Coupled channel approach to the off-shell transition matrix

We propose a coupled-channel separable potential model with the feature that the off-shell elastic transition matrix can be generated directly from the scattering phase shifts. This should be useful in constructing pion-nucleus optical potentials.     

Off-shell behavior of the crossed nucleon pole in pion-nucleus scattering

The contribution of the nucleon-pole piece of the pion-nucleon amplitude to the pion-nucleus optical potential is examined. Straightforward calculations of diagrams which are intermediate between Goldstone and Feynman diagrams produce simple results with well-behaved physical implications. In particular, the direct S-channel nucleon pole contributes to one-body true absorption while the crossed U-channel pole has a novel off-shell behavior and does not contribute to pion true absorption.     

Low-energy pion-nucleus optical potentials

We discuss the local and non-local pion-nucleus optical potentials. We find that the local potential becomes non-local when two-nucleon correlations are included. The two potentials (including correlations) can be made local through a transformation on the wave function. The new local potentials agree up to quadratic terms when expanded in powers of the density. The influence of finite-range correlations and off-shell pion-nucleon form factors is also investigated.     

Improved theoretical pion-nucleus optical potentials

An approach which makes the first order pion-nucleus optical potential theoretically sound is presented. This study should permit higher order improvements to the potential to be more meaningful and the nuclear structure information extracted from pi-nucleus scattering to be more reliable. Based on multiple scattering theory, three optical potentials are constructed and studied in momentum space. These models are the popular Kisslinger potential, the local “Laplacian” potential, and an “improved off-shell potential;” the latter one is derived from absorptive separable pion-nucleon potentials which exactly reproduce on-shell πN scattering. By working in momentum space and explicitly including πN resonances and off-shell effects in the definition of the optical potential, the approach described here is capable of handling any number of pi-nucleon partial waves, is applicable over a very wide energy region, is based on a physical model for off-shell behavior, and is extended easily to include higher order effects. The optical potentials are inserted into two different relativistic wave equations to determine the total cross section and elastic differential cross section for pi-nucleus scattering. It is found that the various models for off-shell πN scattering determine significantly different πC¹² scattering, with the improved off-shell model preferred on theoretical grounds. Also discussed is the importance of properly transforming πN scattering to the pi-nucleus c.m. system, the origin of the shift in the peak position of the π^?C total cross section, and the reason for the increased diffractive nature of the differential cross section at 180 MeV.     

The reaction cross section for low-energy pion-nucleus scattering

It is shown that for low pion energies it is possible to calculate the pion absorption part of the pion-nucleus reaction cross section from a detailed knowledge of the imaginary part of the pion-nucleus optical potential. Such calculations provide tests of various theories of low-energy pion-nucleus dynamics.     

The absorptive pion-nucleus optical potential

We calculate s-wave and p-wave absorptive pion-nucleus optical potentials assuming that a pion is absorbed by a pair of nucleons. Employing a model which takes into account both a single nucleon absorption with nucleon-nucleon correlations and rescattering, we obtain simple analytic expressions for Im B₀ sid Im C₀ of the pion-nucleus optical potential. The off-shell effect on the s-wave pion absorption is examined and shown to be strongly modified by short range correlations. The result for the p-wave absorptive part Im C₀ clearly shows the importance of the tensor correlations. The enhanced nn emission after π^? absorption is shown to be related with a large p-wave πN scattering length a₃₃ via the tensor correlations.     

Subthreshold pion spectra in nucleus-nucleus collision

The energy spectra of neutral pion produced in¹²C-¹²C reaction atE _Lab=60, 74 and 84 MeV/nucleon is calculated within a modified thermal model. The nuclear mean field is taken into account through the Brueckner G-matrix effective interaction. Pion reabsorption is included through different energy and density dependence pion-nucleus optical potentials. A-comparison to the experimental data shows a satisfactory agreement of pion spectra over a wide range of pion energy. Modifications are made to the pion-nucleus optical potential of Meherem et al. in order to describe pion spectra as well as pion-nucleus reactions.     

The low-energy pion-nucleus interaction

We discuss the low-energy pion-nucleus interaction within the framework of the optical model. We find that a consistent expansion of the πN amplitude in momentum is required in order that the low-energy pion-nucleus interaction is physically compatible with a simple compositie picture of the nuclear sacttering process. This introduces an energy-dependent component into the usual local part of the Ericson-Ericson optical potential stemming from the πN s-wave effective range. The size of this component is estimated and the corresponding pion-nucleus effective ranges and volume parameters are determined for uniformly dense nuclei. Finally a modified extrapolation to the bound state energies is considered in the light of these large effective ranges.     

Model dependence of the impulse approximation

We investigate the model dependence of the impulse approximation for proton-nucleus elastic scattering stemming from off-shell and non-local effects. A perturbative treatment allows a direct comparison between off-shell and relativistic effects. Off-shell effects are found to be significantly smaller and unable to account in a natural way for the needed repulsive central potential in the nuclear interior region at proton energies between 300 and 800 MeV.     

A meson-exchange N-N potential

A meson-theoretic model of the intermediate range nucleon-nucleon potential is presented with emphasis placed on the two-pion exchange contribution. The Bethe-Salpeter equation is reduced, by the Blankenbecler-Sugar technique, to a Lippmann-Schwinger equation, from which an approximate nonlocal, energy-dependent potential is obtained. The nucleon-antinucleon pair contribution, which plagues meson-theoretical two-pion calculations, is suppressed by the complex poles of the one-nucleon Green's function. The importance of the retention of the explicit energy dependence of the potential is demonstrated by calculating the off-shell scattering matrices. The potential is presented in a linearized (in energy) form with the core region adjusted to produce a fit to low energy data.     

