Nucleon-nucleon potentials from $\Delta$-full chiral EFT and implications

One of the most fundamental aims in theoretical nuclear physics is to understand nuclear structure and reactions in terms of the basic forces between nucleons. As discussed in numerous review papers ME11 ; EHM09 ; MS16 ; Mac17 ; HKK19 , the nuclear physics community presently perceives chiral effective field theory (EFT) as the authoritative paradigm for the derivation of those forces. This perception is based upon a clearly defined relationship between the fundamental theory of strong interactions, QCD, and chiral EFT via symmetries.

Since a while, it is well established that predictive nuclear structure must include three-nucleon forces (3NFs), besides the usual two-nucleon force (2NF) contribution. The advantage of chiral EFT is that it generates 2NFs and multi-nucleon forces simultaneously and on an equal footing. In the $\Delta$-less theory ME11 , 3NFs occur for the first time at next-to-next-to-leading order (NNLO) and continue to have additional contributions in higher orders. Four-nucleon forces (4NFs) start at next-to-next-to-next-to-leading order (N³LO), but are difficult to implement, which is why they are left out in most present-day calculations. If an explicit $\Delta$-isobar is included in chiral EFT ($\Delta$-full theory ORK94 ; ORK96 ; KGW98 ; KEM07 ), then 3NF contributions start already at next-to-leading order (NLO), which leads to a smoother convergence when advancing from leading order (LO) to NNLO. However, summing up all contributions up to NNLO leads to very similar results for both versions of the theory KEM07 . The convergence of both theories beyond NNLO is expected to be very similar.

In the initial phase, the 3NFs were typically adjusted in $A=3$ and/or the $A=4$ systems and the ab initio calculations were driven up to the oxygen region BNV13 . It turned out that for $A\raisebox{-1.29167pt}{\small$\ \stackrel{{\scriptstyle\textstyle<}}{{\sim}}$ }16$ the ground-state energies and radii are predicted about right, no matter what type of chiral or phenomenological potentials were applied (local, nonlocal, soft, hard, etc.) and what the details of the 3NF adjustments to few-body systems were BNV13 ; Rot11 ; Pia18 ; Lon18 . It may be suggestive to perceive the $\alpha$ substruture of ¹⁶O to be part of the explanation.

The picture changed, when the many-body practitioners were able to move up to medium-mass nuclei (e. g., the calcium or even the tin regions). Large variations of the predictions now occurred depending on what forces were used, and cases of severe underbinding Lon17 as well as of substantial overbinding Bin14 were observed. Ever since the nuclear structure community understands that the ab initio explanation of intermediate and heavy nuclei is a severe, still unsolved, problem.

A seemingly successfull interaction for the intermediate mass region appears to be the force that is commonly denoted by “1.8/2.0(EM)” (sometimes dubbed “the Magic force”) Heb11 ; Heb21 , which is a similarity renormalization group (SRG) evolved version of the N³LO 2NF of Ref. EM03 complemented by a NNLO 3NF adjusted to the triton binding energy and the point charge radius of ⁴He. With this force, the ground-state energies all the way up to the tin isotopes are reproduced perfectly—but with charge radii being on the smaller side Sim17 ; Mor18 . Nuclear matter saturation is also reproduced reasonably well, with a slightly too high saturation density Heb11 . However, these calculations are not consistently ab initio, because the 2NF of “1.8/2.0(EM)” is SRG evolved, while the 3NF is not. Moreover, the SRG evolved 2NF is used like an original force with the induced 3NFs omitted. Still, this force is providing clues for how to get the intermediate and heavy mass region right.

Thus, in the follow-up, there have been attempts to get the medium-mass nuclei under control by means of more consistent ab initio calculations Som20 . Of the various efforts, we will now single out three, which demonstrate in more detail what the problems are.

In Ref. DHS19 , recently developed soft chiral 2NFs EMN17 at NNLO and N³LO were picked up and complemented with 3NFs at NNLO and N³LO, respectively, to fit the triton binding energy and nuclear matter saturation. These forces were then applied in in-medium similarity renormalization group (IM-SRG Her16 ) calculations of finite nuclei up to ⁶⁸Ni predicting underbinding and slightly too large radii Hop19 .

