Session CK: Mini-Symposium on Three Nucleon Forces from Few to Heavier Nucleon Systems

Ab initio calculations of light-nucleus reactions and three-nucleon forces

Achieving a first-principles description of the properties of nuclei as they emerge from the underlying fundamental interactions among the constituent protons and neutrons is a central goal of nuclear theory. An important question is to understand the role of three-nucleon forces. In the past, progress in this area has been mainly driven by nuclear structure calculations, but new capabilities are now emerging that allow us to explore three-nucleon force effects in light-nucleus scattering and reactions. In this talk I will present the state of the art of ab initio calculations for nucleon and deuterium scattering on light nuclei starting from chiral two- and three-body Hamiltonians.   

\textit{Ab initio} calculations of nuclear structure and reactions with chiral two- and three-nucleon interactions

The description of nuclei starting from the constituent nucleons and the realistic interactions among them has been a long-standing goal in nuclear physics. In recent years, a significant progress has been made in developing \textit{ab initio} many-body approaches capable of describing both bound and scattering states in light and medium mass nuclei based on input from QCD employing Hamiltonians constructed within chiral effective field theory. We will present calculations of proton-$^{10}$C scattering and resonances of the exotic nuclei $^{11}$N and $^{9}$He within the no-core shell model with continuum [1]. Also, we will discuss calculations of binding and separation energies of neutron rich isotopes of Ar, K, Ca, Sc and Ti within the self-consistent Gorkov-Green's function approach [2]. \\[4pt] [1] S. Baroni, P. Navratil, and S. Quaglioni, Phys. Rev. Lett. \textbf{110}, 022505 (2013); Phys. Rev. C \textbf{87}, 034326 (2013). \\[0pt] [2] V. Soma, A. Cipollone, C. Barbieri, P. Navratil, and T. Duguet, Phys. Rev. C \textbf{89}, 061301(R) (2014).   

Probing the chiral three-nucleon force effects on many-body nuclear reactions

Understanding of the roles of three-nucleon force (3NF) in nuclear few- and many-body systems is one of the fundamental subjects in nuclear physics. Recently, the 3NF is determined by chiral effective field theory in which two-, three-, and many-nucleon forces are treated consistently and systematically. To investigate the roles of a chiral 3NF on many-body nuclear reactions, we construct a microscopic reaction theory based on the g-matrix interaction including the effects of a chiral 3NF. In the framework, the 3NF effects appear through the density dependence of the g-matrix. I show the first investigation of the chiral 3NF effects on nucleus-nucleus elastic and inelastic scattering, and nucleon-knockout reactions.   

Study of 3NF effects via few-nucleon scattering

Experimentally, one must utirize systems with more than two nucleons($A\ge 3$) to investigate properties of 3NFs. We have performed the measurement of all deuteron analyzing powers for {\it dp} elastic scattering at $250-300$ MeV/nucleon. Serious discrepancies are found between the data and the Faddeev calculations at very backward angles($\theta_{{\rm c.m.}}\ge 120^{\circ}$), which are not explained even taking into account 3NFs. These results indicate some significant dynamical components, e.g. short-range 3NFs, are missing in the calculations. In order to extend the study of 3NF to many-nucleon system we are planning the measurements of {\it p}$+^3$He scattering.   

Three-nucleon force study by the $pd$ breakup measurements at the intermediate energy region

The effects of three-nucleon force (3NF) has been actively studied by using the nucleon-deuteron ($Nd$) scattering states. The differential cross sections of the elastic $Nd$ scattering at the energy below 150 MeV can be well reproduced by the Faddeev calculation based on modern nucleon-nucleon ($NN$) interactions and 3NF. On the other hand, the data at 250 MeV was underestimated by the Faddeev calculations with 3NF by 50\%. For the systematic study to understand the missing picture of 3NF, we performed the measurements of the $pd$ breacup reactions at 170 and 250 MeV. Concerning about the $^2$H($p,pp$)$n$ exclusive breakup reaction at $E_p=$ 250 MeV, the data of the differential cross sections are well reproduced by the Faddeev calculations. However in the case of the $^2$H($p,p$)$pn$ inclusive breakup reaction at $E_p=$ 170 MeV, our data shows large discrepancies between the data and the calculations with 3NFs, which is similar to the results of the $^2$H($p,p$) inclusive breakup reaction at 250 MeV.   

Deriving the nuclear shell model from first principles

The results of an 18-nucleon No Core Shell Model calculation, performed in a large basis space using a bare, $soft$ NN interaction, can be projected into the $0\hbar\omega$ space, i.e., the $sd$-shell. Because the 16 nucleons in the $^{16}$O $core$ are $frozen$ in the $0\hbar\omega$ space, all the correlations of the 18-nucleon system are captured by the two valence, $sd$-shell nucleons. By the projection, we obtain microscopically the $sd$-shell 2-body effective interactions, the core energy and the $sd$-shell s.p. energies. Thus, the input for standard shell-model calculations can be determined microscopically by this approach. If the same procedure is then applied to 19-nucleon systems, the $sd$-shell 3-body effective interactions can also be obtained, indicating the importance of these 3-body effective interactions relative to the 2-body effective interactions. Applications to A=19 and heavier nuclei with different intrinsic NN interactions will be presented and discussed.   

