ELASTIC NUCLEON-DEUTERON SCATTERING WITH THE NUCLEON-NUCLEON OPE-GAUSSIAN FORCE AT E=65 MeV - INTRODUCTORY STUDIES
Y. Volkotrub, R. Skibiński, J. Golak, K. Topolnicki, H. Witała
abstract
We study the elastic nucleon-deuteron (Nd) scattering process at the incoming nucleon laboratory energy E = 6 5 MeV working within the formalism of Faddeev equations. We employ, for the first time in the elastic Nd scattering, the OPE-Gaussian nucleon-nucleon (NN) potential and confirm its high quality by comparing our predictions for the differential cross section with results based on the AV18 potential as well as with available data. We also estimate the theoretical uncertainty of this observable originating from uncertainties of the OPE-Gaussian model parameters. We find the relative uncertainties to be smaller than 0.8% for the differential cross section. The correlations between various parameters of the OPE-Gaussian are also shown.
General operator form of the non-local three-nucleon force
K. Topolnicki
abstract
This paper describes a procedure to obtain the general form of the three-nucleon force. The result is an operator form where the momentum space matrix element of the three-nucleon potential is written as a linear combination of 320 isospin-spin-momentum operators and scalar functions of momenta. Any spatial and isospin rotation invariant three-nucleon force can be written in this way and in order for the potential to be Hermitian, symmetric under parity inversion, time reversal and particle exchange, the scalar functions must have definite transformation properties under these discrete operations. A complete list of the isospin-spin-momentum operators and scalar function transformation properties is given.
Operator form of the three-nucleon scattering amplitude
K. Topolnicki, J. Golak, R. Skibiński, H. Witała
abstract
To extend the applications of the so-called "three-dimensional" formalism to the description of three-nucleon scattering within the Faddeev formalism, we develop a general form of the three-nucleon scattering amplitude. This form significantly decreases the numerical complexity of the "three-dimensional" calculations by reducing the scattering amplitude to a linear combination of momentum-dependent spin operators and scalar functions of momenta. The number and structure of the spin operators is fixed and the scalar functions can be represented numerically using standard methods such as multidimensional arrays. In this paper, we show that all orders of the iterated Faddeev equation can be written in this general form. We argue that calculations utilizing the three-nucleon force will also conform to the same general form. Additionally, we show how the general form of the scattering amplitude can be used to transform the Faddeev equation to make it suitable for numerical calculations using iterative methods.
Modern Chiral Forces Applied to the Nucleon-Deuteron Radiative Capture
R. Skibiński, J. Golak, K. Topolnicki, H. Witała, E. Epelbaum, H. Kamada, H. Krebs, UG. Meissner, A. Nogga
abstract
The chiral nucleon-nucleon interaction with semi-local regularization up to the fifth order of chiral expansion is applied to the nucleon-deuteron radiative capture process. Our theoretical approach is based on the formalism of Faddeev equations and the Siegert theorem is exploited to construct the electromagnetic current operator. The very weak dependence of the differential cross section on values of the regularization parameter is observed. This suggests that the improved chiral two-body interaction is a promising starting point to study electromagnetic processes at low energies.
Three Nucleon Scattering in a 3D Approach at the First Order
K. Topolnicki, J. Golak, R. Skibiński, H. Witała
abstract
The "three dimensional" (3D) approach to few-nucleon calculations, instead of working with partial wave states, uses the momentum degrees of freedom of the nucleons directly. As a result, 3D calculations are equivalent to incorporating all partial waves into the calculation. This equivalence is limited by the available computational resources, nonetheless, the new approach provides an opportunity to construct numerical descriptions of physical systems that were previously unavailable with more traditional methods. We present selected observables for the neutron-deuteron inelastic scattering process calculated using the 3D approach and the traditional partial wave approach. This comparison was carried out using the first order terms of the series expansion of the Faddeev equations and is a strong motivator for a development of a complete 3D calculation that utilizes all orders.
