Off-shell πN amplitudes via a multichannel separable potential

The off-shell pion-nucleon transition matrix is a basic ingredient in theories of pion-nuclear interactions which, in the absence of fundamental theory of πN dynamics, must be obtained by a phenomenological extrapolation from the available on-shell data. As one means of performing such an extrapolation, we explore a multichannel separable potential model with the property that the off-shell elastic scattering amplitude is generated directly from the measured elastic-channel phase shifts. The off-shell πN partial-wave transition amplitudes determined by this procedure are compared with those calculated by Landau and Tabakin using a one-channel absorptive separable potential. We find that the absorptive separable potential approach provides a physically unreasonable off-shell extrapolation at energies where the on-shell amplitude is highly inelastic, and show that the difficulty is a direct consequence of the one-channel nature of that method. The multichannel extrapolation is free of these difficulties.     

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 dependence of the photodisintegration cross section on the off-shell T-matrix

Nucleon-nucleon scattering phase shifts determine the diagonal element of the transition matrix. The off-diagonal elements are not completely arbitrary but have conditions imposed on them by the range and the tail of the potential. Electromagnetic interaction can also be used to place restrictions on the off-diagonal elements. We find that the cross section of the deuteron photodisintegration is sensitive to the off-shell transition matrix. The integrated cross section can be varied by as much as 30 % or more, and the matrix element for the El transition by a factor of 2. While the matrix element for the photodisintegration depends on the off-shell elements of the T-matrix, it cannot be used to discriminate between alternative off-shell T-matrices. We have constructed classes of different off-shell T-matrices, which produce identical photo-disintegration cross sections and other two-body scattering and bound-state properties.     

Importance of the three-body Pauli potential in three-cluster systems

The ground state energies of the 3α system and of theA-α-α system are calculated in the three-cluster fish bone optical model [1], It is found that the three-cluster Pauli potential is very strong in theM-version of the model and weak in the (off-shell transformed)¯M-version. Its influence on theΛ-α-α binding energy and ground state wave function is almost negligible in the¯M-version. The present results indicate that phenomenological three-cluster models with energy-independent two-cluster interactions are not necessarily in contradiction with more microscopic models. It seems to be important, however, to use potentials which contain in their off-shell behaviour both Pauli-principle and saturation character of the nuclear forces.     

The 2H(p, 2p)n reaction at Einc = 45 MeV and its sensitivity to various separable S-wave N-N potentials

This paper presents and analyses the data for the ²H(p, pp)n reaction at E_inc = 44.9 MeV. Kinematic conditions including the quasi-free scattering region and the regions far from quasi-two-body processes are considered. The experimental results are compared with the predictions of several separable S-wave models of the N-N interaction. Potential models that differ only off-shell, as well as models that predict different N-N scattering results are included to help to isolate aspects of the break-up reaction sensitive to off-shell and on-shell differences. The regions far from quasi-free scattering are generally much more sensitive to the off-shell features of the interaction than are quasi-free scattering results. On-shell differences in the potential affect predictions in all regions of phase space with the potentials best representing the free N-N data giving the best overall fit to the data over most regions of phase space.     

Binding energy of triton and scattering lengths in neutron-deuteron collision in Faddeev formalism with local potential

A separable representation for the off-shell two-body t-matrix for a local Hulthén potential is presented, in which deuteron states are chosen as the expansion bases. Using the Faddeev equations with these t-matrices as input, the ground state energy of the triton and doublet and quartet scattering lengths in neutron-deuteron scattering, have been computed. The results have been compared with the experimental findings and the theoretical results of Sitenko et al. obtained in the Sturmian function representation with the same Hulthén potential.     

Elastic scattering of light nuclei through a simple potential model

The phase function method is adapted to deal with the scattering on our proposed interactions for α–α and α–³He systems. The effect of the electromagnetic interaction is included in terms of a screened Coulomb potential. Based on our proposed potential models we present results for α–α and α–³He elastic scattering phase shifts which compare well with more detailed calculations.     

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.     

Phenomenological separable potential of K + N interaction for J≤15/2

A nonlocal separable potential of K ⁺ N interaction in the isospin-0 and isospin-1 S, P, D, ..., J scattering channels is constructed. This potential describes satisfactorily the results of the SP92 phase-shift analysis (R. Arndt) at incident-kaon energies in the interval from 0 to 2.4 GeV. The off-shell amplitudes for kaon-nucleon scattering are calculated. They are shown to differ significantly from the commonly used off-shell amplitudes.     