In a separate study Hut20 , the same 2NFs used in Refs. DHS19 ; Hop19 were employed, but with the 3NFs now adjusted to the triton and ¹⁶O ground-state energies. The interactions so obtained reproduce accurately experimental energies and point-proton radii of nuclei up to ⁷⁸Ni Hut20 . However, when the 2NF plus 3NF combinations of Ref. Hut20 are utilized in nuclear matter, then dramatic overbinding and no saturation at reasonable densities is obtained SM20 .

Obviously, there is a problem with achieving simultaneously reasonable results for nuclear matter and medium mass nuclei: In Refs. DHS19 ; Hop19 , nuclear matter is saturated right, but nuclei are underbound; while in Ref. Hut20 , nuclei are bound accurately, but nuclear matter is overbound.

In recent work by the Gőteborg-Oak Ridge (GO) group Eks18 ; Jia20 , the authors present an NNLO model including $\Delta$-isobars that apparently overcomes the above problem. With this model, the authors obtain “accurate binding energies and radii for a range of nuclei from $A=16$ to $A=132$, and provide accurate equations of state for nuclear matter” Jia20 . However, the accuracy of the $NN$ part of these interactions is not checked against $NN$ data. Another aspect of interest (not investigated in Refs. Eks18 ; Jia20 ) is if the inclusion of $\Delta$-degrees of freedom leads to a higher degree of softness. Note that the successful “Magic” 1.8/2.0(EM) potential is very soft since it is SRG evolved. Moreover, a recent study Lu18 , which investigated the essential elements of nuclear binding using nuclear lattice simulations, has come to the conclusion that proper nuclear matter saturation requires a considerable amout of non-locality in the $NN$ interaction implying a high degree of softness.

Thus, there is a need for a deeper understanding of the elements in the recent model by the GO group Eks18 ; Jia20 , and how they come together to produce the reported nuclear structure predictions. To gain this deeper insight, we will investigate the following issues:

1. 1.

   What are the precision and accuracy of the $\Delta$-full $NN$ potentials developed in Ref. Jia20 ? In the context of chiral EFT, this amounts to asking whether the precision of the $\Delta$-full potentials is consistent with the uncertainty of the chiral order at which they have been derived. And, is the accuracy sufficient for meaningful ab initio predictions? If there are problems with precison and/or accuracy, how does that impact the predictions for nuclear many-body systems?

2. 2.

   Does the inclusion of $\Delta$-isobars increase the smoothness of the interaction and, if so, how does the $\Delta$ degree of freedom accomplish that?

This paper is organized as follows: In Sec. II, we investigate $NN$ potentials based upon $\Delta$-full chiral EFT which, in Sec. III, are applied in nuclear matter. Our conclusions are summarized in Sec. IV.

The attempts to explain nuclear matter saturation have a long history Bet71 ; Mac89 . The modern view is that the 3NF is essential to obtain saturation Bog05 ; Heb21 . In this scenario, the 2NF substantially overbinds nuclear matter, while the 3NF contribution is repulsive and strongly density-dependent leading to saturation at the appropriate energy and density Mac19 . Recent example can be found in the work of Ref. DHS19 , where chiral 2NFs at NNLO and N³LO are complemented with chiral 3NFs of the corresponding orders to saturate nuclear matter around its empirical values.

Besides nuclear matter, there is also the problem of the binding energies of intermediate-mass nuclei. When the 2NF+3NF combinations of Ref. DHS19 were applied in IM-SRG calculations of finite nuclei up to the nickel isotops, underbinding of the ground state energies was obtained Hop19 .

On the other hand, also in Ref. Heb11 , 2NF+3NF combinations were developed; in particular, the force known as 1.8/2.0(EM) or Magic, which saturates nuclear matter properly and reproduces the groundstate energies of nuclei up to the tin region correctly Sim17 ; Mor18 .

What is the difference between the two cases?

As it turns out the crucial difference between the two cases is to be found in the 2NF part of the forces. We demonstrate this in Fig. 3, where we show the 2NF contribution to nuclear matter from the 1.8/2.0(EM) force (denoted by Magic, solid green line). On the other hand, the 2NF contribution to nuclear matter from the 2NFs applied in Ref. Hop19 are located within the shaded band in Fig. 3. Even though in both cases nuclear matter is overbound, Magic overbinds considerable more than the NNLO and N³LO forces from Ref. EMN17 applied in Ref. Hop19 to intermediate-mass nuclei. This shows that a considerable overbinding of nuclear matter by the 2NF is necessary to correctly bind intermediate-mass nuclei SM20 , when 3NFs at NNLO are applied.

