Partial Wave Decomposition Research Articles

A new way to perform partial-wave decompositions of few-nucleon forces

We formulate a general and exact method of partial wave decomposition (PWD) of any nucleon-nucleon (NN) potential and any three-nucleon (3N) force. The approach allows one to efficiently use symbolic algebra software to generate the interaction dependent part of the program code calculating the interaction. We demonstrate the feasibility of this approach for the one-boson exchange BonnB potential, a recent nucleon-nucleon chiral force and the chiral two-pion-exchange three-nucleon force. In all cases very good agreement between the new and the traditional PWD is found. The automated PWD offered by the new approach is of the utmost importance in view of future applications of numerous chiral N3LO contributions to the 3N force in three nucleon calculations.

Poincaré Invariant Three-Body Scattering

Relativistic Faddeev equations for three-body scattering are solved at arbitrary energies in terms of momentum vectors without employing a partial wave decomposition. Relativistic invariance is incorporated withing the framework of Poincaré invariant quantum mechanics. Based on a Malfliet–Tjon interaction, observables for elastic and breakup scattering are calculated and compared to non-relativistic ones.

Two-body scattering without angular-momentum decomposition: Fully off-shell T-matrices

Two-body scattering is studied by solving the Lippmann-Schwinger equation in momentum space without angular-momentum decomposition for a local short-range interaction plus Coulomb. The screening and renormalization approach is employed to treat the Coulomb interaction. Benchmark calculations are performed by comparing our procedure with a configuration space calculation, using the standard partial-wave decomposition, for 12C - 10Be elastic scattering. The fully off-shell T -matrices are also calculated for the final goal of studying the three-body scattering by solving Faddeev/AGS equations.

Use of partial-wave decomposition to identify resonant interference effects in the photoionization–excitation of argon

We have studied simultaneous photoionization and excitation of Ar in the range of incident photon energies between 36.00 and 36.36 eV, where the resonant production of doubly excited neutral Ar states imbedded in the ionization continuum is dominant. By measuring the relative Stokes parameters of the fluorescence from residual Ar+* (3p4 [3P] 4p) ions (2P1/2, 465.8 nm transition; 2P3/2, 476.5 nm; 2D3/2, 472.7 nm; 2D5/2, 488.0 nm; 4P5/2, 480.6 nm; 4D5/2, 514.5 nm) we demonstrate a technique for determining individual partial-wave cross sections in photoionizing collisions. This procedure is shown to be important in sorting out competing dynamical ionization mechanisms, particularly with regard to resonant production of intermediate doubly excited autoionizing states. Comparison with theoretical photoionization cross sections demonstrates that spin–orbit coupling between different states of Ar II needs to be accounted for in the calculations.

Improved Method for Partial-Wave Decomposition of Two-Pion Exchange Three-Nucleon Force

In this article an efficient method to calculate the matrix elements of three-nucleon force is presented. The new method is improved version of partial-wave decomposition of Hüber et al. (Few-Body Syst 22:107, 1997), which simplifies expression to be evaluated as well as permits to reduce computational effort. Proposed method naturally applies to Faddeev-type calculations.

A New Approach to the 3D Faddeev Equation for Three-body Scattering

A novel approach to solve the Faddeev equation for three-body scattering at arbitrary energies is proposed. This approach disentangles the complicated singularity structure of the free three-nucleon propagator leading to the moving and logarithmic singularities in standard treatments. The Faddeev equation 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 an integral equation in five variables, magnitudes of relative momenta and angles. The singularities of the free propagator and the deuteron propagator are now both simple poles in two different momentum variables, and thus can both be integrated with standard techniques.

Faddeev and Glauber calculations at intermediate energies in a model forn+dscattering

Obtaining cross sections for nuclear reactions at intermediate energies based on the Glauber formulation has a long tradition. Only recently the energy regime of a few hundred MeV has become accessible to ab initio Faddeev calculations of three-body scattering. In order to go to higher energies, the Faddeev equation for three-body scattering is formulated and directly solved without employing a partial wave decomposition. In the simplest form the Faddeev equation for interacting scalar particles is a three-dimensional integral equation in five variables, from which the total cross section, the cross sections for elastic scattering and breakup reactions, as well as differential cross sections are obtained. The same observables are calculated based on the Glauber formulation. The first order Glauber calculation and the Glauber rescattering corrections are compared in detail with the corresponding terms of the Faddeev multiple scattering series for projectile energies between 100 MeV and 2 GeV.

