isobars and nuclear saturation
We construct a nuclear interaction in chiral effective field theory with explicit inclusion of the -isobar degree of freedom at all orders up to next-to-next-to-leading order (NNLO). We use pion-nucleon () low-energy constants (LECs) from a Roy-Steiner analysis of scattering data, optimize the LECs in the contact potentials up to NNLO to reproduce low-energy nucleon-nucleon scattering phase shifts, and constrain the three-nucleon interaction at NNLO to reproduce the binding energy and point-proton radius of He. For heavier nuclei we use the coupled-cluster method to compute binding energies, radii, and neutron skins. We find that radii and binding energies are much improved for interactions with explict inclusion of , while -less interactions produce nuclei that are not bound with respect to breakup into particles. The saturation of nuclear matter is significantly improved, and its symmetry energy is consistent with empirical estimates.
pacs:21.30.-x, 21.10.-k, 21.45.-v, 21.60.De
In recent years, ab initio calculation of atomic nuclei with predictive power have advanced from light Kamada et al. (2001); Pieper and Wiringa (2001); Navrátil et al. (2009); Barrett et al. (2013) to medium-mass nuclei Hagen et al. (2014a); Lähde et al. (2014); Hagen et al. (2016); Hergert et al. (2016); Hagen et al. (2016). Such calculations are only as good as their input, i.e. nucleon-nucleon () and three-nucleon () interactions, therefore the quest for more accurate and more precise nuclear potentials is an ongoing endeavor at the forefront of research Epelbaum et al. (2000, 2002); Entem and Machleidt (2003); Shirokov et al. (2004); Hebeler et al. (2011); Ekström et al. (2013); Entem et al. (2015); Epelbaum et al. (2015a); Ekström et al. (2015); Lynn et al. (2016); Carlsson et al. (2016). Here, potentials from chiral effective field theory (EFT) – based on long-ranged pion exchanges and short-ranged contact interactions – play a dominant role Epelbaum et al. (2009); Machleidt and Entem (2011), because they are expected to deliver accuracy (via fit to data) and precision (via increasingly higher orders in the power counting). As it turns out, however, state-of-the-art EFT potentials that are accurate for the lightest nuclei with masses vary considerably in their saturation point for nuclear matter Hebeler et al. (2011) and in their binding energy for heavier nuclei Binder et al. (2014); Carlsson et al. (2016); Simonis et al. (2017).
This sensitivity of the saturation point to the details of the EFT interaction is not well understood Elhatisari et al. (2016) and also puzzling from an EFT perspective. A practical approach to this dilemma consists of constraining EFT potentials to reproduce experimentally determined binding energies and charge radii of nuclei as heavy as oxygen Ekström et al. (2015). In this work, we will follow a different approach and explicitly include the isobar , abbreviated in the following, as a low-energy degree of freedom in addition to pions () and nucleons (). We recall that the - mass-splitting 293 MeV is roughly twice the pion mass ( MeV) and well below the expected breakdown scale of EFT potentials Epelbaum et al. (2009); Machleidt and Entem (2011). Furthermore, the also couples strongly to the system. For these reasons, the early chiral interactions van Kolck (1994); Ordóñez et al. (1994, 1996) included the degree of freedom. Indeed, van Kolck as well as Bernard et al. showed that the low-energy constants (LECs) of the interaction in a -less EFT receive a substantial contribution via resonance saturation. As nuclear interactions from EFT with and without ’s have a similar structure otherwise, only little effort was invested in producing quantitative -full EFT potentials. We refer the reader to the reviews Epelbaum et al. (2009); Machleidt and Entem (2011) for extensive discussions of this topic.