Muon Capture on H-3
J. Golak, R. Skibiński, H. Witała, K. Topolnicki, H. Kamada, A. Nogga, LE. Marcucci
abstract
The mu(-) + (3) H -> nu(mu) + n + n + n capture reaction is studied under full inclusion of final-state interactions with the AV18 nucleon-nucleon potential and the Urbana IX three-nucleon force. We employ the single nucleon weak current operator comprising the dominant relativistic corrections to obtain first estimates of the total capture rates based on realistic forces. Our results are compared with older theoretical predictions.
Role of the Total Isospin 3/2 Component in Three-Nucleon Reactions
H. Witała, J. Golak, R. Skibiński, K. Topolnicki, E. Epelbaum, K. Hebeler, H. Kamada, H. Krebs, UG. Meissner, A. Nogga
abstract
We discuss the role of the three-nucleon isospin amplitude 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 and is driven by the difference between neutron-neutron (proton-proton) and neutron-proton forces. We study the magnitude of that contribution to the elastic scattering and breakup observables, taking the locally regularized chiral NLO nucleon-nucleon potential supplemented by the chiral NLO three-nucleon force. For comparison we employ also the Av18 nucleon-nucleon potential combined with the Urbana IX three-nucleon force. We find that the isospin component is important for the breakup reaction and the proper treatment of charge-independence breaking in this case requires the inclusion of the state with isospin . For neutron-deuteron elastic scattering the contributions are insignificant and charge-independence breaking can be accounted for by using the effective t-matrix generated with the so-called rule.
Muon capture on H-3
J. Golak, R. Skibiński, H. Witała, K. Topolnicki, H. Kamada, A. Nogga, LE. Marcucci
abstract
The mu(-) + H-3 ->nu(mu) + n + n + n capture reaction is studied under full inclusion of final-state interactions. Predictions for the three-body breakup of H-3 are calculated with the AV18 potential, augmented by the Urbana IX three-nucleon force. Our results are based on the single-nucleon weak-current operator comprising the dominant relativistic corrections. This work is a natural extension of our investigations of the mu(-) + He-3 -> nu(mu)+ H-3, mu(-) + He-3 -> nu(mu)+ n + d and mu(-) + He-3 -> nu(mu) + n + n + p capture reactions presented in Golak et al.
The general operator form for the total-momentum-dependent nucleon-nucleon potential
K. Topolnicki, J. Golak, R. Skibiński, H. Witała
abstract
In this paper we describe a procedure to obtain the general operator form of two-nucleon (2N) potentials and apply it to the case of the 2N potential that has an additional dependence on the total momentum of the system. This violates Galilean invariance but terms including the total momentum appear in some relativistic approaches. In operator form, the potential is expressed as a linear combination of a fixed number of known spin-momentum operators and scalar functions of momenta. Since the scalar functions effectively define the potentials, using the operator form significantly reduces the number of parameters that are needed in numerical implementations. The proposed operator form explicitly obeys the usual symmetries of rotational invariance, particle exchange, time reflection and parity.
Testing semilocal chiral two-nucleon interaction in selected electroweak processes
R. Skibiński, J. Golak, K. Topolnicki, H. Witała, E. Epelbaum, H. Krebs, H. Kamada, UG. Meissner, A. Nogga
abstract
The recently developed semilocal improved chiral nucleon-nucleon interaction is used for the first time to study several electromagnetic and weak processes at energies below the pion production threshold. Cross sections and selected polarization observables for deuteron photodisintegration, nucleon-deuteron radiative capture, three-body He-3 photodisintegration, as well as capture rates for decays of the muonic H-2 and He-3 atoms are calculated. The Lippmann-Schwinger and Faddeev equations in momentum space are solved to obtain nuclear states. The electromagnetic current operator is taken as a single nucleon current supplemented by many-body contributions induced via the Siegert theorem. For muon capture processes the nonrelativistic weak current together with the dominant relativistic corrections is used. Our results compare well with experimental data, demonstrating the same quality as is observed for the semiphenomenological Argonne V18 potential. Compared to the older version of the chiral potential with a nonlocal regularization, a much smaller cut-off dependence is found for the state-of-art chiral local interaction employed in this paper. Finally, estimates of errors due to the truncation of the chiral expansion are given.
