Three-nucleon Force Effects Research Articles

Effects of three-nucleon forces in the $$d(\alpha ,\gamma ){^6}$$Li reaction

Effects of three-nucleon forces in the $$d(\alpha ,\gamma ){^6}$$Li reaction

Proton-deuteron elastic scattering at energies 108-392 MeV: An application of spin-independent three-nucleon force potential

Proton-deuteron elastic scattering differential cross section is calculated in the framework of optical limit approximation at the energies 108, 120, 135, 150, 170, 190, 250 and 392 MeV. A spin independent three-nucleon force potential is used to obtain a three-nucleon force profile function. The backward scattering is considered to improve the agreement of theoretical calculations with the experimental data at large scattering angles. The deuteron D-state is included in the deuteron wave function through the fitting parameters. The three-nucleon force effect is not observed at small angles θcm≤60∘ and large angles θcm>140∘. The obtained agreement with the data shows the importance of the three-nucleon force at the minimum, the backward scattering at the considered energies and the applicability of the used potential.

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.

Posterior predictive distributions of neutron-deuteron cross sections

We quantify the posterior predictive distributions (PPDs) of elastic neutron-deuteron ($nd$) scattering cross sections using nucleon-nucleon ($NN$) interactions from chiral effective field theory ($\ensuremath{\chi}\mathrm{EFT}$) up to and including next-to-next-to-next-to-leading order ($\mathrm{N}^{3}\mathrm{LO}$). These PPDs quantify the spread in $nd$ predictions due to the variability of the low-energy constants (LECs) inferred from $NN$ scattering data. We use the wave-packet continuum discretization method to solve the Alt-Grassberger-Sandhas form of the Faddeev equations for elastic scattering. We draw 100 samples from the PPDs of $nd$ cross sections up to 67 MeV in scattering energy, i.e., in the energy region where the effects of three-nucleon forces are expected to be small. We find that the uncertainty about $NN$ LECs inferred from $NN$ scattering data, when assuming uncorrelated errors, does not translate to significant uncertainty in the low-energy $nd$ continuum. Based on our estimates, the uncertainty of $nd$ predictions are dominated by the $\ensuremath{\chi}\mathrm{EFT}$ truncation error, at least below $\mathrm{N}^{3}\mathrm{LO}$. At this order, the 90% credible interval of the PPD and the truncation error are comparable, although both are very small on an absolute scale.

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.

Three-body force effect on Deuteron–Deuteron elastic scattering at medium energies

In the framework of Glauber optical limit approximation taking into account D-state of deuteron and including three-nucleon force effect, the deuteron–deuteron elastic scattering differential cross-section is calculated at momenta [Formula: see text] [Formula: see text] [Formula: see text],GeV/c for incident deuteron. The D-state effect is not enough to obtain an agreement with the experimental data. The inclusion of three-nucleon force is necessary to improve the agreement with the data at momentum transferred squared greater than [Formula: see text] (GeV/c)2.

Elastic scattering of α particle at CERN-ISR energy: Three-nucleon force effect

In the framework of optical limit approximation of Glauber with inclusion of the three-nucleon force effect, we obtained a good agreement with the experimental data of [Formula: see text] elastic scattering differential cross-section at the energies [Formula: see text] and [Formula: see text][Formula: see text]GeV. The relation between the nucleon–nucleon slope parameter and the nucleon–nucleon interaction radius is discussed. In any case, the three-nucleon force is important and must be taken into account for a better agreement with the nucleus–nucleus data.

Modelling overlap functions for one-nucleon removal: role of the effective three-nucleon force

One-nucleon overlap functions, needed for nucleon-removal reaction calculations, are solutions of an inhomogeneous equation with the source term defined by the wave functions of the initial and final nuclear states and interaction between the removed nucleon with the rest. The source term approach (STA) allows the overlaps with correct asymptotic decrease to be modelled while using nuclear many-body functions calculated in minimal model spaces. By properly choosing the removed nucleon interaction the minimum-model-space STA can reproduce reduced values of spectroscopic factors (SFs) extracted from nucleon-removal reactions and predicts isospin asymmetry in the SF reduction. It is well-known that model space truncation leads to the appearance of higher-order induced forces, with three-nucleon force being the most important. In this paper the role of such a force on the source term calculation is studied. Applications to one-nucleon removal from double-magic nuclei show that three-nucleon force improves the description of available phenomenological overlap functions and reduces isospin asymmetry in SFs.