Recently, Piarulli et al. produced minimally non-local EFT potentials at next-to-next-to-next-to leading order (NLO), with ’s included up to next-to-next-to leading order (NNLO), using values for the subleading LECs from Ref. Krebs et al. (2007). Dropping the non-local terms led to the local potentials of Ref. Piarulli et al. (2016). Two different approaches augmented these local potentials with forces up to NNLO. The corresponding diagrams of the force, some diagrams, and the most relevant LECs are shown in Fig. 1. Logoteta et al. adjusted the LECs and of the short-ranged terms to reproduce the saturation point of nuclear matter. However, they did not report results for few-nucleon systems. In contrast, Piarulli et al. adjusted and to reproduce properties of nuclear systems with mass number . Their quantum Monte Carlo calculations yielded accurate results for spectra of light nuclei up to C. We note that the potentials by Logoteta et al. and Piarulli et al. employ values for and that differ in signs and magnitudes.
In this paper we present a systematic construction and comparative analysis of non-local -full and -less EFT potentials at LO, NLO, and NNLO, and report results for light- and medium-mass nuclei, and infinite nucleonic matter. We constrain the relevant short-ranged LECs using experimental data from nuclear systems with mass numbers and use LECs determined in a recent high-precision analysis Siemens et al. (2017) based on the Roy-Steiner equations Hoferichter et al. (2016). We do not include any additional contact operators beyond NNLO in Weinberg power counting. We find that the resulting -full potentials yield accurate charge radii and much improved binding energies for medium mass nuclei, and reproduce the saturation point of symmetric nuclear matter within estimated EFT-truncation errors. Furthermore, estimates of the EFT-truncation errors furnish a discussion of the improved convergence rate of the -full EFT expansion compared to the -less theory.
Ii Optimization of interactions
To isolate the effects of the isobar in the description of the saturation properties of nucleonic matter we compare our results with -less EFT potentials at LO, NLO, and NNLO. Other than the inclusion of the -isobar, the -full and the -less interactions are constructed following identical optimization protocols. For the description of the interaction we build on work van Kolck (1994); Hemmert et al. (1998); Kaiser et al. (1998); Krebs et al. (2007); Epelbaum et al. (2008) and treat the - mass difference as an additional small scale. A power-counting for this approach is provided by the so-called small-scale expansion Hemmert et al. (1998). This is identical to the conventional heavy-baryon formulation of EFT which is already used for including the nucleon mass-scale without any isobars. The -less pion-exchanges in the sector up to NNLO are given in Ref. Entem et al. (2015). The expressions for the contact potentials at LO and NLO are given in e.g. Ref. Machleidt and Entem (2011), and the contributions to the leading and sub-leading -exchanges in the potential are from Ref. Krebs et al. (2007). Charge-independence breaking terms are included in the LO contact LECs as well as the one-pion exchange. Following Ref. Long and Lensky (2011) we remove all contributions that are proportional to the subleading coupling by renormalizing the axial coupling and the subleading couplings . We follow Siemens et al. and use , , and the central Roy-Steiner values of the LECs for the -full and -less potentials up to third order. We recall that -less EFT potentials often employ LECs with values that differ from what is found in scattering, because the absence of ’s strongly renormalize the couplings in the three-nucleon sector Pandharipande et al. (2005). The -full theory is more consistent in this regard and the ’s appear to be more natural in size.
The expressions for the three-nucleon diagrams at NNLO are from Ref. Epelbaum et al. (2002). The NLO -force in the -full theory is given by the well-known Fujita-Miyazawa term Fujita and Miyazawa (1957). This topology is identical in structure to the -less -exchange interaction when using the resonance-saturation values for the relevant LECs
To construct quantitative -full EFT potentials we need to determine the numerical values of the LECs in the LO and NLO contact potentials and the and terms in the interaction at NNLO. To optimize the contact LECs we use a Levenberg-Marquardt algorithm with machine-precise derivatives from automatic differentiation Carlsson et al. (2016). The objective function for the LO and NLO contact LECs consists of the sum of squared differences between the theoretical partial-wave scattering phase shifts and the corresponding values from the the Granada analysis Pérez et al. (2013) up to 200 MeV scattering energy in the laboratory system. At LO, we only use phase shifts up to 1 MeV. The neutron-neutron LEC is constrained to reproduce the effective range expansion in the channel. At NNLO we use the same optimization algorithm to find the and LECs that simultaneously reproduce the binding energy and point-proton radius of He. Although correlated, these observables provide enough information to identify a unique minimum in the - plane that is sufficient for the purpose of comparing the effects in nuclei and nucleonic matter due to the isobar. An extended regression analysis or Bayesian inference approach including additional data from many-nucleon systems or three-nucleon scattering would generate interactions for use in detailed analyses of atomic nuclei or model selection. In this work we focus on the effects of the -isobar in nucleonic matter.