Next, we turn to the nuclear matter properties as predicted by the $\Delta$-full 2NFs discussed in this paper. We apply the particle-particle ladder approximation [Brueckner-Hartree-Fock (BHF)] for nuclear matter. We have compared our nuclear matter results for the Magic 2NF with the results obtained in Ref. Heb11 where many-body perturbation theory (MBPT) is used and obtain the same within $\pm 0.5$ MeV for all densities displayed in Ref. Heb11 . Moreover, we have also compared our BHF results with the MBPT calculations of Ref. DHS19 (for the $\Lambda=450$ MeV potentials) achieving a similar agreement. From this we conclude that our BHF method for nuclear matter is as reliable as the presently more popular MBPT method for soft potentials.

The predictions by the original GO potentials, $\Delta$NNLO(450)_GO and $\Delta$NNLO(394)_GO, are shown in Fig. 3 by the red solid and dashed curves, respectively. These predictions are right on the Magic curve, which explains the results of the GO potentials for nuclei up to $A=132$ Jia20 , similar to what happens with Magic Sim17 ; Mor18 .

On the other hand, in the previous section we have identified serious problems with the accuracy of the GO $NN$ potentials. For that reason, in Sec. II.2.2 we refitted these potentials, generating the Rf versions $\Delta$NNLO(450)_Rf and $\Delta$NNLO(394)_Rf, which are as accurate as expected at NNLO. The nuclear matter predictions by the Rf potentials are shown by the blue solid and dashed curves in Fig. 3, together with their theoretical uncertainties represented by the shaded band. It is seen that the refitted potentials are less attractive than the original GO versions. In fact, their nuclear matter properties are very similar to the ones of the 2NFs used in Ref. Hop19 and, thus, they will most likely not produce the same results as the original GO potentials and, rather, produce underbinding in intermediate-mass nuclei.

The question we wish to address is then why, after refit to proper accuracy, the $\Delta$NNLO potentials lost attraction. As discussed, the main issues with the GO potentials are found in the $P$-waves at low energy. We demonstrate this in Fig 4, where we show, for the three most important $P$-waves, the phase shifts below 100 MeV as predicted by the original GO potentials (red lines) and and the refit versions in comparison with authoritative phase shift analyses.

To further quantify the discrepancies, we provide in Table 6 numerical values for $np$ phase-shifts at 50 MeV lab. energy for the three $P$-waves of interest as predicted by the original GO potentials, the refit potentials, and the phase shift analyses (‘Empirical’). We also provide the NNLO truncation error for the phase shifts, $\Delta\delta_{\rm NNLO}$ foot2 , and state the discrepancies in the fits, $\Delta\delta$, in terms of multiples of the truncation errors, ${\Delta\delta}/{\Delta\delta_{\rm NNLO}}$. For the GO potentials, the discrepancies are on average about ten truncation errors, and can be as large as 44 truncation errors. For the refit potentials, the discrepancies are typically around one truncation error or less – as expected for a properly converging EFT, whose predictions at each order should agree with experiment within the theoretical uncertainty (truncation error) at the given order.

Next, we will explore the impact on nuclear matter from discrepancies in the fit of $NN$ lower partial waves. To investigate this aspect, we show in Table 7 the contributions to the nuclear matter energy around saturation density from distinct partial-wave states. We provide results for the original GO potentials as well as the refit potentials and the Magic potential. Of particular interest are the three $P$-waves that we singled out in Table 6 and Fig. 4. It is seen that, for all three $P$-waves, the contributions from the GO potentials are substatially more attractive than for the other cases. Recalling that, as demonstrated in Fig. 4 and Table 6, all GO potentials overpredict the empirical phase shifts, the increased attraction they generate is not surprising. Thus, while accurately fitted potentials obtain about –7.3 MeV from the three $P$-waves, the GO potentials produce about –10.3 MeV, that is 3 MeV more binding energy per particle. Naturally, this is not a viable source for the additional attraction needed in nuclear structure.