Few-nucleon systems with state-of-the-art chiral nucleon-nucleon forces
S. Binder, A. Calci, E. Epelbaum, RJ. Furnstahl, J. Golak, K. Hebeler, H. Kamada, H. Krebs, J. Langhammer, S. Liebig, P. Maris, UG. Meissner, D. Minossi, A. Nogga, H. Potter, R. Roth, R. Skibiński, K. Topolnicki, JP. Vary, H. Witała
abstract
We apply improved nucleon-nucleon potentials up to fifth order in chiral effective field theory, along with a new analysis of the theoretical truncation errors to study nucleon-deuteron (Nd) scattering and selected low-energy observables in H-3, He-4, and Li-6. Calculations beyond second order differ from experiment well outside the range of quantified uncertainties, providing truly unambiguous evidence for missing three-nucleon forces within the employed framework. The sizes of the required three-nucleon-force contributions agree well with expectations based on Weinberg's power counting. We identify the energy range in elastic Nd scattering best suited to study three-nucleon-force effects and estimate the achievable accuracy of theoretical predictions for various observables.
Orthogonal polynomial approach to calculate the two-nucleon transition operator in three dimensions
K. Topolnicki, J. Golak, R. Skibiński, H. Witała
abstract
We give a short report on the possibility to use orthogonal polynomials (OP) in calculations that involve the two-nucleon (2N) transition operator. The presented work adds another approach to the set of previously developed methods (described in Phys. Rev. C 81, 034006 (2010); Few-Body Syst. 53, 237 (2012); K. Topolnicki, PhD thesis, Jagiellonian University (2014)) and is applied to the transition operator calculated at laboratory kinetic energy 300MeV. The new results for neutron-neutron and neutron-proton scattering observables converge to the results presented in Few-Body Syst. 53, 237 (2012) and to results obtained using the Arnoldi algorithm (Y. Saad, Iterative methods for sparse linear systems (SIAM Philadelphia, PA, USA 2003)). The numerical cost of the calculations performed using the new scheme is large and the new method can serve only as a backup to cross-check the previously used calculation schemes.
Studies of three-nucleon systems with improved chiral forces
R. Skibiński, J. Golak, K. Topolnicki, H. Witała, E. Epelbaum
abstract
Recently developed chiral two-nucleon (2N) potentials are applied to various processes in three-nucleon (3N) systems within the approach based on Faddeev equations. The predictions for the cross sections in elastic nucleon-deuteron (Nd) scattering and for radiative proton-deuteron (pd) capture are similar to results based on the AV] 8 potential. The very good convergence with respect to the chiral expansion and a weak dependence on a regularization parameter are observed. This gives hope that the Hamiltonian combining these 2N potentials with consistent 3N forces will become an important tool to study nuclear structure and processes.
Elastic nucleon-deuteron scattering and breakup with chiral forces
H. Witała, J. Golak, R. Skibiński, K. Topolnicki
abstract
Results on three-nucleon (3N) elastic scattering and breakup below the pion production threshold are discussed. The large discrepancies found between a theory based on numerical solutions of 3N Faddeev equations with standard nucleon-nucleon (NN) potentials only and data point to the need for three-nucleon forces (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 a new generation of chiral NN forces (up to (NLO)-L-4) together with theoretical truncation errors are shown. They support conclusions obtained with standard NN potentials
Break-up channels in muon capture on He-3
J. Golak, R. Skibiński, H. Witała, K. Topolnicki, AE. Elmeshneb, H. Kamada, A. Nogga, LE. Marcucci
abstract
The mu(-) + He-3 -> nu(mu) + n + d and mu(-) + He-3 -> nu(mu) + n + n + p capture reactions are studied under full inclusion of final state interactions with the AV18 nucleon-nucleon potential, augmented by the Urbana IX three-nucleon force, and employing the single nucleon weak current operator. We give first realistic estimates of the total capture rates: 544 s(-1) and 154 s(-1) for the n + d and n + n + p channels, respectively. Our results are compared with the most recent experimental data, finding a rough agreement for the total capture rates, but failing to reproduce the differential capture rates.