To regulate the the interactions we use the usual non-local regulators
in the and interactions, resepectively. Here, and denote the Jacobi momenta in the two-body system and spectator nucleon, respectively, and is the momentum cutoff. The non-local regulator acts multiplicative, i.e.
To explore the sensitivity of the results with respect to changes in the cutoff we employ two common choices, namely MeV and MeV. To regularize the -exchanges in conjunction with non-local regulation we use the standard spectral-function regularization (SFR) Epelbaum et al. (2005) with a cutoff MeV throughout. It should also be pointed out that recent work, e.g. Refs. Valderrama and Arriola (2009); Pavon Valderrama and Ruiz Arriola (2011); Epelbaum et al. (2015b), indicates that a carefully selected local regulation of the long-ranged -exchanges render SFR redundant and yields an improved analytical structure of the scattering amplitude. However, the overall existence of such scheme dependencies Dyhdalo et al. (2016) will persist as long as the chiral interactions cannot be order-by-order renormalized, see e.g Ref. Hoppe et al. (2017) for a recent analysis. The numerical values of the employed LECs and the optimized short-ranged LECs for the -less as well as the -full potentials are given in Tables 1 and 2. For the masses of the pions , proton, neutron, nucleon (), and we use the following values (in MeV): , , , , , and , respectively.
The statistical error from the Roy-Steiner analysis of the scattering data, documented in Ref. Siemens et al. (2017), as well as uncertainties due to the fit of the contact potentials, are not considered any further in this work. When contrasted with the much larger systematic uncertainties due to the truncation of the EFT, such statistical errors presently play a lesser role Carlsson et al. (2016); Ekström et al. (2015); Pérez et al. (2015). It is important to note that although the LECs are extracted from data using a high-precision Roy-Steiner analysis, the corresponding LECs in the -full sector are less precise due to the large uncertainty in the underlying determination of .
To provide a crude estimate of the EFT-truncation uncertainty we follow Refs. Epelbaum et al. (2015a); Furnstahl et al. (2015) and write the EFT expansion for an observable as . Here is the scale of the observable, given e.g. by the LO prediction, are dimensionless expansion coefficients (with in Weinberg power counting), and is the ratio of the typical momentum and the breakdown momentum . The application of Bayes theorem with boundless and uniform prior distribution of the expansion coefficients leads to an expression for the truncation error at order NjLO (: LO, : NLO, : NNLO) according to
see Eq. (36) of Ref. Furnstahl et al. (2015). This estimate is in semi-quantitative agreement with a Bayesian uncertainty quantification of the truncation error. The uncertainty at LO is further constrained to at least the size of the contribution of the higher chiral orders. For the breakdown scale , we start from Ref. Epelbaum et al. (2015a) but use a more conservative estimate of MeV. We also estimate the typical momentum-scale for bound state observables as , and employ (the Fermi momentum) for infinite nucleonic matter, whereas for scattering we extract the momentum scale . We disregard detailed numerical factors in the various possible definitions of the relevant momentum scales for bound states since the estimate in Eq. (1) is only valid up to factors of order unity.
In Figs. 2, 3, and 4 we compare the quality of the scattering phase shifts of the -full and -less interactions with cutoff MeV. The results for the peripheral waves agree well with published interactions that were analyzed in the Born approximation Krebs et al. (2007). The dashed lines show the -less results, order by order from red to green to blue. The full lines show the -full results, and we remind the reader that LO is not affected by the (see Fig. 1).
