Dieter Gromes

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.

Effective Hamiltonian for charmonium and similar two-fermion systems

Abstract We derive the effective Hamiltonian containing the relativistic corrections up to order 1/ c 2 in a Schrodinger equation description of a two-fermion system. For the Bethe-Salpeter kernel which determines the effective potential we admit the most general form consistent with Lorentz covariance, parity, and time-reversal invariance. The resulting general formula is then specialized by using ideas from quantum chromodynamics. An important point is that besides the terms usually considered there may appear additional spin-spin terms with δ-function interaction. We conclude with a discussion of several aspects of elementary particle spectroscopy which lead to stringent restrictions concerning the structure of the kernel.

Linearly rising quark potential and non-strange baryon spectrum

Abstract We investigate the consequences of a linearly rising effective quark-quark two-particle potential within the baryons. We restricted ourselves to a discussion of the nucleonic states. The Schrodinger equation is solved by an approximation technique which makes use of the well-known fact that the three-body problem can be solved for an oscillator potential. The relativistic corrections abstracted from gauge theories are also considered. Some features of the spectrum emerge in a natural way. The degeneracy of the excited oscillator levels is removed and the [70, 1−]1 is pushed up in energy into the region of the second excited band of states. The relativistic corrections are very important and, for usual values of the parameters, the motion of the quarks is relativistic. While the spin-spin splittings come out roughly correct, the spin-orbit splittings become much too large.

Space-time dependence of the gluon condensate correlation function and quarkonium spectra

Abstract We consider the influence of the gluon condensate in QCD on the energy levels of quarkonia, taking into account the x -dependence of 〈 ω |: G μv a ( x ) G aμv (0):| ω 〉. The modification compared to earlier approaches which approxim ated the above vacuum expectation value by a constant is quite sizeable; for the bb system we find that the effect can be essentially described in terms of a local potential.

On the effective quark potential in baryons

Abstract We analyse the splitting of the non-strange memebers of the first excited level [70,1 − ] 1 of baryon resonances. The spin-dependent forces (spin-spin, spin-orbit, tensor) are supposed to arise from the Coulomb term due to one-gluon exchange, from the long-range linearly rising part of the potential, and from additional “hard-core” spin-spin terms which may be generated by higher-order graphs contributing to the qq kernel. For the long range part we either assume that it comes from a superposition of a vector and a scalar kernel of the form ϵ ( γ μ ⊗ γ μ ⊗ 1 + (1 − ϵ )(1 ⊗ 1 ⊗ 1) (+ permutations), or, alternatively, that it arises from a vector exchange with an anomalous moment κ in the quark-gluon vertex. Values of ϵ ≈ 0 or κ ≈ −1 turn out to be favoured. The strong coupling constant and the slope of the linear potential come out in the correct order of magnitude. Very large hard-core spin-spin terms are needed. This fact makes the determination of the effective potential from the underlying theory of quantum chromodynamics as well as the phenomenological analysis of the observed spectra rather problematic.

Poincaré invariance constraints on NRQCD and potential NRQCD

We discuss the constraints induced by the algebra of the Poincare generators on non-relativistic effective field theories. In the first part we derive some relations among the matching coefficients of the HQET (and NRQCD), which have been formerly obtained by use of reparametrization invariance. In the second part we obtain new constraints on the matching coefficients of pNRQCD.

Bethe-salpeter equation with confining kernel; Correct non-relativistic limit and the constant to be added to the linear potential

After showing that the usual way of performing the non-relativistic limit in the BS equation leads to a divergent result in the case of a confining kernel we perform the limit correctly and find that the well-known linear potential will be necessarily accompanied by a negative constant. Under a rather weak naturalness assumption the constant is uniquely determined from the slope of the linear potential. The resulting relation is in agreement with the phenomenology of meson spectroscopy.

Relativistic corrections to the long-range quark antiquark potential electric flux tubes, and area law

The gauge-invariant approach to the potential via the Wilson loop is generalized, replacing the static sources by the appropriate Dirac currents. The resulting expression is expanded with respect to l/m2 and then brought into a suitable form. For the evaluation the only input we employ is the area law for large distances. This allows us to understand how the electric flux tube picture emerges in the case of moving quarks, and why magnetic terms do not contribute to spin-dependent corrections. The only surviving spin dependence is a spin-orbit term like that from a scalar potential, but with the opposite sign.

Field quantization on the surface X2 = constant

Abstract In analogy to non-relativistic quantum field theory where the Galilei invariant plane t = const. is used for a description of bound (and scattering) states we take the hyperboloid x2 = const. which is invariant under homogeneous Lorentz transformations. We give the appropriate complete set of basis functions for the expansion of any scalar and spin 1 2 field on the hyperboloid. The free Klein Gordon and Dirac fields are discussed in detail. In this formulation a multi-particle system can be described by a Lorentz covariant wave function.

Baryon Spectrum and the Forces Between Quarks

Various forms of the potential between quarks in baryons, including both two-body and three-body forces, are analyzed from a theoretical and phenomenological point of view. We give arguments in favor of a long-range logarithmic potential in baryons playing the analogous rôle of the linear one in mesons. With this and a short-range Coulomb interaction a good description of the coarse structure of nonstrange baryon resonances emerges. Relativistic corrections can be consistently implemented, leading to an improved picture which only leaves the problem of spin-spin splittings coming out too small.