Three-nucleon Interactions Research Articles

Large-cutoff behavior of local chiral effective field theory interactions

Interactions from chiral effective field theory have been successfully employed in a broad range of \textit{ab initio} calculations of nuclei and nuclear matter, but it has been observed that most results of few- and many-body calculations experience a substantial residual regulator and cutoff dependence. In this work, we investigate the behavior of local chiral potentials at leading order under variation of the cutoff scale for different local regulators. When varying the cutoff, we require that the resulting interaction produces no spurious bound states in the deuteron channel. We find that, for a particular choice of leading-order operators, nucleon-nucleon phase shifts and the deuteron ground-state energy converge to cutoff-independent plateaus, for all regulator functions we investigate. This observation may enable improved calculations with chiral Hamiltonians that also include three-nucleon interactions.

Optimized chiral N2LO interactions in nuclear matter

We employ modern two- and three-nucleon interactions derived in chiral perturbation theory (ChPT) at next to-next-to leading order (N2LO), to calculate the energy per particle of symmetric nuclear matter and pure neutron matter in the framework of the microscopic Brueckner-Hartree-Fock approach. In particular, we present results concerning two optimized versions at N2LO of chiral potentials ( $ \mathrm{N2LO}_{opt}$ ), fitted to properties of light nuclei. We also employ the recently developed $\mathrm{N2LO}_{sat}$ interaction which has been calculated at the same order of ChPT but fitted in a very different way compared with the $\mathrm{N2LO}_{opt}$ interactions. We find that using these potentials, in general, it is not possible to reproduce all the saturation properties of nuclear matter. In particular the behaviour of the symmetry energy ( $E_{sym}$ predicted by the interactions considered is quite soft. This is shown comparing our results with the empirical constraints on $E_{sym}$ obtained from the data analysis of the excitation energies of isobaric analog states in nuclei and from experimental data of the neutron skin thickness of heavy nuclei. We finally confront our results with similar calculations performed by other research groups using nuclear chiral interactions at various order of ChPT and employing different many-body methods.

Unitary scheme model calculations of the ground and excited state characteristics of 3H and 4He

The ground and excited states of the 3H and 4He nuclei are studied in the framework of group-theoretical methods. Basis functions of the unitary scheme model corresponding to even numbers of quanta of excitation in the range 0 ≤ N ≤ 20 are constructed for the even-parity states of these two nuclei, and bases with 1 ≤ N ≤ 19 are constructed for the odd-parity states of the 4He nucleus. The ground-state and first-excited-state energies and wave functions, the ground S-, P- and D-state probabilities, the root-mean-square radius and the magnetic dipole moment of triton are calculated. Furthermore, for the 4He nucleus, the spectrum and the wave functions, the ground S-, P- and D-state probabilities, the root-mean-square radius and the total integral cross section of the dipole electric transition accompanying the photoabsorption of γ quanta by this nucleus are calculated. The GPT and AV8′ two-body interactions and the UIX three-nucleon interaction are used in these investigations. Moreover, the convergence of the calculations is examined by incrementally extrapolating the nuclear characteristics calculated for N ≤ 20 to N = 30 for both nuclei.

Origin of fine structure of the giant dipole resonance in sd -shell nuclei

A set of high resolution zero-degree inelastic proton scattering data on 24Mg, 28Si, 32S, and 40Ca provides new insight into the long-standing puzzle of the origin of fragmentation of the Giant Dipole Resonance (GDR) in sd-shell nuclei. Understanding is provided by state-of-the-art theoretical Random Phase Approximation (RPA) calculatios for deformed nuclei using for the first time a realistic nucleon-nucleon interaction derived from the Argonne V18 potential with the unitary correlation operator method and supplemented by a phenomenological three-nucleon contact interaction. A wavelet analysis allows to extract significant scales both in the data and calculations characterizing the fine structure of the GDR. The fair agreement supports that the fine structure arises from ground-state deformation driven by alpha clustering.

Auxiliary field diffusion Monte Carlo calculations of light and medium-mass nuclei with local chiral interactions

Quantum Monte Carlo methods have recently been employed to study properties of nuclei and infinite matter using local chiral effective field theory interactions. In this work, we present a detailed description of the auxiliary field diffusion Monte Carlo algorithm for nuclei in combination with local chiral two- and three-nucleon interactions up to next-to-next-to-leading order. We show results for the binding energy, charge radius, charge form factor, and Coulomb sum rule in nuclei with $3\le A\le16$. Particular attention is devoted to the effect of different operator structures in the three-body force for different cutoffs. The outcomes suggest that local chiral interactions fit to few-body observables give a very good description of the ground-state properties of nuclei up to $^{16}$O, with the exception of one fit for the softer cutoff which predicts overbinding in larger nuclei.

