Partial Wave Decomposition Research Articles

Complex conformal spin partial wave expansion of generalized parton distributions and distribution amplitudes

We introduce a new representation of generalized parton distributions and generalized distribution amplitudes that is based on the partial wave decomposition with respect to the complex collinear conformal spin. This decomposition leads us to a versatile parameterization of these non-perturbative functions in terms of conformal moments, which are measurable for integer value on the lattice. This new representation has several advantages: basic properties and crossing relations are automatically implemented, a rather flexible parameterization is possible, the numerical treatment of evolution is simple and analytic approximation of scattering amplitudes can be given. We demonstrate this for simple examples. In particular, phenomenological considerations indicate that the t-dependence of Mellin moments is governed by Regge trajectories. The new representation is vital to push the analysis of deeply virtual Compton scattering to next-to-next-to-leading order.

Three-body scattering at intermediate energies

The Faddeev equation for three-body scattering at arbitrary energies is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial-wave decomposition. In its simplest form, the Faddeev equation for identical bosons, which we are using, is a three-dimensional integral equation in five variables, magnitudes of relative momenta and angles. This equation is solved through Pad\'e summation. Based on a Malfliet-Tjon-type potential, the numerical feasibility and stability of the algorithm for solving the Faddeev equation is demonstrated. Special attention is given to the selection of independent variables and the treatment of three-body breakup singularities with a spline-based method. The elastic differential cross section, semiexclusive $d(N,{N}^{'}$) cross sections, and total cross sections of both elastic and breakup processes in the intermediate-energy range up to about 1 GeV are calculated and the convergence of the multiple-scattering series is investigated in every case. In general, a truncation in the first or second order in the two-body t matrix is quite insufficient.

Inner-shell photoelectron angular distributions from fixed-in-space OCS molecules: comparison between experiment and theory

Photoelectron angular distributions (PADs) for O 1s, C 1s and S 2p1/2, 2p3/2 ionization of OCS molecules have been measured in shape resonance regions. These PAD results are compared with the results for O 1s and C 1s ionization of CO molecules, and multi-scattering Xα (MSXα) calculations. The mechanism of the PAD formation both for parallel and perpendicular transitions differs very significantly in these molecules and a step from a two-centre potential (CO) to a three-centre potential (OCS) plays a principal role in electron scattering and the formation of the resulting PAD. For parallel transitions, it is found that for the S 2p and O 1s ionization the photoelectrons are emitted preferentially in a hemisphere directed to the ionized S and O atom, respectively. In OCS O 1s ionization, the S–C fragment plays the role of a strong ‘scatterer’ for photoelectrons, and in the shape resonance region most intensities of the PADs are concentrated on the region directed to the O atom. The MSXα calculations for perpendicular transitions reproduce the experimental data, but not so well as in the case of parallel transitions. The results of PAD, calculated with different ℓmax on different atomic centres, reveal the important role of the d (ℓ = 2) partial wave for the S atom in the partial wave decompositions of photoelectron wavefunctions.

Bound-State Calculations for Three Atoms Without Explicit Partial Wave Decomposition

A method to calculate the bound states of three atoms without resorting to an explicit partial wave decomposition is presented. The differential form of the Faddeev equations in the total angular momentum representation is used for this purpose. The method utilizes Cartesian coordinates combined with the tensor trick preconditioning for large linear systems and Arnoldi’s algorithm for eigenanalysis. As an example, we consider the He3 system in which the interatomic force has a very strong repulsive core requiring the inclusion of a large number of partial waves to achieve convergence. To improve convergence and stability in the Arnoldi’s algorithm, we modify the system of equations by introducing a background potential that removes the large number of unphysical eigenvalues stemming from the hard core. The introduction of such a modifying potential does not affect the physical spectrum of the system. The results obtained by solving the modified, three-dimensional, Faddeev equations compare favorably with other results in the field.

NUCLEON SPIN AND ITS GAUGE INVARIANCE AND CANONICAL COMMUTATION RELATION

The nucleon spin structure has been shown to be described by a dynamical constituent quark model with ~85% q3 and [Formula: see text] Fock components. As a QCD system there is no gauge invariance and canonical commutation relation simultaneously satisfied decomposition of the nucleon spin operator. To keep gauge invariance but give up the canonical commutation relation of the angular momentum operator will ruin the multiple moments expansion and the partial wave decomposition. A weak gauge invariant condition is proposed where the canonical commutation relation of angular momentum operator is intact and the gauge invariance is only true for special sets of matrix elements instead of the operators.

Partial-wave decomposition of pion and photoproduction amplitudes

Partial wave amplitudes for production and decay of baryon resonances are constructed in the framework of the operator expansion method. The approach is fully relativistically invariant and allow us to perform combined analyses of different reactions imposing directly analyticity and unitarity constraints. All formulas are given explicitly in the form used by the Crystal Barrel collaboration in the (partly forthcoming) analyses of the electro-, photo- and pion induced meson production data.

