Insights into nuclear saturation density from parity violating electron scattering
IInsights into nuclear saturation density from parity violating electron scattering
C. J. Horowitz, ∗ J. Piekarewicz, † and Brendan Reed
1, 3, ‡ Center for the Exploration of Energy and Matter and Department of Physics,Indiana University, Bloomington, IN 47405, USA Department of Physics, Florida State University, Tallahassee, FL 32306, USA Department of Astronomy, Indiana University, Bloomington, Indiana 47405, USA (Dated: October 1, 2020)The saturation density of nuclear matter ρ is a fundamental nuclear physics property that is difficultto predict from fundamental principles. The saturation density is closely related to the interiordensity of a heavy nucleus, such as Pb. Parity violating electron scattering can determine theaverage interior weak charge and baryon densities in
Pb. This requires not only measuring theweak radius R wk but also determining the surface thickness of the weak charge density a . We usethe PREX experimental result for the weak radius of Pb and assume a 10% theoretical uncertaintyin the presently unmeasured surface thickness to obtain ρ = 0 . ± .
010 fm − . Here the 7%error also has contributions from the extrapolation to infinite nuclear matter. These errors can beimproved with the upcoming PREX II results and with a new parity violating electron scatteringexperiment, at a somewhat higher momentum transfer, to determine a . I. INTRODUCTION
The saturation density of nuclear matter ρ is very im-portant for the structure of nuclei. Infinite nuclear mat-ter, a hypothetical uniform system of protons and neu-trons without Coulomb interactions, is expected to havean energy per nucleon that is minimized at ρ . This min-imum describes nuclear saturation and is a fundamentalnuclear-structure property. Furthermore, this value of ρ is an important benchmark that is used to measure evenhigher density matter in astrophysics and in the labora-tory. Nuclear saturation implies that the interior densityof heavy nuclei should be nearly constant and close to ρ . Historically, the semi-empirical mass formula [1, 2]and the liquid drop model [3] describe the nucleus as anincompressible quantum drop at ρ . But why does nu-clear matter saturate? And how can one calculate thesaturation density ρ ? Surprisingly, the answers to thesedeceptively simple questions have proved to be both sub-tle and elusive.Liquid water saturates at a density of 1 g/cm becauseof the size of the water molecules. Does nuclear mattersaturate because of the finite nucleon size and if so, doesthis size explain the value of ρ ≈ .
15 fm − ? The sit-uation is likely more complicated. Nucleons are knownto have repulsive cores because phase shifts for nucleon-nucleon scattering become negative at high energies (seehttp://nn-online.org). However, the core size is too smallto explain the value of ρ [4]. Indeed, nuclear matter cal-culations with only two-nucleon interactions may satu-rate at up to twice the expected density [5]. It is nowbelieved that three- and higher-nucleon interactions areimportant for nuclear saturation and for determining ρ . ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]
Chiral effective field theory (CEFT) provides a system-atic expansion of the strong interaction between nucleonsin powers of the momentum transfer over a suitable chi-ral scale [6–8]. This allows one to calculate the energyof nuclear matter to a given order in a chiral expansion.Note that CEFT includes two-, three-, and many-nucleoninteractions. Under this framework, the empirical satu-ration point (density and energy per nucleon) are wellreproduced within statistical and systematic uncertain-ties [9, 10]. The uncertainty band comes from the trunca-tion of the chiral expansion and from imposing a cutoffat high momentum transfers. Whereas CEFT appearsconsistent with nuclear saturation at ρ , the error bandin present calculations is too broad to make a sharp pre-diction of the actual value of ρ .So if one can not accurately compute ρ from first-principle calculations, can one observe it? Strictly speak-ing, nuclear matter is an infinite system without Coulombinteractions, so observations of ρ must involve an ex-trapolation from measurements in finite nuclei; see forexample [11]. Nevertheless, the interior baryon density ofheavy nuclei is expected to be fairly constant and closeto ρ . Among heavy nuclei, Pb may be particularlyimportant because it is the heaviest stable doubly-magicnucleus. As such, the interior baryon density of
