Recent developments in nuclear theory allow us to make a connection between
quantum chromodynamics (QCD) and low-energy nuclear physics. First, chiral
effective field theory (chiEFT) provides a natural hierarchy to define
two-nucleon (NN), three-nucleon (NNN), and even four-nucleon interactions.
Second, ab initio methods have been developed capable to test these
interactions for light nuclei. In this contribution, we discuss ab initio
no-core shell model (NCSM) calculations for s-shell and p-shell nuclei with NN
and NNN interactions derived within chiEFT.Comment: 6 pages, 6 figures, proceedings of the 20th European Conference on
  Few-Body Problems in Physics (EFB20

A Nogga

A. Nogga

DR Entem

E Epelbaum

J Gasser

J. P. Vary

K Langanke

P Navrátil

P. Navrátil

R Roth

S Quaglioni

S Weinberg

S. Quaglioni

U van Kolck

V. G. Gueorguiev

W. E. Ormand

YC Tang

English

arXiv

Crossref

Recent developments in nuclear theory allow us to make a connection between quantum chromodynamics (QCD) and low-energy nuclear physics. First, chiral effective field theory (chi EFT) provides a natural hierarchy to define two-nucleon (NN), three-nucleon (NNN), and even four-nucleon interactions. Second, ab-initio methods have been developed capable to test these interactions for light nuclei. In this contribution, we discuss ab-initio no-core shell-model (NCSM) calculations for s-shell and p-shell nuclei with NN and NNN interactions derived within chi EFT

Navratil, P.

Gueorguiev, V.G.

Vary, J.P.

Ormand, W.E.

Nogga, A.

Quaglioni, S.

