Harmonic Oscillator Basis Research Articles

Investigation on the stabilities of doubly heavy tetraquark states

In our recent work [S. Noh and W. Park, Nonrelativistic quark model analysis of Tcc, ], the mass and binding energy of Tcc are found to be 3873 MeV and −2 MeV, respectively, which align with the observations reported at LHCb [R. Aaij (LHCb Collaboration), Observation of an exotic narrow doubly charmed tetraquark, ]. Based on our latest quark model approach, we extend our search for other potentially stable configurations of doubly heavy tetraquarks using our nonrelativistic quark model described in Noh and Park, [Nonrelativistic quark model analysis of Tcc, ]. Our numerical calculations indicate that the u¯s¯cb configuration is deeply bound. However, the u¯d¯cb configuration with the isospin symmetry in the light quark sector is relatively less bound. In this study, we emphasize a compulsory aspect for requiring a complete set of three-dimensional harmonic oscillator bases through the discussion of the u¯s¯cb configuration and investigate the essential differences between the u¯s¯cb and u¯d¯cb configurations. Published by the American Physical Society 2024

Moshinsky brackets for a wide range of quantum numbers using generating functions

We used a new Python code to reproduce the brackets for the Moshinsky harmonic oscillator, which was based on the generating function. We made these brackets by transforming the wave functions of two groups of coupled particle harmonic oscillators, Φn1l1,n2l2,Λm1,m2,λ(r→1,r→2) and Φnala,nblb,Λma,mb,λ(r→a,r→b). To convert between the supplied position and momentum coordinates in both frames, we performed orthogonal transformations on nuclei with both low and high angular momentum.In our derivation, we have used the expansion of the generating functions e2p→.r→−p2 and e2cpi.pj in spherical coordinates in terms of harmonic oscillator wave functions. When we modified the Moshinsky brackets for two-coupled oscillator states, we used generating functions with two variables. The number of indices has significantly decreased compared to the oscillator brackets in previous references; this reduction in the program code's iterative process has yielded influential results. Compared to the previous version of the Moshinsky brackets code, the new Python code is easier to use. Our approach utilizes this code to assess Moshinsky brackets across a broad spectrum of quantum numbers. According to the revelation, adding more variables to the generating function makes the number of Moshinsky brackets that work for the higher body interactions increase. Program summaryProgram Title: GFBRACKETSCPC Library link to program files:https://doi.org/10.17632/jvbnwp35rm.1Licensing provisions: MITProgramming language: PythonSupplementary material: AppendixNature of problem: The generating functions were used to obtain a concise representation of the oscillator states for single particles. This was done to formulate and compute the generalized version of the Moshinsky brackets and matrix elements of two-body operators. By enlarging sets of basis oscillator states, it becomes possible to cover interactions involving more than two bodies. This can be easily achieved using the expanded generating function to compute the Moshinsky brackets for high-quantum numbers.Solution method: The Python code requires fewer iterations than similar codes generated by the method of the generating functions.Additional comments including restrictions and unusual features: When calculating Moshinsky brackets for large quantum numbers, restrictions arise. There is no data for quantum numbers greater than 10 in the formation of reaction potential. Additionally, a transformation matrix can be used to switch from single-particle to centre-of-mass coordinates on a two-particle harmonic oscillator basis. This transformation is applicable even when the masses of the two particles are unequal.

Approximate Effective Interaction for Nuclear Matter and Finite Nuclei

In this paper, an approximate effective nucleon-nucleon interaction for nuclear matter and finite studies has been derived using the lowest order constrained variational (LOCV) approach. The LOCV method, a functional minimization procedure, uses a normalization constraint to keep higher-order terms as small as possible. As a first step, two-body matrix elements based on the Reid93 nucleon-nucleon potential were calculated for the nuclear system A = 16 in a harmonic oscillator basis, with the oscillator size parameter <i>ћω </i>= 14.0 MeV, and separated into the central, spin-orbit and tensor channels in conformity with the potentials for Inelastic scattering. Following this, a least squares fitting of the matrix elements to a sum of Yukawa functions was performed to determine the strengths of the effective interaction in the singlet-even, singlet-odd, triplet-even and triplet-odd (Central); tensor-even and tensor-odd (Tensor); spin-orbit-even and spin-orbit-odd (Spin-orbit) channels. Of all the matrix elements, only the triplet-even and tensor-even components, being attractive, are affected by the tensor correlations (<i>a </i>= 0.05); and are shown to exhibit the same trend of variation in conformity with past work, in terms of magnitude, as one goes from the lower-node quantum numbers (n’, n) = (0, 0) to higher ones (n’, n) = (2, 2). When compared with the G-matrix results of previous researchers, the results obtained herein have been found to be in good agreement. This, therefore, gives hope that the new effective interaction promises to be a reliable tool for nuclear matter and nuclear structure studies.