Pbmay provide the finite nucleus observable that is mostclosely related to ρ . In this paper we present a new mea-surement of the interior baryon density of
Pb based onresults from the PREX experiment [29, 38].Unfortunately, we do not have detailed knowledge ofthe neutron density in
Pb; see Ref. [12] and referencescontained therein. The charge density is well measuredso the proton density is accurately known [13]. However,
Pb has 44 excess neutrons, so the neutron density canbe significantly different from the proton density. Giventhis incomplete information, our present best estimateof ρ comes from a variety of empirical nuclear energydensity functionals. These functionals are calibrated tothe binding energies and charge radii of a variety of nu- a r X i v : . [ nu c l - t h ] S e p clei and can then be used to predict ρ , see for exampleRefs. [14, 15]. In particular, Reinhard and Nazarewiczargue that fitting charge radii sharply constrains ρ [16].Alternatively, if one can cleanly measure the interiorneutron density of Pb one should be able to infer ρ with small and quantifiable uncertainties. Often neu-tron densities are determined with strongly interactingprobes [17], such as antiprotons [18, 19], elastic protonscattering [20], heavy-ion collisions [21], elastic pion scat-tering [22], and coherent pion photo production [23]. Onetypically measures cross sections or spin observables thatinvolve the convolution of the neutron density with an ef-fective strong-interaction range for the probe. Althoughthese observables can be measured with small statisticaluncertainties, complexities arising from the strong inter-action introduce significant systematic errors in the ex-tracted neutron densities [12].It is also possible to measure neutron densities, orequivalently weak charge densities, with electroweakprobes using coherent neutrino-nucleus scattering [24–27]or parity violating (PV) electron scattering [28, 29]. Thisis because the weak charge of a neutron is much largerthan that of a proton, so the weak charge density of anucleus is very closely related to its neutron distribu-tion. Compared to strongly interacting probes, parityviolation offers a clean and model-independent way todetermine the weak charge density with much smalleruncertainties (statistical+systematic) than with stronglyinteracting probes. In the last decades significant theo-retical [28, 30–36] and experimental [29, 37] efforts havebeen devoted to improve parity violating electron scat-tering experiments. At Jefferson laboratory, the radiusof the weak charge density of Pb was measured inthe original PREX campaign [29, 38] and is now beingmeasured with increased precision during the follow-upPREX-II campaign [39]. At the same time, CREX willprovide the first electroweak determination of the weakradius of Ca [40].Present parity violating experiments focus on deter-mining the rms radius of the weak charge density R wk from a single measurement at a relatively low momen-tum transfer. Yet additional features of the weak chargedensity ρ wk ( r ) can be revealed by measuring the parityviolating asymmetry A pv at higher momentum transfers.If A pv is measured at several momentum transfers, then acomplete model independent representation of the weakcharge density can be determined [41], either as FourierBessel expansion or as a sum of Gaussians. This is fea-sible for Ca and may require measurements at six orseven momentum transfers. For
Pb, however, this ismore challenging because a determination of ρ wk in thenuclear interior requires a measurement at high momen-tum transfer where the elastic cross section is very small.What is then required to determine the saturation den-sity ρ ? In principle, one could follow these four steps:(a) Determine the entire weak charge density ρ wk ( r ) of Pb; (b) average over ρ wk ( r ) in the interior to obtaina measure of the average weak charge density; (c) com- bine this average weak charge density with an average ofthe experimental charge density to obtain a measure ofthe interior baryon density; (d) extrapolate such a valueto the very closely related saturation density of infinitenuclear matter. Here we combine the first two steps in amanner that dramatically minimizes the need for parityviolating experiments. II. FORMALISM
