General N-Body Theory of Nonrelativistic Quantum Scattering

 The two-Hilbert-space theory of scattering is reviewed with particular reference to its application to nonrelativistic multichannel quantum- mechanical scattering theory. In Part I the abstract assumptions of the theory are collected, transition operators (both on- and off-energy-shell) are defined, the dynamical equations that determine the off-shell transition operators are presented and their real-energy limits examined, and the convergence of sequences of approximate transition operators is established. A section on how to incorporate group symmetries into the formalism reports new work. The material of Part I is relevant to a variety of both classical and quantum scattering systems. In Part II attention is directed specifically to N-body nonrelativistic quantum scattering systems in which the particles interact via short-range pair potentials. A method of constructing approximate transition operators is presented and shown to satisfy all the abstract assumptions of Part I. The dynamical equations that determine the half-on-shell approximate transition operators are shown to be coupled one-dimensional integral equations that have compact kernels and unique solutions when considered as operators on a Hilbert space of H?lder continuous functions. Moreover, the on-shell parts of those approximate transition amplitudes are shown to converge to the exact on-shell amplitudes as the order of the approximation increases. Detailed formulas for the kernels of the integral equations are written down for systems of particles that are distinguishable and for systems containing identical particles. Finally, some important open problems are described. Received July 2, 1999; accepted in final form October 27, 1999     

Connected kernel methods in nuclear reactions: (III). Effective interactions

The recently derived connected kernel equation (CKE) for N-body scattering operators is applied to direct nuclear reactions. A spectral representation is derived for the kernel of the CKE in order to obtain manageable approximations. This allows the kernel to be split into orders corresponding to the propagation of different numbers of bound clusters. By formally solving one part of the kernel at a time, the CKE is written as a hierarchy of nested equations in increasingly many variables. The first equation of this hierarchy is a set of coupled channel Lippmann-Schwinger equations coupling together all two-cluster channels. These equations reduce to the usual coupled channel equations for inelastic scattering and to the coupled channel Born approximation for rearrangement reactions when weak coupling assumptions are made. The second equation of the hierarchy is a two-variable integral equation for the effective interactions appearing in the coupled channel equations. The driving terms and kernel of this integral equation are obtained from the third equation of the hierarchy which is a three-variable integral equation and so forth. The use of the spectral expansion results in a renormalized theory in the sense that the bound state and reaction problems are separated. This permits the inclusion of nuclear models in the theory in a straightforward manner. The hierarchy is applied to a particular example, that of nucleon-nucleus scattering. For this case the hierarchy is truncated at the level allowing no more than three clusters in the continuum. By suppressing exchange and keeping only one-particle transfer and single-nucléon knockout channels, a set of equations for the optical potentials and transfer operators is obtained. These equations provide a three-body treatment of the single scattering approximation to the optical potential. Iteration of the equations yields the usual single scattering approximation in first order including three-body off-shell effects. After suppression of Fermi motion and off-shell effects, the standard impulse approximation is recovered. Modifications of the method for other cases are discussed and other possible applications suggested.     

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.     

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.     

Scattering amplitudes of QCD string in the worldline formalism

I review the derivation of large-N QCD meson scattering amplitudes in the Regge regime, where the effective theory of long strings applies in d = 4. A special attention is payed to the reparametrization path integral which plays a crucial role in the consistency of off-shell amplitudes. I show how the linear Reggeon trajectory is obtained for QCD string in the mean-field approximation, which turns out to be exact for the Nambu-Goto string, and discuss the interrelation with perturbative QCD.     

Eine Integralgleichungsmethode zur Behandlung der Streuung an gebundenen Teilchen

First it is pointed out that various methods known for the treatment of multi-particle scattering problems such as the methods ofLax, Watson, andFaddeev are based on the same type ofT-operator equations with eliminated interactions. They only differ by the identification of the interactions. — Then an integral equation treatment for the scattering of a particle by a system ofn bound particles is developed. If the scattering occurs via local two-body forces, the interaction matrix element splits into that of the two-particle case and a momentum-dependent factor. This fact is used to simplify the scattering equations which then get a mathematical structure similar to that of theT-operator equations discussed at the beginning (however, involving sums of bound states rather than sums of interactions) and which, therefore, can be handled in a similar way. When the interactions are eliminated by means of the two-particle scattering amplitudes, the off-shell energy parameter of these amplitudes may be chosen to be dependent on quantum numbers of the bound system. Such a choice shows indeed to be favorable if one likes to keep only the lowest order approximation of the integral equations. The resulting approximate formula leads, after some further approximations, for resonant scattering to a formula ofLamb, and for weakly energy-dependent amplitudes to a formula ofFermi (being related to the impulse approximation). — The resonant scattering formula is applied to a quantum-mechanical derivation of a method for the determination of nuclear lifetimes which had been proposed on semiclassical arguments byCiocchetti et al. — Finally the method developed for the scattering of a particle by a system of bound particles is extended to collisions between composite particles.     

Variational field theoretic approach to relativistic meson-nucleon scattering

Non-perturbative polaron variational methods are applied, within the so-called particle or world-line representation of relativistic field theory, to study scattering in the context of the scalar Wick-Cutkosky model. Important features of the variational calculation are that it is a controlled approximation scheme valid for arbitrary coupling strengths, the Green functions have all the cuts and poles expected for the exact result at any order in perturbation theory and that the variational parameters are simultaneously sensitive to the infrared as well as the ultraviolet behaviour of the theory. We generalize the previously used quadratic trial action by allowing more freedom for off-shell propagation without a change in the on-shell variational equations and evaluate the scattering amplitude at first order in the variational scheme. Particular attention is paid to the s-channel scattering near threshold because here non-perturbative effects can be large. We check the unitarity of a our numerical calculation and find it greatly improved compared to perturbation theory and to the zeroth order variational results.     

