An optimized chiral nucleon-nucleon interaction at next-to-next-to-leading order

An optimized chiral nucleon-nucleon interaction at next-to-next-to-leading order

A. Ekström Department of Physics and Center of Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA    G. Baardsen Department of Physics and Center of Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway    C. Forssén Department of Fundamental Physics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden    G. Hagen Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA    M. Hjorth-Jensen Department of Physics and Center of Mathematics for Applications, University of Oslo, N-0316 Oslo, Norway National Superconducting Cyclotron Laboratory, Michigan State University, East Lansing, MI 48824, USA Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA    G. R. Jansen Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA    R. Machleidt Department of Physics, University of Idaho, Moscow, ID 83844, USA    W. Nazarewicz Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA Faculty of Physics, University of Warsaw, ul. Hoża 69, 00-681 Warsaw, Poland    T. Papenbrock Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 37996, USA Physics Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA    J. Sarich Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439, USA    S. M. Wild Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439, USA

We optimize the nucleon-nucleon interaction from chiral effective field theory at next-to-next-to-leading order. The resulting new chiral force NNLO yields per degree of freedom for laboratory energies below approximately 125 MeV. In the nucleon systems, the contributions of three-nucleon forces are smaller than for previous parametrizations of chiral interactions. We use NNLO to study properties of key nuclei and neutron matter, and we demonstrate that many aspects of nuclear structure can be understood in terms of this nucleon-nucleon interaction, without explicitly invoking three-nucleon forces.

21.30.-x, 21.10.-k, 21.45.-v, 21.60.-n

Introduction – Interactions from chiral effective field theory (EFT) employ symmetries and the pattern of spontaneous symmetry breaking of quantum chromodynamics van Kolck (1994); Epelbaum et al. (2009); Machleidt and Entem (2011). In this approach, the exchange of pions within chiral perturbation theory yields the long-ranged contributions of the nuclear interaction, while short-ranged components are included as contact terms. The interaction is parametrized in terms of low-energy constants (LECs) that are determined by fit to experimental data. The interactions from chiral EFT exhibit a power counting in the ratio , with being the low-momentum scale being probed and the cutoff, which is of the order of 1 GeV. At next-to-next-to-leading order (NNLO), three-nucleon forces (3NFs) enter, while four-nucleon forces (4NFs) enter at next-to-next-to-next-to-leading order (NLO). For laboratory energies below 125 MeV, the nucleon-nucleon () force exhibits a quality of fit with /datum at NNLO Epelbaum et al. (2004); *epelbaum2002a, while a high-precision potential NLO, with a /datum up to 290 MeV, was obtained by Entem and Machleidt Entem and Machleidt (2003); *ME11.

The 3NFs at NNLO that accompany the current NLO potentials play a pivotal role in nuclear structure calculations Hammer et al. (2013). They determine the ground-state spin of Navrátil et al. (2007), correctly set the drip line in oxygen isotopes Otsuka et al. (2010); Hagen et al. (2012a), and make Ca a doubly magic nucleus Holt et al. (2012); Hagen et al. (2012b). While it might seem surprising that smaller corrections at NNLO are so decisive for basic nuclear structure properties, the 3NF contains spin-orbit and tensor contributions that clearly are important for the currently employed chiral interactions. The contributions of 3NFs at NLO have also been worked out Ishikawa and Robilotta (2007); Bernard et al. (2008); *BEKM11, and there are on-going efforts to compute even higher orders Krebs et al. (2012).

While the quest for higher orders is important, this approach will result in higher accuracy only if the optimization at lower orders was carried out accurately. Thus, it is important and timely to revisit the optimization question. We note in particular that the fits of the currently employed chiral interactions Epelbaum et al. (2002a, b); Entem and Machleidt (2003) date back about a decade and that there has been a considerable recent progress in developing tools for the derivative-free nonlinear least-squares optimization Kortelainen et al. (2010). Furthermore, the quantification of theoretical uncertainties is a long-term objective of nuclear structure theory, and this requires a covariance analysis of the interaction parameters with respect to the experimental uncertainties of the nucleon-nucleon elastic scattering observables; see, for example, Refs. Reinhard and Nazarewicz (2010); Kortelainen et al. (2010). This letter takes the first step toward this goal. We present a state-of-the-art optimization of the chiral EFT interaction at NNLO. This yields a much-improved and a high-precision potential NNLO. The 3NF at NNLO is adjusted to the binding energies in nuclei. We present computations of three-nucleon and four-nucleon bound states, and we employ NNLO to ground states and excited states in B, masses and excited states of oxygen and calcium isotopes, and neutron matter.

