Separable Potential Model Research Articles

Inverse scattering problem with a bare state

In hadron physics, molecular-like multihadron states can interact with compact multiquark states. The latter are modeled as bare states in the Hilbert space of a potential model. In this work, we study several potential models relevant to the bare state, and solve their inverse scattering problems. The first model, called ``cc,'' is a separable potential model. We show that it can approximate (single-channel short-range) $S$-wave near-threshold physics with an error of $\mathcal{O}({\ensuremath{\beta}}^{3}/{M}_{V}^{3})$, where $\ensuremath{\beta}$ sets the maximum momentum of the near-threshold region and ${M}_{V}$ is the typical scale of the potential. The second model, called ``bc,'' serves as the bare-state-dominance approximation, where interaction between continuum states is ignored. Under this model, even though the bare state is always crucial for a bound state's generation, a shallow bound state naturally tends to have a small bare-state proportion. Therefore, we need other quantities to quantify the importance of the bare state. The last model, called ``bcc,'' is a combination of the first two models. This model not only serves as a correction to the bare-state-dominance approximation, but can also be used to understand the interplay between quark and hadron degrees of freedom. This model naturally leads to the presence of a Castillejo-Dalitz-Dyson (CDD) zero. We consider the energy decomposition of a bound state. The potential ratio of the bare-continuum interaction to the continuum self-interaction is proposed to understand how the bound state is generated. Model independently, an inequality for the potential ratio is derived. Based on the model ``bcc,'' the CDD zero can be used to estimate the potential ratio. Finally, we apply these studies to the deuteron, $\ensuremath{\rho}$ meson, and ${D}_{s0}^{*}(2317)$, and analyze their properties.

Near-threshold spectrum from a uniformized Mittag-Leffler expansion: Pole structure of the Z(3900)

We demonstrate how S-matrix poles manifest themselves as the physical spectrum near the upper threshold in the context of the two-channel uniformized Mittag-Leffler expansion, an expression written as a sum of pole terms under an appropriate variable where the S-matrix is made single-valued (uniformization). We show that the transition of the spectrum is continuous as a S-matrix pole moves across the boundaries of the complex energy Riemann sheets and that the physical spectrum peaks at or near the upper threshold when the S-matrix pole is positioned sufficiently close to it on the uniformized plane. There is no essential difference on which sheet the pole is positioned. What is important is the existence of a pole near the upper threshold and the distance between the pole and the physical region, not on which complex energy sheet the pole is positioned. We also point out that when the pole is close to the upper threshold, the complex pole does not have the usual meaning of the resonance. Neither the real part represents the peak energy, nor the imaginary part represents the half width. Subsequently, we try to understand the current status of $Z(3900)$ from the viewpoint of the uniformized Mittag-Leffler expansion reflecting in particular, Phys.Rev.Lett.117, 242001 (2016) in which they concluded that $Z(3900)$ is not a conventional resonance but a threshold cusp. We point out that their results turn out to indicate the existence of S-matrix poles near the $\bar D D^*$ threshold, which is most likely the origin of the peak found in their calculation of the near-threshold spectrum. In order to support our argument, we set up a separable potential model which shares common behavior of poles near the $\bar D D^*$ threshold to the above-mentioned reference and show in our model that the structures near the $\bar D D^*$ threshold are indeed caused by these near-threshold poles.

SU(3) flavor symmetry considerations for the [formula omitted] coupled channels system

We study the impact of SU(3) flavor symmetry breaking on the properties of dynamically generated Λ⁎ states within an effective separable potential model describing the coupled channels K¯N system. The model is based on the chiral meson-baryon Lagrangian at next-to-leading order, and constitutes an improvement over a previous model developed by our group, with its applicability being extended to higher energies covering the Λ(1670) resonance region. It is demonstrated that the ratios of channel couplings to the resonant states can vary dramatically when the flavor breaking is gradually switched off, tracing a path to the restored SU(3) symmetry. We conclude that the couplings determined from physical observables cannot be used to reliably relate a given resonance to a specific flavor multiplet.

