Two-nucleon Forces Research Articles

Implementation of chiral two-nucleon forces to nuclear many-body methods with Gaussian-wave packets

Abstract Many-body methods that use Gaussian-wave packets to describe nucleon spatial distribution have been widely employed for depicting various phenomena in nuclear systems, in particular clustering. So far, however, the chiral effective field theory, a state-of-the-art theory of nuclear force, has not been applied to such methods. In this paper, we give the formalism to calculate the two-body matrix elements of the chiral two-nucleon forces using the Gaussian-wave packets. We also visualize the matrix elements and investigate the contributions of the central and tensor forces. This work is a foothold towards an ab initio description of various cluster phenomena in view of nucleons, pions, and many-nucleon forces.

The nucleon-induced deuteron breakup process as a laboratory for chiral dynamics

The nucleon-induced deuteron breakup reaction is studied within the Faddeev approach at incoming nucleon laboratory energies of 135 and 200 MeV. The chiral semilocal momentum-space (SMS) potential developed up to N4LO+, supplemented by the N2LO three-nucleon interaction, is used. Our investigation is focused on the determination of theoretical uncertainties in a predicted cross section related to its dependence on the value of the cutoff parameter of the regulator. We also compare predictions based on the complete N2LO potential with those based on the two-nucleon force upgraded to the N4LO+ order and augmented with the N2LO three-nucleon force. In addition, we study the three-nucleon force effects predicted by this model of interaction. Our systematic study covers the entire kinematically allowed phase space; however, our main results are obtained when additional restrictions on energies and cross section values are imposed. In such a case, we observe that the dependence of the differential cross sections on the regulator cutoff is moderate at 135 MeV and much stronger at 200 MeV. For the latter energy, it can amount to up to 45% in specific kinematic configurations. Taking into account terms beyond, N2LO in a two-body interaction changes the cross section up to 20% (27%) at E = 135(200) MeV. The inclusion of the three-nucleon force leads to effects of approximately 27% at both energies. We illustrate these dependencies with a few examples of the exclusive cross section as a function of the arc length of the S-curve.

Pion absorption from the lowest atomic orbital in H2, H3 , and He3

The ${\ensuremath{\pi}}^{\ensuremath{-}}+^{2}\mathrm{H}\ensuremath{\rightarrow}n+n, {\ensuremath{\pi}}^{\ensuremath{-}}+^{3}\mathrm{H}\ensuremath{\rightarrow}n+n+n, {\ensuremath{\pi}}^{\ensuremath{-}}+^{3}\mathrm{He}\ensuremath{\rightarrow}n+d$, and ${\ensuremath{\pi}}^{\ensuremath{-}}+^{3}\mathrm{He}\ensuremath{\rightarrow}p+n+n$ capture reactions from the lowest $1S$ atomic orbitals are studied under full inclusion of final state interactions. Our results are obtained with the single-nucleon and two-nucleon transition operators derived at leading order in chiral effective field theory. The initial and final three-nucleon states are calculated with the chiral nucleon-nucleon semilocal momentum space potential up to ${\mathrm{N}}^{4}{\mathrm{LO}}^{+}$, augmented by the consistently regularized chiral ${\mathrm{N}}^{2}\mathrm{LO}$ three-nucleon potential. We found that absorption rates depend strongly on the nuclear pion absorption operator used, and its two-body parts change the rates by a few orders of magnitude. The final state interactions between nucleons generated by the two-nucleon forces are also important, while the three-nucleon interaction plays a visible role only in the ${\ensuremath{\pi}}^{\ensuremath{-}}+^{3}\mathrm{He}\ensuremath{\rightarrow}n+d$ reaction. Our absorption rate for the ${\ensuremath{\pi}}^{\ensuremath{-}}+^{2}\mathrm{H}\ensuremath{\rightarrow}n+n$ process is in good agreement with the experimental data from the hadronic ground-state broadening in pionic deuterium. The capture rates on $^{3}\mathrm{He}$ are also generally consistent with the spectroscopic data within error bars, though our central values are found to be systematically below the data. We show that for the three-body breakup processes the dominant contributions to the absorption rates arise from the quasi-free scattering and final-state interaction kinematical configurations.

Correlations among neutron-proton and neutron-deuteron elastic scattering observables

We employ two models of the nucleon-nucleon force: the OPE-Gaussian as well as the chiral N4LO and N4LO+ interactions with semilocal regularization in momentum space to study correlations among two-nucleon and three-nucleon elastic scattering observables. These models contain a number of free parameters whose values and covariance matrices are evaluated from a fit to the two-nucleon data. Such detailed knowledge of parameters allows us to create, using various sets of statistically generated parameters, numerous versions of these potentials and next apply them to two- and three-nucleon scattering to make predictions of various observables at the reaction energies up to 200 MeV. This permits a systematic analysis of correlations among two-nucleon and three-nucleon observables, basing on a relatively big sample of predictions. We found that most observables in neutron-proton and neutron-deuteron systems are uncorrelated, but there are exceptions revealing strong correlations, which depend on the reaction energy and scattering angle. This information may be useful for precise fixing free parameters of two-nucleon and three-nucleon forces and for understanding dependencies and correlations between potential parameters and observables.

