Potential In Coordinate Space Research Articles

Study of the hidden charm DD¯* interactions in chiral effective field theory

We study the chiral interactions of the hidden charm $D\bar{D}^*$ system within chiral effective field theory. Chiral Lagrangians are constructed by incorporating the chiral symmetry, heavy quark symmetry as well as proper charge conjugation properties of the heavy mesons. The interacting potentials of the $S$-wave $D\bar{D}^*$ are calculated up to second chiral order at 1-loop level, where complete two-pion exchange interactions are included. We further investigate the behaviors of the potentials in coordinate space, as well as their bound state properties. Our studies indicate that there exists a interacting strength ordering among considered four channels: $\text{str.}[0^+(1^{++})]>\text{str.}[0^-(1^{+-})] > \text{str.}[1^+(1^{+-})] > \text{str.}[1^-(1^{++})]$ where str. stands for the strength of the $D\bar{D}^*$ interaction. Moreover, we find that $X(3872)$ can be treated as a good candidate of $0^+(1^{++})$ molecular state. There also tends to form $0^-(1^{+-})$ and $1^+(1^{+-})$ molecular states and we expect the experiments to search for the predicted multi-structures around the $D\bar{D}^*$ mass region.

Fifth force and hyperfine splitting in bound systems

Two recent experimental observations at the ATOMKI Institute of the Hungarian Academy of Sciences (regarding the angular emission pattern of electron-positron pairs from nuclear transitions from excited states in 8Be and 4He) indicate the possible existence of a particle of a rest mass energy of roughly 17 MeV. The so-called X17 particle constitutes a virtual state in the process, preceding the emission of the electron-positron pair. Based on the symmetry of the nuclear transitions ($1^+$ to $0^+$ and $0^-$ to $0^+$), the X17 could either be a vector, or a pseudoscalar particle. Here, we calculate the effective potentials generated by the X17, for hyperfine interactions in simple atomic systems, for both the pseudoscalar as well as the vector X17 hypotheses. The effective Hamiltonians are obtained in a general form which is applicable to both electronic as well as muonic bound systems. The effect of virtual annihilation and its contribution to the hyperfine splitting also is considered. Because of the short range of the X17-generated potentials, the most promising pathway for the observation of the X17-mediated effects in bound systems concerns hyperfine interactions, which, for $S$ states, are given by modifications of short-range (Dirac-delta) potentials in coordinate space. For the pseudoscalar hypothesis, the exchange of one virtual X17 quantum between the bound lepton and the nucleus exclusively leads to hyperfine effects, but does not affect the Lamb shift. Effects due to the X17 are shown to be drastically enhanced for muonic bound systems. Prospects for the detection of hyperfine effects mediated by X17 exchange are analyzed for muonic deuterium, muonic hydrogen, muonium, true muonium ($\mu^+\mu^-$ bound system), and positronium.

SIDES: Nucleon–nucleus elastic scattering code for nonlocal potential

We introduce the package SIDES (Schrödinger Integro-Differential Equation Solver) that solves the integro-differential Schrödinger equation for elastic scattering of a nonlocal optical potential in coordinate space. The code is capable of treating the Coulomb interaction without restrictions. The method is based on previous developments by Jacques Raynal in the DWBA07 code. Elastic scattering observables such as differential and integral cross sections, as well as analyzing power and spin rotation functions for both neutron and proton projectiles are evaluated, with no restriction on the type of nonlocality of the potential nor on the beam energy. The corresponding distorted wavefunctions are calculated as well. The SIDES package includes a Perey–Buck potential generator with two parametrizations. It includes as well local potential parametrizations and allows for mixing local and nonlocal contributions. Benchmarks are performed and discussed. Program summaryProgram Title: SIDESProgram Files doi:http://dx.doi.org/10.17632/cmpjgyrngr.1Licensing provisions: GNU General Public License, Version 2Programming language: FORTRAN-90Nature of problem: The description of nucleon elastic scattering off a target nucleus involves solving the Schrödinger’s wave equation for positive incident energy. The determination of scattering observables calls for accurate treatments of the continuum. The effective coupling between the projectile and the target is accounted for by an optical potential, an operator which is by nature complex, energy-dependent and nonlocal. The coupling becomes long-range in the case of charged projectiles. In a general scenario under nonlocal potentials, Schrödinger’s equation becomes an integro-differential equation.Solution method: SIDES solves the Schrödinger integro-differential equation numerically by matrix inversion using Gibbs, Numerov or a modified Numerov method with a uniform radial mesh in a box. The solution is refined by an iterative procedure until a specified precision is achieved. To obtain elastic scattering observables, the associated phase-shifts are calculated via matching of the numerical solution with its analytic asymptotic behavior.