Optimizing the interaction at NNLO – For the optimization of the chiral interaction we use the Practical Optimization Using No Derivatives (for Squares) algorithm, POUNDerS Kortelainen et al. (2010), as implemented in  Munson et al. (2012). This derivative-free algorithm employs a quadratic model and is particularly useful for computationally expensive objective functions. We optimize the three pion-nucleon () couplings (), and 11 partial wave contact parameters and , while we keep the axial-vector coupling constant , the pion-decay constant , and all masses fixed. In the optimization, we minimize the objective function


where are NNLO phase shifts, are experimental phase shifts from the Nijmegen multi-energy partial-wave analysis Stoks et al. (1993), denotes the parameters of the chiral interaction, and are weighting factors. Note that Eq. (1) is not the with respect to experimental data. The actual is calculated following the POUNDerS optimization. The phase shifts are computed from -matrix inversion, and in the proton-proton () channels we include the Coulomb interaction Vincent and Phatak (1974); Lu et al. (1994). The contact terms are optimized to reproduce the Nijmegen phase shifts for each corresponding partial wave, while keeping the ’s fixed. For the contacts, the weight scales with the third power of the relative momentum , while for the ’s, we employ the uncertainties quoted in the Nijmegen analysis Stoks et al. (1993). This approach can be justified by a physical argument: for the peripheral waves the higher energies still represent longer-range physics, and the need for a pedantic agreement with lower energy phase shifts can be weakened. The couplings , and were simultaneously optimized to the peripheral partial-waves and . Note that the NNLO contact terms do not contribute to orbital angular momenta . We do not include other peripheral waves from the Nijmegen study since they carry extremely small uncertainties, which lead to a very noisy objective function.

Table 1 summarizes the optimization results. Our values should be compared with the couplings as determined from scattering data, where , , and have been obtained Büttiker and Meißner (2000). Thus, POUNDerS yields values for and that agree well with the empirical determination from scattering. The value, however, deviates significantly from its empirical value. The same trend was found in the construction of the NLO Entem and Machleidt (2003) interaction. A detailed statistical sensitivity analysis of the LECs with uncertainty quantification will be presented in Ref. Ekström et al. (2013).

LEC Value LEC Value LEC Value
-0.91863953 -3.88868749 4.31032716
-0.15136604 -0.15214109 -0.15176475
2.40402194 0.92838466 -0.15843418
0.41704554 1.26339076 -0.78265850
0.61814142 -0.67780851
Table 1: Parameters of NNLO at MeV and MeV Epelbaum et al. (2004): (in GeV), (in  GeV), and (in  GeV).

Table 2 shows the /datum for NNLO at various laboratory energy bins. The quality of the fit is particularly good for energies below 125 MeV. For comparison, the NNLO interaction of Ref. Epelbaum et al. (2004) yields /datum of 12–27 in the range  MeV at energies up to 290 MeV.

(MeV) 0–35 35–125 125–183 183–290 0–290
/datum 1.11 1.56 29.26
/datum 0.85 1.17 1.87 6.09 2.95
Table 2: /datum for NNLO at  MeV and MeV Epelbaum et al. (2004) with respect to the and 1999 databases Machleidt (2001). The values without the high-precision data sets Cox et al. (1967); *Jar71 are marked by asterisks.

Around energies of 144 MeV there exist two data sets of differential cross sections with a very high precision (0.5% error) Cox et al. (1967); *Jar71 (47 data points). The total number of data in the energy interval 125–183 MeV is 343. The unusual precision of those 47 data points distorts the /datum for this interval. For this reason, Table 2 also shows the results without the high-precision data.

Two comments are in order. First, the with respect to scattering observables is lower when the phase shifts are weighted with the uncertainties from the Nijmegen analysis. The -waves are accurately reproduced only when going to NLO Entem and Machleidt (2003). Second, the coupled channel is optimized with the additional constraint of reproducing the deuteron binding energy. The remaining deuteron observables, as well as the scattering observables, are predictions and reproduce the experimental values well; see Table 3.