Three-nucleon force effects in inclusive spectra of the neutron-deuteron breakup reaction

We investigate the sensitivity of the non-exclusive nucleon induced deuteron breakup reaction to the three-nucleon interaction and distributions of three-nucleon force effects in inclusive spectra. To this end we solve the three-nucleon Faddeev equation at a number of incoming nucleon laboratory energies using the CD Bonn nucleon-nucleon interaction alone or combined with the 2{\pi}-exchange Tucson-Melbourne three-nucleon force. Based on these solutions energy spectra of an outgoing nucleon, at a specified detection angle as well as spectra integrated over that angle, are calculated. By integrating the spectra at a given angle over the energy of the outgoing nucleon the angular distributions of three-nucleon force effects in the breakup process are additionally obtained. Contrary to elastic nucleon-deuteron scattering, where at higher energies significant three-nucleon force effects were encountered for scattering angles around the minimum of the cross section, for the breakup process only moderate effects are found and they are restricted to forward angles. Results of the present investigation show that the large three-nucleon force effects found for some specific complete breakup configurations are reduced substantially in the incomplete spectra when averaging over contributing complete geometries is performed.

Analyzing powers in $$d(\vec {p}, pp)n$$ at intermediate and large scattering angles at 190 MeV

Understanding of the exact nature of three-nucleon forces is the most challenging topic in the field of nuclear physics. Three-nucleon break-up reaction is a good tool to look into the underlying dynamics of the nuclear force, thanks to its rich kinematical phase space which has different levels of sensitivity to three-nucleon force effects. The recent studies on few-nucleon systems have revealed that the current nuclear force models cannot describe nucleon-deuteron scattering data accurately. In the present work, the analyzing powers of the proton-deuteron break-up reaction obtained using a 190 MeV polarized proton beam will be discussed. We present for the first time the vector analyzing powers for the kinematics in which one of the protons scatters to intermediate and large scattering angles at this energy. The results show a fairly good agreement with various theoretical predictions for both intermediate and large scattering angles of the break-up phase space.

Three-nucleon force effects in the FSI configuration of the d(n, nn)p breakup reaction

We investigated three-nucleon (3N) force effects in the final state interaction (FSI) configuration of the d (n,nn) pd(n,nn)p breakup reaction at the incoming nucleon energy E_nEn= 200 MeV. Although 3N force effects for the elastic nucleon-deuteron scattering cross section at comparable energies are located predominantly in the region of intermediate and backward angles, the corresponding 3N force effects for the integrated FSI configuration breakup cross section are found also at forward scattering angles.

Three-nucleon force effects in nucleon-deuteron scattering at backward angles

Three-nucleon force effects in nucleon-deuteron scattering at backward angles

Exploring three-nucleon forces via three- and four-nucleon scattering

Three-nucleon force effects in three- and four-nucleon scattering are discussed from an experimental point of view.

Vector-analyzing powers in the $$d(\mathop {p}\limits ^{\rightarrow },pp)n$$ and $$d(\mathop {p}\limits ^{\rightarrow },\mathrm{{^{2}He}})n$$ channels at 135 MeV

Vector analyzing powers, $A_x$ and $A_y$, of the proton-deuteron break-up reaction have been measured by using a polarized-proton beam at 135 MeV impinging on a liquid-deuterium target. For the experiment, the Big Instrument for Nuclear-polarization Analysis (BINA) was used at KVI, Groningen, the Netherlands. The data are compared to the predictions of Faddeev calculations using state-of-the-art two- and three-nucleon potentials. Our data are reasonably well described by calculations for the kinematical configurations at which the three-nucleon force (3NF) effect is predicted to be small. However, striking discrepancies are observed at specific configurations, in particular in the cases of symmetric configurations, where the relative azimuthal angle between the two protons is small which corresponds to the $d(p,{\rm{^{2}He}})n$ channel. The vector analyzing powers of these configurations are compared to the proton deuteron elastic scattering to study the spin-isospin sensitivity of the 3NF models. The results are compared to the earlier results of the proton-deuteron break-up reaction at 190 MeV proton-beam energy \cite{Mardanpour2010}. A disagreement is observed for both proton-beam energies between data and calculations which points to a deficiency in the treatment of spin-isospin part of the 3NF.