Properties of He-4 and Li-6 with improved chiral EFT interactions
P. Maris, S. Binder, A. Calci, E. Epelbaum, RJ. Furnstahl, J. Golak, K. Hebeler, H. Kamada, H. Krebs, J. Langhammer, S. Liebig, UG. Meissner, D. Minossi, A. Nogga, H. Potter, R. Roth, R. Skibiński, K. Topolnicki, JP. Vary, H. Witała
abstract
We present recent results for He-4 and Li-6 obtained with improved NN interactions derived from chiral effective field theory up to (NLO)-L-4. The many-body calculations are performed order-by-order in the chiral expansion. At (NLO)-L-3 and (NLO)-L-4 additional renormalization using the Similarity Renormalization Group is adopted to improve numerical convergence of the many-body calculations. We discuss results for the ground state energies, as well as the magnetic moment and the low-lying spectrum of Li-6.
(NLO)-L-3 Chiral Predictions for Spin Observables in Nucleon-Deuteron Elastic Scattering at Low Energies
R. Skibiński, J. Golak, K. Topolnicki, H. Witała
abstract
The chiral next-to-next-to-next-to leading order nuclear forces(1-3) are used to obtain predictions for spin observables in elastic nucleon-deuteron scattering at E = 13 MeV. The three-nucleon force is taken into account with all its complexity, including the short-range part and relativistic corrections. Presented examples of the polarization observables for elastic nucleon-deuteron scattering show visible contributions from these new structures in the three-nucleon potential which emerge for the first time at the next-to-next-to-next-to leading order. However, our results suggest that some modifications of the currently used model of the nuclear forces are necessary.
The Two-Nucleon and Three-Nucleon System in Three Dimensions
K. Topolnicki, J. Golak, H. Witała, R. Skibiński, AE. Elmeshneb
abstract
We present a brief overview of the three-dimensional formalism that is under development in our group. Using the 3D momentum eigenstates of the nucleon directly, instead of relying on the partial wave decomposition of operators involved in the calculations, allows us to use a very direct approach. This in turn enabled us to successfully tackle a large variety of few-body problems. Our calculation of the two nucleon transition operator and bound state can incorporate a very general form of the two-nucleon potential. Calculations of the three-nucleon bound state can include in addition to the two-nucleon potential also a very general operator form of the three-nucleon force. Recently the 3D formalism is also applied to processes that involve electro-weak probes. Carrying out these calculations for a wide spectrum of two-nucleon and three-nucleon potentials using the classical partial wave approach is unpractical due to the complicated spin structure of the operators. Using the 3D formalism, the calculations can be quickly adapted to test new models.
THE CHIRAL LONG-RANGE TWO-PION EXCHANGE ELECTROMAGNETIC CURRENTS IN RADIATIVE NUCLEON-DEUTERON CAPTURE
R. Skibiński, J. Golak, D. Rozpedzik, K. Topolnicki, H. Witała
abstract
The nucleon-deuteron radiative capture process is investigated using the chiral nuclear potentials and the electromagnetic currents developed by the Bochum-Bonn group. While the strong interaction is taken up to the next-to-next-to-leading order, the electromagnetic current consists of a single nucleon current, the leading one-pion exchange one and is supplemented by contributions from the long-range two-pion exchange current at next-to-leading-order. The theoretical predictions for the cross sections as well as analyzing powers show strong dependence on the values of regularization parameters. Only small effects of the three-nucleon force and the long-range two-pion exchange current are observed. The dependence on the choice of regularization parameters results in a big theoretical uncertainty and clearly points to the necessity to include corrections from higher orders of the chiral expansion both for the nuclear forces and currents.
