Chiral Three-nucleon Forces Research Articles

Mass Measurement of Upper fp-Shell N=Z-2 and N=Z-1 Nuclei and the Importance of Three-Nucleon Force along the N=Z Line.

Using a novel method of isochronous mass spectrometry, the masses of ^{62}Ge, ^{64}As, ^{66}Se, and ^{70}Kr are measured for the first time, and the masses of ^{58}Zn, ^{61}Ga, ^{63}Ge, ^{65}As, ^{67}Se, ^{71}Kr, and ^{75}Sr are redetermined with improved accuracy. The new masses allow us to derive residual proton-neutron interactions (δV_{pn}) in the N=Z nuclei, which are found to decrease (increase) with increasing mass A for even-even (odd-odd) nuclei beyond Z=28. This bifurcation of δV_{pn} cannot be reproduced by the available mass models, nor is it consistent with expectations of a pseudo-SU(4) symmetry restoration in the fp shell. We performed abinitio calculations with a chiral three-nucleon force (3NF) included, which indicate the enhancement of the T=1 pn pairing over the T=0 pn pairing in this mass region, leading to the opposite evolving trends of δV_{pn} in even-even and odd-odd nuclei.

Significance of chiral three-nucleon force contact terms for understanding of elastic nucleon-deuteron scattering

We investigate the importance of the three-nucleon (3N) force contact terms in elastic nucleon-deuteron (Nd) scattering by applying the N$^4$LO$^+$ chiral semi-local momentum space (SMS) regularized nucleon-nucleon (NN) chiral potential supplemented by N$^2$LO and all subleading N$^4$LO three-nucleon force (3NF) contact terms. Strength parameters of the contact terms were obtained by least squares fitting of theoretical predictions to cross section and analyzing powers data at three energies of the impinging nucleon. Although the N$^3$LO contributions to the 3N force were completely neglected, the results calculated with the contact terms multiplied by the fitted strength parameters yield an improved description of the elastic Nd scattering observables in a wide range of incoming nucleon energies below the pion production threshold.

Efficient emulator for solving three-nucleon continuum Faddeev equations with chiral three-nucleon force comprising any number of contact terms

We demonstrate a computational scheme which drastically decreases the required time to get theoretical predictions based on chiral two- and three-nucleon forces for observables in three-nucleon continuum. For a three-nucleon force containing N short-range terms all workload is reduced to solving N+1 Faddeev-type integral equations. That done, computation of observables for any combination of strengths of the contact terms is done in a flash. We demonstrate on example of the elastic nucleon-deuteron scattering observables the high precision of the proposed emulator and its capability to reproduce exact results.

Perturbative Treatment of Three-Nucleon Force Contact Terms in Three-Nucleon Faddeev Equations

We present a perturbative approach to solving the three-nucleon continuum Faddeev equation. This approach is particularly well suited to dealing with variable strengths of contact terms in a chiral three-nucleon force. We use examples of observables in the elastic nucleon-deuteron scattering as well as in the deuteron breakup reaction to demonstrate high precision of the proposed procedure and its capability to reproduce exact results. A significant reduction of computer time achieved by the perturbative approach in comparison to exact treatment makes this approach valuable for fine-tuning of the three-nucleon Hamiltonian parameters.

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.

Nuclear multipole responses from chiral effective field theory interactions

We probe nuclear multipole resonances in the framework of the random-phase approximation by using the interaction obtained from the chiral effective field theory. The three-nucleon force is included in a form of the in-medium two-nucleon interaction which was derived from the chiral three-nucleon force. The isoscalar monopole, isoscalar dipole, isovector dipole and isoscalar quadrupole resonances of the closed-shell $^{56,68,78}$Ni have been investigated. The calculations reasonably reproduce the experimental multipole resonances of $^{56,68}$Ni, and well describe the pygmy dipole resonance and dipole polarizability measured in $^{68}$Ni. The multipole resonances of $^{78}$Ni, including pygmy dipole resonance and dipole polarizability, are predicted. The detailed effects of the tensor force and three-body force are analyzed by dissecting the chiral interaction. We find that in general the tensor force effect on electric giant resonances is not as significant as the effect from the three-body force, although the tensor force provides more than half of the binding energy. The effect from three-body force is strong in light nuclei. Particularly, three-body force is crucial for the formation of the pygmy resonance in calculations.