Normal-ordered k-body approximation in particle-number-breaking theories

The reach of ab initio many-body theories is rapidly extending over the nuclear chart. However, dealing fully with three-nucleon, possibly four-nucleon, interactions makes the solving of the A-body Schr\"odinger equation particularly cumbersome, if not impossible beyond a certain nuclear mass. Consequently, ab initio calculations of mid-mass nuclei are typically performed on the basis of the normal-ordered two-body (NO2B) approximation that captures dominant effects of three-nucleon forces while effectively working with two-nucleon operators. A powerful idea currently employed to extend ab initio calculations to open-shell nuclei consists of expanding the exact solution of the A-body Schr\"odinger equation while authorizing the approximate solution to break symmetries of the Hamiltonian. In this context, operators are normal ordered with respect to a symmetry-breaking reference state such that proceeding to a naive truncation may lead to symmetry-breaking approximate operators. The purpose of the present work is to design a normal-ordering approximation of operators that is consistent with the symmetries of the Hamiltonian while working in the context of symmetry broken (and potentially restored) methods. Focusing on many-body formalisms in which U(1) global-gauge symmetry associated with particle number conservation is broken (and potentially restored), a particle-number-conserving normal-ordered k-body (PNOkB) approximation of an arbitrary N-body operator is designed on the basis of Bogoliubov reference states. A numerical test based on particle-number projected Hartree-Fock-Bogoliubov calculations permits to check the particle-number conserving/violating character of a given approximation to a particle-number conserving operator. Using the presently proposed PNOkB approximation, ab initio calculations based on symmetry-breaking and restored formalisms can be safely performed.

Two-Pion Exchange Three-Nucleon Force Effects in Elastic Nucleon–Deuteron Scattering Cross Sections

Differential cross sections for elastic nucleon–deuteron scattering at intermediate energies are calculated by solving the Faddeev equation in coordinate space using interaction models consisting of a two-nucleon potential and three-nucleon potentials based on the exchange of two pions among three nucleons. Attractive effects of the pion-exchange at medium range region in the three-nucleon potential are taken into account by using a larger cutoff mass parameter and by adding phenomenological repulsive three-nucleon potentials as a counterpart. It is shown that these effects increase the cross sections at backward angles, which tends to reduce discrepancies between theoretical calculations and experimental data.

Coulomb Force Effects in Few-Nucleon Systems

Theoretical predictions for sample observables of three-nucleon and four-nucleon reactions are reviewed. The focus is on Coulomb force effects. The calculations are based on the Alt–Grassberger–Sandhas version of the Faddeev equations. The calculations are done in momentum space. The calculational technique used to include the Coulomb repulsion between protons screens the infinite Coulomb tail, renormalizes the results and thereby corrects them for screening. The competition between three-nucleon force and Coulomb force effects as well as the Coulomb domination in special kinematic situations of reactions are discussed. Reactions connected by charge symmetry are reviewed. Special reaction observables are studied, in search for the hadronic violation of charge symmetry in the nuclear interaction and for its competition with the charge-asymmetric Coulomb force.

Three-nucleon force effects in $$A = 4$$ A = 4 scattering

Three-nucleon force effects in $$A = 4$$ A = 4 scattering

Proton-4He elastic scattering at 350–1728MeV energies: Three-body force effect

An approach for the three-body force effect is used to study p-[Formula: see text] elastic scattering at energies 350–1728[Formula: see text]MeV. The multiple scattering theory of Glauber and the optical limit approximation are used in calculations of elastic scattering differential cross-section at 350, 600, 800, 1000 and 1728[Formula: see text]MeV. The inclusion of 2[Formula: see text]-exchange three-nucleon force improves the agreement with the experimental data for both approximations. The results of optical limit approximation without three-nucleon force effect are clearly smaller than the results of multiple scattering theory for [Formula: see text] due to the absence of multiple scattering terms in the first. However, with inclusion of three-nucleon force, the results of both approximations are approximately similar.

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.