Optimized two- and three-nucleon forces at next-to-next-to leading order

In this talk I will present recent developments in optimizing interactions from chiral effective field theory and their impact on nuclear structure calculations. In particular I will demonstrate the importance of simultaneously optimizing nucleon-nucleon and three-nucleon forces at next-to-next-to leading order. The importance of uncertainty quantification and sensitivity analysis in nuclear modeling have been acknowledged recently. Therefore, I will address the extraction of statistical uncertainties of the low-energy coupling constants and the subsequent error propagation to heavier nuclei.   

$A=10$ nuclei and $^{12}$C with SRG evolved chiral three-nucleon interactions

We investigate selected static and transition properties of $A=10$ nuclei and $^{12}$C using {\it ab initio} No-Core Shell Model (NCSM) methods with chiral two- and SRG-evolved three-nucleon interactions. We examine the dependences of observables on the SRG evolution scale and on the model-space parameters. We obtain nearly converged low-lying excitation spectra for $^{12}$C. We compare results of the full NCSM with the Importance Truncated NCSM in large model spaces for benchmarking purposes in $^{12}$C. The agreement of some observables with experiment is improved significantly by the inclusion of 3N interactions, e.g., the B(M1) from the first $(J^\pi,T)=(1^+,1)$ state to the ground state of $^{12}$C. However, in some cases the agreement deteriorates, e.g., for the excitation energy of the first $(1^+,0)$ state, leaving room for improved next-generation chiral Hamiltonians. On the other hand, the excitation energies of $^{10}$C, $^{10}$B,and $^{10}$Be are not as well converged as those of $^{12}$C. In particular the lowest two $(1^+,0)$ states of $^{10}$B are sensitive to both the basis truncation parameters and the 3N interaction.   

Three Dimensional SRG Evolution of the $NN$ Interaction Using Picard Iteration

We solve the similarity renormalization group (SRG) flow equations in a Three Dimensional (3D) helicity representation (without partial wave decomposition) for realistic nucleon-nucleon ($NN$) interactions. During the 3D SRG evolution, the flow equations become extremely stiff for far off diagonal matrix elements (e.g. $|{\bf k}| \gg |{\bf k}^\prime|$). We alleviate this by transforming the differential form of the SRG flow equation into an integral equation that is solved using Picard iteration. The evolved $NN$ interactions are obtained from realistic potentials by solving a single integral equation for total spin $0$ and four coupled integral equations for total spin $1$. We demonstrate the efficiency and accuracy of the Picard integral approach for the Bonn-B and Chiral-N2LO $NN$ potentials. The successful 3D implementation paves the path to consider a 3D evolution of three-nucleon forces.   

Three-body Momentum Representation SRG Evolution and Operator Factorization

The Similarity Renormalization Group (SRG), as applied in nuclear structure and reactions calculations, is a tool to systematically soften nuclear Hamiltonians, including three and higher body terms. It exploits a flow equation that ensures the transformations of the Hamiltonian and other operators are unitary. Previous studies of the SRG evolution of operators, and the corresponding unitary operator, have focused only on one and two body terms in the evolution, neglecting induced few body terms. Using a hyperspherical harmonic momentum representation, we are able to extend such studies to the three-body SRG evolution. We find that similar to the two body sector, the three body unitary operator also factorizes into universal low momentum, and non-universal high momentum functions, analogous to what previous studies has observed for the two body unitary operator.   

Recent development in lattice QCD studies for three-nucleon forces

The direct determination of nuclear forces from QCD has been one of the most desirable challenges in nuclear physics. Recently, a first-principles lattice QCD determination is becoming possible by a novel theoretical method, HAL QCD method, in which Nambu-Bethe-Salpeter (NBS) wave functions are utilized. In this talk, I will focus on the study of three-nucleon forces in HAL QCD method by presenting the recent theoretical / numerical development.   

Chiral 3N forces and Green's function Monte Carlo calculations of light nuclei

In this talk I will present our recent work on Green's function Monte Carlo (GFMC) calculations of light nuclei using local nucleon-nucleon interactions derived from chiral effective field theory (EFT) up to next-to-next-to-leading order (N2LO). GFMC provides important benchmarking capabilities for other methods which rely on techniques to soften the nuclear interaction and also allows for non-perturbative studies of the convergence of the chiral EFT expansion. I will discuss the natural extension of this work to include the consistent three-nucleon (3N) forces at the same order in the chiral expansion. I will discuss our choice of observables to fit the two low-energy constants which enter in the 3N sector at N2LO, present some results for light nuclei, and give some indications of exciting future work which is now possible.