NLO NNLO Exp. Ref.
-7.8188 -7.8174 -7.8196(26) Bergervoet et al. (1988)
-7.8149(29) van der Sanden et al. (1983)
2.795 2.755 2.790(14) Bergervoet et al. (1988)
2.769(14) van der Sanden et al. (1983)
-17.083 -17.825
2.876 2.817
-18.900 -18.889 -18.95(40) Gonzalez Trotter et al. (2006); Chen et al. (2008)
2.838 2.797 2.75(11) Miller et al. (1990)
-23.732 -23.749 -23.740(20) Machleidt (2001)
2.725 2.684 2.77(5) Machleidt (2001)
(MeV) 2.224575 2.224582 2.224575(9) Machleidt (2001)
(fm) 1.975 1.967 1.97535(85) Huber et al. (1998)
(fm) 0.275 0.272 0.2859(3) Machleidt (2001)
(%) 4.51 4.05
Table 3: Scattering lengths and effective ranges (both in fm). The superscripts and for the proton-proton observables refer to nuclear forces and Coulomb-plus-nuclear forces, respectively. , , , and denote the deuteron binding energy, radius, quadrupole moment, and -state probability, respectively. and are calculated without meson-exchange currents and relativistic corrections.

Figure 1 shows some phase shifts of NNLO and compares them with phase shifts from other potentials and partial wave analyses. Apart from the -waves, the phase shifts of NNLO closely agree with those obtained at NLO. Note, however, that these deviations do not spoil the good at laboratory energies below 125 MeV.

Figure 1: (Color online) Computed phase shifts of the optimized NNLO potential of this work (red), the NNLO potential of Ref. Epelbaum et al. (2004) (dashed, blue), and the NLO potential Entem and Machleidt (2003) (green, dotted) compared with the Nijmegen phase shift analysis Stoks et al. (1993) (solid dots) and the VPI/GWU analysis SM99 Arndt et al. (1999) (open circles).
H) He) He) He)
NNLO -8.249 -7.501 -27.759 1.43(8)
NNLO+NNN -8.469 -7.722 -28.417 1.43(8)
Experiment -8.482 -7.717 -28.296 1.467(13)
Table 4: Ground-state energies (in MeV) and point proton radii (in fm) for H, He, and He using the NNLO with and without the NNLO 3NF interaction for and .

Three-nucleon forces also appear at NNLO, and two additional LECs ( and ) enter. These are determined from calculations in the three-nucleon and four-nucleon systems. We find that the binding energies of H, He, and He do not uniquely determine and , and the parametric dependence of both LECs is very similar to those found in previous studies Nogga et al. (2006); Navrátil et al. (2007); Gazit et al. (2009). Therefore, we choose guided by the triton half life Gazit et al. (2009) and obtain from optimization to the binding energies. The resulting point charge radii of He are also in good agreement with experiment; see Table 4.

Performance of NNLO for light- and medium-mass nuclei and neutron matter – In this paper, we apply NNLO to B, isotopes of oxygen and calcium, and neutron matter. The considered systems are particularly interesting because the current chiral interactions at NLO completely fail to describe key aspects of their structure.

To study the ground- and first excited state in B, we carry out no-core shell model (configuration interaction) calculations Barrett et al. (2013) using the bare NNLO in model spaces of up to harmonic oscillator (HO) shells (10 ) above the unperturbed configuration. These model spaces are not large enough to provide fully converged results for the ground state and first excited state of B. Still, the variational upper bounds for the energies are  MeV for the state and  MeV for the state. The energies are very close, in contrast to NLO, which yields a level spacing of about 1.2 MeV between the ground state and the excited state Navrátil et al. (2007).

Chiral interactions at NLO fail to explain the neutron drip-line in oxygen isotopes, and 3NFs have been the key element for understanding the structure of nuclei around Otsuka et al. (2010); Hagen et al. (2012a). Figure 2 shows the experimental ground-state energies of oxygen isotopes and compares the results from coupled-cluster (CC) computations in the triples approximation Kucharski and Bartlett (1998); Taube and Bartlett (2008); Hagen et al. (2010). Our CC calculations employ a Hartree-Fock basis (HF) built from HO shells at  MeV. Because of the “softness” of NNLO, this model space is sufficiently large to converge the ground states and excited states of the nuclei considered. In addition, we performed shell-model (SM) calculations assuming the closed O core with an effective interaction derived from many-body perturbation theory to third order in the interaction and including folded diagrams Hjorth-Jensen et al. (1995). For the SM calculations, the single-particle energies were taken from the experimental O spectrum. In both CC and SM, NNLO results are close to experiment. In contrast, the NLO case requires 3NFs to provide reasonable description of measured values.

Figure 2: (Color online). The ground-state energies of oxygen isotopes obtained in CC with the NNLO and NLO interactions compared with experiment. The inset shows SM results.