Pairing in two-dimensional Fermi gases with a coordinate-space potential

In this work we theoretically study pairing in two-dimensional Fermi gases, a system which is experimentally accessible using cold atoms. We start by deriving the mean-field pairing gap equation for a coordinate-space potential with a finite interaction range, and proceed to solve this numerically. We find that for sufficiently short effective ranges the answer is identical to the zero-range one. We then use Diffusion Monte Carlo to evaluate the total energy for many distinct particle numbers; we employ several variational parameters to produce a good ground-state energy and then use these results to extract the pairing gap across a number of interaction strengths in the strongly interacting two-dimensional crossover. Extracting the gap via the odd-even energy staggering, our microscopic results can be used as benchmarks for other theoretical approaches.

Construction of Breit quark potential in coordinate space and mass splits of meson and quarkonium

Construction of a valid interaction potential function between quarks is a crucial issue in hadronic physics and also one of the frontier issues. Non-relativistic Breit potential is a common model to describe the interaction between quarks. It is used to successfully calculate the bound states of quarks and quark scatterings. These spur people to improve it. As is well known, the full Breit potential function, which includes the color-Coulomb term, the mass term, the orbit-orbit interaction term, the spin-spin interaction term, the spin-orbit interaction term, the tensor force term, and the constant term, contains singularity factors. How to eliminate the singularity factors is the most urgent task for developing Breit potential model. In this paper, we carry out a replacement method to eliminate the singularity factors in the full Breit quark potential function in coordinate space. Except for the color-Coulomb term and the constant term, remaining terms in the Breit quark potential function are all reconstructed. The replacement of (r) 3 e-r/8 is applied to the mass term and the spin-spin interaction term. The replacement of 1/r (1-e-r)/r is applied to the obit-obit interaction term. The replacement of 1/r3[1-(1+r)e-r]/r3 is applied to the spin-obit interaction term and the tensor force term. We calculate mass splits of heavy mesons and quarkonium species by using the reconstructed potential function and test the validity of the reconstructed potential function. The screening mass used in the calculations is not a simple constant but a variable relating to the quark mass mi and mj. It is found that the simple screening-mass expression cannot give the accurate value of B-meson mass, although it may give the mass splits of light mesons. However, the calculated results of the mass splits of the light mesons -, the heavy mesons, c-J/, b-(1s), c0-c2, etc., are highly consistent with the experimental data only when the screening mass is taken to be the Laurent series, =c-3(a+0.512)-3+ c-2(a+0.512)2 +c-1(a+0.512)-1+c0+c1(a+0.512) with respect to the average quark mass a=(mi+mj)/2. In this case, the mass accuracy of other mesons, especially the six D mesons, is improved significantly. Our calculated results indicate that a valid quark potential model, which gives not only the mass values of light mesons accurately but also the mass splits of heavy quarkonium species, is thus constructed in this paper.

