Potential In Coordinate Space Research Articles

Two loop mass effects in the static position space QCD-potential

The perturbatively calculable short distance QCD potential is known to two loops including the effect of massive quarks. Recently, a simple approximate solution in momentum space was utilized to obtain the potential in coordinate space. The latter is important in several respects. A comparison with non-perturbative lattice results is feasible in the overlap regime using light MS masses. This might be even more promising employing the concept of the force between the heavy color singlet sources, which can be easily derived from the potential. In addition, the better than two percent accuracy bottom mass determination from ϒ-mesons is sensitive to massive charm loops at the two loop order. We summarize recent results using exact one loop functions and explicit decoupling parametrizations.

Static QCD potential in coordinate space with quark masses through two loops

The potential between infinitely heavy quarks in a color singlet state is of fundamental importance in QCD. While the confining long distance part is inherently non-perturbative, the short-distance (Coulomb-like) regime is accessible through perturbative means. In this paper we present new results of the short distance potential in coordinate space with quark masses through two loops. The results are given in explicit form based on reconstructed solutions in momentum space in the Euclidean regime. Thus, a comparison with lattice results in the overlap region between the perturbative and non-perturbative regime is now possible with massive quarks. We also discuss the definition of the strong coupling based on the force between the static sources.

Peripheral nucleon-nucleon phase shifts and chiral symmetry

Within the one-loop approximation of baryon chiral perturbation theory we calculate all one-pion and two-pion exchange contributions to the nucleon-nucleon interaction. In fact we construct the elastic NN-scattering amplitude up to and including third order in small momenta. The phase shifts with orbital angular momentum L ≥2 and the mixing angles with J ≥ 2 are given parameter free and thus allow for a detailed test of chiral symmetry in the two-nucleon system. We find that for the D-waves the 2π-exchange corrections are too large as compared with empirical phase shifts, signaling the increasing importance of shorter range effects in lower partial waves. For higher partial waves, especially for G-waves, the model-independent 2π-exchange corrections bring the chiral prediction close to empirical NN phase shifts. We propose to use the chiral NN phase shifts with L ≥ 3 as input in a future phase-shift analysis. Furthermore, we compute the irreducible two-pion exchange NN potentials in coordinate space. They turn out to be of van der Waals type, with exponential screening of two-pion mass range.

THE SPECTRUM AND CONFINEMENT FOR THE BETHE-SALPETER EQUATION

The problem of calculating the mass spectrum of the two-body Bethe-Salpeter equation is studied with no reduction to the three-dimensional (“quasipotential”) equation. The method to find the ground state and excited states for a channel with any quantum numbers is presented. The problem of the confining interaction for the Bethe-Salpeter equation is discussed from the point of view of formal properties of the bound state spectrum, but with only inspiration from QCD. We study the kernel that is nonvanishing at large Euclidean intervals, i.e. RE→∞, which is constructed as a special limiting case of the sum of the covariant one-boson-exchange kernels. In the coordinate space this kernel is just a positive constant and corresponds to the kernel ∝ δ(kE) in the momentum space. When the usual attractive interaction is added, it is found that this kernel is similar in its effect to the nonrelativistic potential in coordinate space, V(r), with V(r→∞)→V∞. The positive real constant V∞ gives the scale that defines the limit of the bound state spectrum compared to the sum of the constituent masses, M < 2m+V∞. At the same time, the self-energy corrections remove the singularities from the propagators of the constituents, i.e. constituents do not propagate as free particles. The combination of these features of the solutions allows an interpretation of this type of interaction as a confining one. The illustrative analytical and numerical calculations are presented for a model of massive scalar particles with scalar interaction, i.e. the “massive Wick model.”

Heavy quarkonia in a covariant Salpeter model

The heavy quarkonia (charmonium c c and bottomonium b b ) are investigated in a covariant formulation of the Salpeter equation. We parameterize confinement alternatively by a linearly rising scalar or a vector interaction kernel and take into account the one-gluon-exchange (OGE) interaction in the instantaneous approximation. Mass spectra as well as leptonic, two-photon, E1 and M1 decay widths are calculated. Our results show that a reasonable description of the experimental data can be obtained with both spin structures for the confining kernel. The relativistic treatment leads to an improved description compared to nonrelativistic results for the two-photon width of the η c and to some extent for the E1-transition widths. However, characteristic deviations indicate that within a relativistic framework confinement is not described adequately by a simple potential in coordinate space.