Clearly, in several partial waves the -full EFT interactions exhibit a faster order-by-order convergence than the corresponding -less formulations. Somewhat surprisingly, the NNLO results at higher scattering energies, in particular for S and selected peripheral waves, such as D, are slightly less accurate than the corresponding -less order. A more involved optimization strategy, such as Bayesian parameter estimation, could further illuminate this point. Nevertheless, the Granada phase shifts fall on the envelope of the estimated truncation errors and the results therefore seem reasonable. Although not shown, the computed phase shifts for the MeV interactions are very similar and exhibit the same features.
Tables 3 and 4 summarize our results for selected bound-state observables in nuclei computed with a Jacobi-coordinate version Navrátil et al. (2000) of the no-core shell model (NCSM) Navrátil et al. (2009); Barrett et al. (2013). All calculations are converged in 41 and 21 major oscillator shells with MeV for and , respectively. The charge radius and binding energy of He were used to constrain the LECs and of the short-ranged three-nucleon force whereas the NCSM results for nuclei are predictions. At NNLO, all results except the binding energy of H, agree with the experimental values within the estimated EFT-truncation errors. The computed point-proton radii were transformed to charge radii using a standard expression, see e.g. Ref. Ekström et al. (2015).
Iii Predictions for medium mass nuclei and nucleonic matter
In this Section we present results for selected finite nuclei and infinite nucleonic matter. For nucleonic matter we present results for both -less and -full interactions, while for finite nuclei we limit the discussion to the -full interactions since the -less interactions produce nuclei that are not bound with respect to breakup into -particles. The computed binding energies and radii of finite nuclei are consistent with our results for the saturation point in symmetric nuclear matter.
iii.1 Finite nuclei
The many-body calculations for finite nuclei are performed with the coupled-cluster (CC) method Kümmel et al. (1978); Bishop (1991); Bartlett and Musiał (2007); Hagen et al. (2014a). We employ the translationally invariant Hamiltonian
Here, denotes the total kinetic energy and the kinetic energy of the center of mass. As the Hamiltonian (2) does not reference the center-of-mass coordinate, the ground-state wave function is a product of an intrinsic and a Gaussian center-of-mass wave function Hagen et al. (2009, 2010); Jansen (2013); Morris et al. (2015); Hergert et al. (2016). The CC method yields a similarity transformed Hamiltonian whose ground state is the product state corresponding to a closed-shell nucleus. In the coupled-cluster singles and doubles (CCSD) approximation, typically accounting for about 90% of the correlation energy, the ground-state is orthogonal to all 1-particle–1-hole (-) and - excitations. In addition to the CCSD approximation we include leading-order - excitations perturbatively by employing the -CCSD(T) method Taube and Bartlett (2008); Hagen et al. (2010); Binder et al. (2013). This approximation typically captures about 99% of the correlation energy. We employ a model space of 15 oscillator shells with MeV, and a cutoff for the maximum excitation energy of three nucleons interacting via the three-nucleon potential . This potential enters the CC calculations in the normal-ordered two-body approximation Hagen et al. (2007); Roth et al. (2012) in the Hartree-Fock basis.
To asses the impact of the -isobar in finite nuclei we calculated the binding energies and charge radii for He, O, and Ca order-by-order, i.e. at LO, NLO, and NNLO. Figure 5 shows the results using the -full interactions with a momentum cutoff MeV. The ground-state energies are , , and MeV and , , and MeV at LO, NLO, and NNLO, respectively. The charge radii are , , and fm and , , and fm at LO, NLO, and NNLO, respectively. Before we analyze the results, we estimate the systematic uncertainties due to the truncation of the EFT. Again we follow Refs. Epelbaum et al. (2015a); Binder et al. (2016), use Eq. (1) and set the momentum scale for our low-energy observables. The predicted charge radii are accurate at each order within uncertainties. Already at NLO, which is independent of the sub-leading -exchange LECs , we obtain an accurate description of both radii and binding energies of He, O and Ca. At NNLO, the charge radii also exhibit a first sign of convergence in terms of the chiral expansion. Binding energies exhibit a nearly identical order-by-order increase in precision but somewhat underbind nuclei at NNLO. These results demonstrate that the -isobar can play an important role also in low-energy nuclear structure and nuclear saturation Hebeler et al. (2011); Ekström et al. (2015).
