Elastic scattering and the K matrix: (II). Phase shifts,resonances and few-level approximations

The K-matrix expansions given in part I are applied to the scattering by a square-well potential. Their convergence proves very satisfactory from a physical point of view since few-level approximations allow very good approximations to the phase shifts and cross sections. It also appears that all the complex poles of K_l and the real ones with positive residue should undoubtedly be associated with physical resonances. As for the real ones with negative residue, i.e. the echo poles, they are obviously unrelated to resonances, but they provide a very good parametrization of the background part of the scattering. The time delay is given a major role in the argument. The possibility of having double poles is also discussed and sum rules are given for the energies and residues of the poles.     

Separable energy-dependent approximation to local potentials

An extension of the single-pole separable approximation of a two-body t-matrix in which the effects of several poles are included is made. The simple form for the t-matrix derived from a single separable potential is retained. However, the separable potential is constructed using an energy-dependent superposition of the states corresponding to the various poles. The energy dependence is chosen so as to obtain the correct residue of both the on-shell and off-shell t-matrices at each of these poles, while preserving unitarity. The formalism is specialized to the case of s-wave scattering from an attractive square well. Comparison to the exact s-wave cross section gives good results.     

Interplay of quark and meson degrees of freedom in a near-threshold resonance

We investigate the interplay of quark and meson degrees of freedom in a physical state representing a near-threshold resonance for the case of a single continuum channel. We demonstrate that such a near-threshold resonance may possess quite peculiar properties if both quark and meson dynamics generate weakly coupled near-threshold poles in the S -matrix. In particular, the scattering t -matrix may possess zeros in this case. We also discuss possible implications for production reactions as well as studies within lattice QCD.     

The P-matrix and a matching radius model for pion-hadron scattering

A matching radius model of pion-hadron scattering is developed by assuming the continuity of the axial current. This model is applied to the simplified situation of a pion scattering off a static empty bag to study the relationship between P-matrix poles and the bag model primitives. It is found that only under extremely simplified assumptions can the P-matrix poles be identified with the bag model primitives.     

K −-Proton atomic transitions

The relation between theK ^}-P scattering length and the X-ray spectrum for the 2p → 1s electromagnetic transition inK ^?-P atoms is examined. A coupled-channel potential model is used to explicitly calculate the energy of theS-matrix pole in the 1s channel, which is then compared with the energy obtained from the scattering lengths via the standard equation. The X-ray spectrum is calculated and compared with the Lorentzian shape associated with the complex energy of theS-matrix pole. In addition, theK ^?p branching ratios are compared at threshold and at the complexS-matrix pole energy.     

P-Matrix approach and parameters of low-energy scattering in the presence of a Coulomb potential

The scattering of two charged strongly interacting particles is described on the basis of the P-matrix approach. In the P matrix, it is proposed to isolate explicitly the background term corresponding to purely Coulomb interaction, whereby it becomes possible to improve convergence of the expansions used and to obtain a correct asymptotic behavior of observables at high energies. The expressions for the purely Coulomb background P matrix, its poles and residues, and purely Coulomb eigenfunctions in the P-matrix approach are obtained. The nuclear-Coulomb parameters of the low-energy scattering of two charged hadrons are investigated on the basis of this approach combined with the method for isolating the background P matrix. Simple explicit expressions for the nuclear-Coulomb scattering length and effective range in terms of the residual P matrix are derived. For models of short-range strong interaction, these expressions give a general form of nuclear-Coulomb parameters for low-energy scattering. Specific applications of the general expressions derived in this study are exemplified by considering, on the basis of these expressions, some exactly solvable models of strong interaction, including the hard-core model, and, for these models, the nuclear-Coulomb parameters for low-energy scattering at arbitrary values of the orbital angular momentum are found explicitly for the first time. In particular, the nuclear-Coulomb scattering length and effective range are obtained explicitly for the boundary-condition model, the model of a hard-core delta-shell potential, the Margenau model, and the model of square-well hard-core potential.     

On the Connection between Particles and Poles of the S-Matrix

In axiomatic S-matrix theory it is usually assumed that stable particles give rise to simple poles of the S-matrix for real negative energies while unstable particles give rise to poles close to the real axis on an unphysical sheet of the energy Riemann surface. The stable particle — pole association has been known for a long time not to be always true. For example in potential scattering what is relevant in this case in fact is not the S-matrix but the Jost function. The zeroes of this function for real negative energies are in fact in one-to-one correspondence with the bound states, while the correspondence may break down for the poles of the S-matrix. On the other hand it has recently been pointed out that there also is in general no connection between unstable particles and poles of the S-matrix.     