We propose, rather than to determine the full density, asimple representation of ρ wk ( r ) using a symmetrized two-parameter Fermi function that is then used to performthe interior average. That is, we model ρ wk ( r ) as [42, 43] ρ wk ( r, c, a ) = ρ sinh( c/a )cosh ( r/a ) + cosh( c/a ) , (1)where c is the half-density radius, a the surface diffuse-ness, and the normalization constant is ρ = 3 Q wk πc ( c + π a ) ⇒ (cid:90) d rρ wk ( r, c, a ) = Q wk . (2)Here the total weak charge of a nucleus with N neutronsand Z protons is Q wk = Q n N + Q p Z , where (includingradiative corrections [44, 45]) Q n = − . Q p = 0 . Pb, Q wk = − . R = 1 Q wk (cid:90) r ρ wk ( r ) d r = 35 c + 75 ( πa ) . (3)We propose to use ρ in Eq.(2) as the measure of theaverage interior weak charge density, which for clarity werewrite in terms of the weak radius R wk rather than c : ρ = 27 Q wk π (5 R − π a ) (cid:112) R − π a . (4)Given that we are interested only in the average den-sity ρ rather than on the full density, PV experimentsneed only to determine the weak radius R wk and the sur-face thickness a . The existing PREX and PREX II [39]measurements are primarily sensitive to R wk , so an addi-tional PV experiment at a somewhat higher momentumtransfer could determine a [43]. We will describe this ex-periment in a forthcoming paper.We illustrate our procedure in Fig. 1, which shows theexperimental charge density of Pb along with a SFermifunction fit that yields: c ch = 6 . a ch = 0 . R ch = 5 . ρ = 0 . r (fm) ρ (f m - ) ρ ch Pb - ρ wk FIG. 1: The experimental charge density of
Pb [13] (redcircles) and the corresponding SFermi function fit (solid redline). Also shown is the weak charge density as predictedby the FSUGold interaction [46] (blue circles) along with aSFermi function fit (solid black line). fm − . This is our measure of the average interior chargedensity of Pb. Figure 1 also shows a model weakcharge density as predicted by the FSUGold relativisticmean field interaction [46] and the corresponding SFermifunction fit. The SFermi functions—which average overshell oscillations—are seen to be very good represen-tations of both the (electromagnetic) charge and weakcharge densities. Note that we are not proposing to usemodel predictions for the weak charge density but rather,a SFermi function with both parameters R wk and a de-termined from experiment.We now combine the average interior weak and chargedensities to obtain an estimate of the average interiorbaryon density ρ . That is, ρ b = ρ n + ρ p = 1 Q n (cid:16) ρ − Q p ρ (cid:17) + ρ = 1 Q n ρ + (cid:18) − Q p Q n (cid:19) ρ = − (1 . ρ + (1 . ρ . (5)The final step is to extrapolate the interior baryon den-sity ρ b to the closely related saturation density of infinitenuclear matter ρ . We define an extrapolation factor f ex as the saturation density of infinite nuclear matter ρ over the average interior density of Pb : f ex = ρ ρ b . (6)We expect f ex ≈
1. We estimate f ex by considering a vari-ety of relativistic and nonrelativistic energy density func-tionals (EDFs). For each EDF one calculates point pro-ton ρ p ( r ) and neutron ρ n ( r ) densities and then computesthe weak density by folding these point-nucleon densitieswith a dipole nucleon form factor of radius r p = 0 .
84 fm that accounts for the finite nucleon size. Next, one fitsSFermi functions to the model weak and charge densitiesto obtain ρ , ρ , and ultimately ρ b from Eq.(5). Com-paring this value of ρ b to the prediction for the saturationdensity ρ yields f ex for that particular EDF.Results are plotted in Fig. 2 for the following non-relativistic Skyrme functionals: SIII [47], SLY4, SLY5,SLY7, and SKM* [48], SV-min [15], UNEDF0 [49], andUNEDF1 [50]. We also include results for the followingrelativistic functionals: FSUGold [46], IUFSU [51], NL3[52], FSUGarnet, RMF012, 022, 028 and 032 [53]. We R n -R p (fm) f e x Rel(Y p =1/2)Rel(Y p =82/208)NRel(Y p =1/2)NRel(Y p =82/208)UNEDF0UNEDF1SV-minSKM*SLY5SLY4SLY7SIIIRMF012IUFSUFSUGarnet FSUGoldRMF022 NL3RMF028 RMF032 FIG. 2: The extrapolation factor f ex defined in Eq.(6) asa function of the the neutron skin thickness of Pb for anumber of nonrelativistic and relativistic EDFs. Shown withtriangles is the extrapolation factor (cid:101) f ex to asymmetric nuclearmatter with the same ratio of neutrons to protons as Pb. see that f ex is indeed close to one for all of the modelsthat have been considered. However, if one looks in moredetail, f ex for relativistic models is in general very closeto one with a slight increase with increasing neutron skin(neutron minus proton radius R n − R p ). This is likely re-lated to the density dependence of the symmetry energywhich increases with increasing neutron skin. Most of thenonrelativistic models that we consider predict f ex ≈ . γ = 1 forthe density dependent term t ρ γ , predicts f ex ≈ .
