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We present selected results of theoretical investigations of three-nucleon systems which have been pursued in Kraków for more than thirty years. The Kraków–Bochum group has gathered a lot of experience related to investigations of elastic nucleon–deuteron scattering and nucleon-induced deuteron breakup processes. These investigations are based on rigorous solutions of the 3N Faddeev equations in momentum space and aim to understand the properties of two- and three-nucleon forces. Since the late 1980s, very many different models of nuclear potentials, including several generations of the forces derived within chiral effective field theory, have been put to stringent tests. Beside pure three-nucleon reactions, our Faddeev framework has been used to describe many electroweak processes, where the initial-state or final-state interactions among three-nucleons are very important. Our theoretical results need to be confronted with precision experimental data, so collaboration with many experimental groups all over the world is crucial for our research., 45th Congress of Polish Physicists, September 13–18, 2019, Kraków (Cracow), Poland

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., 43 pages, 13 figures

We build a relativistic model to perform calculations of exclusive, semi-exclusive and inclusive unpolarized cross sections and various polarization observables in electron and neutrino scattering experiments with deuteron targets. The strong interaction dynamics is defined by an explicit dynamical unitary representation of the Poincar\'e group, where representations of space translations and rotations in the interacting and non-interacting representations are the same. The Argonne V18 potential is used to construct a relativistic nucleon-nucleon interaction reproducing the experimental deuteron binding energy and nucleon-nucleon scattering observables. Our formalism does not include the pion production channel and neglects two-body contributions in the electromagnetic as well as in the weak nuclear current operator. We show that it is applicable to processes at kinematics, where the internal two-nucleon energy remains below the pion production threshold but the magnitude of the three-momentum transfer extends at least to several GeV., Comment: 59 pages, 29 figures

We apply a recently developed momentum-space semilocally regularized chiral potential to the $^{2}\mathrm{H}$ and $^{3}\mathrm{He}$ photodisintegration processes and to the (anti)neutrino induced deuteron breakup reactions. Specifically, the differential cross section, the photon analyzing power, and the final proton polarization have been calculated for deuteron photodisintegration at photon energies of 30 and 100 MeV. For $^{3}\mathrm{He}$ photodisintegration, predictions for the semi-inclusive and exclusive differential cross sections are presented for photon energies up to 120 MeV. The total cross section is calculated for (anti)neutrino disintegration of the deuteron for (anti)neutrino energies below 200 MeV. Predictions based on the Argonne V18 potential and on an older chiral force with regularization applied in coordinate space are used for comparison. Using the fifth-order chiral nucleon-nucleon potential supplemented with dominant contributions from the sixth order allows us to obtain converged predictions for observables in the investigated reactions. Our results based on the newest semilocal chiral potentials show even smaller cutoff dependence and a somewhat faster convergence for the considered electroweak observables than the previously reported ones with a coordinate-space regulator, which make them a promising tool to study electroweak processes.

W. Parol, A. Kozela, K. Bodek, A. Deltuva, M. Eslami-Kalantari, J. Golak, N. Kalantar-Nayestanaki, G. Khatri, St. Kistryn, J. Kubo´s, P. Kulessa, A. Magiera, H. Mardanpour, J. G. Messchendorp, I. Mazumdar, R. Skibiński, I. Skwira-Chalot, A. Ramazani-Moghaddam-Arani, D. Rozpędzik, H. Witała, B. Włoch, A. Wrońska, Differential cross sections for the deuteron breakup 1H(d, pp)n reaction were measured for a large set of 243 geometrical configurations at the beam energy of 80 MeV/nucleon. The cross-section data are normalized by the luminosity factor obtained on the basis of a simultaneous measurement of the elastic-scattering channel and the existing cross-section data for this process. The results are compared with the theoretical calculations modeling nuclear interactions with and without taking into account the three-nucleon force (3NF) and the Coulomb interaction. In the validated region of the phase space, both the Coulomb force and 3NF play an important role in providing a good description of the data. There are also regions in which description improvements due to the inclusion of 3NF are insufficient.