Multiple scattering, optical potential, and N-body approaches to elastic two-fragment collisions

Two-fragment elastic scattering problems are often studied using multiple scattering theories such as those due to Watson along with Feshbach-type optical potentials. These conventional methods are re-examined, rephased, and generalized using the language and techniques of contemporary N-particle scattering theory. A special realization of the latter theory is developed which is especially useful for relating the older and newer methods. This is facilitated by maintaining the same off-shell continuations of the scattering operators in both approaches. In particular, a set of connected-kernel scattering integral equations is introduced which provides a consistent N-particle framework for the calculation of that definition of the optical potential possessing the Feshbach off-shell continuation. These equations exhibit a multiple-scattering substructure and therefore allow the systematic generalization of some of the usual low-order approximations.     

The πNN vertex function and off-mass-shell πN scattering

We have extended the dispersion relation calculation of the πNN vertex function to include off-shell terms in the πN scattering matrix. These off-shell terms are constrained by the current algebra and PCAC results, and contribute to the $\text{N}\text{N}\text{→ ππ}$ s-wave. As such, they add to the kinematic off-shell effects which Durso, Jackson and VerWest found in the p-terms. The off-shell terms increase the calculated Goldberger-Treiman discrepancy from 0.02 to 0.03, bringing it into agreement with the field theory value of Jones and Scadron (0.035). The calculated discrepancy remains smaller than the experimental value of 0.06±0.01.     

Hulthén potential models for <Emphasis Type="BoldItalic">α−α</Emphasis> and <Emphasis Type="BoldItalic">α−</Emphasis>He<Superscript><Emphasis Type="BoldItalic">3</Emphasis></Superscript> elastic scattering

Simple Hulthén-type potential models are proposed to treat the α?α and \(\alpha {-} \text {He}^{3}\) elastic scattering. The merit of our approach is examined by computing elastic scattering phases through the judicious use of the phase function method. Reasonable agreements in scattering phase shifts are obtained with the standard data.     

Off-energy-shell continuation of the πN transition matrix element

General relationships of the fully off-shell T-matrix element for the central forces are derived. The problem of finding the fully off-shell T-matrix element can be reduced to finding a real two-parameter function φ(p, k). It can be shown in the case of no bound state that if φ(p, k) is separable, the whole problem becomes equivalent to the inverse scattering problem. An approximate method to include inelastic effects is proposed and the πN off-shell T-matrix element is calculated.     

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.     

Superfluid nuclear matter calculations

We present a method to calculate nuclear matter properties in the superfluid phase. The method is based on the use of self-consistent off-shell nucleon propagators in the T-matrix equation. Such a complete treatment of the spectral function is required below and around T_c due to a pseudogap formation in the spectral function. In the superfluid phase we introduce the anomalous self-energy in the fermion propagators and in the T-matrix equation, consistently with the strong coupling BCS equations. The equations for the nucleon spectral function include both a contribution of condensed and scattering pairs. The method is illustrated by numerical calculations. Above T_c pseudogap formation is visible in the spectral function and below T_c a superfluid gap also appears.     

Binding effects in the optical potential of pionic atoms

We calculate the contribution from nucleon binding to the coefficient Re B₀ in the optical potential of pionic atoms. The relevant Feynman graphs are deduced for nuclear matter. They depend on off-shell values for pion-nucleon scattering lengths a_πN and for the nucleon-binding potential V_NA. Off-shell effects are found to be very important: A pole model for a_πN, extrapolated off-shell, increases binding effects by a factor of three over the use of on-shell values. Two simple models are proposed for the off-shell continuation of V_NA. One leads to attractive, the other to repulsive contributions to Re B₀.     

Off-shell extension of the π-nucleon transition matrix and separable π-nucleon potentials

The off-shell extension of the P-wave π-nucleon transition matrix given by the Chew-Low model is compared with the one obtained from a separable potential. It is shown that the two off-shell extensions are rather different; π-deuteron scattering results are compared for the two cases.     

Scattering length and effective range in closed form for the Coulomb plus Yamaguchi potential

From the known expression for the off-shell T-matrix corresponding to the potential consisting of the sum of the Coulomb potential and the Yamaguchi potential, the physical scattering amplitude can be derived in a satisfactory way. We derive simple exact closed formulae for the scattering length a_cs and the effective range r_cs from this amplitude. These are compared with approximate formulae derived by Harrington. Also a few numerical calculations are reported and compared with results obtained by Harrington and by Ali et al.