Coordinate-space Calculations Research Articles

Finite-size instabilities in finite-range forces

It has been recently shown that some Gogny finite-range interactions suffer from finite-size instabilities in coordinate-space calculations [Eur. Phys. J. A 55, 150 (2019)]. We confirm this finding by using the Hartree-Fock (HF) method in the quasilocal approximation to finite-range forces. The use of the quasilocal approximation substantially simplifies the calculations as compared with those including the exact exchange contribution to the energy and HF fields. The quantity most affected by the finite-size instabilities in the coordinate-space calculations is the spatial density at the origin that wildly oscillates as the HF iterative process proceeds. In addition to the recent ${\mathrm{D1M}}^{*}$ parametrization of the Gogny force, we find that the D1M parametrization also shows this deficiency in several nuclei. We find that the harmonic-oscillator basis with its ultraviolet cutoff provides converged results in a wide and realistic range of basis sizes. This result serves as a justification of the numerous calculations with D1M and ${\mathrm{D1M}}^{*}$ in finite nuclei that show no trace of instability.

Coordinate space calculation of two- and three-loop sunrise-type diagrams, elliptic functions and truncated Bessel integral identities

We integrate three-loop sunrise-type vacuum diagrams in D0=4 dimensions with four different masses using configuration space techniques. The finite parts of our results are in numerical agreement with corresponding three-loop calculations in momentum space. Using some of the closed form results of the momentum space calculation we arrive at new integral identities involving truncated integrals of products of Bessel functions. For the non-degenerate finite two-loop sunrise-type vacuum diagram in D0=2 dimensions we make use of the known closed form p-space result to express the moment of a product of three Bessel functions in terms of a sum of Clausen polylogarithms. Using results for the nondegenerate two-loop sunrise diagram from the literature in D0=2 dimensions we obtain a Bessel function integral identity in terms of elliptic functions.

Precision of finite-difference representation in 3D coordinate-space Hartree-Fock-Bogoliubov calculations

The current work plans to study the accuracy due to FD approximation to the 3D nuclear HFB problem. By (1) taking the wave functions solved in harmonic oscillator (HO) basis, (2) representing the HFB problem in coordinate space using FD method, the current work carefully evaluates the error due to box discretization by examining the deviation of the resulted HFB matrix, the total energies in the coordinate space, from those calculated with HO method, the latter of which is free from numerical error within its model configuration. To estimate how the error (given by the box discretization schemes suggested above) accumulates with self-consistent iterations, self-consistent HF and HFB calculations (with two-basis method) has been carried out for doubly magic nuclei, $^{40}$Ca, $^{132}$Sn, and $^{110}$Mo. The resulted total energies are compared with those of HO basis, and 3D coordinate space calculations in literatures. The analysis shows that, for grid spacing $\le$0.6\,fm, the off-diagonal elements of the resulted HFB matrix elements (M.E.) are extremely small ($<$1\,keV). The resulted quasi-particle (q.p.) spectra differ from those of HO calculations by a few keV. Self-consistent HF and HFB calculations within the current FD method with the above box discretizatioin schemes give results similar to those calculations of existing HO basis, and coordinate space method.

Weakly bound nuclei: Exotic ground state properties and collective excitations

Weakly bound nuclei are open quantum many-body systems and can have exotic structures and dynamics in contrast to our knowledges of stable nuclei. Currently the related studies are among the central topics in nuclear physics and are important for further improvements of nuclear theory and understandings of element enrichment processes in astrophysical environments. Since the improvements of intensity and quality of radioactive nuclear beams are very difficult, theoretical studies will be very helpful for planning experiments and the interpretation of results. Here we review recent developments of continuum density functional theory and studies of weakly bound nuclei. The theoretical challenge is to self-consistently describe the weak-binding effects, exotic shapes, continuum effects and dynamics together. To this end, our main work is to solve the Hartree-Fock-Bogoliubov equation in non-spherical coordinate- spaces and then to describe the ground state properties as well as collective excitations of weakly bound nuclei. In the 2D lattice, we solved the HFB equation with the B-spline techniques. More recently, we solved the HFB equation in the 3D coordinate-space with the adaptive multi-resolution multi-wavelet techniques, which is novel and much more efficient. Based on our calculations, we discussed the exotic egg-like deformed halo structures, the core-halo deformation decoupling, and the role of continuum contributions. We also see pairing density distributions have remarkable deformed halo structures. The deformed cores are not so favorable for the formation of 2 n -halos, in particular heavy 2 n -halos. The non-resonant continuum rather than the resonant continuum is mainly responsible for halo features. With the quasiparticle blocking calculations, we also studied the deformed 1 n -halo nuclei, in which the pairing density is unusual due to the spin polarization. In addition, we developed the deformed continuum FAM-QRPA calculations in coordinate spaces and studied the emergent soft monopole collective excitations. The soft collective modes are unique in weakly bound nuclei and are useful for studies of nuclear properties in extreme conditions. Compared to the conventional QRPA diagonalization method, the FAM-QRPA solves the QRPA equation iteratively and is more efficient. Otherwise the deformed QRPA calculations with continuum would be very challenging. We see the pairing density halos and non-resonant continuum play an important role in these soft collective excitations. Our studies involved large coordinate-space calculations in Tianhe supercomputers to take into account shape deformations, large spatial extensions and continuum spectra. Otherwise, calculations within small coordinate spaces would produce a coarse continuum discretization and even false resonance peaks. Finally we give some persectives on studies of weakly bound nuclei and our plans. Since weakly bound nuclei exhibit few-body physics in many-body systems, the conventional shell-structure picture breaks down in this respect. Theorefore new many-body techniques, time-dependent 3D dynamics and improved effective interactions are very desirable. The studies of weakly bound nuclei as open quantum systems that involve threshold effects and continuum contributions can have broad interdisciplinary interests.