Singular inverse square potential in coordinate space with a minimal length

The problem of a particle of mass m in the field of the inverse-square potential α∕r2 is studied in quantum mechanics with a generalized uncertainty principle, characterized by the existence of a minimal length. Using the coordinate representation, for a specific form of the generalized uncertainty relation, we solve the deformed Schrödinger equation analytically in terms of confluent Heun functions. We explicitly show the regularizing effect of the minimal length on the singularity of the potential. We discuss the problem of bound states in detail and we derive an expression for the energy spectrum in a natural way from the square integrability condition; the results are in complete agreement with the literature.

Planar density of vacuum charge induced by a supercritical Coulomb potential

Analytical expressions for the planar density of an induced vacuum charge are obtained in a strong Coulomb potential in coordinate space. Treatment is based on a self-adjoint extension approach for constructing of the Green's function of a charged fermion in an external electromagnetic field. Induced vacuum charge density is calculated and analyzed in subcritical and supercritical Coulomb potentials for massless and massive fermions. We argue that the virtual and so-called real vacuum polarizations contribute in an induced vacuum charge in a supercritical Coulomb potential. The behavior of the polarization vacuum charge density is investigated at long and short distances from the Coulomb center. The induced vacuum charge has a screening sign. Screening of a Coulomb impurity in graphene is briefly discussed. The real vacuum polarization charge density that acquires the quantum electrodynamics vacuum in a supercritical Coulomb potential due to the real vacuum polarization is calculated. It is shown that the vacuum charge densities essentially differ in massive and massless cases. We expect that our results can, as a matter of principle, be tested in graphene with a supercritical Coulomb impurity.

Description of light nuclei in pionless effective field theory using the stochastic variational method

We construct a coordinate-space potential based on pionless effective field theory with a Gaussian regulator. Charge-symmetry breaking is included through the Coulomb potential and through two- and three-body contact interactions. Starting with the effective field theory potential, we apply the stochastic variational method to determine the ground states of nuclei with mass number $A\leq 4$. At next-to-next-to-leading order, two out of three independent three-body parameters can be fitted to the three-body binding energies. To fix the remaining one, we look for a simultaneous description of the binding energy of $^4$He and the charge radii of $^3$He and $^4$He. We show that at the order considered we can find an acceptable solution, within the uncertainty of the expansion. We find that the EFT expansion shows good agreement with empirical data within the estimated uncertainty, even for a system as dense as $^4$He.

Next-to-leading order effective field theory ΛN → NN potential in coordinate space

The potential in coordinate space for the ΛN→NN weak transition, which drives the weak decay of most hypernuclei, is derived within the effective field theory formalism up to next-to-leading order. This coordinate space potential allows us to discuss how the different contributions to the potential add up at the different scales. Explicit expressions are given for each two-pion-exchange diagram contributing to the interaction. The potential is also reorganized into spin and isospin operators, and the coefficient for each operator is given in analytical form and represented in coordinate space. The relevance of explicitly including the mass differences among the baryons appearing in the two-pion-exchange diagrams is also discussed in detail.

Single-particle resonances in a deformed Dirac equation

Single-particle bound and resonant states are obtained by solving the Dirac equation with quadrupole-deformed Woods-Saxon potential in coordinate space with the coupled-channel approach. Taking the m{sup {pi}}=1/2{sup +} resonant states at deformation {beta}=0.1 as examples, the roles of their spherical components have been investigated based on the behaviors of the eigenphases and the corresponding probabilities weighted by the scalar spherical potential. It is shown that the realization of the m{sup {pi}}=1/2{sup +} resonances is supported mainly by the lnot =0 components, and the mixture of the s{sub 1/2} component can lead to the disappearance of some resonances at finite energy. The dominance of the lnot =0 component (d{sub 3/2}) in the small-energy region guarantees the continuation of a certain resonance to the corresponding bound state.