The degree of freedom also impacts the stability of nclei with respect to breakup into alpha particles. At LO, O and Ca are not stable with respect to alpha emission. Similar results were observed in pionless EFT Stetcu et al. (2007); Contessi et al. (2017); Bansal et al. (2017) and nuclear lattice EFT Elhatisari et al. (2016). However, the modifies the -exchanges between nucleons, and we observe that the -full interactions at NLO and NNLO yield nuclei that are stable with respect to alpha emission. This is in stark contrast to results we obtained here using the -less NLO and NNLO interactions at cutoff MeV, and to those of Ref. Carlsson et al. (2016).
Table 5 summarizes binding energies, radii, and also the neutron skins of nuclei with closed subshells up to Ca. Note that the lack of a spin-orbit (LS) force at LO results in energy-degeneracies that hamper CC calculations of non LS-closed nuclei. Therefore, we can obtain EFT truncation-errors only for O and Ca using Eq. (1). For Ca we predict a neutron skin of fm at NLO and NNLO, consistent with the recent ranges - fm and - fm from Ref. Birkhan et al. (2017) and Ref. Hagen et al. (2016), respectively.
|NLO||NNLO||Exp. Wang et al. (2012)||NLO||NNLO||Exp. Angeli and Marinova (2013)||NLO||NNLO||NLO||NNLO||NLO||NNLO|
Figure 6 shows the charge form-factor at NLO and NNLO, compared to NNLO Ekström et al. (2015) and data. The charge form-factor is obtained by a Fourier transform of the intrinsic charge density Giraud (2008); Hagen et al. (2016), and agrees with data for momentum transfers up to about fm. Also for this quantity, the NLO results indicate an improved convergence of chiral expansion compared to the -less formulation.
We also computed spectra of various nuclei. These explorations exhibited mixed results: While the low-lying states in O were in good agreement with data, O is bound at NNLO with respect to O by about 0.5MeV, and the state in O is too low. We believe that these shortcomings should not distract from the main results reported in this work: accurate saturation properties at NLO in the -full EFT. We speculate that finer details such as spectra will require us to go to higher order in the interaction (as was done, e.g., in Ref. Piarulli et al. (2017) by including NLO contacts), or to vary the -full couplings within their somewhat more generous uncertainty limits due to the rather poorly known coupling , or to also use data of heavier nuclei in the optimization of the interaction. The interactions constructed in this work serve as excellent starting points for such endeavors.
iii.2 Nucleonic matter
We turn to the CC calculations of nuclear matter using -full and -less interactions up to NNLO. We follow Ref. Hagen et al. (2014b) and employ a Hamiltonian . The basis is a discrete lattice in momentum space corresponding to periodic boundary conditions in a cubic box of length in position space, and the discrete lattice momenta are given by , with , and . We used as the maximum number of lattice points. The CC calculations were carried out at the doubles excitation level (-) with perturbative triples (-) corrections [CCD(T)]. Due to translational invariance, there are no - excitations. We use “closed-shell” lattice configurations with 66 neutrons for neutron matter, and 132 nucleons for symmetric nuclear matter. These nucleon numbers exhibit only small finite-size effects Gandolfi et al. (2009); Hagen et al. (2014b). The CCD(T) calculations were performed with the normal-ordered two-body approximation for the interaction Hagen et al. (2007); Roth et al. (2012), i.e. the three-nucleon force enters the normal-ordered Hamiltonian as 0-body, 1-body and 2-body interactions; summing over 3, 2, and 1 particles in the reference state, respectively. All results are well converged for at all considered densities, i.e. fm To gauge the quality of the normal-ordered two-body approximation, we also included the “residual” interaction (i.e. those that generate - excitations when acting on the reference) in perturbation theory. We found that the residual contribution is negligible for neutron matter, and small (0.2-0.3 MeV per nucleon) in symmetric nuclear matter. This suggests that the normal-ordered two-body approximation for the three-nucleon force is sufficiently precise for the -full interactions considered in this work.