Coherent photoproduction of pions on spin-zero nuclei in a relativistic,non-local model

A relativistic and non-local model for the coherent photoproduction of pions on spin-zero nuclei is presented. The production operator is derived from an effective Lagrangian model that contains all the degrees of freedom known to contribute to the elementary process. Using the framework of DWIA, the matrix elements of this operator are evaluated using relativistic bound state wave functions. Final state interactions are accounted for via a pion-nucleus optical potential. The effects of in-medium modifications of the production operator are discussed. We give results for ¹²C and ⁴⁰Ca, and compare our calculations to the data of the A2 collaboration at MAMI.     

Impulse-approximation correction in pion-nucleus optical potential

The local density approximation is used in this paper to calculate the first-order pion-nucleus optical potential. TheπN scattering matrix in nuclear medium is computed by employing separableπN scattering matrix. This nuclear-mediumπN scattering matrix, which includes impulse-approximation correction is then used to construct the pion-nucleus optical potential.π-¹²C elastic scattering results obtained by using this potential are compared with the impulse-approximation potential results.     

Chiral dynamics of deeply bound pionic atoms

We present and discuss a systematic calculation, based on two-loop chiral perturbation theory, of the pion-nuclear s-wave optical potential. A proper treatment of the explicit energy dependence of the off-shell pion self-energy together with (electromagnetic) gauge invariance of the Klein-Gordon equation turns out to be crucial. Accurate data for the binding energies and widths of the 1s and 2p levels in pionic 205Pb and 207Pb are well reproduced without need for a notorious "missing repulsion" in the pion-nuclear s-wave optical potential. The connection with the in-medium change of the pion decay constant is clarified.     

The inhomogeneous Schrödinger equation for the off-shell wave function

A method has been developed for calculating the off-shell wave function by solving the inhomogeneous Schrödinger equation with allowance for the nuclear and Coulomb interactions. The off-shell wave function makes it possible to construct the off-shell scattering amplitude in order to solve the problems for three or more particles. An important application of the method is the Trojan Horse calculations of nuclear reactions that are important in nuclear astrophysics. Specific calculations are performed for neutron and proton scattering on the ⁷Be nucleus. The Woods-Saxon potential is used and the spin-orbital interaction is taken into account.     

Off-shell corrections to the glauber theory of nuclear multiple scattering

We study the first-order off-shell corrections to Glauber's multiple scattering model. These consist of a direct and a rescattering term, and are obtained from the eikonal corrections studied in a previous paper by a simple substitution. For an independent particle nuclear model the direct term becomes identical to an expression previously derived by Moniz. We show that the rescattering term is in general not small compared to the direct term. For high incident energy a comparison is made with other corrections to the Glauber model. An optical potential is proposed which consists of the Glauber potential and its off-shell, eikonal, Fresnel and spatial correlation corrections.     

Test of effective cluster interactions by pion scattering

The superposition of effective interactions is studied by a cluster approach to the pion-nucleus potential with particular emphasis on the treatment of the Pauli principle between the clusters. The model is based on the formalism of Kerman, McManus, and Thaler and constructs the pion-nucleus interaction by a superposition procedure of pion-cluster interactions in the framework of the fish-bone model. The second-order contribution due to excitations of the relative motion between the clusters has been taken into account as well as true absorption. A test calculation of ^+_20 scattering with the cluster approach reproduces the gross structure of the differential elastic cross section and the total non-elastic cross section of conventional calculations quite well. Furthermore, it is confirmed that the three-body Pauli interaction is negligible also in the case of a realistic and rather complicated pion-nucleus interaction thus justifying the usual folding approach. The rotational bands contribute about 10% to the cross sections.     

Nucleon-nucleon on-shell and off-shell contributions to the triton binding energy in an energy-dependent potential model formulation

An energy-dependent nucleon-nucleon (NN) potential model is presented which allows an unambiguous separation of NN on-shell and off-shell contributions to trinucleon observables. A potential of the form $\text{V(E; p, p′) = λ (E)}\text{V}\text{(p, p′)}$ is constructed at negative energies by comparison with various hermitian potentials. The latter comparison suggests a symmetric choice of λ(E) with respect to E = 0. It is shown that the force model constructed according to the present study is more reasonable and less ambiguous than the hermitian potential model, for which the negative-energy dependence of the corresponding t-matrix is generated in an arbitrary way by the accidental choice of the off-shell behavior of the potential. The pertinent shortcomings of the conventional potential model are elucidated by their inconsistent predictions for two- and three-nucleon systems. In the case of the proposed energy-dependent potential model the on-shell information is decoupled from the off-shell behavior at positive and negative energies. This allows one to eliminate the two-body off-shell contributions within the three-body equations in a continuous process by taking the range of forces to zero while keeping at each step the two-body on-shell information unaltered.     

Threshold anomaly in non-central forces

The behaviour of the threshold anomaly for non-central potentials, which account for collective excitations in heavy-ion collisions, is investigated. It is shown that the non-central potentials should exhibit an energy dependence at energies in the vicinity of the Coulomb barrier. This energy dependence is, however, different from that of the elastic optical potential, occurring at lower energies. It if further shown that there are corrections to the traditional collective model such that, if the transition potential is expressed as the derivative of the optical potential, the corresponding deformation length will be complex and energy-dependent. Simple model calculations are presented.     

Shifts of the peaks in pion-nucleus scattering near the Δ(1231 MeV) and a new effective optical potential

We present a different kind of calculation of a complex index of refraction or equivalently, a complex effective optical potential, for pion-nucleus scattering near the first pion-nucleon resonance where the elementary interaction is strong and varying with energy. Therefore certain dynamical effects of the medium upon the potential become important.