Ab Initio Calculations Of Three-Nucleon Scattering
H. Witała, J. Golak, R. Skibiński, K. Topolnicki
abstract
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 (NLO)-L-4) together with theoretical truncation errors are shown.
Low-energy neutron-deuteron reactions with (NLO)-L-3 chiral forces
J. Golak, R. Skibiński, K. Topolnicki, H. Witała, E. Epelbaum, H. Krebs, H. Kamada, UG. Meissner, V. Bernard, P. Maris, J. Vary, S. Binder, A. Calci, K. Hebeler, J. Langhammer, R. Roth, A. Nogga, S. Liebig, D. Minossi
abstract
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.
Calculations of three-nucleon reactions with (NLO)-L-3 chiral forces: achievements and challenges
H. Witała, J. Golak, R. Skibiński, K. Topolnicki
abstract
We discuss the application of the chiral (NLO)-L-3 forces to three-nucleon reactions and point to the challenges which will have to be addressed. Present approaches to solve three-nucleon Faddeev equations are based on a partial-wave decomposition. A rapid increase of the number of terms contributing to the chiral three-nucleon force when increasing the order of the chiral expansion from (NLO)-L-2 to (NLO)-L-3 forced us to develop a fast and effective method of automatized partial-wave decomposition. At low energies of the incoming nucleon below approximate to 20 MeV, where only a limited number of partial waves is required, this method allowed us to perform calculations of reactions in the three-nucleon continuum using (NLO)-L-3 two- and three-nucleon forces. It turns out that inclusion of consistent chiral interactions, with relativistic 1/m corrections and short-range 2 pi-contact term omitted in the (NLO)-L-3 three-nucleon force, does not explain the long standing low energy A(y)-puzzle. We discuss problems arising when chiral forces are applied at higher energies, where large three-nucleon force effects are expected. It seems plausible that at higher energies, due to a rapid increase of a number of partial waves required to reach convergent results, a three-dimensional formulation of the Faddeev equations which avoids partial-wave decomposition is desirable.
Break-up channels in muon capture on He-3 (vol 90, 024001, 2014)
J. Golak, R. Skibiński, H. Witała, K. Topolnicki, AE. Elmeshneb, H. Kamada, A. Nogga, LE. Marcucci
abstract
Break-up channels in muon capture on He-3
J. Golak, R. Skibiński, H. Witała, K. Topolnicki, AE. Elmeshneb, H. Kamada, A. Nogga, LE. Marcucci
abstract
The mu(-) + H-2 -> nu(mu)+ n + n, mu(-) + He-3 -> nu(mu)+ H-3, mu(-) + He-3 -> nu(mu)+ n + d, and mu(-) + He-3 -> nu(mu) + n + n + p capture reactions are studied with various realistic potentials under full inclusion of final-state interactions. Our results for the two- and three-body break-up of He-3 are calculated with a variety of nucleon-nucleon potentials, among which is the AV18 potential, augmented by the Urbana IX three-nucleon potential. Most of our results are based on the single-nucleon weak-current operator. As a first step, we tested our calculation in the case of the mu(-) + H-2 -> nu(mu)+ n + n and mu(-) + He-3 -> nu(mu) + H-3 reactions, for which theoretical predictions obtained in a comparable framework are available. Additionally, we have been able to obtain for the first time a realistic estimate for the total rates of the muon capture reactions on 3He in the break-up channels: 544 s(-1) and 154 s(-1) for the n + d and n + n + p channels, respectively. Our results are compared with the most recent experimental data, finding a rough agreement for the total capture rates, but failing to reproduce the differential capture rates.