Classifying Near-Threshold Enhancement Using Deep Neural Network

One of the main issues in hadron spectroscopy is to identify the origin of threshold or near-threshold enhancement. Prior to our study, there is no straightforward way of distinguishing even the lowest channel threshold-enhancement of the nucleon-nucleon system using only the cross-sections. The difficulty lies in the proximity of either a bound or virtual state pole to the threshold which creates an almost identical structure in the scattering region. Identifying the nature of the pole causing the enhancement falls under the general classification problem and supervised machine learning using a feed-forward neural network is known to excel in this task. In this study, we discuss the basic idea behind deep neural network and how it can be used to identify the nature of the pole causing the enhancement. The applicability of the trained network can be explored by using an exact separable potential model to generate a validation dataset. We find that within some acceptable range of the cut-off parameter, the neural network gives high accuracy of inference. The result also reveals the important role played by the background singularities in the training dataset. Finally, we apply the method to nucleon-nucleon scattering data and show that the network was able to give the correct nature of pole, i.e. virtual pole for $${}^1S_0$$ partial cross-section and bound state pole for $${}^3S_0$$ .

Classifying the pole of an amplitude using a deep neural network

Most of the exotic resonances observed in the past decade appear as a peak structure near some threshold. These near-threshold phenomena can be interpreted as genuine resonant states or enhanced threshold cusps. Apparently, there is no straightforward way of distinguishing the two structures. In this work, we employ the strength of deep feed-forward neural network in classifying objects with almost similar features. We construct a neural network model with scattering amplitude as input and the nature of a pole causing the enhancement as output. The training data is generated by an S-matrix satisfying the unitarity and analyticity requirements. Using the separable potential model, we generate a validation data set to measure the network's predictive power. We find that our trained neural network model gives high accuracy when the cutoff parameter of the validation data is within 400--800 MeV. As a final test, we use the Nijmegen partial wave and potential models for nucleon-nucleon scattering and show that the network gives the correct nature of the pole.

Quasibound state instead of ΛΛnn bound state * * This work has been partially funded by COFAA-IPN (México), by Ministerio de Economía, Industria y Competitividad and EU FEDER (FPA2016-77177)

The coupled system was studied to investigate whether the inclusion of channel coupling is able to bind the system. We use a separable potential three-body model of the coupled system and a variational four-body calculation with realistic interactions. Our results exclude the possibility of a bound state by a large margin. Instead, we found a quasibound state above the threshold.

Pion-eta scalar-isovector 3-coupled channel amplitude fitted to branching ratios and threshold plus subthreshold parameters

The low energy (below 2 GeV) πη channel interaction amplitude becomes an object of interest mainly because of the search for exotic mesons in just beginning to collect data detector GlueX in JLab. Finding and interpretation of expected weak signals from these states require a comparison with a very accurate amplitude containing standard ( qq ) states i.e. a 0 (980) and a 0 (1450). The main problem in the determination of such amplitude is a total absence of data about the phases and inelasticities in the elastic and inelastic region. In addition, it is necessary to take into account the next two coupled higher channels - KK and πη ′. Presented here amplitude is based on separable potential model (working very well for the scalar-isoscalar ππ interactions) with only 9 free parameters. To determine such 3-coupled channel amplitude, the following information has been taken into account: experimental branching ratios and positions of both a 0 resonances, theoretical couplings, scattering length from ChPT and value of squared radius of the πη form factor. Phase shifts, inelasticities and cross sections in all single and crossed channels are presented.