An integral transform of Green’s function, off-shell Jost solution and T-matrix for Coulomb-Yamaguchi potential in coordinate representation

By exploiting the theory of ordinary differential equations together with certain properties of higher transcendental functions, a useful analytical expression for the integral transform of the Green’s function for motion in Coulomb-Yamaguchi potential is derived via the r-space approach. This integral transform is applied to construct an analytical expression for off-shell Jost solution in its ‘maximal reduced form’ involving confluent and Gaussian hypergeometric functions. Corresponding Jost functions automatically follow from this solution. Finally, as another application of the off-shell Jost solution, the off-shell T-matrix is calculated by using a modified relation between off-shell physical wave function and T-matrix which does not involve the potential explicitly, thereby avoiding certain difficult integrals, and expressed it in terms of rational functions and simple hypergeometric functions which is in exact agreement with the results given previously by other authors.      

Nuclear relaxation via paramagnetic impurities

In the first part of this study, the kinetics of nuclear relaxation via paramagnetic impurities for systems of arbitrary space dimension d (including fractal ones) is calculated under the assumptions that are commonly adopted at present for three-dimensional objects. In the second part, a new mean-field-type theory is formulated that reproduces all the results of the first part for integral values of d and which is intended for describing the process at longer times, when the kinetics calculated in the first part is not correct and when a crossover to Balagurov-Vaks asymptotic behavior begins to develop. Solutions to the equations of the new theory are constructed at integral values of d. In order to obtain these solutions, a method is developed for calculating the low-energy and long-wave asymptotic behavior of the off-shell T matrix for potential scattering in the case of singular repulsive potentials.     

The polyakov path integral over bordered surfaces

The geometrical approach to the functional integral over Faddeev-Popov ghost fields is developed and applied to construct the BRST extension of the off-shell closed string amplitudes in the constant curvature gauge. In this gauge the overlap path integral for off-shell amplitudes is evaluated. It leads to the nonlocal sewing procedure generating all off-shell amplitudes from the cubic interaction vertex. The general scheme of the reconstruction of a covariant closed string field theory from the off-shell amplitudes is discussed within the path integral framework.     

Inelastic shadowing effects in multiple scattering

The projectile-nucleon scattering amplitudes used as input into multiple scattering theories of projectile-nucleus scattering naturally include the effects of coupling to inelastic (i.e., production) channels. We employ a multichannel separable potential to describe the projectile-nucleon interaction and show that within the fixed nucleon framework we can obtain the nuclear elastic scattering amplitude. This includes terms outside the conventional formalisms, corresponding to intermediate propagation in the inelastic channels both above and below inelastic threshold. We refer to this as inelastic shadowing. In a two-channel approximation, we show that knowledge of the projectile-nucleon elastic scattering phase shifts plus specification of the inelastic threshold energy are sufficient to determine the off-shell coupled-channel transition matrix, implying that the nuclear amplitude can be calculated within this model without any detailed information about the inelastic channels. We study this solution quantitatively for some model problems and for pion scattering, with the general result that inelastic shadowing can be significant whenever the elementary interaction has important channel coupling. For pion scattering in the energy regime characterized by strongly absorptive resonances, we find, for example, that the effect of inelastic shadowing is much more important than that due to two-nucleon correlations.     

-Body Quantum Scattering Theory in Two Hilbert Spaces: -Body Integral Equations

Scattering for a nonrelativistic system of distinguishable and spinless particles interacting via short-range pair potentials is considered. Half-on-shell integral equations (the CG equations) are proposed, the solutions of which determine approximate scattering amplitudes that converge to the exact scattering amplitude. It is proved, under mild H?lder integrability assumptions, that these apparently singular equations actually have a compact kernel for real energies and, consequently, a unique solution. The CG equations have a structure that is much simpler than the Yakubovskii equations and similar to that of coupled-reaction-channel equations. The driving terms look like distorted-wave Born integrals and nonorthogonality integrals. However, there is no restriction to channels with only two asymptotic bound clusters and for all channels, no matter how many bound clusters, appropriate boundary conditions are exactly satisfied. This work completes the establishment of a rigorous mathematical link between the solutions of the half-on-shell CG equations and the on-shell transition operators defined in time-dependent multichannel scattering theory, and it provides for the first time a rigorous theoretical basis for practical calculations of scattering amplitudes for certain problems with . Received October 27, 1997; accepted for publication December 29, 1997     

The scattering amplitude for four off-shell tachyons from functional integrals

We use functional integral techniques to calculate the scattering amplitude for four open off-shell tachyons in Witten's string field theory and show that the residues of the first three poles agree with those obtained using oscillator methods.     

Two-body T-matrix continued off the energy shell

A generalized parametrization scheme, which treats both half-shell and off-shell scattering on equal footing, is derived for the two-particle transition operator. The appropriate theory in respect of this is developed by using a wave function approach to off-shell scattering and the computational procedure to be followed is demonstrated by means of a model calculation.     

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.     

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.     

Approximation of off-shell scattering operators for non-lorentzian line shape calculations

A previously derived approximation for the matrix elements of the Fano relaxation operator m(ω), which gives rise to non-Lorentzian line wings, is examined. This approximation is expressed in terms of an off-the-energy shell scattering operator. Calculation of this operator by a second-order Born expansion is unsuitable because this procedure limits the radiation frequency dependence to the same order. The main part of this paper is devoted to expressing the off-shell scattering operator in the time-ordered formalism. In this form, a scattering operator approximation suggested by Dillonet al.⁽⁶⁾ can be applied which is not restrictive to the frequency dependence.     

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.