Juelich Shared Electronic Resources

arXiv:0712.1207v1  [nucl-th]  7 Dec 2007Few-Body Systems 0, 1–6 (2014)Few-BodySystemsc© by Springer-Verlag 2014Printed in AustriaLight nuclei from chiral EFT interactionsP. Navra´til1∗, V. G. Gueorguiev1, J. P. Vary1,2, W. E. Ormand1, A. Nogga3,S. Quaglioni11Lawrence Livermore National Laboratory, L-414, P.O. Box 808, Livermore, CA 94551, USA2Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA3Forschungszentrum Ju¨lich, Institut fu¨r Kernphysik (Theorie), D-52425 Ju¨lich, GermanyAbstract. Recent developments in nuclear theory allow us to make a con-nection between quantum chromodynamics (QCD) and low-energy nuclearphysics. First, chiral effective field theory (χEFT) provides a natural hierar-chy to define two-nucleon (NN), three-nucleon (NNN), and even four-nucleoninteractions. Second, ab initio methods have been developed capable to testthese interactions for light nuclei. In this contribution, we discuss ab initiono-core shell model (NCSM) calculations for s-shell and p-shell nuclei withNN and NNN interactions derived within χEFT.1 IntroductionThe major outstanding problem in nuclear physics is to calculate propertiesof finite nuclei starting from the basic interactions among nucleons. There aretwo aspects to this problem. First, the basic interactions among nucleons arecomplicated, they are not uniquely defined and there is evidence that more thanjust two-nucleon forces are important. Second, the nuclear many-body problemis very difficult to solve. This is a direct consequence of the complex nature ofthe inter-nucleon interactions.Interactions among nucleons are governed by QCD. In the low-energy regimerelevant to nuclear structure, QCD is non-perturbative, and, therefore, hardto solve. Thus, theory has been forced to resort to models for the interaction.New theoretical developments, however, allow us connect QCD with low-energynuclear physics through promising bridge of χEFT [1].2 Ab initio no-core shell modelIn the ab initio NCSM, we consider a system of A point-like non-relativistic nu-cleons that interact by realistic NN or NN+NNN interactions. Unlike in standard∗E-mail address:navratil1@llnl.gov2 Light nuclei from chiral EFT interactionsshell model calculations, in the NCSM there is no inert core, all the nucleons areconsidered active. Therefore the “no-core” in the name of the approach. Besidesthe employment of realistic NN or NN+NNN interactions, two other major fea-tures characterize the NCSM: i) the use of an harmonic oscillator (HO) basistruncated by a chosen maximal HO excitation energy Nmax~Ω relative to theunperturbed ground state of the A-nucleon system; ii) the use of effective in-teractions. The reason behind the choice of the HO basis is the fact that this isthe only basis that allows to use single-nucleon coordinates and consequently thesecond-quantization representation without violating the translational invarianceof the system. The powerful techniques based on the second quantization anddeveloped for standard shell model calculations can then be utilized. Thereforethe “shell model” in the name of the approach. As a downside, one has to facethe consequences of the incorrect asymptotic behavior of the HO basis. The sec-ond feature comes as a consequence of the basis truncation. In order to speedup convergence with the basis enlargement, we construct an effective interactionfrom the original realistic NN or NN+NNN potentials by means of a unitarytransformation. The effective interaction depends on the basis truncation and byconstruction recovers the original realistic NN or NN+NNN interaction as thesize of the basis approaches infinity. In principle, one can also perform calcu-lations with the unmodified, “bare”, original interactions. Such calculations arethen variational with respect to the HO frequency Ω and the basis truncationparameter Nmax.3 Light nuclei from chiral EFT interactionsCurrently the most promising approach to the construction of accurate inter-nucleon forces from QCD is the χEFT. The χEFT predicts, along with the NNinteraction at the leading order, an NNN interaction starting at the 3rd order(next-to-next-to-leading order or N2LO) [1, 2], and even an NNNN interactionstarting at the 4th order (N3LO) [3]. The details of QCD dynamics are containedin parameters, low-energy constants (LECs), not fixed by the symmetry. Theseparameters can be constrained by experiment. A crucial feature of χEFT is theconsistency between the NN, NNN and NNNN parts. As a consequence, at N2LOand N3LO, except for two LECs, assigned to two NNN diagrams, the potentialis fully constrained by the parameters defining the NN interaction.We adopt the potentials of the χEFT at the orders presently available, theNN at N3LO of Ref. [4] and the NNN interaction at N2LO [2]. We use ab initioNCSM calculations in two ways. One of them is the determination of the LECsassigned to two NNN diagrams, cD and cE [5] that