The remaining extra attraction by the GO potentials comes from the ${}^{3}S_{1}$ state, and is on average $\approx 1.5$ MeV as compared to the corresponding Rf potentials. This additional gain in binding energy is, again, linked to unsatisfactory description of phase parameters, in this case the $\epsilon_{1}$ mixing parameter, cf. the $\epsilon_{1}$ frame in Fig. 2. The explantion of this effect is somewhat involved. Note that the $\epsilon_{1}$ parameter is proportional to the strength of the nuclear tensor force. For states in which the tensor force has a dominant role (like the ${}^{3}S_{1}$–${}^{3}D_{1}$–$\epsilon_{1}$ system), the $\widehat{T}$-matrix, Eq. (32), is approximately given by:

where $\widehat{V}_{C}$ denotes the central force and $\widehat{V}_{T}$ the tensor force. The on-shell $\widehat{T}$-matrix is related to the phase-shifts (and observables) of $NN$ scattering. Thus, potentials that fit the same phase shifts produce the same on-shell $\widehat{T}$-matrix elements. However, that does not imply that the potentials are the same. As evident from the above equation, the $\widehat{T}$-matrix is essentially the sum of two terms: the central force term, $\widehat{V}_{C}$, and the second order in $\widehat{V}_{T}$. A potential with a strong $\widehat{V}_{T}$ will produce a large (attractive) second order term and, hence, go along with a weaker (attractive) central force; as compared to a weak tensor force potential, where the lack of attration by the second order term has to be compensated by a stronger (attractive) central force.

Now, when we enter nuclear matter, we encounter particle-particle ladder graphs represented by the $\widehat{G}$-matrix:

which—similarly to what happened above to the $\widehat{T}$-matrix—for states where the tensor force rules, can be approximated by:

The $\widehat{G}$-matrix equation differs from the $\widehat{T}$-matrix equation in two ways: First, the Pauli projector, $Q_{P}$, which prevents scattering into occupied states and, thus, cuts out the low-momentum spectrum. Second, the single-particle spectrum in nuclear matter, which enhances the energy denominator, thereby decreasing the integrand. Using a simple parametrization of the single particle energies in nuclear matter, this effect comes down to simply replacing the free nucleon mass, $M_{N}$, by the effective mass $M_{N}^{\star}<M_{N}$.

Both medium effects reduce the size of the (attractive) integral term and, thus, are repulsive. The larger $V_{T}$ and the second order $V_{T}$ term, the larger the repulsive effects. Thus, large tensor force potentials undergo a larger reduction of attraction from these medium effects than weak tensor force potentials. This explains the well-known fact that $NN$ potentials with a weaker tensor force yield more attractive results when applied in nuclear few- and many-body systems as compared to their strong tensor force counterparts.

The GO potentials have a very weak tensor force, which explains their relatively large ${}^{3}S_{1}$ contribution (cf. Table 7). In fact, the tensor force is excessively weak, as can be inferred from the underpredicted $\epsilon_{1}$ parameter (cf. Fig. 2). To agree with the empirical information within the truncation error, the tensor force has to be stronger, like in the case of the Rf potentials, leading to less binding energy in nuclear matter.

At this point of our discussion, a word is in place about what laboratory energies of $NN$ scattering are most relevant for predictions in many-body systems. In $P$-waves, about 95% of the contributions to the $\widehat{G}$-matrix, Eq. (8), comes from the Born term. In the sum of the energy contributions from zero to the Fermi-momentum $k_{F}$, the average relative momentum is $\bar{p}=\sqrt{0.3}\,k_{F}$ which, for $k_{F}=1.35$ fm^-1, yields $\bar{p}=146$ MeV/c, equivalent to $T_{\rm lab}=45$ MeV. Thus, for $P$-waves, the phase shifts around $T_{\rm lab}=50$ MeV are most relevant for nuclear matter predictions at saturation density. This fact is particularly evident from the ${}^{3}P_{0}$ phase shifts shown in Fig. 4. In this figure, it is clearly seen that the phase shift predictions around 50 MeV by the GO potentials are substantially too large, meaning too attractive. On the other hand, the ${}^{3}P_{0}$ phase shifts above 100 MeV by the same potentials (cf. Fig. 2) are too low, implying too repulsive. But from Table 7 we know that the GO potentials generate a ${}^{3}P_{0}$ contribution that is too attractive. The conclusion then is that phase shifts below 100 MeV are the most relevant ones for many-body predictions at normal densities – the reason why we chose $T_{\rm lab}=50$ MeV for the discussion of $P$-wave phase shift in Table 6.