Chiral three-nucleon force and continuum for dripline nuclei and beyond

Three-nucleon force and continuum play important roles in reproducing the properties of atomic nuclei around driplines. Therefore it is valuable to build up a theoretical framework where both effects can be taken into account to solve the nuclear Schrödinger equation. To this end, in this letter, we have expressed the chiral three-nucleon force within the continuum Berggren representation, so that bound, resonant and continuum states can be treated on an equal footing in the complex-momentum space. To reduce the model dimension and computational cost, the three-nucleon force is truncated at the normal-ordered two-body level and limited in the sd-shell model space, with the residual three-body term being neglected. We choose neutron-rich oxygen isotopes as the test ground because they have been well studied experimentally, with the neutron dripline determined. The calculations have been carried out within the Gamow shell model. The quality of our results in reproducing the properties of oxygen isotopes around the neutron dripline shows the relevance of the interplay between three-nucleon force and the coupling to continuum states. We also analyze the role played by the chiral three-nucleon force, by dissecting the contributions of the 2π exchange, 1π exchange and contact terms.

Density-dependent NN interaction from subsubleading chiral 3N forces: Intermediate-range contributions

From the subsubleading chiral three-nucleon force [intermediate-range contributions, published in Phys. Rev. C\,87, 054007 (2013)] a density-dependent NN-interaction $V_\text{med}$ is derived in isospin-symmetric nuclear matter. Following the division of the pertinent 3N-diagrams into two-pion-one-pion exchange topology and ring topology, one evaluates for these all selfclosings and concatenations of nucleon-lines to an in-medium loop. In the case of the $2\pi 1\pi$-exchange topology, the momentum- and $k_f$-dependent potentials associated with the isospin-operators ($1$ and $\vec\tau_1 \!\cdot\! \vec\tau_2$) and five independent spin-structures require at most one numerical integration. For the more challenging (concatenations of the) ring diagrams proportional to $c_{1,2,3,4}$, one ends up with regularized double-integrals $\int_0^\lambda dr\,r \int_0^{\pi/2} d\psi$ from which the $\lambda^2$-divergence has been subtracted and the logarithmic piece $\sim \ln (m_\pi/\lambda)$ is isolated. The derived semi-analytical results are most helpful to implement the subsubleading chiral 3N-forces into nuclear many-body calculations.

Density-dependent NN interaction from subleading chiral three-nucleon forces: Long-range terms

We derive from the subleading contributions to the chiral three-nucleon (3N) force [long-range terms; published by V. Bernard et al., Phys. Rev. C 77, 064004 (2008)] a density-dependent two-nucleon interaction ${V}_{\text{med}}$ in isospin-symmetric, spin-saturated nuclear matter. Following the division of the pertinent 3N diagrams into two-pion-exchange topology, two-pion-one-pion--exchange topology, and ring topology, we evaluate for these all self-closings and concatenations of nucleon lines to an in-medium loop. The momentum and ${k}_{f}$-dependent potentials associated with the isospin operators (1 and ${\stackrel{P\vec}{\ensuremath{\tau}}}_{1}\ifmmode\cdot\else\textperiodcentered\fi{}{\stackrel{P\vec}{\ensuremath{\tau}}}_{2}$) and five independent spin structures are expressed in terms of functions which either are given in closed analytical form or require at most one numerical integration. In the same way we treat the $2\ensuremath{\pi}$-exchange 3N force at sub-subleading order. Our results for ${V}_{\text{med}}$ are most helpful to implement long-range subleading chiral 3N forces into nuclear many-body calculations.

Momentum distributions and short-range correlations in the deuteron and He3 with modern chiral potentials

We study momentum distributions and short-range correlation probabilities in $A$=2 and $A$=3 systems. First, we show results with phenomenological and meson-theoretic two- and three-nucleon forces to verify consistency with previous similar studies. We then apply most recent high-quality chiral nucleon-nucleon potentials up to fifth order in the chiral expansion together with the leading chiral three-nucleon force. Predictions are examined in the context of a broader discussion of short-range correlation probabilities extracted from analyses of inclusive electron scattering data, addressing the question of whether modern interactions can be reconciled with the latter.