Davidson basis in the shell model for 2p, 2h, and pn nuclei

In this study, we replace the widely used harmonic oscillator basis with the Davidson basis for the shell model computation. The new foundation was introduced and its application to the shell model computation was explained. A detailed discussion is given on the method of calculation and utilization. Then, three cases two-particle, 18O, and two-hole, 38Ar, and proton-neutron nuclei, 6Li, and 18F, that illustrate various scenarios are examined. In these two scenarios, both Davidson bases and harmonic oscillator are used to replicate energy levels and a few electric transition probabilities. We assess each case’s accuracy by comparing the theoretical predictions with the experimental data.

Direct ab initio calculation of the 4He nuclear electric dipole polarizability

The calculation of nuclear electromagnetic sum rules by directly solving for numerous eigenstates in a large basis is numerically challenging and has not been performed for A>2 nuclei. With the significant progress of high performance computing, we show that calculating sum rules using numerous discretized continuum states obtained with the ab initio no-core shell model in the harmonic oscillator basis is achievable numerically. Specifically, we calculate the 4He electric dipole (E1) polarizability, that is an inverse energy weighted sum rule, employing the Daejeon16 NN interaction. We demonstrate that the calculations are numerically tractable as the dimension of the basis increases and are convergent. Our results for the 4He electric dipole polarizability are consistent with the most recent experimental data and are compared with those of other theoretical studies employing different techniques and various interactions.

Covariant density functional theory for nuclear fission based on a two-center harmonic oscillator basis

Covariant density functional theory for nuclear fission based on a two-center harmonic oscillator basis

Converging many-body perturbation theory for ab-initio nuclear structure: Brillouin-Wigner perturbation series for closed-shell nuclei

Convergence aspects of nuclear many-body perturbation theory for ground states of closed-shell nuclei are explored using a Brillouin-Wigner formulation with a new vertex function enabling high-order calculations. A general formalism for Hamiltonian partitioning and a convergence criterion for the perturbation series are proposed. Analytical derivations show that with optimal partitioning, the convergence criterion for ground states can always be satisfied. This feature attributes to the variational principle and does not depend on the choice of an internucleon interaction or a many-body basis. Numerical calculations of the ground state energies of He4 and O16 with Daejeon16 and a bare N3LO potential in both harmonic-oscillator and Hartree-Fock bases confirm this finding.

An approach to calculate Franck-Condon factors involving anharmonic potentials using harmonic oscillator bases

In this contribution a discrete variable representation (DVR) approach to calculate Franck-Condon factors (FCFs) is presented. The method is illustrated using a harmonic oscillator basis. The advantage of this approach is that it is possible to calculate FCFs associated with general potentials based on the knowledge of the harmonic oscillator theory. First the coordinate and momentum representations are introduced. Based on this concept, the DVR is proposed, which allows the Franck-Condon factors to be calculated in a simple form. This method takes advantage of the momentum as generator of translations to include the displacement of the potentials. The calculation of the FCFs is exemplified for transitions involving the ground state X1Σg+ and the excited state a 1Π g of the diatomic molecule 14N2 both modelled with Morse potentials.

Distribution amplitude and decay constant of 1S and 2S state light mesons in the light-front quark model