Deeply quasi-bound state in single- and double- $\bar{K}$ K ¯ nuclear clusters

New calculations of the quasi-bound state positions in $K^{-}K^{-}pp$ kaonic nuclear cluster are performed using non-relativistic four-body Faddeev-type equations in AGS form. The corresponding separable approximation for the integral kernels in the three- and four-body kaonic clusters is obtained by using the Hilbert-Schmidt expansion procedure. Different phenomenological models of $\bar{K}N-\pi\Sigma$ potentials with one- and two-pole structure of $\Lambda$(1405) resonance and separable potential models for $\bar{K}$-$\bar{K}$ and nucleon-nucleon interactions, are used. The dependence of the resulting four-body binding energy on models of $\bar{K}N-\pi\Sigma$ interaction is investigated. We obtained the binding energy of the $K^{-}K^{-}pp$ quasi-bound state $\sim$ 80-94 MeV with the phenomenological $\bar{K}N$ potentials. The width is about $\sim$ 5-8 MeV for the two-pole models of the interaction, while the one-pole potentials give $\sim$ 24-31 MeV width.

Chirally motivated separable potential model for ηN amplitudes

We analyze the $\eta N$ interaction using a coupled channel separable potential model that implements the chiral symmetry. The model predicts an $\eta N$ stattering length $\Re a_{\eta N} \approx 0.7$ fm and in-medium subthreshold attraction most likely sufficient to generate $\eta$-nuclear bound states. The energy dependence of the $\eta N$ amplitude and pole content of the model are discussed. An idea of the same origin of the baryon resonances $N^{\star}(1535)$ and $N^{\star}(1650)$ is presented.

Strong binding and shrinkage of single and double <span style="text-decoration: overline;"><i>K</i></span> nuclear systems (<i>K</i><sup>−</sup><i>pp</i>, <i>K</i><sup>−</sup><i>ppn</i>, <i>K</i><sup>−</sup><i>K</i><sup>−</sup><i>p</i> and <i>K</i><sup>−</sup><i>K</i><sup>−</sup><i>pp</i>) predicted by Faddeev-Yakubovsky calculations

Non-relativistic Faddeev and Faddeev-Yakubovsky calculations were made for K−pp, K−ppn, K−K−p and K−K−pp kaonic nuclear clusters, where the quasi bound states were treated as bound states by employing real separable potential models for the K−-K− and the K−-nucleon interactions as well as for the nucleon-nucleon interaction. The binding energies and spatial shrinkages of these states, obtained for various values of the interaction, were found to increase rapidly with the interaction strength. Their behaviors are shown in a reference diagram, where possible changes by varying the interaction in the dense nuclear medium are given. Using the Λ(1405) ansatz with a PDG mass of 1405 MeV/c2 for K−p, the following ground-state binding energies together with the wave functions were obtained: 51.5 MeV (K−pp), 69 MeV (K−ppn), 30.4 MeV (K−K−p) and 93 MeV (K−K−pp), which are in good agreement with previous results of variational calculation based on the Akaishi-Yamazaki coupled-channel potential. The K−K−pp state has a significantly increased density where the two nucleons are located very close to each other, in spite of the inner NN repulsion. Relativistic corrections on the calculated non-relativistic results indicate substantial lowering of the bound-state masses, especially of K−K−pp, toward the kaon condensation regime. The fact that the recently observed binding energy of K−pp is much larger (by a factor of 2) than the originally predicted one may infer an enhancement of the interaction in dense nuclei by about 25% possibly due to chiral symmetry restoration. In this respect some qualitative accounts are given based on “clearing QCD vacuum” model of Brown, Kubodera and Rho.

Separable potential model for meson-baryon interaction beyond theSwave

A model for low-energy meson-baryon interaction in the strange sector is presented. The interaction is described in terms of separable potentials with multiple partial waves considered. A general solution of the Lippmann-Schwinger equation for the scattering of spin-0 and spin-1/2 particles is derived. Next, the general framework is applied to the $\overline{K}N$ sector in a simple model with only the $S$ and $P$ waves taken into account. The separable potential is designed to match the chiral perturbation theory at the lowest nontrivial order. It is shown that although a simple model with three free parameters works well for the $S$ wave, it fails to reproduce the $P$-wave features of kaon-nucleon physics. Most importantly, the $P$-wave interaction is too weak to express a resonant behavior that could be identified as $\ensuremath{\Sigma}(1385)$ resonance.