Neutron and nuclear matter properties with chiral three-nucleon forces

The equation of state of neutron and symmetric matter is calculated within the lowest order constrained variational (LOCV) method using the Argonne V18 (AV18) nucleon–nucleon interaction supplemented by the local, coordinate space versions of the chiral three-nucleon force (TNF) at next-to-next-to-leading order (N2LOL). We present the first results of studying the nuclear equation of state in the LOCV framework by employing a chiral TNF. It is shown that although some calculated values of saturation properties lie well in the empirically determined ranges, all of the saturation quantities can not be predicted simultaneously. We also study other possible parametrizations of the N2LOL TNF by fitting the low-energy constants (LECs) to nuclear saturation point. It is found that by employing the proposed sets of LECs, LOCV method is able to successfully produce the equilibrium properties of symmetric nuclear matter.

The radius of the canonical-mass neutron star and chiral effective field theory

We calculate neutron star masses and radii from equations of state based on recent high-quality chiral nucleon–nucleon potentials up to fifth order of the chiral expansion and the leading chiral three-nucleon force. Our focus is on the radius of a 1.4 M⊙ neutron star, for which we report predictions that are consistent with the most recent constraints. We also show the full M(R) relations up to their respective maximum masses. Beyond the densities for which microscopic predictions are derived from chiral forces, the equations of state are obtained via polytropic extrapolation. However, the radius of a 1.4 M⊙ neutron star is nearly insensitive to the high-density extrapolation.

Density-dependent NN interaction from subleading chiral 3N forces: Short-range terms and relativistic corrections

We derive from the subleading contributions to the chiral three-nucleon force (short-range terms and relativistic corrections, published in Phys. Rev. C84, 054001 (2011)) a density-dependent two-nucleon interaction $V_\text{med}$ in isospin-symmetric nuclear matter. The momentum and $k_f$-dependent potentials associated with the isospin operators ($1$ and $ \vec\tau_1\!\cdot\!\vec\tau_2$) and five independent spin-structures are expressed in terms of loop functions, which are either given in closed analytical form or require at most one numerical integration. Our results for $V_\text{med}$ are most helpful to implement subleading chiral 3N-forces into nuclear many-body calculations.

Effects of chiral three-nucleon forces on 4He-nucleus scattering in a wide range of incident energies

It is a current important subject to clarify properties of chiral three-nucleon forces (3NFs) not only in nuclear matter but also in scattering between finite-size nuclei. Particularly for the elastic scattering, this study has just started and the properties are not understood in a wide range of incident energies ($E_{\rm in}$). We investigate basic properties of chiral 3NFs in nuclear matter with positive energies by using the Brueckner-Hartree-Fock method with chiral two-nucleon forces of N$^{3}$LO and 3NFs of NNLO, and analyze effects of chiral 3NFs on $^{4}$He elastic scattering from targets $^{208}$Pb, $^{58}$Ni and $^{40}$Ca over a wide range of $30 \lesssim E_{\rm in}/A_{\rm P} \lesssim 200$ MeV by using the $g$-matrix folding model, where $A_{\rm P}$ is the mass number of projectile. In symmetric nuclear matter with positive energies, chiral 3NFs make the single-particle potential less attractive and more absorptive. The effects mainly come from the Fujita-Miyazawa 2$\pi$-exchange 3NF and slightly become larger as $E_{\rm in}$ increases. These effects persist in the optical potentials of $^{4}$He scattering. As for the differential cross sections of $^{4}$He scattering, chiral-3NF effects are large in $E_{\rm in}/A_{\rm P} \gtrsim 60$ MeV and improve the agreement of the theoretical results with the measured ones. Particularly in $E_{\rm in}/A_{\rm P} \gtrsim 100$ MeV, the folding model reproduces measured differential cross sections pretty well. Cutoff ($\Lambda$) dependence is investigated for both nuclear matter and $^{4}$He scattering by considering two cases of $\Lambda=450$ and $550$ MeV. The uncertainty coming from the dependence is smaller than chiral-3NF effects even at $E_{\rm in}/A_{\rm P}=175$ MeV.