Studying meson structures is essential for gaining insights into the nonperturbative nature of Quantum Chromodynamics (QCD). This study will focus on calculating the decay constant and distribution amplitudes (DAs) of unflavored light mesons (π and ρ) in the 1S and 2S states using the Light Front Quark Model. This study utilizes the QCD-motivated Hamiltonian, taking into account both contact and smeared spin-spin interactions. The two lowest harmonic oscillator bases are employed in this work to achieve improved results for the 2S states. The study found the optimal mixing parameter in basis expansion coefficients to be θ = 10°. Light meson properties, including the mass spectrum, decay constant, and twist-2 DAs, are then predicted using model parameters fixed through the variational principle. While the contact spin-spin interaction yields poor accuracy, the results from the smeared interaction generally agree well with experimental data and other theoretical models with $ f_{\pi}^{1S}=130 $ MeV and $ f_{\rho}^{1S}=210 $ MeV in the mixed state. Unlike the case for the 1S state, it should be noted that the properties of the 2S state are sensitive to the mixing parameter θ. In addition, we observe that the decay constant for ρ(2S) is $ f_{\rho}^{2S}=116 $ MeV. While for π(2S) the decay constant is extremely small with the value of $ f_{\pi}^{2S}=0.9 $ MeV, which is mainly due to the dynamical chiral symmetry breaking.

Exploring the parameter space of an endohedral atom in a cylindrical cavity.

Endohedral fullerenes, or endofullerenes, are chemical systems of fullerene cages encapsulating single atoms or small molecules. These species provide an interesting challenge of Potential Energy Surface determination as examples of non-covalently bonded, bound systems. While the majority of studies focus on C60 as the encapsulating cage, introducing some anisotropy by using a different fullerene, e.g., C70 can unveil a double well potential along the unique axis. By approximating the potential as a pairwise Lennard-Jones (LJ) summation over the fixed C cage atoms, the parameter space of the Hamiltonian includes three tunable variables: (M, ɛ, σ) representing the mass of the trapped species, the LJ energy, and length scales respectively. Fixing the mass and allowing the others to vary can imitate the potentials of endohedral species trapped in more elongated fullerenes. We choose to explore the LJ parameter space of an endohedral atom in C70 with ɛ ∈ [20, 150 cm-1], and σ ∈ [2.85, 3.05 Å]. As the barrier height and positions of these wells vary between [1, 264 cm-1] and [0.35, 0.85 Å] respectively, using a 3D direct product basis of 1D harmonic oscillator (HO) wavefunctions centred at the origin where there is a local maximum is unphysical. Instead we propose the use of a non-orthogonal basis set, using 1D HO wavefunctions centred in each minimum and compare this to other choices. The ground state energy of the X@C70 is tracked across the LJ parameter space, along with its corresponding nuclear translational wavefunctions. A classification of the wavefunction characteristics, namely the prolateness and "peanut-likeness" based on its statistical moments is also proposed. Excited states of longer fullerenes are assigned quantum numbers, and the fundamental transitions of Ne@C70 are tracked across the parameter space.

Doubly-charmed pentaquark in a quark model with a complete set of harmonic oscillator bases

Doubly-charmed pentaquark in a quark model with a complete set of harmonic oscillator bases

NuHamil : A numerical code to generate nuclear two- and three-body matrix elements from chiral effective field theory

The applicability of nuclear ab initio calculations has rapidly extended over the past decades. However, starting research projects is still challenging due to the required numerical expertise in the generation of underlying nuclear interaction matrix elements and many-body calculations. To ease the first issue, in this paper we introduce the numerical code NuHamil to generate the nucleon-nucleon (NN) and three-nucleon (3N) matrix elements expressed in a spherical harmonic-oscillator basis, inputs of many-body calculations. The ground-state energies for the selected doubly closed shell nuclei are calculated with the no-core shell-model (NCSM) and in-medium similarity renormalization group (IMSRG). The code is written in modern Fortran, and OpenMP+MPI hybrid parallelization is available for the 3N matrix-element calculations.

Triaxially deformed relativistic Hartree–Bogoliubov theory in Woods–Saxon basis

A triaxially deformed relativistic Hartree–Bogoliubov theory in the Woods–Saxon basis is developed with the aim of treating the triaxial deformation, pairing correlations and continuum in a unified way. In order to consider the triaxial deformation, the deformed potentials are expanded in terms of spherical harmonic functions in the coordinate space. In order to take the pairing correlations into account and treat the continuum properly, by using the Dirac Woods–Saxon basis, which has correct asymptotic behavior, the relativistic Hartree–Bogoliubov equation with triaxial deformation is solved. The formalism of triaxially deformed relativistic Hartree–Bogoliubov theory in Woods–Saxon basis is presented. Taking an axially deformed nucleus 24Ne and a triaxially deformed nucleus 76Ge as examples, the numerical checks are performed. A weakly bound nucleus 112Ge is taken as an example to carry out the necessary converge checks for the numerical parameters. In addition, the ground-state properties of even–even germanium isotopes are investigated. The evolutions of two-neutron separation energy, deformation, root-mean-square radii and density distribution with mass number are analyzed. The comparison between the calculations from the relativistic Hartree–Bogoliubov theory based on harmonic-oscillator basis and the triaxially deformed relativistic Hartree–Bogoliubov theory in Woods–Saxon basis is performed. It is found that the neutron drip line is extended from 114Ge to 118Ge in the triaxially deformed relativistic Hartree–Bogoliubov theory in Woods–Saxon basis.