Now we consider the heavy isotopes of calcium. Here, Ca is doubly magic, Ca exhibits a soft subshell closure, and Ca is predicted to have an even softer subshell closure Hagen et al. (2012b). A signature of shell closure is the location of the first state. We employed CC equation-of-motion methods within the singles and doubles approximation Hagen et al. (2010); Jansen et al. (2011) to compute the first state in the calcium isotopes. Figure 3 shows that NLO fails to describe the location of the first state in Ca. In contrast, NNLO yields Ca as a doubly magic nucleus and predicts subshell closures in Ca.

Figure 3: (Color online). The first state in selected calcium isotopes obtained in CC with the NNLO and NLO interactions compared with experiment.

The NNLO overbinds the calcium isotopes by about 1 MeV per nucleon. In particular Ca are overbound by 1.03 MeV, 1.06 MeV, and 1.04 MeV per nucleon, respectively. That is, the excess energy per nucleon is fairly constant; hence, NNLO reproduces binding energy differences, such as neutron-separation energies and low-lying excited states, rather well.

The complete description of nuclei at NNLO also requires 3NFs. We computed the first state in O and in Ca with the 3NF compatible with the NNLOinteraction. The matrix elements of the 3NF are expensive computationally, and we must at present limit their calculation to three-body energies up to . (Recall that we employ major harmonic oscillator shells for the interaction.) We also used the normal ordered two-body approximation for the 3NF Hagen et al. (2007); Binder et al. (2013) with respect to a HF reference. With the restriction of , we were not able to obtain fully converged results for the binding energies of oxygen and calcium isotopes. However, excitation energies relative to the ground state converge somewhat better. Our results for the first state in O and in Ca are 2.3(3) MeV, 3.5(5) MeV and 4.8(7) MeV, respectively. We estimate the uncertainty by varying in the interval 16–22 MeV. The results obtained by using NNLO interaction alone yields 2.5 MeV, 5.0 MeV, and 4.5 MeV in O and Ca, respectively. These preliminary results suggest that the 3NFs may not dramatically change the results that were obtained with the NNLO interaction alone.

It is instructive to compare the predictions of NNLO and NLO for the neutron matter equation of state at sub-saturation densities with the results of ab-initio calculations of Refs. Tews et al. (2013). Figure 4 shows that the performance of NNLO is on par with the EGM results of Ref. Tews et al. (2013), which take into account the effects of 3NFs and 4NFs. The predictions of NLO deviate from other results at higher densities.

Figure 4: (Color online). Energy per nucleon for neutron matter for NNLO and NLO Entem and Machleidt (2003). The calculations used the CC method with the inclusion of particle-particle ladders and a continuous single-particle spectrum. The shaded area (EGM) shows uncertainty bands for NLO chiral effective field theory calculations of Ref. Tews et al. (2013), including 3NFs.

Conclusions – We constructed the new chiral EFT interaction NNLO at next-to-next-to-leading order using the optimization tool POUNDerS in the phase-shift analysis. The optimization of the low-energy constants in the -sector at NNLO yields a /datum of about one for laboratory scattering energies below 125 MeV. The NNLO interaction yields very good agreement with binding energies and radii for nuclei. Key aspects of nuclear structure, such as excitation spectra, the position of the neutron drip line in oxygen, shell-closures in calcium, and the neutron matter equation of state at sub-saturation densities, are reproduced by NNLO interaction alone, without resorting to 3NFs. We performed the initial calculation of the first states in O and Ca with NNLO supplemented by a 3NF and found effects of 3NFs to be small and good agreement with experimental excitation energies. The precise role of 3NFs in medium-mass nuclei, the quantification of theoretical uncertainties, and optimizations at higher-order chiral interactions will be addressed in forthcoming investigations.

We thank M. P.  Kartamyshev, B. D. Carlsson, and H. T. Johansson for discussions and related code development. This work was supported by the Research Council of Norway under contract ISP-Fysikk/216699; by the Office of Nuclear Physics, U.S. Department of Energy (Oak Ridge National Laboratory), under Grant Nos. DE-FG02-03ER41270 (University of Idaho), DE-FG02-96ER40963 (University of Tennessee), DE-AC02-06CH11357 (Argonne), and DE-SC0008499 (NUCLEI SciDAC collaboration); by the Swedish Research Council (dnr 2007-4078), and by the European Research Council (ERC-StG-240603). Computer time was provided by the Innovative and Novel Computational Impact on Theory and Experiment (INCITE) program. This research used resources of the Oak Ridge Leadership Computing Facility located in the Oak Ridge National Laboratory, which is supported by the Office of Science of the Department of Energy under Contract No. DE-AC05-00OR22725, and used computational resources of the National Center for Computational Sciences, the National Institute for Computational Sciences, and the Notur project in Norway.


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