Deuteron photodisintegration up to 200 MeV: (II). Comparison with experimental data

We use the model of deuteron photodisintegration presented in the previous paper to calculate the reaction observables and to compare theory and experiment up to 200 MeV lab photon energy. We describe how the parameters which control the 1πE, Δ and 1ϱE contributions are fixed by means of a best fit to selected experimental data. We show that the data on the forward cross section cannot be fitted using the parametrised Paris and Moscow potentials. Using one of the versions of the Bonn coordinate-space potential, we obtain a rather satisfactory global fit to the whole body of data up to 200 MeV. The result is a conceptually simple picture of the reaction in this energy range, in which the calculated observables are completely dominated by the impulse approximation, 1πE in leading order and the contributions of Δ and 1ϱE to the M (1) amplitudes with a singlet final state.

Three-body reactions for a local potential in coordinate space

The three-body theory of reactions is formulated for local potentials in coordinate space on the basis of the Faddeev equation. The wave function is expressed as a sum of all possible channel wave functions. In solving, the physically important pole terms are taken as the zeroth order terms. The higher order terms are treated as the perturbation. The kernels that appear in the perturbational series are regular at the origin and confined in a finite region of space, thus enabling us to calculate the breakup process numerically. We can expect that this perturbational series is rapidly convergent. The equation is finally reduced to a coupled set of integral equations for the elastic and breakup amplitudes.

Modifications of a two-nucleon potential due to solitary waves in self-interacting spin-zero meson fields

AλΦ4 solitary-wave propagator, a propagator constructed from exact particular solutions (solitary-wave solutions) to a nonlinear field equation is used in the usual derivation of a two-nucleon potential. The field equation is a Klein-Gordon equation modified by the addition of a cubic self-interaction term (λΦ3) and describes massive spinless persistently self-interacting systems. The solitary-wave propagator has poles at integral multiple [(2n + 1)m] mass states. Contributions from these poles to the two-nucleon potential in co-ordinate space appear as a superposition of Yukawa and exponential terms with residues that depend on the self-interaction coupling constant (λ). In the limit of vanishing λ, the solitary-wave exchange potential (SWEP) reduces to the standard OPEP. For a two-nucleon in ans-state, the SWEP is very strongly attractive at intermediate ranges (0.5 ≲mr ≲ 1.5) and smoothly gives rise to a repulsive core at very short ranges (mr ≲ 0.5). At very large ranges (mr ≳ 1.5) the SWEP approaches the OPEP tail. Overall, the SWEP exhibits characteristic features of OBEP and phenomenological potential models, but requires only a small number of parameters even in contrast to OBEP.

Optical-model calculations with a realistic nucleon-nucleon interaction: (I). Theory

The real parts of the optical potentials for the elastic scattering of 65 MeV protons on 16O, 17O and 18O are calculated in an impulse approximation. The prescription used is derived from three-body theory, considering the system as consisting of the incident nucleon, an active bound nucleon, and an inert core. The effective interaction resulting is a fully off-shell T-matrix and is both non-local and energy dependent. Reid's soft-core potential is used to generate the requisite T-matrices. Antisymmetry between the incident and active nucleon is included and all relative two-nucleon partial waves with l ≦ 3 are retained. The optical potential is constructed by assuming a pure shell-model target and summing the state of the active nucleon through the single-particle states. The resulting non-locality is investigated explicitly and is found to be negligible. A coordinate space potential is constructed. When compared to the real parts of phenomenologically generated optical potentials, our potentials are found to be similar in the surface ( r ≈ 2.5 fm) but deeper in the interior. When our parameter-free real well is combined with phenomenological imaginary and spin-orbit wells, high-quality fits to the experimental data are obtained, comparable to those obtained when all three wells are chosen phenomenologically. Off-energy shell effects and the energy dependence are investigated.

High-Energy Optical Potential and Nuclear Correlations

The high-energy elastic scattering of nucleons by nuclei is studied using the optical-potential operator provided by Watson's multiple-scattering theory. The main motivation is to investigate the possibility of detecting two-body correlations in terms of an optical potential. We have obtained an expression for the second-order potential in coordinate space which explicitly exhibits its nonlocality in terms of the two-body correlations arising from the Pauli principle and the repulsive core of the nucleon-nucleon interaction. We have performed a self-consistent analysis of the elastic scattering of 1-GeV protons on $^{4}\mathrm{He}$ using a reasonable nucleon-nucleon scattering amplitude, the one-body density obtained from elastic electron scattering, and a parametrization for the two-body density consistent with the one-body density and dependent upon a parameter identifiable as a correlation length. We have also attempted to discriminate among several sets of nucleon-nucleon phase shifts at 630 MeV. The scattering of 1-GeV protons by $^{12}\mathrm{C}$ and $^{16}\mathrm{O}$ is studied in order to investigate the relative importance of Pauli and short-range correlations. We also critically compare our work with that of several others.