Equation of state of asymmetric nuclear matter using re-projected nucleon–nucleon potentials

In this paper, we have calculated the equation of state of asymmetric nuclear matter using the lowest order constrained variational approach and Argonne family potentials with and without three-nucleon interaction (TNI) contribution. In particular, we have used the AV18 potential and the re-projected potentials, AV8′, and AV6′. We have also calculated the saturation properties of symmetric nuclear matter, and the nuclear symmetry energy using AV18+TNI, AV8′+TNI and AV6′+TNI potentials. The inclusion of TNI has modified the agreement with experiment. We have also made a comparison between our results and those of other many-body calculations.

Microscopic optical potentials derived from ab initio translationally invariant nonlocal one-body densities

We derive a microscopic optical potential for intermediate energies using ab initio translationally invariant nonlocal one-body nuclear densities computed within the no-core shell model (NCSM) approach utilizing two- and three-nucleon chiral interactions as the only input. The optical potential is derived at first-order within the spectator expansion of the non-relativistic multiple scattering theory by adopting the impulse approximation and using the same chiral nucleon-nucleon interaction as that used to compute densities. The ground state local and nonlocal densities of 4,6,8He, 12C, and 16O are calculated and applied to optical potential construction. The differential cross sections and the analyzing powers for the elastic proton scattering off of these nuclei are then calculated for different values of the incident proton energy. The impact of nonlocality and the COM removal is discussed. The use of nonlocal densities has a substantial impact on the differential cross sections and improves agreement with experiment in comparison to results generated with the local densities especially for light nuclei.

Properties of Nuclei up to A=16 using Local Chiral Interactions.

We report accurate quantum MonteCarlo calculations of nuclei up to A=16 based on local chiral two- and three-nucleon interactions up to next-to-next-to-leading order. We examine the theoretical uncertainties associated with the chiral expansion and the cutoff in the theory, as well as the associated operator choices in the three-nucleon interactions. While in light nuclei the cutoff variation and systematic uncertainties are rather small, in ^{16}O these can be significant for large coordinate-space cutoffs. Overall, we show that chiral interactions constructed to reproduce properties of very light systems and nucleon-nucleon scattering give an excellent description of binding energies, charge radii, and form factors for all these nuclei, including open-shell systems in A=6 and 12.

Nuclear matter calculations with chiral interactions

Using two-nucleon and three-nucleon interactions derived in the framework of chiral perturbation theory (ChPT) with and without the explicit Δ isobar contributions, we calculate the energy per particle of symmetric nuclear matter and pure neutron matter employing the microscopic Brueckner–Hartree–Fock approach. Specifically, we present nuclear matter calculations using the new fully local in coordinate-space two-nucleon interaction at the next-to-next-to-next-to-leading-order (N3LO) of ChPT with Δ isobar intermediate states (N3LOΔ) recently developed by Piarulli et al. [1] supplemented with a local N2LO three-nucleon interaction with explicit Δ isobar degrees of freedom. We show that for this combination of two- and three-nucleon interactions it is possible to obtain a good saturation point of symmetric nuclear matter. We also calculate the nuclear symmetry energy and compare our results with the available empirical constraints on this quantity.

Quantum Monte Carlo calculations of weak transitions in A=6–10 nuclei

Ab initio calculations of the Gamow-Teller (GT) matrix elements in the $\beta$ decays of $^6$He and $^{10}$C and electron captures in $^7$Be are carried out using both variational and Green's function Monte Carlo wave functions obtained from the Argonne $v_{18}$ two-nucleon and Illinois-7 three-nucleon interactions, and axial many-body currents derived from either meson-exchange phenomenology or chiral effective field theory. The agreement with experimental data is excellent for the electron captures in $^7$Be, while theory overestimates the $^6$He and $^{10}$C data by $\sim 2\%$ and $\sim 10\%$, respectively. We show that for these systems correlations in the nuclear wave functions are crucial to explain the data, while many-body currents increase by $\sim 2$--$3\%$ the one-body GT contributions. These findings suggest that the longstanding $g_A$-problem, i.e., the systematic overprediction ($\sim 20 \%$ in $A\le 18$ nuclei) of GT matrix elements in shell-model calculations, may be resolved, at least partially, by correlation effects.

Δ isobars and nuclear saturation

We construct a nuclear interaction in chiral effective field theory with explicit inclusion of the $\Delta$-isobar $\Delta(1232)$ degree of freedom at all orders up to next-to-next-to-leading order (NNLO). We use pion-nucleon ($\pi N$) low-energy constants (LECs) from a Roy-Steiner analysis of $\pi N$ scattering data, optimize the LECs in the contact potentials up to NNLO to reproduce low-energy nucleon-nucleon scattering phase shifts, and constrain the three-nucleon interaction at NNLO to reproduce the binding energy and point-proton radius of $^{4}$He. For heavier nuclei we use the coupled-cluster method to compute binding energies, radii, and neutron skins. We find that radii and binding energies are much improved for interactions with explicit inclusion of $\Delta(1232)$, while $\Delta$-less interactions produce nuclei that are not bound with respect to breakup into $\alpha$ particles. The saturation of nuclear matter is significantly improved, and its symmetry energy is consistent with empirical estimates.