99 thatis close to the prediction of most relativistic models. Incontrast, all other Skyrme forces (shown in Fig. 2) havesmaller values for γ and yield significantly larger f ex .The extrapolation from ρ b in Pb to ρ involves threeeffects. First, surface tension—which is absent in an in-finite system—increases the density of lead and tends tomake f ex <
1. Second, Coulomb interactions which areignored in infinite nuclear matter reduce the density oflead making f ex >
1. To some extent, effects from sur-face tension and Coulomb interaction cancel out restoring f ex ≈
1. Finally, one is extrapolating in isospin from theneutron rich lead nucleus to symmetric nuclear matter, as ρ is the saturation density of symmetric nuclear matter.To explore the consequences of the extrapolation inisospin, we define (cid:101) ρ as the saturation density of asym-metric nuclear matter with a proton fraction identical tothat of Pb, namely, Y p = 82 / (cid:39) .
39. It is a sim-ple matter to calculate (cid:101) ρ for all EDFs included in Fig.2.Note that to a very good approximation (cid:101) ρ is given by [54] (cid:101) ρ ρ = 1 − LK α + O ( α ) , (cid:16) α = 1 − Y p (cid:17) , (7)where K is the incompressibility coefficient of symmetricmatter and L the slope of the symmetry energy. Fol-lowing Eq.(6) we define in analogy (cid:101) f ex = (cid:101) ρ /ρ b . Valuesfor (cid:101) f ex are shown in Fig. 2 using up and down trian-gles. For relativistic functionals, f ex ≈ (cid:101) f ex < (cid:101) ρ decreaseswith increases L , a quantity that is strongly correlated to R n − R p . In contrast, for nonrelativistic functionals (cid:101) f ex ≈ Ca [55, 56], micro-scopic coupled cluster calculations for
Pb may becomefeasible in the near future. This could provide a micro-scopic determination of f ex that is more closely connectedto chiral two- and three-nucleon forces. Until then, weuse all models in Fig. 2 to infer the following limit: f ex ≈ . ± . . (8)That is, the extrapolation to infinite nuclear matter in-troduces a ∼
3% uncertainty in the inferred value of ρ .In summary, PV experiments can determine both theradius R wk and surface thickness a of the weak chargedensity of Pb, from which the average weak density ρ is calculated using Eq.(4). The known charge density ρ is then added to ρ in Eq.(5) to obtain ρ b . This, inturn, is extrapolated to ρ using Eqs.(6) and (8). III. RESULTS
We present a first estimate of ρ based on the exist-ing PREX result of R wk = 5 . ± .
181 fm [38]. Unfor-tunately, at present there is no electroweak experimentthat constrains the surface thickness a . Thus, we providea conservative theoretical estimate for a . Considering allEDFs in Fig. 2 yields a surface thickness in the 0.58 fm(SIII) to 0.632 fm (RMF032) range. We arbitrarily se-lect the UNEDF0 result to define the central value andassign a very conservative 10% error that more than cov-ers the theoretical range; that is, a = 0 . ± .
062 fm.A future PV experiment at a slightly larger momentum transfer to constrain a would allow a direct experimentaldetermination of the interior weak density.Adopting the PREX value for R wk , our theoretical as-sumption for a , and Eqs.(4) and (5) yields, ρ b = 0 . ± . ± . − , (9)were the first error is from the PREX error in R wk whilethe second error corresponds to our assumed 10% un-certainty in a . The last step is to multiply this resultby f ex = 1 . ± .
03 to get our present estimate for thesaturation density of nuclear matter: ρ = 0 . ± . ± . ± . − , (10)where the last error is due to the uncertainty in f ex .Adding all three errors in quadrature gives a total un-certainty of 7% that is dominated by the error in R wk .That is, ρ = 0 . ± .
010 fm − . (11)Our result is consistent, although somewhat lower,than the phenomenological estimate of ρ = 0 . ± .
007 fm − claimed in Ref. [9] based on some selected den-sity functionals—yet fully consistent with ρ = 0 . ± .
001 fm − predicted by a relativistic EDF calibrated us-ing exclusively physical observables [57]. Note that an al-ternative procedure that uses a Helm-type [38, 58] weakcharge density instead of a SFermi function yields a con-sistent, yet slightly lower density than Eq.(11).How accurately can ρ be measured in the near future?The PREX II campaign has completed data taking with agoal of measuring R wk to 1%. Figure 3 shows an examplebaryon density for Pb assuming a SFermi weak chargedensity with R wk = 5 .
826 fm (central PREX value [38])and a = 0 .
62 fm. We have added the charge density asper Eq.(5). The error band in Fig. 3 includes a 1% errorin R wk and a 10% error in a added in quadrature. Thistotal error corresponds to ± .
004 fm − in ρ b or about a2.5% error in ρ that is comparable to our assumed 3%error in f ex .There is strong motivation for an additional parity vi-olating electron scattering experiment to measure thesurface thickness a . Both PREX and PREX II wereperformed at a momentum transfer of q ≈ .