A recent paper by P. Reinert et al. [Eur. Phys. J. A54, 86 (2018)] showed a possibility of improving chiral nucleon-nucleon potential models by employing a semi-local regularization in momentum space. The authors derived nucleon-nucleon potentials with this kind of regularization completely up to the fifth order of the chiral expansion and considered additionally some contact interactions which appear at the sixth order. Such a chiral interaction has never been applied to study electromagnetic processes in two- or three-nucleon systems. Here we continue our research of photodisintegration processes, now using the improved chiral force. In particular, we discuss our predictions for the deuteron photodisintegration reaction in the photon energy range up to 100 MeV. The results of our calculations reveal that the new potential yields predictions characterized by a weaker dependence on the regularization parameter and faster convergence with respect to the chiral expansion order compared to the older chiral potentials.

A chiral nucleon–nucleon potential has been recently further improved by employing a semi-local regularization in momentum space. As such, a potential has not been tested yet in an electromagnetic process, we apply it to the deuteron photodisintegration reaction with photon energies up to 100 MeV. Results of our calculations show a weaker dependence of a selected observable on the cut-off parameter and a faster convergence with respect to the chiral expansion order compared to older chiral potentials.

We report the first measurement of the (e,e^{'}p) three-body breakup reaction cross sections in helium-3 (^{3}He) and tritium (^{3}H) at large momentum transfer [⟨Q^{2}⟩≈1.9 (GeV/c)^{2}] and x_{B}1 kinematics, where the cross section should be sensitive to quasielastic (QE) scattering from single nucleons. The data cover missing momenta 40≤p_{miss}≤500 MeV/c that, in the QE limit with no rescattering, equals the initial momentum of the probed nucleon. The measured cross sections are compared with state-of-the-art ab initio calculations. Overall good agreement, within ±20%, is observed between data and calculations for the full p_{miss} range for ^{3}H and for 100≤p_{miss}≤350 MeV/c for ^{3}He. Including the effects of rescattering of the outgoing nucleon improves agreement with the data at p_{miss}250 MeV/c and suggests contributions from charge-exchange (SCX) rescattering. The isoscalar sum of ^{3}He plus ^{3}H, which is largely insensitive to SCX, is described by calculations to within the accuracy of the data over the entire p_{miss} range. This validates current models of the ground state of the three-nucleon system up to very high initial nucleon momenta of 500 MeV/c.

The elastic nucleon–deuteron (N d) scattering process at nucleon laboratory energies up to 200 MeV can be used to study the three-nucleon interaction (3N). In order to draw final conclusions based on the comparison of theoretical predictions and data, an estimation of theoretical uncertainties is necessary. We focus here on the statistical uncertainties of theoretical predictions. We use the One-Pion-Exchange Gaussian (OPEGaussian) nucleon–nucleon (NN) potential and compare our predictions for the 3N observables with results based on the AV18 potential and results obtained with chiral potentials as well as with available data. We give examples of polarization observables at E = 135 MeV and discuss magnitudes of some theoretical uncertainties.

Response functions, differential cross sections and total cross sections for several (anti)neutrino induced reactions on 2H, 3He and 3H are calculated in momentum space for (anti)neutrino energies up to 160 MeV, using the AV18 nucleon-nucleon potential and a single-nucleon weak current operator. This work is a continuation of our investigations presented in J. Golak et al. [Phys. Rev. C 98, 015501 (2018)]., Comment: 23 pages, 16 figures in 60 files

We study the elastic nucleon–deuteron scattering process at incoming-nucleon laboratory energies up to E=200 MeV working within the formalism of Faddeev equations. We focused here on the computation and systematic analysis of the correlation coefficients among various observables in nucleon-nucleon and nucleon–deuteron elastic scattering. As a result, we obtained pairs of correlated/uncorrelated observables or sets of very weakly correlated observables. The knowledge, if some observables are or are not correlated, would impact future methods of fixing free parameters of the two- and many-body potentials, and could possibly help determine observables which should be measured to increase the precision of potential parameters’ determination.

The JISP16 nucleon–nucleon potential has been applied to explore the nucleon-induced deuteron breakup process at the incoming nucleon laboratory energy E=65 MeV. We have found that the JISP16 model generally gives predictions for the differential cross section and the nucleon analysing power, which are similar to the ones obtained with the standard realistic nucleon–nucleon AV18 potential. However, there are some regions of the phase space, for which the cross section and the analysing power differ by approximately 50%. These kinematical configurations could be used to refit the JISP16 potential parameters.