must be determined in A ≥ 3systems. cD (cE) is the strength of the NN -pi-N (NNN) contact term. The otheris testing predictions of the chiral NN and NNN interactions for light nuclei.It is important to note that our NCSM results through A = 4 are fullyconverged in that they are independent of the Nmax cutoff and the ~Ω HOenergy. This is demonstrated in Fig. 1, where convergence of the 4He ground-stateenergy using χEFT interactions with and without the NNN terms is shown. Forheavier systems, we characterize the approach to convergence by the dependenceP.Navra´til, V. G. Gueorguiev, J. P. Vary, W. E. Ormand, A. Nogga and S. Quaglioni 34 6 8 10 12 14 16 18 20Nmax-30-29-28-27-26-25-24-23-22-21-20-19-18E [MeV]NN 36 bareNN 36 effNN 28 effNN+NNN 36 bareNN+NNN 36 eff4HeFigure 1. 4He ground-state energy depen-dence on the size of the basis. The HO frequen-cies of ~Ω = 28 and 36 MeV were employed.Results with (thick lines) and without (thinlines) the NNN interaction are shown. Thefull lines correspond to calculations with three-body effective interaction, the dashed lines tocalculations with the bare interaction.-2 0 2 4 6 8 10 12 14 16cD-1-0.8-0.6-0.4-0.200.20.40.60.8c E3H cD-cE curveAverage cD-cE curve3He cD-cE curve-2 0 2 4 6 8 10 12 14-28.8-28.6-28.4-28.2-284 He E gs [MeV]4He Exp-2 0 2 4 6 8 10 12 141.641.661.681.71.724 He r c [fm](a)(b)Figure 2. Relations between cD and cE forwhich the binding energy of 3H (8.482 MeV)and 3He (7.718 MeV) are reproduced. (a) 4Heground-state energy along the averaged curve.(b) 4He charge radius rc along the averagedcurve. Dotted lines represent the rc uncer-tainty due to the uncertainties in the protoncharge radius.of results on Nmax and ~Ω.Fig. 2 shows the trajectories of the two LECs cD − cE that are determinedfrom fitting the binding energies of the A = 3 systems. Separate curves areshown for 3H and 3He fits, as well as their average. There are two points wherethe binding of 4He is reproduced exactly. We observe, however, that in the wholeinvestigated range of cD − cE , the calculated4He binding energy is within a fewhundred keV of experiment. Consequently, the determination of the LECs in thisway is likely not very stringent. We therefore investigate the sensitivity of thep-shell nuclear properties to the choice of the cD − cE LECs. First, we maintainthe A = 3 binding energy constraint. Second, we limit ourselves to the cD valuesin the vicinity of the point cD ∼ 1 since the values close to the point cD ∼ 10overestimate the 4He radius.While most of p-shell nuclear properties, e.g. excitation spectra, are not verysensitive to variations of cD in the vicinity of the cD ∼ 1 point, we were ableto identify several observables that do demonstrate strong dependence on cD.For example, the 6Li quadrupole moment that changes sign depending on thechoice of cD. In Fig. 3, we display the ratio of the B(E2) transitions from the10Bground state to the first and the second 1+0 state. This ratio changes by severalorders of magnitude depending on the cD variation. This is due to the fact thatthe structure of the two 1+0 states is exchanged depending on cD. From Figs. 2and 3, we can see that for cD < −2 the4He radius underestimate experimentwhile for cD > 0 the lowest two 1+ states of 10B are reversed. We therefore selectcD = −1 for our further investigation.We present in Fig. 4 the excitation spectra of 10B as a function of Nmaxfor the chiral NN interaction alone. The convergence with increasing Nmax isquite reasonable for the low-lying states. Similar convergence rates are obtained4 Light nuclei from chiral EFT interactionsFigure 3. Dependence of the 10BB(E2;3+1 0 → 1+1 0)/B(E2;3+1 0 → 1+2 0)ratio on cD, with cE constrained by the A = 3binding energy fit, for different basis sizes.The HO frequency of ~Ω = 14 MeV wasemployed.Ω Ω Ω Ω−3−2−101234567893+1+0+ 11+2+3+2+ 12+4+2+ 11+3+2+1+0+ 12+3+4+2+ 12+ 110B     NN             Ω =15 MeV    h¯6h¯ 4h¯ 2h¯ 0h¯ExpE(MeV)Figure 4. 10B excitation spectra as functionof the basis space size Nmax at ~Ω = 15 MeVusing the chiral NN interaction compared toexperiment. The isospin of the states not ex-plicitly depicted is T=0.for other p−shell nuclei that we investigate. A remarkable feature of the 10Bresults is the observation that the chiral NN interaction alone predicts incorrectground-state spin of 10B. The experimental value is 3+0, while the calculated oneis 1+0. On the other hand, once we also include the chiral NNN interaction in theHamiltonian, which is actually required by the χEFT, the correct ground-statespin is predicted. Further, once we select the cD value as discussed above, i.e.cD = −1, we also obtain the two lowest 1+0 states in the experimental order.We display in Fig. 5 the natural parity excitation spectra of four nuclei inthe middle of the p−shell with both the NN and the NN+NNN interactionsfrom χEFT. The results shown are obtained in the largest basis spaces achievedto date for these nuclei with