Low-energy three-body scattering in a hyperspherical adiabatic basis

We propose a hyperspherical adiabatic formalism for the calculation of the 3-to-3S-matrix at low energy, for repulsive potentials, and use it then in a model calculation. That is for McGuire's model (3 particles in one dimension subject to repulsive delta-function interactions), we use analytical expressions for the hyperspherical adiabatic basis, the adiabatic coupling matrix elements, and eigenpotentials to obtain the first terms of the exactS-matrix analytically, in an expansion in powers of the wave number. We were able to associate the definite powers ofq in the expansion of theS-matrix to the corresponding inverse powers of in the expansions of the adiabatic eigenpotentials and coupling matrix elements. We investigate the effect of making the usual approximations found in the literature (extreme and uncoupled adiabatic approximations), when calculating the diagonal and off-diagonalS-matrix elements. Finally, we show that the coupled adiabatic equations uncouple as the energy goes to zero.     

Extraction of Astrophysical Cross Sectionsin the Trojan-Horse Method

 The Trojan-horse method has been proposed to extract S-matrix elements of a two-body reaction at astrophysical energies from a related reaction with three particles in the final state. This should be useful in cases where the direct measurement of the two-body reaction at the necessary low energies is experimentally difficult. The formalism of the Trojan-horse method for nuclear reactions is developed in detail from basic scattering theory including spin degrees of freedom of the nuclei and we specify the necessary approximations. The energy dependence of the three-body reaction is determined by characteristic functions that represent the theoretical ingredients for the method. In a plane-wave Born approximation of the T-matrix the differential cross section assumes a simple structure. Received August 31, 1999; revised June 14, 2000; accepted for publication June 30, 2000     

Theory of the Trojan-Horse method

The Trojan-Horse method is an indirect approach to determine the energy dependence of S factors of astrophysically relevant two-body reactions. This is accomplished by studying closely related three-body reactions under quasi-free scattering conditions. The basic theory of the Trojan-Horse method is developed starting from a post-form distorted wave Born approximation of the T-matrix element. In the surface approximation the cross-section of the three-body reaction can be related to the S-matrix elements of the two-body reaction. The essential feature of the Trojan-Horse method is the effective suppression of the Coulomb barrier at low energies for the astrophysical reaction leading to finite cross-sections at the threshold of the two-body reaction. In a modified plane wave approximation the relation between the two- and three-body cross-sections becomes very transparent. The appearing Trojan-Horse integrals are studied in detail.     

Classical and quantum pumping in closed systems

Pumping of charge (Q) in a closed ring geometry is not quantized even in the strict adiabatic limit. The deviation form exact quantization can be related to the Thouless conductance. We use the Kubo formalism as a starting point for the calculation of both the dissipative and the adiabatic contributions to Q. As an application we bring examples for classical dissipative pumping, classical adiabatic pumping, and in particular we make an explicit calculation for quantum pumping in case of the simplest pumping device, which is a three site lattice model. We make a connection with the popular S-matrix formalism which has been used to calculate pumping in open systems.     

Study of theS-matrix near bound states in the continuum

The behaviour ofS-matrix for potentials generating bound states in continuum in the neighbourhood of the positive bound state energies is studied. It is shown that unlike the case of usual negative energy bound states, theS-matrix does not have a pole at the positive bound state energy but becomes unity at the energy corresponding to bound states in continuum. Calculations ofS-waveS-matrix for a local potential constructed by Stillinger and Herrick and a separable nonlocal potential constructed by the present authors verify these results. Our results indicate that the bound states embedded in continuum constructedvia the von Neumann and Wigner procedure cannot be interpreted as resonances with zero width.     