The need of a better understanding of the weak interaction and the electro-weak unification triggered in the last decades an interest in neutrino scattering off nuclei. Here, we present a study on neutrino-induced reactions off 2H. The cross sections are calculated via the nuclear response functions, which contain the information about the nuclear structure and interactions., 45th Congress of Polish Physicists, September 13–18, 2019, Kraków (Cracow), Poland

Results on three-nucleon (3N) reactions below the pion production threshold are discussed with an emphasis on the need for a three-nucleon force (3NF). The large discrepancies found between a theory based on numerical solutions of 3N Faddeev equations with standard NN potentials only and data point to the need for 3NF’s. This is supported by the fact that another possible reason for the discrepancies, relativistic effects, turned out to be small. Results for new generation of chiral NN forces with theoretical truncation errors are shown.

Deuteron photodisintegration, three-body photodisintegration of 3He as well as muon capture on the deuteron and 3He are theoretically analyzed by means of improved chiral nucleon-nucleon interactions. To this end we solve the Schrödinger and Faddeev equations in the momentum space. While our electromagnetic current operator implicitly receives implicitly many-body contributions via the Siegert theorem, the weak current is taken in the form of the nonrelativistic single nucleon current supplemented by the dominant relativistic corrections. Such models of the current operators and the wave functions generated with the new chiral potentials lead to reasonable predictions for the cross sections and capture rates studied in this contribution. Our results show very welcome features of the new forces and prove their usefulness in studies of electroweak processes with few nucleons.

We compare three methods to calculate the nucleon-nucleon t-matrix based on the three-dimensional formulation of J. Golak et al., Phys. Rev. C 81, 034006, (2010). In the first place we solve a system of complex linear inhomogeneous equations directly for the t-matrix. Our second method is based on iterations and a variant of the Lanczos algorithm. In the third case we obtain the t-matrix in two steps, solving a system of real linear equations for the k-matrix expansion coefficients and then solving an on-shell equation, which connects the scalar coefficients of the k- and t-matrices. A very good agreement among the three methods is demonstrated for selected nucleon-nucleon scattering observables using a chiral next-to-next-to-leading-order neutron-proton potential. We also apply our three-dimensional framework to the demanding problem of proton-proton scattering, using a corresponding version of the nucleon-nucleon potential and supplementing it with the (screened) Coulomb force, taken also in the three-dimensional form. We show converged results for two different screening functions and find a very good agreement with other methods dealing with proton-proton scattering., Comment: 18 pages, 10 figures (54 eps files)

A recently developed formulation for a direct treatment of the equations for two- and three-nucleon bound states as set of coupled equations of scalar functions depending only on vector momenta is extended to three-nucleon scattering. Starting from the spin-momentum dependence occurring as scalar products in two- and three-nucleon forces together with other scalar functions, we present the Faddeev multiple scattering series in which order by order the spin-degrees can be treated analytically leading to 3D integrations over scalar functions depending on momentum vectors only. Such formulation is especially important in view of awaiting extension of 3N Faddeev calculations to projectile energies above the pion production threshold and applications of chiral perturbation theory 3N forces, which are to be most efficiently treated directly in such three-dimensional formulation without having to expand these forces into a partial wave basis., 25 pages, 0 figures

We propose a novel approach to incorporate the proton-proton (pp) Coulomb force into the three-nucleon (3N) Faddeev calculations. The main new ingredient is a 3-dimensional screened pp Coulomb t-matrix obtained by a numerical solution of the 3-dimensional Lippmann-Schwinger (LS) equation. We demonstrate numerically and provide analytical insight that the elastic proton-deuteron (pd) observables can be determined directly from the resulting on shell 3N amplitude increasing the screening radius. The screening limit exists without the need of renormalisation not only for observables but for the elastic pd amplitude itself.

We present an approach to solve the three-nucleon Faddeev equation which avoids the complicated singularity pattern going with the moving logarithmic singularities of the standard approach. Very good agreement of the new and old approaches in the application to nucleon-deuteron elastic scattering and the breakup reaction is found.

Results on three-nucleon (3N) elastic scattering below the pion production threshold are discussed with an emphasis on the need for a three-nucleon force (3NF). The large discrepancies found between a theory based on numerical solutions of 3N Faddeev equations with (semi)phenomenological NN potentials only and data point to the need for 3NF’s. This notion is supported by the fact that another possible reason for the discrepancies in elastic nucleon-deuteron (Nd) scattering, relativistic effects, turned out to be small. Results for new generation of chiral NN forces (up to N4LO) together with theoretical truncation errors are shown.