In Fig. 7 we compare the results for the energy per nucleon at different densities in symmetric nuclear matter and pure neutron matter using -full and -less interactions with a momentum cutoff 450 MeV at LO, NLO, and NNLO. The saturation points in symmetric matter at NLO and NNLO shift towards considerably more realistic values upon inclusion of the . This observation is consistent with our results for finite nuclei. For the EFT truncation uncertainty we use Eq. 1, a relevant momentum scale , and the breakdown momentum MeV. The uncertainties also make the accelerated convergence and consistency of the -full expansion more apparent. We note that our breakdown scale is rather conservative. For MeV the truncation-error bands of the -full and -less NNLO(450) interactions no longer overlap in the region of the empirical saturation density. We also note that nuclear matter does not saturate at LO in the range of densities we studied, and we remind the reader once more that the does not enter at this chiral order.
Figure 8 shows the difference between the equations of state for neutron matter and symmetric nuclear matter for the MeV cutoff. At the saturation density (), indicated as vertical lines for the different orders, this yields the symmetry energy (). Our results are fm, MeV, and MeV at NLO, and fm, MeV, and MeV at NNLO. The estimated EFT truncation error for is very small at NNLO because its central value and lower and upper bounds have essentially the same saturation point. The estimated EFT truncation error for is the maximum difference between the energies per particle in neutron matter and symmetric nuclear matter, at the saturation point. This uncertainty also decreases with increasing order. Finally, the estimated uncertainty in the slope () of the symmetry energy is taken from the ranges of slopes of at its upper and lower values. It is large even at NNLO and reflects that the slope in neutron matter exhibits a greater variance at NNLO than at NLO, see Fig. 7. We note that our predictions for the symmetry energy and its density derivative at NLO and NNLO are consistent with the recent estimates of Ref. Tsang et al. (2012); Tews et al. (2016).
We presented results for selected finite nuclei and infinite nucleonic matter using optimized interactions from EFT with explicit -isobar degree-of-freedom. We optimized both -full and -less interactions order-by-order in the power counting up to NNLO, for two different cutoffs, and with LECs from a recent Roy-Steiner analysis of scattering. The contact potentials up to NNLO were adjusted to phase shifts, while the short-ranged parts of the interactions were constrained by energy and radius data on He. We emphasize that the only differences between the -full and -less interactions are due to the explicit inclusion of the isobar. In a detailed comparison, we found that radii in nuclei up to Ca are accurate within EFT-truncation error estimates, and that binding energies – while improving order-by-order in precision – somewhat underbind heavier nuclei. The saturation point in nuclear matter is consistent with data within EFT error estimates. Our results also show that the inclusion of -isobars in the nuclear interaction can address the long-standing problem regarding nuclear saturation. This work therefore provides a valuable starting point for constructing more refined -full EFT interactions, also at higher chiral orders, with improved uncertainty estimates.
Acknowledgements.We would like to thank Hermann Krebs for valuable discussions and input on the manuscript, Kai Hebeler for providing us with matrix elements in Jacobi coordinates for the three-nucleon interaction, and Gustav Jansen for providing us with the code that transforms three-nucleon matrix elements to the laboratory system. This material is based upon work supported by the Swedish Research Council under Grant No. 2015- 00225, and the Marie Sklodowska Curie Actions, Cofund, Project INCA 600398, the U.S. Department of Energy, Office of Science, Office of Nuclear Physics under Award Numbers DEFG02-96ER40963 (University of Tennessee), DE-SC0008499 and DE-SC0018223 (NUCLEI SciDAC collaboration), and the Field Work Proposal ERKBP57 at Oak Ridge National Laboratory. 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, the Swedish National Infrastructure for Computing (SNIC).
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