The one loop $$ \overline {\text{MS}} $$ static potential in the Gribov-Zwanziger lagrangian

We compute the static potential in the Gribov-Zwanziger Lagrangian as a function of the Gribov mass, gamma, in the MSbar scheme in the Landau gauge at one loop. The usual gauge independent one loop perturbative static potential is recovered in the limit as gamma -> 0. By contrast the Gribov-Zwanziger static potential contains the term gamma^2/(p^2)^2. However, the linearly rising potential in coordinate space as a function of the radial variable r does not emerge due to a compensating behaviour as r -> infty. Though in the short distance limit a dipole behaviour is present. We also demonstrate enhancement in the propagator of the bosonic localizing Zwanziger ghost field when the one loop Gribov gap equation is satisfied. The explicit form of the one loop gap equation for the Gribov mass parameter is also computed in the MOM scheme and the zero momentum value of the renormalization group invariant effective coupling constant is shown to be the same value as that in the MSbar scheme.

KK¯molecules with momentum-dependent interactions

It is shown that the momentum-dependent kaon-antikaon interactions generated via vector-meson exchange from the standard SU{sub V}(3)xSU{sub A}(3) interaction Lagrangian lead to a nonlocal potential in coordinate space that can be incorporated without approximation into a nonrelativistic version of the Bethe-Salpeter wave equation containing a radial-dependent effective kaon mass appearing in a fully symmetrized kinetic energy operator, in addition to a local potential. Estimates of the mass and decay widths of f{sub 0}(980) and a{sub 0}(980), considered as KK molecules of isospin 0 and 1, as well as for K{sup +}K{sup -} atomic bound states (kaonium) are presented and compared with previous studies of a similar nature. It is argued that without a better knowledge of hadronic form factors it is not possible to distinguish between the molecular versus elementary particle models for the structure of the light scalar mesons.

Solving the Dirac Equation with Nonlocal Potential by Imaginary Time Step Method

The imaginary time step (ITS) method is applied to solve the Dirac equation with the nonlocal potential in coordinate space by the ITS evolution for the corresponding Schrödinger-like equation for the upper component. It is demonstrated that the ITS evolution can be equivalently performed for the Schrödinger-like equation with or without localization. The latter algorithm is recommended in the application for the reason of simplicity and efficiency. The feasibility and reliability of this algorithm are also illustrated by taking the nucleus 16O as an example, where the same results as the shooting method for the Dirac equation with localized effective potentials are obtained.

Meson spectra and mass splitting of $ \eta_{c}^{}$ - J/ $ \psi$ calculated with a regularized quark potential

The complete Breit potential contains the terms of spin-spin, spin-orbit, orbit-orbit, and tensor force interactions which become singular at short distance. Most of previous calculations of the non-relativistic potential quark model considered only the spin-spin interaction and substituted the \( \delta\)(r) -function by the Gaussian or Yukawa potential in coordinate space. Recently, a method to regularize the Breit potential consists of subtracting terms that cancel the singularity at the origin but leave the intermediate- and long-distance behavior unchanged. Motivated by this work we regularize the Breit potential by multiplying the singular terms in momentum space identically by the form factor [\( \mu^{2}_{}\)/(q2 + \( \mu^{2}_{}\))]2 of the momentum transfer q , where the screened mass μ increases with the reduced mass of the meson. With the regularized Breit potential we calculate the masses of 30 common mesons and the new \( \eta_{b}^{}\) meson. We find that the calculated masses from light to heavy mesons agree well with experimental data. The inclusion of such a dependence of the reduced mass in the potential regularization improves the spin-spin splittings of \( \eta_{c}^{}\) -J/\( \psi\) and \( \eta_{b}^{}\) - \( \Upsilon\)(1S) . The spin-orbit and tensor force interactions in the Breit potential lead to the splittings of \( \chi_{{c0}}^{}\) , \( \chi_{{c1}}^{}\) , and \( \chi_{{c2}}^{}\) .