2N and 3N Systems in a Three Dimensional Formalism
K. Topolnicki, J. Golak, R. Skibiński, AE. Elmeshneb, H. Witała, A. Nogga, H. Kamada
abstract
We present an overview of our framework used to treat two- and three-nucleon (2N, 3N) systems employing three dimensional momentum eigenstates. Using a three dimensional formalism instead of the classical partial wave approach is an attractive alternative for a number of reasons, the most prominent being the very direct way of performing calculations. With the use of our tools it is possible to produce a working numerical realization of calculations in only a couple of steps from the fundamental (Schrodinger or Lippmann-Schwinger) equations. The FORTRAN implementations of the most complicated parts of the calculations are generated automatically by software that was written in our group. Additionally, at higher energies, three dimensional calculations avoid problems arising from slow convergence of partial wave decomposition based techniques. Our approach utilizes a very general form of the 2N and 3N forces and has been successfully used to obtain results for the 2N transition operator as well as for the 2N and 3N bound states (Golak et al. in Phys Rev C 81:034006, 2010; Few-Body Syst 53:237, 2012a; Few-Body Syst, 2012b).
Muon induced deuteron disintegration in three-dimensions
K. Topolnicki, J. Golak, R. Skibiński, LE. Marcucci, H. Witała, AE. Elmeshneb
abstract
We present a three-dimensional (3D) description of muon induced deuteron disintegration. This reaction is treated as the decay of the muonic atom with the muon initially on the lowest K shell. Our aim is to calculate the total and differential decay rates. We work in momentum space and use 3D momentum eigenstates directly. This approach allowed us to calculate the appropriate nuclear matrix elements, necessary building blocks for the differential decay rate, in a single step. For contrast - in classical calculations many partial-waves have to be taken into account. We achieved a very good agreement between the 3D and partial-wave methods for calculations that involve single-nucleon currents. Our result for the total decay rate is also in agreement with experimental values, though these are not very precise. This success motivates us to also include two-nucleon current contributions that include the meson exchange currents. Additionally, our formalism can also be applied to other, so far poorly described, processes like: mu +(3) He -> + nu + n + d or mu +(3) He -> nu + n + n + p.
Three-nucleon reactions with chiral dynamics
H. Witała, J. Golak, R. Skibiński, K. Topolnicki
abstract
Faddeev calculations using the chiral three-nucleon force at next-to-next-to-next-to-leading-order show that this force is not able to provide an explanation for the low-energy A(y) puzzle. Also the large discrepancies between data and theory for the symmetric-space-star and for the neutron-neutron quasi-free-scattering cross sections in low energy neutron-deuteron breakup cannot be explained by that three-nucleon force. The discrepancy for the neutron-neutron quasi-free-scattering cross section seems to require a modification of the S-1(0) neutron-neutron force.
A Three-Dimensional Treatment of the Three-Nucleon Bound State
J. Golak, K. Topolnicki, R. Skibiński, W. Glockle, H. Kamada, A. Nogga
abstract
Recently a formalism for a direct treatment of the Faddeev equation for the three-nucleon bound state in three dimensions has been proposed. It relies on an operator representation of the Faddeev component in the momentum space and leads to a finite set of coupled equations for scalar functions which depend only on three variables. In this paper we provide further elements of this formalism and show the first numerical results for chiral NNLO nuclear forces.
Deuteron Disintegration in Three Dimensions
K. Topolnicki, J. Golak, R. Skibiński, AE. Elmeshneb, W. Glockle, A. Nogga, H. Kamada
abstract
We compare results from traditional partial wave treatment of deuteron electro-disintegration with a new approach that uses three-dimensional formalism. The new framework for the two-nucleon (2N) system using a complete set of isospin-spin states made it possible to construct simple implementations that employ a very general operator form of the current operator and 2N states.
Calculations of Three-Nucleon Reactions
H. Witała, J. Golak, R. Skibiński, K. Topolnicki, H. Kamada, E. Epelbaum, W. Glockle, H. Krebs, WN. Polyzou, A. Nogga
abstract
Faddeev calculations using the chiral three-nucleon force in next-to-next-to-next-to-leading-order show that this force is too weak to provide an explanation for the low-energy A (y) puzzle. The large discrepancy between data and theory for the neutron-neutron quasi-free-scattering cross section in low energy neutron-deuteron breakup requires a modification of the neutron-neutron force. We discuss the consequences that a bound state of two neutrons has on neutron-deuteron scattering observables. At higher energies we compare the solutions of the non-relativistic three-nucleon Faddeev equations with three-nucleon force included to the solutions of its Poincar, invariant version.