General structure of a two-body operator for spin-12particles

A direct derivation of the operator structure for two spin-$\frac{1}{2}$ particles is presented subject to invariance under basic symmetries and Galilean frame transformation. The partial wave decomposition for coefficient functions, valid on- and off-shell, is explicitly deduced. The momentum transfer representation and angular momentum decomposition for general spin-dependent potentials are obtained.

Direct calculation of the scattering amplitude without partial-wave analysis. II. Inclusion of exchange

The development of a practical method of accurately calculating the full scattering amplitude, without making a partial-wave decomposition is continued. The method is developed in the context of electron-hydrogen scattering, and here exchange is dealt with by considering $e\penalty1000-\hskip0pt\mathrm{H}$ scattering in the static exchange approximation. The Schr\"odinger equation in this approximation can be simplified to a set of coupled integro-differential equations. The equations are solved numerically for the full scattering wave function. The scattering amplitude can most accurately be calculated from an integral expression for the amplitude; that integral can be formally simplified, and then evaluated using the numerically determined wave function. The results are essentially identical to converged partial-wave results.

Operator form ofH3(He3)and its spin structure

An operator form of the three-nucleon $(3N)$ bound state is proposed. It consists of eight operators formed out of scalar products in relative momentum and spin vectors, which are applied on a pure $3N$ spin $1∕2$ state. Each of the operators is associated with a scalar function depending only on the magnitudes of the two relative momenta and the angle between them. The connection between the standard partial wave decomposition of the $3N$ bound state and the operator form is established, and the decomposition of these scalar function in terms of partial wave components and analytically known auxiliary functions is given. That newly established operator form of the $3N$ bound state exhibits the dominant angular and spin dependence analytically. The scalar functions are tabulated and can be downloaded. As an application the spin dependent nucleon momentum distribution in a polarized $3N$ bound state is calculated to illustrate the use of the new form of the $3N$ bound state.

An iterative procedure for the diffraction of water waves by multiple cylinders

Diffraction of water waves by an array of vertical cylinders of circular cross section is studied. In order to account for first order interaction among the cylinders, the body boundary condition is satisfied for each cylinder considering the scattered wave field from other cylinders in an iterative way. After each iteration, coefficients in the partial wave decomposition of the wave potential are modified. Convergence is fast for the whole range of frequencies and for a large number of bodies, compared with exact algebraic methods of Linton and Evans [J. Fluid Mech. 46 (1990) 549] and Kagemoto and Yue [J. Fluid Mech. 166 (1) (1986) 189].

Phenomenology of thepp→ppηreaction close to threshold

The recent high statistics measurement of the pp -> pp eta reaction at an excess energy Q=15.5 MeV has been analysed by means of partial wave decomposition of the cross section. Guided by the dominance of the final state 1S0 pp interaction (FSI), we keep only terms involving the FSI enhancement factor. The measured p-p and p-eta effective mass spectra can be well reproduced by lifting the standard on-shell approximation in the enhancement factor and by allowing for a linear energy dependence in the leading 3P0->1S0,s partial wave amplitude. Higher partial waves seem to play only a marginal role.

Solving potential scattering equations without partial wave decomposition

Considering two-body integral equations we show how they can be dimensionally reduced by integrating exactly over the azimuthal angle of the intermediate momentum. Numerical solution of the resulting equation is feasible without employing a partial-wave expansion. We illustrate this procedure for the Bethe-Salpeter equation for pion-nucleon scattering and give explicit details for the one-nucleon-exchange term in the potential. Finally, we show how this method can be applied to pion photoproduction from the nucleon with $\pi N$ rescattering being treated so as to maintain unitarity to first order in the electromagnetic coupling. The procedure for removing the azimuthal angle dependence becomes increasingly complex as the spin of the particles involved increases.

Model Study of Three-Body Forces in the Three-Body Bound State

The Faddeev equations for the three-body bound state with two- and three-body forces are solved directly as three-dimensional integral equation. The numerical feasibility and stability of the algorithm, which does not employ partial wave decomposition is demonstrated. The three-body binding energy and the full wave function are calculated with Malfliet-Tjon-type two-body potentials and scalar Fujita-Miyazawa type three-body forces. The influence of the strength and range of the three-body force on the wave function, single particle momentum distributions and the two-body correlation functions are studied in detail. The extreme case of pure three-body forces is investigated as well.