The photon analyzing power for the photodisintegration of the deuteron was measured for seven gamma-ray energies between 2.39 and 4.05 MeV using the linearly polarized gamma-ray beam of the High-Intensity Gamma-ray Source at the Duke Free-Electron Laser Laboratory. The data provide a stringent test of theoretical calculations for the inverse reaction, the neutron-proton radiative capture reaction at energies important for Big-Bang nucleosynthesis. Our data are in excellent agreement with potential model and effective field theory calculations.

We solve three-nucleon Faddeev equations with nucleon-nucleon and three-nucleon forces derived consistently in the framework of chiral perturbation theory at next-to-next-to-next-to-leading order in the chiral expansion. In this first investigation we include only matrix elements of the three-nucleon force for partial waves with the total two-nucleon (three-nucleon) angular momenta up to 3 (5/2). Low-energy neutron-deuteron elastic scattering and deuteron breakup reaction are studied. Emphasis is put on Ay puzzle in elastic scattering and cross sections in symmetric-space-star and neutron-neutron quasi-free-scattering breakup configurations, for which large discrepancies between data and theory have been reported., 22 pages, 7 figures

We present a precise measurement of double-polarization asymmetries in the ^{3}He[over →](e[over →],e^{'}d) reaction. This particular process is a uniquely sensitive probe of hadron dynamics in ^{3}He and the structure of the underlying electromagnetic currents. The measurements have been performed in and around quasielastic kinematics at Q^{2}=0.25(GeV/c)^{2} for missing momenta up to 270 MeV/c. The asymmetries are in fair agreement with the state-of-the-art calculations in terms of their functional dependencies on p_{m} and ω, but are systematically offset. Beyond the region of the quasielastic peak, the discrepancies become even more pronounced. Thus, our measurements have been able to reveal deficiencies in the most sophisticated calculations of the three-body nuclear system, and indicate that further refinement in the treatment of their two-and/or three-body dynamics is required.

A series of measurements have been performed at KVI to obtain the vector analyzing power Ay Of the H-2((p) over right arrow, pd) reaction as a function of incident beam energy at energies of 120, 135, 150, and 170 MeV. For all these measurements, a range of upsilon (c.m) from 30 degrees to 170 degrees has been covered. The purpose of these investigations is to observe possible spin-dependent effects beyond two-nucleon forces. When compared to the,predictions of Faddeev calculations, based on two-nucleon forces only, significant deviations are observed at all energies and at center-of-mass angles between 70 degrees and 130 degrees. The addition of present-day three-nucleon forces does not improve the description of the data, demonstrating the still insufficient understanding of the properties of three-nucleon systems.

We employ the chiral nucleon-nucleon potential derived using the method of unitary transformation up to next-to-next-to-leading order (NNLO) to study bound and scattering states in the two-nucleon system. The predicted partial wave phase shifts and mixing parameters for higher energies and higher angular momenta beyond the ones which are fitted are mostly well described for energies below 300 MeV. Various deuteron properties are discussed. We also apply the next-to-leading order (NLO) potential to 3N and 4N systems. The resulting 3N and 4N binding energies are in the same range what is found using standard NN potentials. Experimental low-energy 3N scattering observables are also very well reproduced like for standard NN forces. Surprisingly the long standing Ay-puzzle is resolved at NLO. The cut-off dependence of the scattering observables is rather mild., Comment: LaTeX2e, 8 pages; invited talk presented at the XVIIth European Conference on Few-Body Problems in Physics, Evora, Portugal, September 2000; to be published in the Proceedings

Indications for three nucleon force (3NF) effects in PJd elastic scattering and in specific kinematical configurations of the deuteron breakup reaction became motivation for complex studies of the three nucleon system dynamics in the dp breakup process over a broad region of phase-space. A dedicated experiment is aimed at precise determination of cross sections and both vector and tenser analyzing powers for the H-1((p) over right arrow pp)n breakup reaction at 130 MeV deuteron beam energy. The experiment, employs a position-sensitive detection system covering a large solid angle and allowing for excellent particle identification. The first data taking run has been completed and the accumulated data are being analyzed. The results will be compared with the rigorous calculations within the Faddeev formalism, including various nucleon-nucleon potentials and 3NF models.