On the Repulsive Core and the Nonstatic Effects of Nuclear Forces in Momentum Space

Nucleon-nucleon scattering due to the exchange of 'lC, (J), P and an isoscalar scalar meson is calculated for spin singlet states in momentum space. We are mainly concerned with the problerri whether or not the (J)-meson exchange can be the origin of the repulsive core by treating the nonstatic effects of heavy mesons as accurately as possible. The ai-nucleon coupling constant obtained in the ISO state by momentum-space calculations can be taken smaller than that obtained from the static coordinate-space potential, but is still· too large to reconcile with small values inferred from the higher partial wave fit. In the framework of the one-boson-exchange model, the non-static effects are attractive and cannot produce the repulsive effects in the ISO state. Comparison with the results ob­ tained from the velocity-dependent potential shows that such attractive nonstatic effects can be simulated by the effect of a p2-dependent term.

Theory and Calculation of Scattering with the Bethe-Salpeter Equation

The Bethe-Salpeter equation studied in this paper describes the interaction of two scalar particles via the exchange of a third scalar particle in the ladder approximation. The properties of the Green's function and the potential in coordinate space are shown to permit a Wick rotation to an imaginary time variable, without appeal to information not contained in the original equation. The resulting four-dimensional (Euclidean) wave equation has a solution which grows exponentially for large time-like distances but behaves as an ordinary Schr\"odinger scattering wave for large space-like distances. A modification of the Schwinger variational principle is used to obtain, with a modest use of computing machinery, scattering phase shifts for various angular momenta and for energies below the inelastic threshold. The success of these calculations indicates that the Bethe-Salpeter equation can be accepted as a powerful and practical tool for the study of strong-interaction dynamics.

Singular Logarithmic Potentials in Coordinate Space

In previous works we have studied the problem of the determination of the Jost function for singular repulsive potentials behaving near the origin like inverse powers. The key was to define ``new Jost solutions'' which, still being asymptotically ingoing (or outgoing) waves, tend to constant (Jost functions) near the origin. It was shown from the perturbation expansion in coordinate space of these ``new Jost solutions'' that we can construct the Jost functions by connecting the radial coordinate r and the order of the perturbation expansion p. More precisely, if we introduce an r(p) dependence, the Jost function is the limit of convergent sequences provided r(p) goes to zero less rapidly than a given limiting dependence rL(p). In this paper, working in coordinate space, the same method is extended for two other families of singular potentials: firstly, we consider the case in which the most singular part of the potentials behaves like G2(log r−1)βr−2n, (n ≥ 1, β arbitrary) near the origin; secondly, we study exponentially singular repulsive potentials of finite range. It is found that the more singular is supposed to be the potential, the higher becomes the available limiting dependence.

Singular potentials in co-ordinate space

With the help of the Laplace transform previous works have shown the possibility of useful approximate methods for the Jost functions in the case of Yukawa generalized potentials, where the most singular term near the origin is likeG2r-2n (n≥ 1, G2>0 for n>l). Now, we want to study directly in co-ordinate space the same problem of the construction of the Jost function for families of singular potentials where the most singular term near the origin is repulsive. In this first paper we consider the same family of short-range potentials, as previously made with the use of Laplace transform, and also a family of arbitrary power potentials more singular than r-2. The key (as previously made) is to define « new Jost solutions » asymptotically ingoing (or outgoing) and going to constant (Jost function) near the origin. If we consider the perturbation expansion of these new Jost functions we find, that if we connect in a certain manner the order of the perturbationp and the radial co-ordinate r, then the Jost function is the limit of convergent sequences. More explicitly, we find that there exists a limiting dependencer L (p)=constantp(1/n−1))−ɛ′, (e′> 0 arbitrary small) such that ifr(p) goes to zero less rapidly thanrL(p) these sequences are convergent. We have never used arbitrary cut-off or hard-core procedure and only whenp is infinite one has that theser(p) dependences are 0.