475 fm − and are primarily sensitive to the weak radius. Instead,a new experiment near q ≈ .
78 fm − is sensitive to a [43]. Following [59] we have calculated the parity vi-olating asymmetry A pv for elastic electron scattering in-cluding Coulomb distortions [30]. We find that the loga-rithmic derivative of A pv with respect to log( a ) is about0.53 at q = 0 .
78 fm − . Therefore a 5% measurement of A pv can constrain a to 10%. We will discuss this possibleexperiment in more detail in a forthcoming paper. IV. CONCLUSIONS
In conclusion, the saturation density of nuclear matter ρ is a fundamental nuclear physics property that is diffi- r (fm) ρ (f m - ) ρ ch Pb - ρ wk ρ b FIG. 3: Theoretical prediction for the baryon density of
Pb. The error band assumes that R wk is measured to 1%and the surface thickness is constrained to 10%, see text fordetails. The corresponding curve for the weak charge densityis also shown. Finally, the experimental charge density [13] isdisplayed along with a SFermi fit. cult to predict from chiral effective field theory. Becauseof nuclear saturation, ρ is closely related to the interiordensity of a heavy nucleus. We emphasize that the aver-age interior baryon density of Pb is an experimentallyobservable quantity that can be determined with parityviolating electron scattering. We used the existing PREX results for the weak radius to obtain a first measurementof the interior baryon density of
Pb. We then extrap-olated this result to infinite nuclear matter and obtained ρ = 0 . ± .
011 fm − . The quoted 7% error has contri-butions from the PREX error on the weak radius, uncer-tainty in a theoretical estimate of the surface thickness a , and the error in extrapolating to infinite nuclear mat-ter. These errors can be improved with the upcomingPREX II results and with a new parity violating electronscattering experiment—at a somewhat higher momen-tum transfer—to determine the surface thickness of theweak density. This will allow an accurate determinationof ρ that is very closely related to the experimentallymeasured interior baryon density of Pb. As a resultof the parity violating measurements, the theoretical as-sumptions necessary to extract ρ will be both reducedand clarified. Acknowledgements
We thank Witek Nazarewicz, Concettina Sfienti, DickFurnstahl, and Zach Jaffe for helpful discussions. Thismaterial is based upon work supported by the U.S.Department of Energy Office of Science, Office of Nu-clear Physics under Awards DE-FG02-87ER40365 (In-diana University), DE-FG02-92ER40750 (Florida StateUniversity), and DE-SC0018083 (NUCLEI SciDAC Col-laboration). [1] C. F. von Weizsacker, Zeitschrift fur Physik (in German)
431 (1935).[2] W. D. Myers and W. J. Swiatecki, Nucl. Phys.
632 (1930).[4] H. A. Bethe, Annual Review of Nuclear Science , 93(1971).[5] B. D. Day, Phys. Rev. Lett., , 226 (1981).[6] Steven Weinberg, Phys. Let. B
