Generalization and applications of the Sasakawa theory

The Sasakawa theory of scattering is phrased in the form of a Fredhohn reduction technique for integral equations possessing a fixed-point singularity in their kernels. This permits the generalization of this theory to a large variety of scattering integral equations. Some specific applications include the two-particle off-shell and multichannel scattering problems. In the first instance a rank-three approximation to the fully off-shell transition matrix is derived which is exact on and half-off shell, satisfies off-shell unitarity, and which possesses no unphysical singularities. In the second problem it is shown how the method leads to the generation of a unitary approximation to the multichannel amplitudes.     

A renormalized equation for the three-body system with short-range interactions

We study the three-body system with short-range interactions characterized by an unnaturally large two-body scattering length. We show that the off-shell scattering amplitude is cutoff independent up to power corrections. This allows us to derive an exact renormalization group equation for the three-body force. We also obtain a renormalized equation for the off-shell scattering amplitude. This equation is invariant under discrete scale transformations. The periodicity of the spectrum of bound states originally observed by Efimov is a consequence of this symmetry. The functional dependence of the three-body scattering length on the two-body scattering length can be obtained analytically using the asymptotic solution to the integral equation. An analogous formula for the three-body recombination coefficient is also obtained.     

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.     

Recursion relations and scattering amplitudes in the light-front formalism

The fragmentation functions and scattering amplitudes are investigated in the framework of light-front perturbation theory. It is demonstrated that, the factorization property of the fragmentation functions implies the recursion relations for the off-shell scattering amplitudes which are light-front analogs of the Berends–Giele relations. These recursion relations on the light-front can be solved exactly by induction and it is shown that the expressions for the off-shell light-front amplitudes are represented as a linear combinations of the on-shell amplitudes. By putting external particles on-shell we recover the scattering amplitudes previously derived in the literature.     

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.     

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 extension function from a generalised phase-function method

It is shown that the Combined-variable-phase-off-shell scattering theory or the generalised phase-function method provides efficient algorithms to calculate the off-shell extension function which plays a role in the theories of three-particle scattering. In order to assert this we have chosen to work with a model potential for theα-α interaction. The functions which occur in the context of our approach exhibit certain interesting features with regard to effect of the potential in producing on-and off-shell quantities like the phase shift and quasi-phase. Interpolating off-shell extension function is seen to exhibit a discontinuity where the phase function goes through a zero.     

Off-shell sensitivity,repulsive correlations and the pion-nucleus optical potential

Repulsive nucleon-nucleon correlations tend to reduce the dependence of pion-nucleus elastic scattering upon the off-shell pion-nucleon dynamics. However, optical potential calculations can in practice be quite sensitive to the particular choice of off-shell model parameters. It is argued that this sensitivity results from the nature of the optical potential as a one-body operator which introduces extra off-shell dependence not found in the physical many-body process itself. Thus, one must be very careful in any attempt to extract correlation or off-shell information, or to predict pion-nucleus phase shifts, by means of an optical potential theory. Results of model calculations are presented for purposes of illustration.     

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.     

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.     

Generalized optical theorems for the reconstruction of Green's function of an inhomogeneous elastic medium

This paper investigates the reconstruction of elastic Green's function from the cross-correlation of waves excited by random noise in the context of scattering theory. Using a general operator equation-the resolvent formula-Green's function reconstruction is established when the noise sources satisfy an equipartition condition. In an inhomogeneous medium, the operator formalism leads to generalized forms of optical theorem involving the off-shell T-matrix of elastic waves, which describes scattering in the near-field. The role of temporal absorption in the formulation of the theorem is discussed. Previously established symmetry and reciprocity relations involving the on-shell T-matrix are recovered in the usual far-field and infinitesimal absorption limits. The theory is applied to a point scattering model for elastic waves. The T-matrix of the point scatterer incorporating all recurrent scattering loops is obtained by a regularization procedure. The physical significance of the point scatterer is discussed. In particular this model satisfies the off-shell version of the generalized optical theorem. The link between equipartition and Green's function reconstruction in a scattering medium is discussed.     

Two-Nucleon Scattering Without Partial Waves Using a Momentum Space Argonne V18 Interaction

We test the operator form of the Fourier transform of the Argonne V18 potential by computing selected scattering observables and all Wolfenstein parameters for a variety of energies. These are compared to the GW-DAC database and to partial wave calculations. We represent the interaction and transition operators as expansions in a spin-momentum basis. In this representation the Lippmann–Schwinger equation becomes a six channel integral equation in two variables. Our calculations use different numbers of spin-momentum basis elements to represent the on- and off-shell transition operators. This is because different numbers of independent spin-momentum basis elements are required to expand the on- and off-shell transition operators. The choice of on and off-shell spin-momentum basis elements is made so that the coefficients of the on-shell spin-momentum basis vectors are simply related to the corresponding off-shell coefficients.     

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.     

Low Energy D(p, p)D Elastic Scattering Including Coulomb Force

The motivation of this paper is to obtain the three-body amplitudes for the Coulomb potential plus a nuclear force in momentum space. Not only the two-body off-shell nuclear amplitude but also the two-body off-shell Coulomb amplitude is important in the three-body calculation. For calculating the Coulomb amplitude, the modified Coulomb potential whose Fourier transformation is analytically equivalent to the pure configuration space Coulomb potential, is introduced. In addition, the decisive screening range parameter is also utilized instead of the screening range. The modified Coulomb potential plus the parameter is called decisive modified Coulomb potential. The three-body proton-deuteron elastic scattering is calculated by using the proper two-body off-shell amplitude for the decisive modified Coulomb potential.     

Off-energy-shell variations of two-nucleon transition matrix and three-nucleon problem

For a schematic three-nucleon problem, we derive approximate analytic expressions for the functional derivatives of measurable three-particle quantities with respect to off-shell variations of the triplet-s, two-nucleon transition matrix. Those quantities include neutron-deuteron scattering lengths, trinucleon binding energies, and the ³He charge form-factor minimum; correlations between off-shell changes in the latter two are discussed. We indicate how results of this kind may be used to decide whether or not a given set of discrepancies between calculated and experimental three-nucleon observables can be reconciled in terms of off-shell variations of a nonretarded hermitean two-nucleon interaction. The treatment is not restricted to special classes of phase-shift equivalent potentials or phase-shift preserving transformations but instead makes use of a systematic parameterization of off-shell variations in terms of symmetric rational approximants of increasing order.     

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.     

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.     

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.