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     

Sensitivity of the three-body calculations to different effects

The bound state of few-body systems in light nuclei is studied as a three-body problem. The three-body problem is solved following the different approaches of the Faddeev formalism as well as the unitary pole approximation. Separable approximations are introduced to reduce the three-body problem to a set of coupled integral equations. Numerical calculations are carried out for the resulting integral equations and the separable expansion. In the present work, we calculate the ground-state binding energy of the bound three-nucleon system³H. The main interest of the present work is to investigate the sensitivity of the three-body binding energy to different effects in the problem. For this reason, we study the dependence of the three-body binding energy of different forms of local and separable two-body potentials, on the effective range of the two-body potentials, and on the percent of theD state in the deuteron wave function. Also, we test the sensitivity of the three-body binding energy to the considered number of terms from the separable expansion.     

Threshold singularities of theS-matrix and convergence of Haag-Ruelle approximations

We establish a one-to-one correspondence between the continuity properties of theS-matrix at the 2-particle threshold and the rate of convergence of the Haag-Ruelle approximations ψ(t) for asymptotic 2-particle states ψ with smooth wavefunctions. It turns out that the norm distance ∥ψ?ψ(t)∥ approaches 0 liktt ^?5/4 if theS-matrix has the normal threshold singularities and liket ^?3/4 in the exceptional case where the threshold has “absorbed” a bound state. These connections are valid both in relativistic quantum field theory and in non-relativistic models with short range interaction.     

Topics in the S-matrix theory of massless particles

We discuss how massless particle reactions may be incorporated into standard S-matrix theory. The crucial element for doing so is a low-energy zero. Examples of reactions where such zeros occur are weak interaction processes involving neutrinos, chirally symmetric massless pion scattering, and two-photon exchange between neutral systems. These zeros make two-body unitarity a good approximation for sufficiently low energy despite the coalescence of multiparticle thresholds. Through two-body unitarity, these zeros produce lines of zeros in the absorptive parts and double spectral functions. These lines of zeros are the S-matrix analog of the requirement of an infrared finite field theory. Not only do they produce finite total cross sections at finite energies, but they also allow both upper and lower bounds to be derived for these cross sections at high energies. This upper bound is our main result. If a plausible smoothness assumption is made, we find σ_tot <s^? (where ? is arbitrarily small). In particular, the experimentally observed linear rise of the neutrino proton cross section cannot continue indefinitely.     

Bound states and scattering from bound states in the light-front field theory dynamics

Starting from the Weinberg rules we derive a covariant form of the relativistic Schrödinger equation and formulate the bound state problem in the light-front field theory dynamics. We present an explicit rule for embodying the two-body subsystem in the three-body space and demonstrate that the cluster decomposition property is explicitly preserved in the light front field theory dynamics. As an application of these results we write amplitudes forπd→nN ^*, πd→πpn, andπd→πd, in the impulse approximation, in terms of the internal bound state wave functions and two-body reducedt-matrix elements.     

Anderson impurity model: Vertex corrections within the finite U non-crossing approximation

We revise the simplest possible approximations to solve numerically the vertex equations for the single impurity Anderson model (SIAM) within the finite U non-crossing approximation (UNCA), considering the self-energies at lowest order in the 1/N diagrammatic expansion. We introduce an approximation to the vertex corrections that includes the double energy dependence and compare it with an approximation (NCAf²v) that neglects a second energy argument. Finally, we analyse the influence of the different approximations on the estimated Kondo scale for simple electronic models.     

A coordinate-space study of kowalski's three-term separable approximation for the fully off-shell two-nucleon T-matrix

Using elementary coordinate-space methods, we show that a three-term separable approximate fully off-shell T-matrix proposed by Kowalski can be reduced to a simpler expression. This T-matrix incorporates off-shell unitarity exactly, is exact half off the energy shell, and is free from the spurious poles that arise in the Noyes approximation. However, numerical tests employing the wave-function model of Picker, Redish, and Stephenson show that for realistic ¹S_o interactions, the Noyes approximation is more accurate than Kowalski's approximation except near the spurious pole at 250 MeV. We give a plausible explanation of this result.     