A modified K-matrix approach to many-channel processes

The definition of a K-matrix, that does not have the direct channel singularities of the scattering amplitude, is not unique. Different choices lead to different integral equations for K with different physical content. We discuss the choice where the intermediate states have a simple off-shell behaviour for each partial wave. Crossing leads to an identification of the singularities of the K-matrix elements and to an iterative method of finding the scattering amplitude starting from a “bare” world.     

The pion mass difference in threshold pion reactions on light nuclei

We investigate the effect of the pion mass difference in scattering and photoproduction of pions on light nuclei near threshold. The effect is found to be most easily described in a nuclear K-matrix formalism where the nuclear K-matrix elements are shown to be independent of the mass difference.     

On the Regge-pole description of nucleus-nucleus scattering

The properties of the Regge poles of theS-matrix for scattering of strongly-absorbed nuclear particles are considered. Simple formulae are obtained for describing the Regge trajectories in terms of the nuclear radius, the quasi stationary levels in the combined nuclear-Coulomb-potential and the widths of these levels. The predictions of these formulae are compared with the Regge trajectories obtained previously, for a Woods-Saxon potential, and with those required to fit¹⁶O-¹²C backward scattering.     

The time-dependent mean-field S-matrix theory in the adiabatic limit

To elucidate the spirit of the recently developed time-dependent mean-field S-matrix theory we apply it to low-energy elastic collisions and solve the corresponding temporally non-local mean-field equation under the assumption that the relative motion proceeds adiabatically. Neglecting the exchange processes this assumption leads to a pure potential scattering picture. The phase shift extracted from the corresponding mean-field S-matrix coincides with the WKB result of potential scattering.     

Application of K-matrix CMSR'S to single charged pion photoproduction

The background terms, required in addition to Regge poles with some definite properties, in order to describe the high energy behavior of the t-channel single pion photoproduction amplitudes F_t^(?)(v, t), i = 1, …, 4, are investigated in the framework of a reggeized K-matrix model. To this end continuous moment sum rules (CMSR's) are applied, not to the full amplitudes, but just to the two-particle K-matrix amplitudes assumed to show pure Regge pole asymptotic behavior. The K-matrix amplitudes are defined by taking into account in the Heitler equation the elastic (πN) intermediate state alone. Such a definition corresponds to a weak-cut model with only final-state interactions included. It turns out that strong background contributions are still present on the high energy side of the K-matrix CMSR's. The Regge cut contributions generated within our formalism are found to be too weak to account for the background terms in the full amplitudes. This is presumably to a great extent due to the neglect of the (?N) diffraction-dissociation intermediate state in the definition of the K-matrix amplitudes.     

Reduction of the Lane-Robson “calculable” reaction theory to standard R-matrix form

General dynamical equations derived from the Lane-Robson calculable reaction formalism are cast into a form amenable to standard R-matrix treatment, permitting the resonance content of the equations to be made explicit. Formulae are given which enable the collision matrix and the amplitudes of physical eigenfunctions to be calculated directly from the R-matrix with or without the isolation of resonance contributions. The present methodology permits a significant reduction of effort in numerical investigations of the energy dependence inherent in dynamical models of the nucleus. The formalism is illustrated by calculational results obtained from a potential model fitted to ¹⁶O + n scattering data.     

Coupled-channel R-matrix method on a Lagrange mesh

The coupled-channel R-matrix method on a Lagrange mesh is a very simple approximation of the R-matrix method with a basis. The mesh points are zeros of shifted Legendre polynomials. Bound-state energies and scattering matrices are easily calculated with small numbers of potential values at mesh points. A test with an exactly solvable two-channel potential provides an excellent accuracy over a broad energy range with only 30 mesh points. The efficiency of the method is illustrated for a single channel on α + α scattering and for two channels on the deuteron ground-state energy and on nucleon-nucleon scattering.     

A P-matrix analysis of nucleon-nucleon scattering below 1 GeV inspired by the quark compound bag model

The quark compound bag model was suggested recently as a dynamical model for the hadronic interaction, taking quark degrees of freedom into account. Important parameters of this model are the positions and residues of the Jaffe-Low P-matrix poles. They can be extracted from the experimental data on the S-matrix. The P-matrix is determined for several nucleon-nucleon channels. A realistic long-range interaction is included in this analysis which introduces a small model dependence in P. The Wigner-Breit condition on the P-matrix can be used to detect deficiencies of existing phase-shift analyses at low energies.     

Long-range tail produced by the CDD ambiguity in a static model

An aspect of a potential-model analogy of the customary CDD ambiguity is studied. Starting from the idea that the known CDD ambiguity in theS-matrix produces an ambiguity in the determination of potentials, the influence of CDD poles upon the long-range behaviour of potentials in the relativistic (Klein-Gordon) static model is demonstrated. Generation of the long-range tail by CDD poles is investigated in terms of a scattering inversion-problem formalism, based on theN/D equations extended off the mass shell.     