The breakup reaction has been investigated in the past at several tandem accelerator energies below 20 MeV in a number of “classical” kinematical configurations. These data are compared with predictions of four modern high-precision nucleon-nucleon potentials, calculated for 2H(n,np)n, i.e., without inclusion of the Coulomb interaction. The results show a remarkably good agreement between data and theory in some situations such as the final-state interaction, indicating the smallness of three-nucleon-force effects as well as the insignificance of the Coulomb force for this particular configuration. Other configurations such as the symmetric space-star or quasi-free scattering are not well described. The discrepancies cannot be removed by including into three-nucleon Faddeev calculations the 2π-exchange Tucson-Melbourne three-nucleon force with the cutoff parameter adjusted individually for each nucleon-nucleon interaction to fit the experimental triton binding energy. In contrast to quasi-free or symmetric space-star scattering configurations a comparison with corresponding neutron-deuteron data shows that this discrepancy cannot be traced back to the missing Coulomb interaction.

Developments in spin-polarized internal targets for storage rings have permitted measurements of 197 MeV polarized protons scattering from vector polarized deuterons. This work presents measurements of the polarization observables A_y, iT_11, and C_y,y in proton-deuteron elastic scattering. When compared to calculations with and without three-nucleon forces, the measurements indicate that three-nucleon forces make a significant contribution to the observables. This work indicates that three-body forces derived from static nuclear properties appear to be crucial to the description of dynamical properties., 8 pages 2 figures Latex, submitted to Phys. Rev. Letters

The n-n final-state interaction (FSI) was investigated via the 2H(n, np)n reaction at 25 MeV, using a geometry which enables the simultaneous observation of n-p quasifree (QFS) scattering. The data were analyzed with Monte Carlo simulations based on rigorous Faddeev calculations with realistic nucleon-nucleon potentials. The value of a(nn) deduced from the absolute yield in the FSI peak is -16.27+/-0. 40 fm while the relative data, normalized in the QFS region, give -16.06+/-0.35 fm. Thus our results differ from the "recommended" value of a(nn) = -18.5+/-0.3 fm by more than 5 standard deviations.

Polarization observables in low-energy proton-deuteron radiative capture have been calculated using the Bonn-B nucleon-nucleon potential. Their sensitivity to the changes of the \(\) nucleon-nucleon force components has been studied. Modifications of these forces, which drastically diminished the permanent discrepancy between the data and theory for the vector-analyzing power in low-energy nucleon-deuteron elastic scattering, have only small effects on the pd capture observables.

New vector analyzing-power data on (p) over right arrow+ d elastic scattering at E-p = 150 and 190 MeV have been measured. These an presented together with existing data and with recent (d) over right arrow + p vector and tensor analyzing power data at E-d = 270 MeV. The strong negative extremum of both vector analyzing powers A(y)(p) and A(y)(d) at theta(c.m.) similar or equal to 80 degrees-120 degrees is underestimated by Faddeev calculations using modern NN forces. Inclusion of the Tucson-Melbourne 3N force shifts the minima upwards, but with conflicting results for p A(y)(p), and leading to a good description for A(y)(d). An A(y)(p) puzzle, previously thought to exist at energies E-N less than or equal to 30 MeV only, appears to exist also at intermediate energies.

This paper reports high-accuracy cross-section data taken for identical kinematic conditions of the neutron-proton $(\mathrm{np})$ and neutron-neutron $(\mathrm{nn})$ final-state interactions in the ${}^{2}\mathrm{H}(n,nnp)$ reaction at an incident mean neutron energy of 13.0 MeV. These data were analyzed with rigorous three-nucleon calculations to determine the ${}^{1}{S}_{0}$ $\mathrm{np}$ and $\mathrm{nn}$ scattering lengths, ${a}_{\mathrm{np}}$ and ${a}_{\mathrm{nn}}$. Our results are ${a}_{\mathrm{nn}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}18.7\ifmmode\pm\else\textpm\fi{}0.6\mathrm{fm}$ and ${a}_{\mathrm{np}}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\ensuremath{-}23.5\ifmmode\pm\else\textpm\fi{}0.8\mathrm{fm}$. The value obtained for ${a}_{\mathrm{nn}}$ in the present work is in agreement with that from ${\ensuremath{\pi}}^{\ensuremath{-}}d$ measurements but disagrees with values obtained from earlier neutron-deuteron breakup studies.