The Role of the Isospin 3/2 Component in Elastic Neutron-Deuteron Scattering and in the Deuteron Breakup Reaction

We discuss the importance of the three-nucleon isospin [Formula: see text] component in elastic neutron-deuteron scattering and in the deuteron breakup reaction. The contribution of this amplitude originates from charge-independence breaking of the nucleon-nucleon potential. We study the magnitude of that contribution to the elastic scattering and breakup observables, taking the Av18 nucleon-nucleon potential alone or combined with the Urbana IX three-nucleon force as well as the locally regularized chiral N4LO nucleon-nucleon potential alone or supplemented by the chiral N2LO three-nucleon force. We find that the isospin [Formula: see text] component is important for the breakup reaction and the proper treatment of charge-independence breaking in this case requires the inclusion of the [Formula: see text] state with isospin [Formula: see text]. For neutron-deuteron elastic scattering the [Formula: see text] contributions are insignificant and charge-independence breaking can be accounted for by neglecting [Formula: see text] component and using the effective t-matrix generated with the so-called [Formula: see text] rule.

Erratum: Effects of a chiral three-nucleon force on nucleus-nucleus scattering [Phys. Rev. C 90 , 051601(R) (2014)

Erratum: Effects of a chiral three-nucleon force on nucleus-nucleus scattering [Phys. Rev. C <b>90</b> , 051601(R) (2014)

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.

Operator evolution forab initioelectric dipole transitions ofHe4

A goal of nuclear theory is to make quantitative predictions of low-energy nuclear observables starting from accurate microscopic internucleon forces. A major element of such an effort is applying unitary transformations to soften the nuclear Hamiltonian and hence accelerate the convergence of ab initio calculations as a function of the model space size. The consistent simultaneous transformation of external operators, however, has been overlooked in applications of the theory, particularly for nonscalar transitions. We study the evolution of the electric dipole operator in the framework of the similarity renormalization group method and apply the renormalized matrix elements to the calculation of the $^{4}\mathrm{He}$ total photoabsorption cross section and electric dipole polarizability. All observables are calculated within the ab initio no-core shell model. We find that, although seemingly small, the effects of evolved operators on the photoabsorption cross section are comparable in magnitude to the correction produced by including the chiral three-nucleon force and cannot be neglected.

Chiral three-nucleon forces and the evolution of correlations along the oxygen isotopic chain

The impact of three-nucleon forces (3NFs) along the oxygen chain is investigated for the spectral distribution for attachment and removal of a nucleon, spectroscopic factors and matter radii. We employ self-consistent Green's function (SCGF) theory which allows a comprehensive calculation of the single particle spectral function. For the closed subshell isotopes, $^{14}$O, $^{16}$O, $^{22}$O, $^{24}$O and $^{28}$O, we perform calculations with the Dyson-ADC(3) method. The remaining open shell isotopes are studied using the newly developed Gorkov-SCGF formalism up to second order. We produce plots for the full-fledged spectral distributions. The spectroscopic factors for the dominant quasiparticle peaks are found to depend very little on the leading order (NNLO) chiral 3NFs. The latters have small impact on the calculated matter radii, which, however are consistently obtained smaller than experiment. Similarly, single particle spectra tend to be diluted with respect to experiment. This effect might hinder, to some extent, the onset of correlations and screen the quenching of calculated spectroscopic factors. The most important effects of 3NFs is thus the fine tuning of the energies for the dominant quasiparticle states, which govern the shell evolution and the position of driplines. Although present chiral NNLO 3NFs interactions do reproduce the binding energies correctly in this mass region, the details of the nuclear wave function remain at odd with the experiment showing too small radii and a too dilute single particle spectrum, similar to what already pointed out for larger masses. This suggests a lack of repulsion in the present model of NN+3N interactions which is mildly apparent already for masses in the A=14--28 range.

Unified description of ^{6}Li structure and deuterium-^{4}He dynamics with chiral two- and three-nucleon forces.

We provide a unified abinitio description of the ^{6}Li ground state and elastic scattering of deuterium (d) on ^{4}He (α) using two- and three-nucleon forces from chiral effective field theory. We analyze the influence of the three-nucleon force and reveal the role of continuum degrees of freedom in shaping the low-lying spectrum of ^{6}Li. The calculation reproduces the empirical binding energy of ^{6}Li, yielding an asymptotic D- to S-state ratio of the ^{6}Li wave function in the d+α configuration of -0.027, in agreement with a determination from ^{6}Li-^{4}He elastic scattering, but overestimates the excitation energy of the 3^{+} state by 350keV. The bulk of the computed differential cross section is in good agreement with data. These results endorse the application of the present approach to the evaluation of the ^{2}H(α,γ)^{6}Li radiative capture, responsible for the big-bang nucleosynthesis of ^{6}Li.