Strength functions for fragmented doorway states

Coupling a strongly excited “doorway state” to weak “hallway states” distributes its strength into micro-resonances seen in differential cross sections taken with very good energy resolution. The distribution of strength is shown to be revealed by reduced widths of the K-matrix rather than by the imaginary part of poles of the S-matrix. Different strength functions (SF) constructed by averaging the K-matrix widths are then investigated to determine their dependences on energy and on parameters related to averages of microscopic matrix elements. A new sum rule on the integrated strength of these SF is derived and used to show that different averaging procedures actually distribute the strength differently. Finally, it is shown that the discontinuous summed strength defines spreading parameters for the doorway state only in strong coupling, where it approximates the indefinite integral of the continuous SF of MacDonald-Mekjian-Kerman-De Toledo Piza. A new method of “parametric continuation” is used to relate a discontinuous sliding box-average, or a finite sum, of discrete terms to a continuous function.     

Analytic Structure and Power-Series Expansion of the Jost Matrix

For the Jost-matrix that describes the multi-channel scattering, the momentum dependencies at all the branching points on the Riemann surface are factorized analytically. The remaining single-valued matrix functions of the energy are expanded in the power-series near an arbitrary point in the complex energy plane. A systematic and accurate procedure has been developed for calculating the expansion coefficients. This makes it possible to obtain an analytic expression for the Jost-matrix (and therefore for the S-matrix) near an arbitrary point on the Riemann surface (within the domain of its analyticity) and thus to locate the resonant states as the S-matrix poles. This approach generalizes the standard effective-range expansion that now can be done not only near the threshold, but practically near an arbitrary point on the Riemann surface of the energy. Alternatively, The semi-analytic (power-series) expression of the Jost matrix can be used for extracting the resonance parameters from experimental data. In doing this, the expansion coefficients can be treated as fitting parameters to reproduce experimental data on the real axis (near a chosen center of expansion E ₀) and then the resulting semi-analytic matrix S(E) can be used at the nearby complex energies for locating the resonances. Similarly to the expansion procedure in the three-dimensional space, we obtain the expansion for the Jost function describing a quantum system in the space of two dimensions (motion on a plane), where the logarithmic branching point is present.     

Two-Body T-Matrices Without Angular-Momentum Decomposition: Energy and Momentum Dependences

The two-body T-matrix is calculated directly as function of two vector momenta for different Malfliet-Tjon-type potentials. At a few hundred MeV projectile energy the total amplitude is quite a smooth function showing only a strong peak in forward direction. In contrast, the corresponding partial-wave contributions, whose number increases with increasing energy, become more and more oscillatory with increasing energy. The angular and momentum dependence of the full amplitude is studied and displayed on as well as off the energy shell as function of positive and negative energies. The behaviour of the T-matrix in the vicinity of bound-state poles and resonance poles in the second energy sheet is studied. It is found that the angular dependence of T exhibits very characteristic properties in the vicinity of those poles, which are given by the Legendre function corresponding to the quantum number either of the bound state or the resonance (or virtual) state. This behaviour is illustrated along numerical examples. Received May 29, 1997; revised October 17, 1997; accepted for publication December 28, 1997     

Scalar Glueball: Analysis of the (IJ PC = 00++)-wave

Basing on the previously performed K-matrix analysis of experimental data, we investigate, in the framework of the propagator matrix (D-matrix) technique, the 1100–1900 MeV mass region, where overlapping resonances f ₀(1300), f ₀(1500), f ₀ (1530 ± ₂₅₀ ⁹⁰ ), and f ₀(1780) are located. Necessary elements of the D-matrix technique are developed. The D-matrix analysis confirms previous K-matrix results: in the region 900–1900 MeV five scalar/isoscalar states are located. Four of them are members of the two qq?-nonets, while one state is an extra for the qq? systematics, being a good candidate for the lightest scalar glueball. The D-matrix analysis shows that this extra state, a candidate for the lightest scalar glueball, is dispersed, due to a mixing with qq?- states, over three resonances: f ₀(1300), f ₀(1500), and f ₀ (1530 ± ₂₅₀ ⁹⁰ ). The broad resonance f ₀ (1530 ± ₂₅₀ ⁹⁰ ) is a descendant of the lightest glueball carrying about 50% of the gluonium component, the rest of the gluonium is shared between f ₀(1300) and f ₀(1500). The broad resonance is formed by accumulating a large part of the neighbouring resonance widths that elaborates some type of a trap for the adjoining states.