Momentum-Space Calculation of 4He Triatomic System with Realistic Potential

The system of three 4He atoms with realistic interactions is studied in the momentum-space framework. It is demonstrated that short and long-range difficulties encountered in the coordinate-space calculations can be reliably resolved in the momentum-space calculations. Well-converged and accurate results are obtained for the ground and excited trimer energies, atom-dimer scattering length, phase shifts, inelasticity parameters, and elastic and breakup cross sections. A significant correction to previously published results is found for the elastic cross section at very low energy.

Separable representation of proton-nucleus optical potentials

Recently, a new approach for solving the three-body problem for (d,p) reactions in the Coulomb-distorted basis in momentum space was proposed. Important input quantities for such calculations are the scattering matrix elements for proton- and neutron-nucleus scattering. We present a generalization of the Ernst-Shakin-Thaler scheme in which a momentum space separable representation of proton-nucleus scattering matrix elements can be calculated in the Coulomb basis. The viability of this method is demonstrated by comparing S-matrix elements obtained for p$+^{48}$Ca and p$+^{208}$Pb for a phenomenological optical potential with corresponding coordinate space calculations.

Coordinate-space approach to vacuum polarization

The vacuum-polarization correction for bound electrons or muons is examined. The objective is to formulate a framework for calculating the correction from bound-state quantum electrodynamics entirely in coordinate space, including the Uehling potential which is usually isolated and treated separately. Pauli-Villars regularization is applied to the coordinate-space calculation and the most singular terms are shown to be eliminated, leaving the physical correction after charge renormalization. The conventional derivation of the Uehling potential in momentum space is reviewed and compared to the coordinate-space derivation.

Computation of the soft anomalous dimension matrix in coordinate space

We complete the coordinate space calculation of the three-parton correlation in the two-loop massive soft anomalous dimension matrix. The full answer agrees with the result found previously by a different approach. The coordinate space treatment of renormalized two-loop gluon exchange diagrams exhibits their color symmetries in a transparent fashion. We compare coordinate space calculations of the soft anomalous dimension matrix with massive and massless eikonal lines and examine its nonuniform limit at absolute threshold.

Benchmark calculation for proton-deuteron elastic scattering observables including the Coulomb interaction

Two independent calculations of proton-deuteron elastic scattering observables including Coulomb repulsion between the two protons are compared in the proton laboratory energy region between 3 and 65 MeV. The hadron dynamics is based on the purely nucleonic charge-dependent AV18 potential. Calculations are done in both coordinate space and momentum space. The coordinate-space calculations are based on a variational solution of the three-body Schr\"odinger equation using a correlated hyperspherical expansion for the wave function. The momentum-space calculations proceed via the solution of the Alt-Grassberger-Sandhas equation using the screened Coulomb potential and the renormalization approach. Both methods agree to the order of 1% on all observables, showing the reliability of both numerical techniques in that energy domain. At energies below the three-body breakup threshold, the coordinate-space method remains favored; at energies higher than 65 MeV, the momentum-space approach seems to be more efficient.

One-loop anomalous couplings and matching conditions in the nonlinear realization of the [formula omitted] Higgs model

We study the renormalization of the nonlinear realization of the SU ( 2 ) Higgs model in the modified minimal subtraction renormalization scheme. We propose that the effective field method with truncated operator series is trustworthy even when the Higgs boson is relatively light. Using the technique of background field method in the coordinate space, we derive the matching conditions at tree and one-loop order, both in the regions where the Higgs boson is heavy and where it is light. We obtain the complete one-loop anomalous couplings up to O ( p 4 ) in the SU ( 2 ) Higgs model. We observe that the contribution of the gauge bosons and the Goldstone bosons to the anomalous couplings at the one-loop level is significant. By establishing the correspondence between our coordinate space calculation and the momentum space calculation that exists in the literature, we find agreement in all the matching conditions in the heavy Higgs boson limit.