Electric Dipole Moments of Light Nuclei and the Implications for CP Violation

A definitive measurement of an electric dipole moment (EDM) would likely imply new physics beyond the standard model. Although the standard model strong interaction term could theoretically produce an EDM of any size, that it is constrained by the current neutron EDM limit to be some 10 orders of magnitude smaller than 1 suggests that the electroweak sector and CP violation will be the source of a measurable EDM. The weak interaction standard model EDM is itself orders of magnitude smaller than contemporary experiments can measure. Direct measurement of the neutron EDM lies in the next decade; measurement of the proton EDM could well come first. A BNL proposal for an electrostatic storage ring measurement lies in the offing. Unless the EDM proves to be an isoscalar, one will need other measurements to separate the isoscalar, isovector, and isotensor components. Measurement of a nuclear EDM will be required: 2H, 3H, or 3He being the simplest nuclear systems. A storage ring measurement of the triton EDM could be accomplished in a manner analogous to that proposed for the proton. However, the deuteron EDM measurement offers certain advantages, even though the experiment would be more complex, involving electric and magnetic fields, than that required for the proton and triton. The COSY facility in the Forschungszentrum Juelich is almost an ideal facility to house such an experiment; one could also measure in the same ring the EDM for the proton and He. The deuteron is the one nucleus for which exact model calculations can easily be performed. We briefly explore the model dependence of deuteron EDM calculations. Using a separable potential formulation of the Hamiltonian, we examine the sensitivity of the deuteron EDM to variations in the nucleon–nucleon interaction, including contemporary potential models, and we explore the dependence upon intermediate state multiple scattering in the 3P1 channel. We investigate the tensor force contribution to the model results and examine the effects of short-range repulsion that characterize realistic, contemporary potential models of the deuteron. Because one-pion exchange dominates the EDM calculation, separable potential model calculations appear to provide an adequate description of the deuteron EDM until such time as a measurement of better than 10 % is achieved.

Model dependence of theH2electric dipole moment

Background: Direct measurement of the electric dipole moment (EDM) of the neutron is in the future; measurement of a nuclear EDM may well come first. The deuteron is one nucleus for which exact model calculations are feasible. Purpose: We explore the model dependence of deuteron EDM calculations. Methods: Using a separable potential formulation of the Hamiltonian, we examine the sensitivity of the deuteron EDM to variation in the nucleon-nucleon interaction. We write the EDM as the sum of two terms, the first depending on the target wave function with plane-wave intermediate states, and the second depending on intermediate multiple scattering in the ${}^{3}$$P$${}_{1}$ channel, the latter being sensitive to the off-shell behavior of the ${}^{3}$$P$${}_{1}$ amplitude. Results: We compare the full calculation with the plane-wave approximation result, examine the tensor force contribution to the model results, and explore the effect of short-range repulsion found in realistic, contemporary potential models of the deuteron. Conclusions: Because one-pion exchange dominates the EDM calculation, separable potential model calculations will provide an adequate description of the $^{2}\mathrm{H}$ EDM until such time as a measurement better than $10%$ is obtained.

Separable potential model for K-N interactions at low energies

The effective separable meson-baryon potentials are constructed to match the equivalent chiral amplitudes up to the second order in external meson momenta. We fit the model parameters (low energy constants) to the threshold and low energy $K^{-}p$ data. In the process, the $K^{-}$-proton bound state problem is solved exactly in the momentum space and the 1s level characteristics of the kaonic hydrogen are computed simultaneously with the available low energy $K^{-}p$ cross sections. The model is also used to describe the $\pi \Sigma$ mass spectrum and the energy dependence of the $K^{-}n$ amplitude.