Poincaré invariant three-body scattering at intermediate energies

The Faddeev equation for three-nucleon scattering, based on an exactly Poincar\'e invariant formulation of quantum mechanics, is solved for projectile energies up to 2 GeV. As in the nonrelativistic three-body problem, the three-body dynamics is determined, up to three-body interactions, by the two-body dynamics and cluster properties. The two-body interactions are determined, up to a unitary scattering equivalence, by two-body scattering data, which in our application are generated by a nonrelativistic Malfliet-Tjon interaction. The Faddeev equation is directly solved in a kinematic momentum representation without employing a partial-wave decomposition. The solution of the Faddeev equation is generated using Pad\'e summation, and the numerical feasibility and stability of the solution is demonstrated. Scattering observables for elastic and breakup scattering are calculated for projectile energies in the intermediate energy range up to 2 GeV, and compared with their nonrelativistic counterparts. The convergence of the multiple scattering series is investigated as a function of the projectile energy in different scattering observables and configurations. The complementary roles of kinematic and dynamical contributions to our Poincar\'e invariant model are investigated. Approximations to the two-body interaction embedded in the three-particle space are compared with the exact treatment.

Three-nucleon bound state in a spin-isospin dependent three dimensional approach

A spin-isospin-dependent three-dimensional approach based on momentum vectors for formulation of the three-nucleon bound state is presented in this article. The three-nucleon Faddeev equations with two-nucleon interactions are formulated as a function of vector Jacobi momenta, specifically the magnitudes of the momenta and the angle between them with the inclusion of the spin-isospin quantum numbers, without employing a partial wave decomposition. As an application the spin-isospin-dependent Faddeev integral equations are solved with Bonn-B potential. Our result for the Triton binding energy with the value of $\ensuremath{-}8.152$ MeV is in good agreement with the achievements of the other partial wave based methods.

Relativistic effects in exclusive pd breakup scattering at intermediate energies

The relativistic Faddeev equation for three-nucleon scattering is formulated in momentum space and directly solved in terms of momentum vectors without employing a partial wave decomposition. Relativistic invariance is achieved by constructing a dynamical unitary representation of the Poincaré group on the three-nucleon Hilbert space. The exclusive breakup reaction at 508 MeV is calculated based on a Malfliet–Tjon type two-body interaction and the cross sections are compared to measured cross sections at this energy. We find that the magnitude of the relativistic effects can be quite large and depends on the configurations considered. In spite of the simple nature of the model interaction, the experimental cross sections are in surprisingly good agreement with the predictions of the relativistic calculations. We also find that although for specific configurations the multiple scattering series converges rapidly, this is in general not the case.

Towards a fitting procedure for deeply virtual Compton scattering at next-to-leading order and beyond

Combining dispersion and operator product expansion techniques, we derive the conformal partial wave decomposition of the virtual Compton scattering amplitude in terms of complex conformal spin to twist-two accuracy. The perturbation theory predictions for the deeply virtual Compton scattering (DVCS) amplitude are presented in next-to-leading order for both conformal and modified minimal subtraction scheme. Within a conformal subtraction scheme, where we exploit predictive power of conformal symmetry, the radiative corrections are presented up to next-to-next-to-leading order accuracy. Here, because of the trace anomaly, the mixing of conformal moments of generalized parton distributions (GPD) at the three-loop level remains unknown. Within a new proposed parameterization for GPDs, we then study the convergence of perturbation theory and demonstrate that our formalism is suitable for a fitting procedure of DVCS observables.

First order relativistic three-body scattering

Relativistic Faddeev equations for three-body scattering at arbitrary energies are formulated in momentum space and in first order in the two-body transition operator directly solved in terms of momentum vectors without employing a partial wave decomposition. Relativistic invariance is incorporated within the framework of Poincar\'e invariant quantum mechanics and presented in some detail. Based on a Malfliet-Tjon-type interaction, observables for elastic and breakup scattering are calculated up to projectile energies of 1 GeV. The influence of kinematic and dynamic relativistic effects on those observables is systematically studied. Approximations to the two-body interaction embedded in the three-particle space are compared to the exact treatment.

The Ar 3 molecular system

We employ the three-dimensional Faddeev equations in the total angular momentum representation to investigate the contributions stemming from the various long-range three-body forces to the low-lying vibrational states of the Ar3 molecule. The resulting large matrices were handled using the Arnoldi's algorithm and a background potential which can eliminate the plethora of unphysical eigenvalues close to unity. We demonstrated that our method is suited for calculations in relatively heavy trimers without resorting to an explicit partial wave decomposition.

Three-body elastic and inelastic 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. For identical bosons this results in a three-dimensional integral equation in five variables, magnitudes of relative momenta and angles. The cross sections for both elastic and breakup processes in the intermediate energy range up to about 1 GeV are calculated based on a Malfliet-Tjon type potential, and the convergence of the multiple scattering series is investigated.