Hadronic structure on the light front. IV. Heavy and light baryons

This work is a continuation in our series of papers, that addresses quark models of hadronic structure on the light front. The chief focus of this paper is the quantum-mechanical solution of the three-quark model Hamiltonian, describing baryons. In Jacobi coordinates, we use a harmonic oscillator basis for the transverse directions. For the longitudinal momentum fractions $x_i$, the pertinent basis follows from quantum mechanics in a "triangular cup" potential, which we solve exactly. We calculate the masses and light front wave functions for the flavor symmetric $\frac 32^+$ baryons $bbb,ccc,sss,uuu$.

Machine learning light hypernuclei

We employ a feed-forward artificial neural network to extrapolate at large model spaces the results of ab-initio hypernuclear No-Core Shell Model calculations for the Λ separation energy BΛ of the lightest hypernuclei, Λ3H, Λ4H and Λ4He, obtained in computationally accessible harmonic oscillator basis spaces using chiral nucleon-nucleon, nucleon-nucleon-nucleon and hyperon-nucleon interactions. The overfitting problem is avoided by enlarging the size of the input dataset and by introducing a Gaussian noise during the training process of the neural network. We find that a network with a single hidden layer of eight neurons is sufficient to extrapolate correctly the value of the Λ separation energy to model spaces of size Nmax=100. The results obtained are in agreement with the experimental data in the case of Λ3H and the 0+ state of Λ4He, although they are off of the experiment by about 0.3 MeV for both the 0+ and 1+states of Λ4H and the 1+ state of Λ4He. We find that our results are in excellent agreement with those obtained using other extrapolation schemes of the No-Core Shell Model calculations, showing this that an ANN is a reliable method to extrapolate the results of hypernuclear No-Core Shell Model calculations to large model spaces.

Ab initio no-core shell-model description of C10–14 isotopes

We present a systematic study of the $^{10--14}\mathrm{C}$ isotopes within the ab initio no-core shell-model theory. We apply four different realistic nucleon-nucleon $(NN)$ interactions: (i) the charge-dependent Bonn 2000 (CDB2K) potential; (ii) the inside nonlocal outside Yukawa (INOY) potential; (iii) the next-to-next-to-next-to-leading order $({\mathrm{N}}^{3}\mathrm{LO})$ potential; and (iv) the optimized next-to-next-to-leading order $({\mathrm{N}}^{2}{\mathrm{LO}}_{\mathrm{opt}})$ potential. We report the low-lying energy spectra of both positive- and negative-parity states for $^{10--14}\mathrm{C}$ isotopes and investigate the level structures. We also calculate electromagnetic properties such as transition strengths, quadrupole moments, and magnetic moments. The dependence of point-proton radii on the harmonic-oscillator frequency and basis space is shown. We present calculations of the translation invariant one-body density matrix in the no-core shell-model and discuss isotopic trends in the density distribution. The maximum basis space reached is $10\ensuremath{\hbar}\mathrm{\ensuremath{\Omega}}$ for $^{10}\mathrm{C}$ and $8\ensuremath{\hbar}\mathrm{\ensuremath{\Omega}}$ for $^{11--14}\mathrm{C}$, with a maximum M-scheme dimension of $1.3\ifmmode\times\else\texttimes\fi{}{10}^{9}$ for $^{10}\mathrm{C}$. We found that, while the INOY interaction gives the best description of the ground-state energies, the ${\mathrm{N}}^{3}\mathrm{LO}$ interaction best reproduces the point-proton radii.