the NNN interactions, Nmax = 6 (6~Ω). Overall,the NNN interaction contributes significantly to improve theory in comparisonwith experiment. This is especially well-demonstrated in the odd mass nuclei forthe lowest few excited states. The case of the ground state spin of 10B and itssensitivity to the presence of the NNN interaction discussed also in Fig. 4. isclearly evident. We note that the 10B results shown in Fig. 4 were obtained withthe HO frequency of ~Ω = 15 MeV, while those in Fig. 5 with ~Ω = 14 MeV. Aweak HO frequency dependence of the Nmax = 6 results is apparent.Further details on calculations presented in this section were published inRef. [6]. These calculations demonstrate that the chiral NNN interaction makessubstantial contributions to improving the spectra and other observables. How-ever, there is room for further improvement in comparison with experiment. Weused a strength of the 2pi-exchange piece of the NNN interaction, which is consis-tent with the NN interaction that we employed (i.e. from Ref. [4]). This strengthis somewhat uncertain (see e.g. Ref. [5]). Therefore, it will be important to studythe sensitivity of our results with respect to this strength. Further on, it will beinteresting to incorporate sub-leading NNN interactions and also four-nucleoninteractions, which are also order N3LO [3]. Finally, it will be useful to extendthe basis spaces to Nmax = 8 (8~Ω) for A > 6 to further improve convergence.P.Navra´til, V. G. Gueorguiev, J. P. Vary, W. E. Ormand, A. Nogga and S. Quaglioni 5048NN+NNN Exp NN3 + 3 +1 +1 +0 +; 1 0 +; 11 +1 +2 +2 +3 +3 +2 +; 12 +; 12 +2 +4 +4 +2 +; 12 +; 1hΩ=140481216NN+NNN  Exp NN 3/2- 3/2- 1/2- 1/2- 5/2- 5/2- 3/2- 3/2- 7/2- 7/2- 5/2- 5/2- 5/2- 5/2- 1/2-; 3/2 1/2-; 3/20481216NN+NNN  Exp NN0 + 0 +2 +2 +1 +1 +4 +4 +1 +; 1 1 +; 12 +; 1 2 +; 10 +; 10 +; 10481216NN+NNN  Exp NN 1/2-  1/2- 3/2- 3/2- 5/2- 5/2- 1/2- 1/2- 3/2- 3/2- 7/2- 7/2- 3/2-; 3/2 3/2-; 3/210B11B 12C 13CFigure 5. States dominated by p-shell configurations for 10B, 11B, 12C, and 13C calculated atNmax = 6 using ~Ω = 15 MeV (14 MeV for10B). Most of the eigenstates are isospin T=0 or1/2, the isospin label is explicitly shown only for states with T=1 or 3/2. The excitation energyscales are in MeV.4 Beyond nuclear structure with chiral EFT interactionsA shortcoming of the ab initio NCSM is its incorrect description of long-rangecorrelations and its lack of coupling to continuum due to the expansion of theeigenstates in a finite HO basis. If we want to build upon the ab initio NCSMto microscopically describe nuclear reactions we can proceed in two ways. First,we can rely on techniques such as the Lorentz Integral Transform that reducethe continuum problem to a bound-state-like problem. Using this approach, weperformed 4He photo-disintegration cross section calculations using the χEFTinteractions [7]. We demonstrated a sizeable effect of the chiral NNN interactionon the cross section. Second, we can augment the ab initio NCSM basis by ex-plicitly including cluster states and solve for their relative motion while imposingthe proper boundary conditions. This approach, applicable to a wide range of re-actions as well as to weakly bound states, is in the spirit of the resonating groupmethod (RGM) [8]. In our approach, we use the ab initio NCSM wave functionsfor the clusters involved and effective interactions derived from realistic NN (andeventually also from NNN) potentials. As an example, a converged calculationof the n+4He 2S1/2 phase shift using the χEFT NN interaction is presented inFig. 6. More details on this approach are given in Ref. [9].5 ConclusionsThe ab initio NCSM evolved into a powerful many-body technique. Presently,it is the only method capable to use interactions derived within the χEFT forsystems of more than four nucleons. Among its successes is the demonstrationof the importance of the NNN interaction for nuclear structure. Applicationsto nuclear reactions with a proper treatment of long-range properties are underdevelopment. Extension to heavier nuclei is achieved through the importance-truncated NCSM [10]. Within this approach, ab initio calculations for nuclei as6 Light nuclei from chiral EFT interactions0 1 2 3 4 5 6 7 8 9 10-90-60-300468101214n+4He Nmaxh¯Ω = 19 MeV2S1/2E [MeV]δ[deg]N3LO NNExpt.Figure 6. The calculated 2S1/2 phase shift for the n+4He system compared with experimentaldata. The χEFT NN potential of Ref. [4] was used in model spaces up to the 14~Ω.heavy as 40Ca become possible.Acknowledgement. This work performed under the auspices of the U.S. Department of Energyby Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344. Supportfrom U.S. DOE/SC/NP (Work Proposal Number SCW0498) and the Department of Energyunder Grant DE-FC02-07ER41457 is acknowledged.References1. S. Weinberg, Physica 96A, 327 (1979); Phys. Lett. B 251, 288 (1990); Nucl.Phys. B363, 3 (1991); J. 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