The story is different for states where the tensor force plays a dominant role, like in the coupled ${}^{3}S_{1}$–${}^{3}D_{1}$–$\epsilon_{1}$ system, where the integral term in Eq. (9) makes a large contribution. Note that the integration extends from around $k_{F}$ (due to Pauli blocking) to the cutoff region of the potential. Thus, the relative momenta involved are $p\raisebox{-1.29167pt}{\small$\ \stackrel{{\scriptstyle\textstyle>}}{{\sim}}$ }k_{F}$, equivalent to $T_{\rm lab}\raisebox{-1.29167pt}{\small$\ \stackrel{{\scriptstyle\textstyle>}}{{\sim}}$ }150$ MeV for $k_{F}=1.35$ fm^-1, which explains why the $\epsilon_{1}$ mixing parameter needs to be considered for energies of 150 MeV or even higher (cf. Table 6).

Finally, a comment on the many-body predictions by Magic is in order. As seen in Table 7, the $P$-wave contributions from Magic are essentially the same as the ones from the properly fitted Rf potentials, namely –7.31 MeV from the three $P$-waves of special interest. What sets the Magic potential apart from all the others is the exceptionally large ${}^{3}S_{1}$ contribution – note that the $\epsilon_{1}$ predictions by Magic are identical to the ones by the N³LO potential of Ref. EM03 , which are right on the data up 300 MeV. The extraordinarily nonlocal nature of Magic due to its similarity renormalization group (SRG) evolution is the source of the additional attraction that shows up in nuclear structure. This has the consequence that the second order $V_{T}$ term in Eq. (9) is unusually small and, consequently, the central force, $V_{C}$, unusually large and attractive, giving rise to the very large, attractive ${}^{3}S_{1}$ contribution by Magic. This degree of nonlocality can, presently, not be achieved by any original chiral potential, no matter if $\Delta$-full or $\Delta$-less and, therefore, these potentials cannot generate ${}^{3}S_{1}$ contributions as large as the Magic one. Making up for this by incorrect, extra attractive $P$-waves is not a valid solution.

To summarize, when the three $P$-waves and the $\epsilon_{1}$ parameter of the GO potentials are corrected to obtain a realistic fit, the favorable predictions for intermediate-mass nuclei are very likely to disappear, as did the extra attraction in nuclear matter.

We have closely investigated chiral $NN$ potentials at NNLO including $\Delta$-isobar degrees of freedom and have come to the following conclusions:

1. 1.

   The $\Delta$-full $NN$ potentials at NNLO constructed by the Gőteborg-Oak Ridge (GO) group Jia20 are up to 40 times outside the theoretical error of chiral EFT at NNLO and are, therefore, inconsistent with the EFT that the potentials are intended to be based upon. In line with this fact, these potentials reproduce the $NN$ data with a very large $\chi^{2}$. This is unacceptable based on contemporary precision standards.

2. 2.

   The predictions by the GO $NN$ potentials for the energy per nucleon in nuclear matter are very attractive, similar to the predictions by the 1.8/2.0(EM) $NN$ potential of Ref. Heb11 , also known as ‘Magic’. The extremely attractive nature of both the GO and the Magic potentials is the reason for the favorable reproduction of the energies (and radii) of intermediate-mass nuclei, which have proven to be a problem in ab initio nuclear structure physics. However, the extra attraction in the GO potentials which brings them to the level of Magic can be traced to incorrect $P$-wave and $\epsilon_{1}$ mixing parameters.

3. 3.

   When all phase parameters, including the $P$-wave and the $\epsilon_{1}$-mixing parameters, are fitted within the NNLO truncation error, then the extra attraction disappears and the nuclear matter predictions become very similar to the ones by $NN$ potentials constructed within the $\Delta$-less theory. Thus, we find claims that $\Delta$-full potentials lead to more attraction in nuclear many-body systems to be incorrect.

4. 4.

   The extraordinarily attractive nature of Magic is due to its high degree of nonlocality which, in turn, is due to its SRG construction. This degree of nonlocality is not achieved by chiral $NN$ potentials, no matter if $\Delta$s are included or excluded, because all two-pion exchange (2PE) contributions in both version of the theory are local (at least up to NNLO, see appendix) and nonlocality is generated only by the regulator function, which adds only moderate nonlocality.

5. 5.

   The problem with a microscopic description of intermediate mass nuclei with realistic chiral nuclear forces remains, unfortunately, unsolved.

This work was supported in part by the U.S. Department of Energy under Grant No. DE-FG02-03ER41270 (R.M. and Y.N.), by Ministerio de Ciencia e Innovación under Contract No. PID2019-105439GB-C22/AEI/10.13039/501100011033 and by EU Horizon 2020 research and innovation program, STRONG-2020 project, under grant agreement No 824093 (D.R.E.).