Towards a variational Jordan–Lee–Preskill quantum algorithm

Rapid developments of quantum information technology show promising opportunities for simulating quantum field theory in near-term quantum devices. In this work, we formulate the theory of (time-dependent) variational quantum simulation of the dimensional quantum field theory including encoding, state preparation, and time evolution, with several numerical simulation results. These algorithms could be understood as near-term variational quantum circuit (quantum neural network) analogs of the Jordan–Lee–Preskill algorithm, the basic algorithm for simulating quantum field theory using universal quantum devices. Besides, we highlight the advantages of encoding with harmonic oscillator basis based on the Lehmann—Symanzik—Zimmermann reduction formula and several computational efficiency such as when implementing a bosonic version of the unitary coupled cluster ansatz to prepare initial states. We also discuss how to circumvent the ‘spectral crowding’ problem in the quantum field theory simulation and appraise our algorithm by both state and subspace fidelities.

Dmscatter: A fast program for WIMP-nucleus scattering

Recent work [1,2], using an effective field theory framework, have shown the number of possible couplings between nucleons and the dark-matter-candidate Weakly Interacting Massive Particles (WIMPs) is larger than previously thought. Inspired by an existing Mathematica script that computes the target response [2], we have developed a fast, modern Fortran code, including optional OpenMP parallelization, along with a user-friendly Python wrapper, to swiftly and efficiently explore many scenarios, with output aligned with practices of current dark matter searches. A library of most of the important target nuclides is included; users may also import their own nuclear structure data, in the form of reduced one-body density matrices. The main output is the differential event rate as a function of recoil energy, needed for modeling detector response rates, but intermediate results such as nuclear form factors can be readily accessed. Program summaryProgram Title: dmscatterCPC Library link to program files:https://doi.org/10.17632/6fm7d4m2py.1Developer's repository link:https://github.com/ogorton/dmscatterLicensing provisions: MITProgramming languages: Fortran, PythonNature of problem: Simulating the event rate of nuclear recoils from collisions with dark matter for a variety of different nuclear targets and different nucleon-dark matter couplings.Solution method: The event rate is an integral over the product of the nuclear and dark matter response functions, weighted by the expected dark matter flux. To compute the nuclear response, reduced one-body nuclear density matrices are coupled to multipole expansion of operators computed analytically in a harmonic oscillator basis. The nuclear structure input, in the form of one-body densities matrices, are supplied by outside code; we provide a library of the most prominent targets.

Sub Coulomb barrier d+ 208Pb scattering in the time-dependent basis function approach

We employ the non-perturbative time-dependent basis function (tBF) approach to study the scattering of the deuteron on 208Pb below the Coulomb barrier. We obtain the bound and discretized scattering states of the projectile, which form the basis representation of the tBF approach, by diagonalizing a realistic Hamiltonian in a large harmonic oscillator basis. We find that the higher-order inelastic scattering effects are noticeable for sub barrier scatterings with the tBF method. We have successfully reproduced experimental sub Coulomb barrier elastic cross section ratios with the tBF approach by considering only the electric dipole (E1) component of the Coulomb interaction between the projectile and the target during scatterings. We find that the correction of the polarization potential to the Rutherford trajectory is dominant in reproducing the data at very low bombarding energies, whereas the role of internal transitions of the deuteron projectile induced by the E1 interaction during the scattering becomes increasingly significant at higher bombarding energies.

BCS solutions and effective quark energies of the QCD Hamiltonian in the Coulomb gauge

The search of theoretical approximations adequate to the description of the non-perturbative regime of QCD, that is the low-energy portion of the hadron spectrum, requires the adoption of notions more frequently applied to other, more conventional, quantum many body systems, like the atomic nucleus, solid state systems, etc. The identification and the use of effective degrees of freedom is one of the notions whose limitations we shall explore in this work. To start with the choice of the basis to be adopted to represent effective fermionic degrees of freedom is a crucial first step towards the goal of constructing an effective Hamiltonian from which the low energy portion of the hadronic spectrum may emerge as its solution. Here, we propose the use of the harmonic oscillator basis and emphasize its advantages compared to the basis of free momentum states. The starting Hamiltonian is the effective Coulomb plus linear potential, which we have used in previous calculations. We proceed by pre-diagonalizing the Hamiltonian to build a single particle spectrum, namely: the spectrum of effective quark degrees of freedom, starting from arbitrary chosen quark masses, which are later on re-normalized. Then, quark-pair correlations are described in the context of the well-known Bogoliubov transformations. The resulting quasiparticle states are then used to construct meson-like states. The dependence of the results upon the parameters which enter in the calculations is explored in detail.