Deuteron–deuteron elastic scattering with three-nucleon force effect in complete Glauber multiple scattering theory

In the framework of Glauber multiple scattering theory including the three-nucleon force, we can well describe the experimental measurements of deuteron–deuteron elastic scattering differential cross-section at momenta [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] GeV/c and center of mass energies [Formula: see text] and [Formula: see text] GeV. Due to relatively small radius of three-nucleon force it has no effect in the forward direction where [Formula: see text] (GeV/c)2 at any energy. The two-nucleon force is enough to describe the data where the single scattering terms are dominant. At medium energy, the effect of three-nucleon force is clear for [Formula: see text] (GeV/c)2. At the same time, for intersecting storage rings (ISR) energies, the three-nucleon force effect is clearly observed at the minimum of the differential cross-section and for the other values of [Formula: see text] (GeV/c)2 is, in general, small.

Effect of Nucleon Dressing on Triton Binding Energy

The effect of nucleon dressing by pions, on the binding energy of three nucleons interacting via two-body forces, is calculated for the first time within a conventional nuclear physics approach. It is found that the dressing increases the binding energy of the triton by an amount approximately in the range from 0.3 MeV to 0.9 MeV, depending on the model used for dressing. This suggests that nucleon dressing may help explain the underestimation of the triton binding energy in previous calculations using only two-nucleon forces.

Temperature effects on the neutron matter equation of state obtained from chiral effective field theory

Temperature effects on the neutron matter equation of state (EoS) are investigated in the framework of chiral effective field theory. Recently, state-of-the-art chiral two-nucleon forces are applied from third to fifth order in the chiral expansion together with chiral three-nucleon forces, allowing for a determination of the truncation error of the theoretical predictions. The thermodynamic quantities considered include the chemical potential, the internal energy, the entropy, and the free energy. In general, good order-by-order convergence of all predictions is observed. As to be expected, temperature effects are largest at low density. The temperature dependence of the chiral three-nucleon force turns out to be weak.

Studying the coherent channel of neutral current ν-nucleus interaction

Original cross-sections calculations for neutral current neutrino scattering on 40Ar isotope are performed in the context of the quasi-particle random phase approximation (QRPA) by utilizing realistic two-nucleon forces. The incoming neutrino energy range adopted, εν ≤ 100 MeV, covers the supernova neutrinos, the low-energy beta-beam-neutrinos and the pion-muon stopped neutrino-beams existing or planned to be conducted at future neutron spallation sources. Subsequently, are the original cross sections convoluted with various supernova neutrino-energy distributions such as the two-parameter Fermi-Dirac and the power law distributions. The folded cross sections are obtained for various values of the parameters of these neutrino energy distributions corresponding to different supernova scenarios. One of the main purposes of this work is to explore the response of the 40Ar isotope as supernova neutrino detector.

The Tjon band in Nuclear Lattice Effective Field Theory

We explore the lattice spacing dependence in Nuclear Lattice Effective Field Theory for few-body systems up to next-to-next-to leading order in chiral effective field theory including all isospin breaking and electromagnetic effects, the complete two-pion-exchange potential and the three-nucleon forces. We calculate phase shifts in the neutron-proton system and proton-proton systems as well as the scattering length in the neutron-neutron system. We then perform a full next-to-next-to-leading order calculation with two-nucleon and three-nucleon forces for the triton and helium-4 and analyse their binding energy correlation. We show how the Tjon band is reached by decreasing the lattice spacing and confirm the continuum observation that a four-body force is not necessary to describe light nuclei.

Propagation of Uncertainties of the Nucleon-Nucleon Potential to the Neutron-Deuteron Elastic Scattering Cross Section

Faddeev formalism is used to study the propagation of theoretical uncertainties from the two-nucleon force to three-nucleon scattering observables. Predictions are obtained with the One-Pion-Exchange Gaussian interaction, for which correlations between its parameters are known. Within the Monte Carlo approach we are able to estimate uncertainties of three-nucleon observables arising from the imprecise knowledge of the One-Pion-Exchange Gaussian potential parameters. We found that the uncertainties of this type are small for three-nucleon elastic scattering cross section at investigated here energies up to the nucleon laboratory energy of 200 MeV. They remain smaller than the dominant theoretical uncertainty arising from using various models of nucleon-nucleon interactions. We also compare the above-mentioned results with other types of theoretical uncertainties, that is with the ones stemming from order truncation errors and regulator dependencies, present in calculations based on the chiral interaction.