288 (1990).[7] H.-W. Hammer, Sebastian Konig, and U. van Kolck, Rev.Mod. Phys. , 1(2011).[9] C. Drischler, K. Hebeler, and A. Schwenk, Phys. Rev.Lett. , 042501 (2019).[10] D. Lonardoni, I. Tews, S. Gandolfi, J. Carlson, Phys.Rev. Research , 022033 (2020).[11] P.-G. Reinhard, M. Bender, W. Nazarewicz, and T.Vertse, Phys. Rev. C , 014309 (2006).[12] M. Thiel , C. Sfienti, J. Piekarewicz, C. J. Horowitz, andM. Vanderhagen, J. Phys. G: Nucl. Part. Phys. ,495 (1987).[14] Michael Bender, Paul-Henri Heenen, and Paul-GerhardReinhard, Rev. Mod. Phys. , 121 (2003). [15] P. Kl¨upfel, P.-G. Reinhard, T. J. B¨urvenich, and J. A.Maruhn, Phys. Rev. C et al. , Phys. Rev. C , 015803 (2012).[18] A. Trzci´nska et al. , Phys. Rev. Lett. , 082501 (2001).[19] B. K(cid:32)los, et al. , Phys. Rev. C , 014311 (2007).[20] J. Zenihiro, et al. , Phys. Rev. C , 044611 (2010).[21] Lie-Wen Chen, Che Ming Ko, and Bao-An Li, Phys. Rev.C , 064309 (2005).[22] E. Friedman, Nucl. Phys. A896,
46 (2012).[23] C. M. Tarbert, et al. , Phys. Rev. Lett. , 242502(2014).[24] COHERENT Collaboration: D. Akimov, J. B. Albert,P. An, C. Awe, P. S. Barbeau, B. Becker, V. Belov, A.Brown, A. Bolozdynya, B. Cabrera-Palmer, et al., Sci-ence , 1123 (2017).[25] COHERENT Collaboration: D. Akimov, J.B. Albert,P. An, C. Awe, P.S. Barbeau, B. Becker, V. Belov,M.A. Blackston, L. Blokland, A. Bolozdynya, et al.,arXiv:2003.10630 (2020).[26] P. S. Amanik and G. C. McLaughlin, J. Phys. G: Nucl.Part. Phys. et al. , Phys. Rev. C , 024612 (2012).[28] T.W. Donnelly, J. Dubach, and Ingo Sick, Nucl. Phys. A503,
589 (1989). [29] S. Abrahamyan, et al. , Phys. Rev. Lett. , 112502(2012).[30] C. J. Horowitz, Phys. Rev. C , 3430 (1998).[31] D. Vretenar, P. Finelli, A. Ventura, G. A. Lalazissis, andP. Ring, Phys. Rev. C , 064307 (2000).[32] Tiekuang Dong, Zhongzhou Ren, and Zaijun Wang,Phys. Rev. C , 064302 (2008).[33] Jian Liu, Zhongzhou Ren, Chang Xu, and Renli Xu,Phys. Rev. C , 054321 (2013).[34] C. J. Horowitz, Phys. Rev. C , 045503 (2014).[35] O. Moreno and T. W. Donnelly, Phys. Rev. C , 015501(2014).[36] Junjie Yang, Jesse A. Hernandez, and J. PiekarewiczPhys. Rev. C , 054301 (2019).[37] Toshio Suzuki, Phys. Rev. C , 2815 (1994).[38] C. J. Horowitz, et al. , Phys. Rev. C , 032501(R) (2012).[39] The PREX-II proposal, unpublished, available athttp://hallaweb.jlab.org/parity/prex[40] The CREX proposal, unpublished, available athttp://hallaweb.jlab.org/parity/prex[41] Zidu Lin, C. J. Horowitz, Phys. Rev. C , 014313(2015).[42] D. W. Sprung and J. Martorell, J. Phys. A , 6525(1997).[43] J. Piekarewicz, A. R. Linero, P. Giuliani, and E. Chicken,Phys. Rev. C , 034316 (2016).[44] J. Erler, A. Kurylov, and M. J. Ramsey-Musolf, Phys.Rev. D , 016006 (2003).[45] K. Nakamura, et al. (Particle Data Group) J. Phys. G , 075021 (2010).[46] B. G. Todd-Rutel and J. Piekarewicz, Phys. Rev. Lett. , 122501 (2005).[47] M. Beiner, H. Flocard, Nguyen van Giai, and P. Quentin,Nucl. Phys. A238,
29 (1975).[48] E. Chabanat, P Bonche, P Haensel, J. Meyer, and R.Schaeffer, Nuc. Phys. A
231 (1998).[49] M. Kortelainen, T. Lesinski, J. More, W. Nazarewicz, J.Sarich, N. Schunck, M. V. Stoitsov, and S. Wild, Phys.Rev. C
540 (1997).[53] Wei-Chai Chen and J. Piekarewicz, Phys. Lett. B,
284 (2015).[54] J. Piekarewicz and M. Centelles, Phys. Rev. C , 054311(2009).[55] G. Hagen, T. Papenbrock, D. J. Dean, and M. Hjorth-Jensen, Phys. Rev. C , 034330 (2010).[56] G. Hagen, A. Ekstrom, C. Forssen, G. R. Jansen, W.Nazarewicz, T. Papenbrock, K. A. Wendt, S. Bacca, N.Barnea, B. Carlsson, C. Drischler, K. Hebeler, M. Hjorth-Jensen, M. Miorelli, G. Orlandini, A. Schwenk and J.Simonis, Nature Physics
186 (2016).[57] W.-C. Chen and J. Piekarewicz, Phys. Rev. C , 044305(2014).[58] R. H. Helm, Phys. Rev. , 1466 (1956).[59] C. J. Horowitz, S. J. Pollock, P. A. Souder, R. Michaels,Phys. Rev. C63