Franz F. Schöberl

Bound states of quarks

Abstract This review consists of two parts, the phenomenology of non-relativistic potential models and the theoretical understanding of the forces between quarks. The first part reports on the description of hadrons as bound states of quarks by non-relativistic potential models. It starts with a brief sketch of the way in which information on the interquark potential may be gained from quantum chromodynamics. Some general theorems related to the potential-model approach are proven. The significance of the treatment of relativistically moving constituents by an effective non-relativistic Schrodinger equation is discussed. A brief survey of the motivations for various proposed potential models is given. Finally, the application of the developed theoretical framework is illustrated by a few selected examples. The second part starts with a review of the approach via the Bethe-Salpeter equation. Although this has not led to a breakthrough in our understanding, it has played an important role in the past and is still indispensable for several questions. The next topic is the Wilson loop approach, together with its extensions for the inclusion of spin-dependent corrections. Here great progress has been made in the last few years. Connections with perturbation theory, lattice gauge theory, and the non-trivial QCD vacuum are exploited at the end.

Solving the Schrodinger equation for bound states with Mathematica 3.0

Using Mathematica 3.0, the Schrodinger equation for bound states is solved. The method of solution is based on a numerical integration procedure together with convexity arguments and the nodal theorem for wave functions. The interaction potential has to be spherically symmetric. The solving procedure is simply defined as some Mathematica function. The output is the energy eigenvalue and the reduced wave function, which is provided as an interpolated function (and can thus be used for the calculation of, e.g., moments by using any Mathematica built-in function) as well as plotted automatically. The corresponding program schroedinger.nb can be obtained from franz.schoeberl@univie.ac.at.

A simultaneous and systematic study of meson and baryon spectra in the quark model

Abstract We show that the whole quarkonium spectrum is nicely reproduced by a single nonrelativistic q − q potential V q q = V (scalar) + V (vector) . The baryon spectrum can also be reproduced remarkably well if we use the same potential with slightly different weight V qq = 1 2 [V (scalar) + 2 3 V (vector) ] .

SEMIRELATIVISTIC TREATMENT OF BOUND STATES

This review discusses several aspects of the semirelativistic description of bound states by the spinless Salpeter equation (which represents the simplest equation of motion incorporating relativistic effects) and, in particular, presents or recalls some very simple and elementary methods which allow us to derive rigorous statements on the corresponding solutions, that is, on energy levels as well as wave functions.

DISCRETE SPECTRA OF SEMIRELATIVISTIC HAMILTONIANS

We review various attempts to localize the discrete spectra of semirelativistic Hamiltonians of the form (defined, without loss of generality but for definiteness, in three spatial dimensions) as entering, for instance, in the spinless Salpeter equation; every Hamiltonian in this class of operators consists of the relativistic kinetic energy , where β > 0 allows for the possibility of more than one particles of mass m, and a spherically symmetric attractive potential V(r), r ≡ |x|. In general, accurate eigenvalues of a nonlocal Hamiltonian operator can only be found by the use of a numerical approximation procedure. Our main emphasis, however, is on the derivation of rigorous semianalytical expressions for both upper and lower bounds to the energy levels of such operators. We compare the bounds obtained within different approaches and present relationships existing between the bounds.

Energy bounds for the spinless Salpeter equation: harmonic oscillator

We study the eigenvalues nl of the Salpeter Hamiltonian H = β(m2 + p2)1/2 + vr2, v>0, β>0, in three dimensions. By using geometrical arguments we show that, for suitable values of P, here provided, the simple semiclassical formula nl≈min r>0{v(Pnl/r)2 + β(m2 + r2)1/2} provides both upper and lower energy bounds for all the eigenvalues of the problem.

Variational approach to the spinless relativistic Coulomb problem.

By application of a straightforward variational procedure, we derive a simple, analytic upper bound on the ground-state energy eigenvalue of a semirelativistic Hamiltonian for (one or two) spinless particles which experience some Coulomb-type interaction.

QUARK-ANTIQUARK BOUND STATES: RELATIVISTIC VERSUS NONRELATIVISTIC POINT OF VIEW

This brief review discusses various aspects of the description of hadrons as bound states of quarks in the simpler example of mesons, which are regarded as being composite of a (constituent) quark and an antiquark. It is shown how a recently derived relativistic virial theorem may be used to cast some light on the interplay between the (semi)relativistic and (maybe only effectively) nonrelativistic treatment of bound states. Furthermore, a new relativistic approach to fermion-antifermion bound states is sketched and applied to predict analytically a few selected general features of the meson spectrum.

Energy bounds for the spinless Salpeter equation

We study the spectrum of the Salpeter Hamiltonian H=βm2+p2+V(r), where V(r) is an attractive central potential in three dimensions. If V(r) is a convex transformation of the Coulomb potential −1/r and a concave transformation of the harmonic-oscillator potential r2, then both upper and lower bounds on the discrete eigenvalues of H can be constructed, which may all be expressed in the form E=minr>0[βm2+P2/r2+V(r)] for suitable values of P here provided. At the critical point r=r the relative growth to the Coulomb potential h(r)=−1/r must be bounded by dV/dh<2β/π.

Quark core contribution to the electric polarizability of hadrons

Abstract The quark core contributions to the electric polarizability of the nucleon, the δ-particle and several mesons are calculated in the framework of a non-relativistic potential model. For the nucleon the electric polarizability due to the valence quarks amounts to к = 3.1 × 10 −4 fm 3 , which is in good agreement with recent estimates in the chiral quark model. The quark core polarizability of light mesons turns out to be two orders of magnitude smaller.