Practical method for treating the Coulomb force in momentum space

We propose a practical numerical method for treating the Coulomb interaction in momentum space. The Coulomb potential is regularized by expanding its inner part $r&lt;~R$ as the superposition of the Gaussian functions. This smooth-cutoff Coulomb potential decreases rapidly in momentum space without oscillation. In addition, the partial-wave decomposition can be done analytically without numerical integration. First, the phase shifts are calculated with the regularized Coulomb plus short-range strong potentials. Then, the correct phase shifts or the wave functions can be reconstructed with the aid of coordinate-space calculation from the asymptotic region inward to R. Another possibility is to calculate the logarithmic derivative at R directly by the Fourier transform, which is matched to the point-Coulomb wave functions ${F}_{L}$ and ${G}_{L}.$ This method is examined for calculating the phase shifts of proton-nucleus elastic scattering and found to be accurate over wide energy regions.

Exact numerical solution of the spinless Salpeter equation for the Coulomb potential in momentum space.

The spinless Salpeter equation with the Coulomb potential is solved exactly in momentum space and is shown to agree very well with a coordinate-space calculation. The agreement is so good that it clearly establishes that the procedure discussed can be used to reliably calculate fine-structure effects in any bound-state system without relying on approximations. The procedure works no matter how large the coupling constant. An equation, called the spinless Salpeter equation with retardation, is introduced and this is also able to be solved exactly.

Resonating group method in momentum space.

A method for solving the resonating group method equation in momentum space is presented. To examine the numerical accuracy of this method, the phase shifts of \ensuremath{\alpha}-\ensuremath{\alpha} and $^{16}\mathrm{\ensuremath{-}}^{16}$O scattering are calculated and are compared with those by the conventional coordinate-space calculation. It is shown that the momentum-space method provides us with an efficient and useful mean to treat the nucleus-nucleus scattering problem, especially for heavy systems.

Elastic scattering of π− from 12C at 29 MeV

We have measured the differential cross section for π − elastically scattered from 12 C at T ̄ π=29 MeV . The scattered π − were detected with pion range telescopes. The measured angular distribution is compared with a momentum space optical potential calculation. Derived phase shifts are compared with those obtained from π + data and from momentum and coordinate space calculations.

Nonstatic Nuclear Forces of the Triplet State in Momentum Space

The nucleon-nucleon scattering in the uncoupled triplet states is described by the one­ boson-exchange model in conjunction with the Schrodinger equation in momentum representa­ tion. The nonstatic matrix elements due to the exchange of rc, (J), p and an isoscalar scalar meson are obtained. The nonstatic nuclear forces in momentum space are discussed in detail in connection with the LS force and the velocity-dependent term in coordinate space. Also the behavior of 3P0 phase shift at 25"'-'50 MeV is especially discussed. The meson coupling constants are adjusted to fit the experimentally determined phase shifts. The phase shifts obtained from our momentum-space calculations are in reasonable quantitative agreement with those obtained from the phase-shift analysis. § I. Introduction Analysis of the experimental data on nucleon~nucleon scattering at high energies has made it clear that there exists a strong nonstatic force. Investigation of the nonstatic effects such as the LS, the quadratic LS and the velocity­ dependent forces, is indispensable in order to determine nuclear forces. The semi­ phenomenological potentials with the above~mentioned nonstatic effects have already been obtained by several authors. 1 ) However, it is an open question whether the nonstatic effects are determined uniquely at small inter~nucleon distances. In an ordinary procedure to obtain the nonstatic nuclear forces, we first derive the nuclear potential in momentum representation, then transform it into coordinate representation. In this procedure, however, some of the nonstatic effects are left out of account. In order to give a complete description of the nonstatic interaction, the recoil effects and the retardation effects must be taken into account. Thus, we start with relativistic covariant interaction and treat the problem in momentum space throughout so that we are able to take the nonstatic effects and nonlocality into account as accurately as possible. Momentum~space calculations in conjunction with the Schrodinger equation have not been performed throughly compared to coordinate-space calculations. However, momentum-space calculations are advantageous in dealing with the nonstatic effects and nonlocality, and make it possible to obtain essential understanding of the nonstatic nuclear forces 2 a), 2 b) (hereafter we refer to Ref. 2a) as I). In I, we analyzed nuclear forces in the singlet states in order to discuss the problems of the repulsive core