The2H Electric Dipole Moment in a Separable Potential Approach

Measurement of the electric dipole moment (EDM) of 2 H or of 3 He may well come prior to the cov- eted measurement of the neutron EDM. Exact model calculations for the deuteron are feasible, and we explore here the model dependence of such deuteron EDM calculations. We investigate in a separable potential approach the relationship of the full model calculation to the plane wave approximation, correct an error in an early poten- tial model result, and examine the tensor force aspects of the model results as well as the e ect of the short range repulsion found in the realistic, contemporary potential model calculations of Liu and Timmermans. We conclude that, because one-pion exchange dominates the EDM calculation, separable potential model calculations should provide an adequate picture of the 2 H EDM until better than 10% measurements are achieved.

Np → ηd near threshold and the s -wave ηN amplitude

We calculate the process np → ηd near threshold using a separable potential model of the coupled ηN - πN - ππN subystems, and a relativistic three-body calculation for the ηd scattering amplitude. The ππN channels are represented by an effective σN channel, and we compare the case where the σ and π masses are related by mσ = 2mπ and no width is considered, to another where the mass and width of the σ -meson are taken from ππ scattering data. The np → ηd cross-section can be well described up to about 60MeV by models where the real part of the ηN scattering length lies between 0.4≤Re(aηN)≤0.6 fm which allows us to determine the s -wave ηN scattering amplitude for -60≤E≤60 MeV.

The solvable three-dimensional rank-two separable potential model: partial-wave scattering

In this paper, we study the ℓth partial-wave scattering for particles subjected to a non-local rank-two separable potential. Analytic expressions for the scattering amplitude, bound and resonance states, phase shifts and time delays are obtained and numerical illustrations are provided.

A study of the cross sections and the spin observables for proton-12C elastic scattering in momentum space in the intermediate energy region

The differential cross sections and the spin observables of p-12C elastic scattering in the whole intermediate energy region are calculated using the KMT multiple scattering theory with microscopic momentum space optical potential which incorporates the spin dependence of the NN and p-12C interactions. The theory takes into consideration the antisymmetrized NN amplitudes, nucleon recoil and binding energy shifts, Lorentz invariant angle transformations, t matrices with on-shell behavior from NN phase shifts,and off-shell behavior from a realistic separable potential model. Qualitative agreement is found in differential cross sections and polarization data for 200Ep<1000 MeV.

Relativistic field-theoretical equations for the three-body multichannel πN and γN scattering reactions and propagator of the Δ-resonance

A new form of the relativistic three-body equations for the coupled πN and γN scattering reactions with three particle final states ππN and γπN is suggested. These equations are derived in the framework of the time-ordered three dimensional field theory. The solutions of the considered equations satisfy the unitarity condition and are exactly gauge invariant. The form of these three-body equations is does not depend on the choice of the model of Lagrangian and is also the same for the formulations with and without quark degrees of freedom. The effective potentials of the suggested equations are defined by the vertex functions with two on-mass shell particles. It is emphasized that these input vertex functions can be constructed from experimental data. Special attention is given to the construction of the propagator of the Δ-resonance in the framework of the separable πN potential model. The strong dependence of the multichannel πN and γN cross sections on the form of the Δ-resonance propagator[2] is discussed. The used formulation of the relativistic three-dimensional and three-body equations allows us to overcome a number of approximations which are usually used by practical calculations of the πN and γN scattering reactions.

Solvable separable nonlocal potential models

We introduce a generalized Yamaguchi rank-one separable potential with three parameters which has the separable Yukawa or central Yamaguchi potentials as limiting cases. The three parameters can be obtained, when possible, from the set of experimental data (binding energy, scattering length and effective range) by solving a simple cubic equation. We also discuss a rank-two separable potential with one of the strengths infinite. For both potentials the solutions can be expressed analytically.