Neutrinoless double- β decay matrix elements in light nuclei

We present the first ab initio calculations of neutrinoless double beta decay matrix elements in $A=6$-$12$ nuclei using Variational Monte Carlo wave functions obtained from the Argonne $v_{18}$ two-nucleon potential and Illinois-7 three-nucleon interaction. We study both light Majorana neutrino exchange and potentials arising from a large class of multi-TeV mechanisms of lepton number violation. Our results provide benchmarks to be used in testing many-body methods that can be extended to the heavy nuclei of experimental interest. In light nuclei we have also studied the impact of two-body short range correlations and the use of different forms for the transition operators, such as those corresponding to different orders in chiral effective theory.

Spin–isospin excitation of 3He with three-proton final state

Spin-isospin excitation of ${}^3$He nucleus by proton-induced charge-exchange reaction, ${}^3\mathrm{He}(p,n)ppp$, at forward neutron scattering angle is studied in a plane wave impulse approximation (PWIA). In PWIA, cross sections of the reaction is written in terms of proton-neutron scattering amplitudes and response functions of the transition from ${}^{3}$He to three-proton state by spin-isospin transition operators. The response functions are calculated with realistic nucleon-nucleon potential models using a Faddeev three-body method. Calculated cross sections agree with available experimental data in substance. Possible effects arising from the uncertainty of proton-neutron amplitudes and three-nucleon interactions in three-proton system are examined.

Quantum Monte Carlo calculations of light nuclei with local chiral two- and three-nucleon interactions

Local chiral effective field theory interactions have recently been developed and used in the context of quantum Monte Carlo few- and many-body methods for nuclear physics. In this work, we go over detailed features of local chiral nucleon-nucleon interactions and examine their effect on properties of the deuteron, paying special attention to the perturbativeness of the expansion. We then turn to three-nucleon interactions, focusing on operator ambiguities and their interplay with regulator effects. We then discuss the nuclear Green's function Monte Carlo method, going over both wave-function correlations and approximations for the two- and three-body propagators. Following this, we present a range of results on light nuclei: Binding energies and distribution functions are contrasted and compared, starting from several different microscopic interactions.

Erratum: Nuclear and neutron matter G -matrix calculations with a chiral effective field theory potential including effects of three-nucleon interactions [Phys. Rev. C 88 , 064005 (2013)

Erratum: Nuclear and neutron matter <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>G</mml:mi></mml:math> -matrix calculations with a chiral effective field theory potential including effects of three-nucleon interactions [Phys. Rev. C <b>88</b> , 064005 (2013)

Analyzing the Fierz rearrangement freedom for local chiral two-nucleon potentials

Chiral effective field theory is a framework to derive systematic nuclear interactions. It is based on the symmetries of quantum chromodynamics and includes long-range pion physics explicitly, while shorter-range physics is expanded in a general operator basis. The number of low-energy couplings at a particular order in the expansion can be reduced by exploiting the fact that nucleons are fermions and therefore obey the Pauli exclusion principle. The antisymmetry permits the selection of a subset of the allowed contact operators at a given order. When local regulators are used for these short-range interactions, however, this "Fierz rearrangement freedom" is violated. In this paper, we investigate the impact of this violation at leading order (LO) in the chiral expansion. We construct LO and next-to-leading order (NLO) potentials for all possible LO-operator pairs and study their reproduction of phase shifts, the ${}^4$He ground-state energy, and the neutron-matter energy at different densities. We demonstrate that the Fierz rearrangement freedom is partially restored at NLO where subleading contact interactions enter. We also discuss implications for local chiral three-nucleon interactions.