Study of theηπ0spectrum and search for aJPC=1−+exotic meson

A partial wave analysis ~PWA! of the hp 0 system ~where h!gg) produced in the charge exchange reaction p 2 p!hp 0 n at an incident momentum of 18 GeV/c is presented as a function of the hp 0 invariant mass mhp 0 and momentum transfer squared t p2!hp , from the incident p 2 to the outgoing hp 0 system. S, P, and D waves were included in the PWA. The a0(980) and a2(1320) states are clearly observed in the overall hp 0 effective mass distribution as well as in the amplitudes associated with S wave and D waves, respectively, after partial wave decomposition. The observed distributions in moments ~averages of spherical harmonics! were compared to the results from the PWA and the two are consistent. The distribution in t p2!hp for individual D waves associated with natural and unnatural parity exchange in the t channel are consistent with Regge phenomenology. Of particular interest in this study is the P wave since this leads to an exotic J PC 51 21 for the hp system. A P wave is present in the data; however, attempts to describe the mass dependence of the amplitude and phase motion with respect to the D wave as a Breit-Wigner resonance are problematic. This has implications regarding the existence of a reported exotic J PC 51 21 meson decaying into hp 0 with a mass near 1.4 GeV/c 2 .

THE PROTON-DEUTERON BREAK-UP PROCESS IN A THREE-DIMENSIONAL APPROACH

The pd break-up amplitude in the Faddeev scheme is calculated by employing a three-dimensional method without partial wave decomposition (PWD). In the first step and in view of higher energies only the leading term is evaluated and this for the process d(p,n)pp. A comparison with the results based on PWD reveals discrepancies in the cross section around 200 MeV. This indicates the onset of a limitation of the partial wave scheme. Also around 200 MeV relativistic effects are clearly visible and the use of relativistic kinematics shifts the cross section peak to where the experimental peak is located. The theoretical peak height, however, is wrong and calls first of all for the inclusion of rescattering terms, which are shown to be important in a nonrelativistic full Faddeev calculation in PWD.

Numerical implementation of three-body forces in bound state Faddeev calculations in three dimensions

The Faddeev equations for the three-body bound state are solved directly as thre e-dimensional integral equations without employing partial wave decomposition. Two-body forces of the Malfliet-Tjon type and simple spin independent genuine three-body forces are considered for the calculation of the three-body binding energy.

Hydrogen-induced modifications to the electronic structure of intermetallic compounds

First principles electronic structure calculations are reported for hydrogen absorbing intermetallic compounds belonging to an important class of Laves phases: the superconducting AB 2 compounds (with A=Zr, Hf; B=V). The C-15 Laves phases Zr x Hf 1− x V 2 have a maximum superconducting critical temperature of T c max=10 K for x=0.5. The large value of the electron–phonon coupling constant of these intermetallic compounds is mainly associated with the existence of high peaks in the density of states (DOS) at the Fermi energy, E F. In contrast, the hydrogen-rich phases are not superconducting. We analyze the modifications in the electronic states at E F due to the formation of hydrides and discuss the observed changes in the superconducting properties, the magnetic susceptibility and the electronic specific heat coefficient. The metal–hydrogen interactions are analyzed in terms of partial wave decomposition of the DOS. These results are used to interpret the available photoemission and X-ray emission spectra.

Integral equations of scattering in one dimension

A self-contained discussion of integral equations of scattering is presented in the case of centrally symmetric potentials in one dimension, which will facilitate the understanding of more complex scattering integral equations in two and three dimensions. The present discussion illustrates in a simple fashion the concept of partial-wave decomposition, Green’s function, Lippmann–Schwinger integral equations of scattering for wave function and transition operator, optical theorem, and unitarity relation. We illustrate the present approach with a Dirac delta potential.

Relativistic NN scattering equations without partial-wave decomposition

Relativistic quasi-potential equations describing NN scattering are compared. Within the spectator formalism a cancellation is seen to occur between retardation and negative-energy effects.

The onset of the flexural mode in fluid loaded elastic shells revisited, and its value in object identification in both the frequency and time domain

The onset of the excitation of flexural resonances for fluid loaded evacuated elastic shells produces a striking event. This issue has been interpreted theoretically in 1988 by means of a partial wave decomposition which showed a very narrow-peaked subsonic wave and a broad-peaked weak wave that begins at the speed of sound of the entraining fluid and increases. The narrow peaks are identified with subsonic water-borne waves that resonate in the fluid along the surface of an elastic object, and the broad peaks correspond to the inception of flexural waves. They exist over a small interval in wave number at the point when the flexural wave begins to couple with the fluid. Here we examine the pulse solution. With increasing frequency the partial waves change phase (a 90 degree phase change) and the overlapping flexural waves transition from a partially coherent constructive signal to one that is partially destructive. This leads to an envelope or hump effect, also called mid-frequency enhancement; it is a function of shell thickness as well as material property. We demonstrate how this effect may be employed to identify a submerged elastic shell in either the pulse or frequency solution.