory of the Strong Interactions, Quantum Chromodynamics (QCD), is known since decades we still lack a fundamental understanding of the corresponding physics. As a phenomenon confinement is easily described. On one hand, representing the Strong Interaction by a local Quantum Field Theory (i.e. by QCD) necessitates to introduce fundamental fields with a new quantum number, namely quarks and gluons with some “colour“. The advantage of this approach is twofold: It provides a mathematical framework, and it orders the plethora of hadrons into a clearly arranged pattern. On the other hand, quarks and gluons have never been detected as particles, i.e. nobody has ever seen quarks and gluons making a track in a detector. The confinement hypothesis can therefore be formulated as: the colour-neutral hadrons, being a kind of bound states of coloured quarks and gluons, are the only strongly interacting particles, no “coloured“ particles exist. This hypothesis has been extremely successful. The colour-charge version of ionization does plainly not occur. Even more, the concept of mutual forces by mutual polarization, the van-der-Waals forces, also does not have a colour-charge analogue. Thus as a phenomenon confinement seems to be plain and simple. As a theoretical concept confinement is astonishingly hard to put into precise terms. Even the question how to obtain a concise definition of ‘charge’ did undergo some severe discussions when trying to find a theoretically unequivocal definition of confinement. E.g. the Wilson loop provides an order parameter only in the absence of fundamental charges, i.e. quarks. Despite all efforts such an order parameter has not been found in the real world with light quarks, a satisfactory and detailed description of the underlying mechanisms of confinement stays elusive. The fact that for charges in higher representations there are common aspects with the Higgs mechanism complicates the issue even further. The situation is not drastically different when addressing dynamical Chiral Symmetry Breaking (χSB) and the UA(1) anomaly. As phenomena they are clearly identifiable, the first because of the relatively small pion mass and several patterns in the interaction of pions with themselves and other hadrons. The latter because of the large η mass. When it comes to theoretically understanding χSB we also lack a lot of basic knowledge. We know that dynamical χSB comes along with the dynamical generation of “constituent“quark masses (which, however, depend on the momentum of the quarks). One may explain dynamical χSB and the UA(1) anomaly with two seemingly different approaches. One approach starts by considering quark zero modes in topologically non-trivial field configurations. A non-vanishing