Three-nucleon interaction in heavy-ion collisions at intermediate energies

The effect of a three-nucleon ($3N$) interaction is studied for the production of high energy protons in heavy-ion collisions in the incident energy range of 44 to $400\phantom{\rule{0.16em}{0ex}}A\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$. The $3N$ interaction is incorporated into the antisymmetrized molecular dynamics transport model of Ono [A. Ono, Phys. Rev. C 59, 853 (1999)] as a $3N$ collision term, following a diagram of three consecutive binary collisions. For the theoretical $^{40}\mathrm{Ar}+^{51}\mathrm{V}$ reaction studies, no contribution from the $3N$ collisions is observed for high energy proton production at the incident energy of $44A\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$. However when the incident energy increases, the contribution increases gradually. At $200A\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$ and above, the contribution is observed as distinctly harder energy slopes in the proton energy spectra. The model is applied to the available Bevalac data for $^{40}\mathrm{Ar}+^{40}\mathrm{Ca}$ at $42,92$, and $137\phantom{\rule{0.16em}{0ex}}A\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$. The experimental proton energy spectra are reasonably well reproduced at angles $\ensuremath{\theta}\ensuremath{\ge}{70}^{\ensuremath{\circ}}$ for all three incident energies, showing negligible $3N$ contributions at $42A\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$ and significant contributions at $137A\phantom{\rule{0.16em}{0ex}}\mathrm{MeV}$ at the large laboratory angles. Good agreement at these large angles, where the $3N$ collision is a major mechanism to produce such protons, strongly indicates for the first time the importance of the $3N$ interaction in intermediate heavy ion reactions in a full transport calculation. The possible relation between the $3N$ collision term and the short range and the tensor interactions is suggested.

Three-Nucleon Forces and Triplet Pairing in Neutron Matter

The existence of superfluidity of the neutron component in the core of a neutron star, associated specifically with triplet $P-$wave pairing, is currently an open question that is central to interpretation of the observed cooling curves and other neutron-star observables. Ab initio theoretical calculations aimed at resolving this issue face unique challenges in the relevant high-density domain, which reaches beyond the saturation density of symmetrical nuclear matter. These issues include uncertainties in the three-nucleon (3N) interaction and in the effects of strong short-range correlations -- and more generally of in-medium modification of nucleonic self-energies and interactions. A survey of existing solutions to the gap equations in the triplet channel shows that the separate or combined impacts of 3N forces, coupled channels, and mass renormalization range from moderate to strong to devastating, thus motivating a detailed analysis of the competing effects. In the present work we track the effects of the 3N force and in-medium modifications in the representative case of the $^3P_2$ channel, based on the Argonne V18 two-nucleon (2N) interaction supplemented by 3N interactions of the Urbana IX family. Sensitivity of the results to the input interaction is clearly demonstrated, while consistency issues arise with respect to the simultaneous treatment of 3N forces and in-medium effects. We consider this pilot study as the first step towards a systematic and comprehensive exploration of coupled-channel $^3P F_2$ pairing using a broad range of 2N and 3N interactions from the current generation of refined semi-phenomenological models and models derived from chiral effective field theory.

Variational calculation of the ground state of closed-shell nuclei up to A=40

Variational calculations of ground-state properties of $^4$He, $^{16}$O, and $^{40}$Ca are carried out employing realistic phenomenological two- and three-nucleon potentials. The trial wave function includes two- and three-body correlations acting on a product of single-particle determinants. Expectation values are evaluated with a cluster expansion for the spin-isospin dependent correlations considering up to five-body cluster terms. The optimal wave function is obtained by minimizing the energy expectation value over a set of up to 20 parameters by means of a nonlinear optimization library. We present results for the binding energy, charge radius, one- and two-body densities, single-nucleon momentum distribution, charge form factor, and Coulomb sum rule. We find that the employed three-nucleon interaction becomes repulsive for $A\geq16$. In $^{16}$O the inclusion of such a force provides a better description of the properties of the nucleus. In $^{40}$Ca instead, the repulsive behavior of the three-body interaction fails to reproduce experimental data for the charge radius and the charge form factor. We find that the high-momentum region of the momentum distributions, determined by the short-range terms of nuclear correlations, exhibit a universal behavior independent of the particular nucleus. The comparison of the Coulomb sum rules for $^4$He, $^{16}$O, and $^{40}$Ca reported in this work will help elucidate in-medium modifications of the nucleon form factors.

Saturation with chiral interactions and consequences for finite nuclei

We explore the impact of nuclear matter saturation on the properties and systematics of finite nuclei across the nuclear chart. Using the ab initio in-medium similarity renormalization group (IM-SRG), we study ground-state energies and charge radii of closed-shell nuclei from $^4$He to $^{78}$Ni, based on a set of low-resolution two- and three-nucleon interactions that predict realistic saturation properties. We first investigate in detail the convergence properties of these Hamiltonians with respect to model-space truncations for both two- and three-body interactions. We find one particular interaction that reproduces well the ground-state energies of all closed-shell nuclei studied. As expected from their saturation points relative to this interaction, the other Hamiltonians underbind nuclei, but lead to a remarkably similar systematics of ground-state energies. Extending our calculations to complete isotopic chains in the $sd$ and $pf$ shells with the valence-space IM-SRG, the same interaction reproduces not only experimental ground states but two-neutron-separation energies and first excited $2^+$ states. We also calculate radii with the valence-space IM-SRG for the first time. Since this particular interaction saturates at too high density, charge radii are still too small compared with experiment. Except for this underprediction, the radii systematics is, however, well reproduced. Our results highlight the importance of nuclear matter as a theoretical benchmark for the development of next-generation chiral interactions.