Total S-matrix and adiabatic amplitudes for the three-body problem

The three-body quantum scattering problem reduced by the expansion of the wavefunction over the specially constructed basis to a two-body problem is considered. The asymptotics of this basis, as well as the solutions of the effective two-body equations are derived. A total S-matrix for 2 (2, 3) processes is expressed in terms of adiabatic amplitudes and vice versa.     

Dimers, Effective Interactions, and Pauli Blocking Effects in a Bilayer of Cold Fermionic Polar Molecules

We consider a bilayer setup with two parallel planes of cold fermionic polar molecules when the dipole moments are oriented perpendicular to the planes. The binding energy of two-body states with one polar molecule in each layer is determined and compared to various analytic approximation schemes in both coordinate- and momentum-space. The effective interaction of two bound dimers is obtained by integrating out the internal dimer bound state wave function and its robustness under analytical approximations is studied. Furthermore, we consider the effect of the background of other fermions on the dimer state through Pauli blocking, and discuss implications for the zero-temperature many-body phase diagram of this experimentally realizable system.     

Positronium formation by electron capture from hydrogen-like ions

Two different impulse approximations are provided for the problem of fast positron collisions with hydrogen-like ions. One of the impulse approximations is formulated by making a consistent expansion of the scattering wave function in powers of the weak interaction to the strong. The other impulse approximation is formulated by making a consistent expansion of theT-matrix in powers of the weak interaction to the strong. In this impulse approximation, the opposite limits of the target nuclear charge tending to zero and to infinity are examined. Differential and angle-integrated cross sections are computed for ground-state positronium formation from hydrogen within the impulse approximations. The full peaking approximation is employed in the evaluation of theT-matrix. By 300 eV, the impulse approximations for the angle-integrated cross section are in close agreement with the strong potential Born and the exact second Born calculations.     

The nuclear Yukawa model on a lattice

We present the results of the quantum field theory approach to the nuclear Yukawa model obtained by standard lattice techniques. We have considered the simplest case of two identical fermions interacting via a scalar meson exchange. Calculations have been performed using Wilson fermions in the quenched approximation. We found the existence of a critical coupling constant above which the model cannot be numerically solved. The range of the accessible coupling constants is below the threshold value for producing two-body bound states. Two-body scattering lengths have been obtained and compared to the non-relativistic results.     

Can QCD Be a Perturbation Theory of Hadrons?

We have found that the S-matrix for atoms and hadrons depends on a gauge as the elementary particles are off mass-shell in the bound states. The S-matrix for bound states one should to construct by the projection of the Belinfante energy-momentum tensor on the Gauss equation solution for the time component with the time-axis chosen as the eigenvector of the bound state total momentum operator. We have shown that this QCD Hamiltonian determined in infrared region by the rising potential ansatz, besides the parton model in the specific gauge, contains also the nonrelativistic potential model for heavy quarkonia, the chiral Lagrangians for light quarkonia with their spectrum, the glueball physics, and the small effective coupling constant in the whole region of transversal momenta.     

Diffraction model analysis of pion-12C elastic scattering at 800 MeV/c: Optical potential by inversion

Elastic scattering of 800 MeV/c pions by¹²C has been studied in the diffraction model with a view to determine pion optical potential by the method of inversion. Finding an earlier diffraction model analysis to be deficient in some respects, we propose a Glauber model based parametrization for the elasticS-matrix and show that it provides an exceedingly good fit to the pion-carbon data. The proposed elasticS-matrix gives a closed expression for the pion-¹²C optical potential by the method of inversion in the high energy approximation.     

Multiphoton detachment from a zero-range potential revisited

Direct detachment of an electron bound by a regularized zero-range potential is reinvestigated. For the definition of the S-matrix amplitude for detachment, the wave function of the electron after the laser pulse has passed through is projected on an exact scattering state of the zero-range potential rather than a plane-wave state, which is normally used. The exact scattering state generates an additional contribution to the amplitude, which leads to an isotropic term in the angle-resolved energy spectrum.