Neutron-proton scattering at next-to-next-to-leading order in Nuclear Lattice Effective Field Theory

We present a systematic study of neutron-proton scattering in Nuclear Lattice Effective Field Theory (NLEFT), in terms of the computationally efficient radial Hamiltonian method. Our leading-order (LO) interaction consists of smeared, local contact terms and static one-pion exchange. We show results for a fully non-perturbative analysis up to next-to-next-to-leading order (NNLO), followed by a perturbative treatment of contributions beyond LO. The latter analysis anticipates practical Monte Carlo simulations of heavier nuclei. We explore how our results depend on the lattice spacing a, and estimate sources of uncertainty in the determination of the low-energy constants of the next-to-leading-order (NLO) two-nucleon force. We give results for lattice spacings ranging from a = 1.97 fm down to a = 0.98 fm, and discuss the effects of lattice artifacts on the scattering observables. At a = 0.98 fm, lattice artifacts appear small, and our NNLO results agree well with the Nijmegen partial-wave analysis for S-wave and P-wave channels. We expect the peripheral partial waves to be equally well described once the lattice momenta in the pion-nucleon coupling are taken to coincide with the continuum dispersion relation, and higher-order (N3LO) contributions are included. We stress that for center-of-mass momenta below 100 MeV, the physics of the two-nucleon system is independent of the lattice spacing.

Neutron properties from light nuclei

We review the achievements of the project B.6 . Topics addressed include pion photoproduction of three-nucleon systems, electromagnetic and axial currents in chiral nuclear EFT, aspects of EFTs with spin-3/2 fields, nuclear structure and dynamics from the pionless EFT, the development and applications of high-precision two-nucleon forces and pion production in nucleon-nucleon collisions.

Contributions of chiral two- and three-nucleon interactions to closed shell nuclei

Using two-nucleon force alone is usually inadequate to interpret nuclear systems’ experimental data. We adopt a chiral N 3 LO two-nucleon potential V 2N with the inclusion of an in-medium three-nucleon (NNN) force V 3N to calculate binding energies of closed-shell nuclei. The matrix elements of low momentum nucleon-nucleon potential Vlow−k obtained from integrating the high momentum part of a realistic potentials is inputted in the particle-particle hole-hole ring diagram calculation to study nuclear properties. Nuclear binding energies are accurately reproduced. Without this three-nucleon force, the nuclear binding energy is too weak, as already known. The correction from ring diagrams of order higher than 1 can not be ignored.

Chiral nucleon-nucleon forces in nuclear structure calculations

Realistic nuclear potentials, derived within chiral perturbation theory, are a major breakthrough in modern nuclear structure theory, since they provide a direct link between nuclear physics and its underlying theory, namely the QCD. As a matter of fact, chiral potentials are tailored on the low-energy regime of nuclear structure physics, and chiral perturbation theory provides on the same footing two-nucleon forces as well as many-body ones. This feature fits well with modern advances in ab-initio methods and realistic shell-model. Here, we will review recent nuclear structure calculations, based on realistic chiral potentials, for both finite nuclei and infinite nuclear matter.

Microscopic calculations based on chiral two- and three-nucleon forces for proton- andHe4-nucleus scattering

We investigate the effects of chiral three-nucleon force (3NF) on proton scattering at 65 MeV and $^{4}$He scattering at 72 MeV/nucleon from heavier targets, using the standard microscopic framework composed of the Brueckner-Hartree-Fock (BHF) method and the $g$-matrix folding model. For nuclear matter, the $g$ matrix is evaluated from chiral two-nucleon force (2NF) of N$^{3}$LO and chiral 3NF of NNLO by using the BHF method. Since the $g$ matrix thus obtained is numerical and nonlocal, an optimum local form is determined from the on-shell and near-on-shell components of $g$ matrix that are important for elastic scattering. For elastic scattering, the optical potentials are calculated by folding the local chiral $g$ matrix with projectile and target densities. This microscopic framework reproduces the experimental data without introducing any adjustable parameter. Chiral-3NF effects are small for proton scattering, but sizable for $^{4}$He scattering at middle angles where the data are available. Chiral 3NF, mainly in the 2$\pi$-exchange diagram, makes the folding potential less attractive and more absorptive for all the scattering.