H-3 at Next-to-Next-to-Next-to Leading Order of the Chiral Expansion
R. Skibiński, J. Golak, K. Topolnicki, H. Witała, E. Epelbaum, W. Glockle, H. Krebs, H. Kamada, A. Nogga
abstract
The chiral three-nucleon force (3NF) at next-to-next-to-next-to leading order ((NLO)-L-3) is used to calculate the triton wave function and the doublet nucleon-deuteron scattering length. This allows us to fix the values of the low-energy constants which are free parameters of the theory. The obtained values of these parameters, the expectation values of the kinetic energy, two- and three-body potentials and individual contributions of different parts of 3NF are given.
Different Methods for the Two-Nucleon T-Matrix in the Operator Form
J. Golak, R. Skibiński, H. Witała, K. Topolnicki, W. Glockle, A. Nogga, H. Kamada
abstract
We compare three methods to calculate the nucleon-nucleon t-matrix based on the three-dimensional formulation of 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.
Recent Developments of a Three-dimensional Description of the NN System
R. Skibiński, J. Golak, D. Rozpedzik, K. Topolnicki, H. Witała, W. Glockle, A. Nogga, E. Epelbaum, H. Kamada, C. Elster, I. Fachruddin
abstract
A recently developed three-dimensional formulation of nucleon-nucleon (NN) scattering is briefly presented. Here the NN t-matrix is represented by six spin-momentum operators accompanied by six scalar functions of momentum vectors. A numerical example for the NN scattering cross section is given.
The Tucson-Melbourne three-nucleon force in the automatized partial-wave decomposition
R. Skibiński, J. Golak, K. Topolnicki, H. Witała, H. Kamada, W. Glockle, A. Nogga
abstract
A recently developed procedure for a partial-wave decomposition of a three-nucleon force is applied to the pi-pi, pi-rho and rho-rho components of the Tucson-Melbourne three-nucleon potential. The resulting matrix elements for the pi-pi and pi-rho components are compared with the values obtained using the standard approach to the partial-wave decomposition, in which the pi-rho expressions for the matrix elements are also derived and presented. Several numerical tests and results for the triton binding energy and the correlation function prove the reliability and efficiency of the new method.
Two-nucleon systems in three dimensions
J. Golak, W. Glockle, R. Skibiński, H. Witała, D. Rozpedzik, K. Topolnicki, I. Fachruddin, C. Elster, A. Nogga
abstract
A recently developed formulation for treating two- and three-nucleon bound states in a three-dimensional formulation based on spin-momentum operators is extended to nucleon-nucleon scattering. Here the nucleon-nucleon T-matrix is represented by six spin-momentum operators accompanied by six scalar functions of momentum vectors. We present the formulation and provide numerical examples for the deuteron and nucleon-nucleon scattering observables. A comparison to results from a standard partial-wave decomposition establishes the reliability of this formulation.
A new way to perform partial-wave decompositions of few-nucleon forces
J. Golak, D. Rozpedzik, R. Skibiński, K. Topolnicki, H. Witała, W. Glockle, A. Nogga, E. Epelbaum, H. Kamada, C. Elster, I. Fachruddin
abstract
We formulate a general and exact method of partial-wave decomposition (PWD) of any nucleon-nucleon (NN) potential and any three-nucleon (3N) force. The approach allows one to efficiently use symbolic algebra software to generate the interaction-dependent part of the program code calculating the interaction. We demonstrate the feasibility of this approach for the one-boson exchange BonnB potential, a recent nucleon-nucleon chiral force and the chiral two-pion-exchange three-nucleon force. In all cases very good agreement between the new and the traditional PWD is found. The automated PWD offered by the new approach is of the utmost importance in view of future applications of numerous chiral N3LO contributions to the 3N force in three-nucleon calculations.