THE OLD AND THE NEW OF PARITY-VIOLATING TWO-PION-EXCHANGE NN POTENTIAL

We consider the parity-violating two-pion-exchange potential obtained from the covari-ant formalism in the past and the state-of-the-art effective field theory approach. We discuss the behavior of the potential in coordinate space and its application to the parity-violating asymmetry in [Formula: see text] at threshold.

EffectiveK¯Ninteraction based on chiral SU(3) dynamics

The effective $\overline{K}N$ interaction based on chiral SU(3) coupled-channel dynamics is derived and its extrapolation below the $\overline{K}N$ threshold is studied in detail. Starting from the coupled-channel scattering equations, we eliminate the channels other than $\overline{K}N$ and obtain an effective interaction in the single $\overline{K}N$ channel. An equivalent local potential in coordinate space is constructed such as to reproduce the full scattering amplitude of the chiral SU(3) coupled-channel framework. We discuss several realistic chiral SU(3)-based models in comparison to reach conclusions about the uncertainties involved. It turns out that, in the region relevant to the discussion of deeply bound $\overline{K}$-nuclear few-body systems, the resulting energy-dependent, equivalent local potential is substantially less attractive than the one suggested in previous purely phenomenological treatments.

Dominance of low-ℓcomponent in weakly bound deformed single-neutron orbits

Calculating single-particle (Nilsson) levels in axially symmetric quadrupole-deformed potentials in coordinate space, the structure of weakly bound neutron orbits is studied in the absence of pair correlation. It is shown that in the wave functions of ${\ensuremath{\Omega}}^{\ensuremath{\pi}}=1∕{2}^{+}$ orbits, where $\ensuremath{\Omega}$ expresses the projection of the particle angular momentum along the symmetry axis, the $\ensuremath{\ell}=0\phantom{\rule{0.3em}{0ex}}({s}_{1∕2})$ component becomes overwhelmingly dominant as the binding energy of the orbits approaches zero, irrespective of the size of the deformation and the kind of Nilsson orbits. Consequently, all ${\ensuremath{\Omega}}^{\ensuremath{\pi}}=1∕{2}^{+}$ levels become practically unavailable for both deformation and many-body pair correlation, when the levels approach continuum or lie in the continuum.

Analytic solution to wave packet dynamics in a laser field: the case of linear chirp

An analytical expression for the nonadiabatic transition probability is derived for the case that two linear potentials in coordinate space are coupled by a linearly chirped time-dependent laser field. This constitutes a solution to the wave packet dynamics in the high energy approximation. Numerical calculations are carried out to demonstrate the applicability of the formula.

A Predictive Model of Nucleon-Nucleus Scattering Cross Sections

Nucleon total reaction and neutron total cross sections as well as angular variables including spin observables from 25 to 300 MeV for all stable nuclei from 6Li to 238U have been predicted that are in good agreement with measured data. Those predictions have been made using non-local, energy dependent, and complex optical potentials in coordinate space formed by full folding of effective nucleon-nucleon interactions with realistic nuclear ground state densities. By inverse kinematics the same model prescription describes exotic (radioactive) nuclei scattering from hydrogen as a target and the results reveal the extended (neutron) distributions such nuclei can have.

Chiral2π-exchangeNNpotentials: Relativistic1/M2corrections

We calculate in baryon chiral perturbation theory the relativistic ${1/M}^{2}$ corrections to the leading order two-pion exchange diagrams. We give explicit expressions for the corresponding one-loop $\mathrm{NN}$ amplitudes in momentum space. The resulting isovector central and isoscalar spin-spin and tensor $\mathrm{NN}$ amplitudes involve nonstatic terms proportional to the squared nucleon center-of-mass momentum ${p}^{2}.$ From the two-pion exchange box diagrams we obtain an isoscalar quadratic spin-orbit $\mathrm{NN}$ amplitude. We give also analytical expressions for the corresponding $\mathrm{NN}$ potentials in coordinate space. The diagrammatic results presented here make the chiral $\mathrm{NN}$ potential complete at next-to-next-to-next-to-leading order.