Toward order-by-order calculations of the nuclear and neutron matter equations of state in chiral effective field theory

We calculate the nuclear and neutron matter equations of state from microscopic nuclear forces at different orders in chiral effective field theory and with varying momentum-space cutoff scales. We focus attention on how the order-by-order convergence depends on the choice of resolution scale and the implications for theoretical uncertainty estimates on the isospin asymmetry energy. Specifically we study the equations of state using consistent NLO and N2LO (next-to-next-to-leading order) chiral potentials where the low-energy constants cD and cE associated with contact vertices in the N2LO chiral three-nucleon force are fitted to reproduce the binding energies of 3H and 3He as well as the beta-decay lifetime of 3H. At these low orders in the chiral expansion there is little sign of convergence, while an exploratory study employing the N3LO two-nucleon force together with the N2LO three-nucleon force give first indications for (slow) convergence with low-cutoff potentials and poor convergence with higher-cutoff potentials. The consistent NLO and N2LO potentials described in the present work provide the basis for estimating theoretical uncertainties associated with the order-by-order convergence of nuclear many-body calculations in chiral effective field theory.

Accurate nuclear radii and binding energies from a chiral interaction

With the goal of developing predictive ab-initio capability for light and medium-mass nuclei, two-nucleon and three-nucleon forces from chiral effective field theory are optimized simultaneously to low-energy nucleon-nucleon scattering data, as well as binding energies and radii of few-nucleon systems and selected isotopes of carbon and oxygen. Coupled-cluster calculations based on this interaction, named NNLO$_{\rm sat}$, yield accurate binding energies and radii of nuclei up to $^{40}$Ca, and are consistent with the empirical saturation point of symmetric nuclear matter. In addition, the low-lying collective $J^\pi=3^-$ states in $^{16}$O and $^{40}$Ca are described accurately, while spectra for selected $p$- and $sd$-shell nuclei are in reasonable agreement with experiment.

Roles of chiral three-nucleon forces in nucleon–nucleus scattering

We investigate the effects of chiral three-nucleon force (3NF) at NNLO level on nucleon–nucleus (NA) elastic scattering, using the standard framework based on the Brueckner–Hartree–Fock method for nuclear matter and the g-matrix folding model for NA elastic scattering. The optical potential in nuclear matter calculated from chiral two-nucleon force (2NF) at N3LO level is found to be close to that from Bonn-B 2NF, whereas the Melbourne g-matrix is known as a practical effective nucleon–nucleon interaction constructed by localizing the g-matrices calculated from Bonn-B 2NF. As the first attempt to estimate chiral-3NF effects on NA scattering, the effects are simply introduced by multiplying the local Melbourne g-matrix by the ratio of the optical potential in nuclear matter calculated from chiral 2NF+3NF to that from chiral 2NF. For NA elastic scattering on various targets at 65 MeV, chiral 3NF makes the folding potential less attractive and more absorptive. The novel property for the imaginary part is originated in the enhancement of tensor correlations due to chiral 3NF (mainly the 2π-exchange diagram). The two effects are small for differential cross sections and vector analyzing powers at the forward and middle angles where the experimental data are available. If backward measurements are made, the data will reveal the effects of chiral 3NF.

Effects of a chiral three-nucleon force on nucleus-nucleus scattering

We investigate the effects of chiral NNLO three-nucleon force (3NF) on nucleus-nucleus elastic scattering, using a standard prescription based on the Brueckner-Hartree-Fock method and the g-matrix folding model. The g-matrix calculated in nuclear matter from the chiral N3LO two-nucleon forces (2NF) is close to that from the Bonn-B 2NF. Because the Melbourne group have already developed a practical g-matrix interaction by localizing the nonlocal g-matrix calculated from the Bonn-B 2NF, we consider the effects of chiral 3NF, in this first attempt to study the 3NF effects, by modifying the local Melbourne g-matrix according to the difference between the g-matrices of the chiral 2NF and 2NF+3NF. For nucleus-nucleus elastic scattering, the 3NF corrections make the folding potential less attractive and more absorptive. The latter novel effect is due to the enhanced tensor correlations in triplet channels. These changes reduce the differential cross section at the middle and large angles, improving the agreement with the experimental data for 16O-16O scattering at 70 MeV/nucleon and 12C-12C scattering at 85 MeV/nucleon.

Three-nucleon forces

In this paper, the role of three-nucleon forces in ab initio calculations of nuclear systems is investigated. The difference between genuine and induced many-nucleon forces is emphasized. Induced forces arise in the process of solving the nuclear many-body problem as technical intermediaries toward calculationally converged results. Genuine forces make up the Hamiltonian. They represent the chosen underlying dynamics. The hierarchy of contributions arising from genuine two-, three- and many-nucleon forces is discussed. Signals for the need of the inclusion of genuine three-nucleon forces are studied in nuclear systems, technically best under control, especially in three-nucleon and four-nucleon systems. Genuine three-nucleon forces are important for details in the description of some observables. Their contributions to observables are small on the scale set by two-nucleon forces.