Four-Body Bound-State Calculations in Three-Dimensional Approach

The four-body bound state with two-body interactions is formulated in Three-Dimensional approach, a recently developed momentum space representation which greatly simplifies the numerical calculations of few-body systems without performing the partial wave decomposition. The obtained three-dimensional Faddeev-Yakubovsky integral equations are solved with two-body potentials. Results for four-body binding energies are in good agreement with achievements of the other methods.

Structure of low-lying 12C resonances

The hyperspherical adiabatic expansion is combined with complex scaling and used to calculate low-lying nuclear resonances of $^{12}$C in the $3\alpha$-model. We use Ali-Bodmer potentials and compare results for other potentials $\alpha-\alpha$ with similar $^{8}$Be-properties. A three-body potential is used to adjust the $^{12}$C-resonance positions to desired values extending the applicability of the method to many-body systems decaying into three $\alpha$-particles. For natural choices of three-body potentials we find 14 resonances below the proton separation threshold, i.e. two $0^{+}$, three $2^{+}$, two $4^{+}$, one of each of $1^{\pm}$, $2^{-}$, $3^{\pm}$, $4^{-}$, and $6^{+}$. The partial wave decomposition of each resonance is calculated as function of hyperradius. Strong variation is found from small to large distance. Connection to previous experimental and theoretical results are discussed and agreements as well as disagreements are emphasized.

Four-nucleon scattering:Ab initiocalculations in momentum space

The four-body equations of Alt, Grassberger, and Sandhas are solved for $n\text{\ensuremath{-}}$$^{3}\mathrm{H}$ scattering at energies below three-body breakup threshold using various realistic interactions including one derived from chiral perturbation theory. After partial wave decomposition the equations are three-variable integral equations that are solved numerically without any approximations beyond the usual discretization of continuum variables on a finite momentum mesh. Large number of two-, three-, and four-nucleon partial waves are considered until the convergence of the results is obtained. The total $n\text{\ensuremath{-}}$$^{3}\mathrm{H}$ cross section data in the resonance region is not described by the calculations which confirms previous findings by other groups. Nevertheless the numbers we get are slightly higher and closer to the data than previously found and depend on the choice of the two-nucleon potential. Correlations between the ${A}_{y}$ deficiency in $n\text{\ensuremath{-}}d$ elastic scattering and the total $n\text{\ensuremath{-}}$$^{3}\mathrm{H}$ cross section are studied.

Eikonal approximation in AdS/CFT: Conformal partial waves and finite N four-point functions

We introduce the impact parameter representation for conformal field theory correlators of the form A ∼ 〈 O 1 O 2 O 1 O 2 〉 . This representation is appropriate in the eikonal kinematical regime, and approximates the conformal partial wave decomposition in the limit of large spin and dimension of the exchanged primary. Using recent results on the two-point function 〈 O 1 O 1 〉 shock in the presence of a shock wave in anti-de Sitter, and its relation to the discontinuity of the four-point amplitude A across a kinematical branch cut, we find the high spin and dimension conformal partial wave decomposition of all tree-level anti-de Sitter Witten diagrams. We show that, as in flat space, the eikonal kinematical regime is dominated by the T-channel exchange of the massless particle with highest spin ( graviton dominance). We also compute the anomalous dimensions of the high spin O 1 O 2 composites. Finally, we conjecture a formula re-summing crossed-ladder Witten diagrams to all orders in the gravitational coupling.

Solution of the Spectator Equation for Relativistic NN Scattering Without Partial Wave Expansion

We developed a numerical method, based on the Pade summation, to solve the covariant spectator equation without partial wave decomposition, and applied it to the NN system. We present a general analytical formula for the polar angle integration, and a new prescription to handle the singularities present in the kernel of the spectator equation, due to the anti-symmetrization condition. The convergence of the partial wave-decomposition series is tested as a function of the energy. The on- and off-mass-shell amplitudes are calculated. The NN model was fitted to the np differential cross section, up to 320 MeV laboratory energy.

Exact partial wave expansion of optical beams with respect to an arbitrary origin

Using an analytical expression for an integral involving Bessel and Legendre functions, we succeed in obtaining the partial wave decomposition of a general optical beam at an arbitrary location relative to the origin. We also showed that solid angle integration will eliminate the radial dependence of the expansion coefficients. The beam shape coefficients obtained are given by an exact expression in terms of single or double integrals. These integrals can be evaluated numerically on a short time scale. We present the results for the case of a linear-polarized Gaussian beam.