Exploring effects of tensor force and its strength via neutron drops
RRIKEN-QHP-490RIKEN-iTHEMS-Report-21
Exploring effects of tensor force and its strength via neutron drops
Zhiheng Wang ( 王 之 恒 ), Tomoya Naito ( 内 藤 智 也 ),
2, 3
Haozhao Liang ( 梁 豪 兆 ),
2, 3 and Wen Hui Long ( 龙 文 辉 )
1, 4 School of Nuclear Science and Technology, Lanzhou University, Lanzhou 730000, China Department of Physics, Graduate School of Science,The University of Tokyo, Tokyo 113-0033, Japan RIKEN Nishina Center, Wako 351-0198, Japan Joint department for nuclear physics, Lanzhou University and Institute of Modern Physics,Chinese Academy of Sciences, Lanzhou 730000, China (Dated: January 18, 2021)The tensor-force effects on the evolution of the spin-orbit splittings in the neutron drops areinvestigated within the framework of the relativistic Hartree-Fock theory. For fair comparisons onthe pure mean-field level, the results of the relativistic Brueckner-Hartree-Fock calculation with theBonn A interaction are adopted as meta-data. Through a quantitative analysis, we certify that the π -pseudovector ( π -PV) coupling affects the evolutionary trend through the tensor force embedded.The strength of the tensor force is explored by enlarging the strength f π of the π -PV coupling. It isfound that weakening the density dependence of f π is slightly better than enlarging it with a factor.We thus provide a semiquantitative support for the renormalization persistency of the tensor forcewithin the framework of density functional theory. This will serve as an important guidance forthe further development of the relativistic effective interactions with particular focus on the tensorforce. I. INTRODUCTION
The nuclear tensor force is one of the most impor-tant components of the bare nucleon-nucleon interac-tion [1, 2]. In recent decades, the effects of the tensorforce in the nuclear mediums have also been intensivelyinvestigated, after being neglected for a long time. Ithas been shown that the tensor force plays crucial rolesin the shell-structure evolution [3–25], spin-isospin ex-citations [26–31], and the giant resonances [32–34]. Inspite of such achievements, there are still open questionsabout the properties of the in-medium tensor force, i.e.,the effective tensor force. Among the most attractive andchallenging problems is the constraint of the strength ofthe tensor force [2]. To achieve this goal, one needs tofind the observables that are sensitive to the effectivetensor force [2, 8, 35–37]. By fitting to these observablesbased on certain many-body theories, one can expect topin down the sign and strength of the tensor force.The nuclear density functional theory (DFT) [38–43]is currently the only candidate that can be applied to al-most the whole nuclear chart except for very light nuclei.Within both the nonrelativistic and relativistic DFT, theparameters of the effective interactions are usually deter-mined by fitting to the bulk nuclear properties, such asthe masses and radii of the finite nuclei, as well as theempirical knowledge of the infinite nuclear matter. How-ever, these bulk properties are found to be, in general, notsensitive to the tensor components in the effective inter-actions. In particular, by adding the Fock term of thepion exchange, which is one of the most important car-riers of the tensor force in the relativistic nuclear forces,on top of the conventional relativistic mean-field (RMF)theory, Lalazissis et al. [44] found that the bulk proper-ties of spherical finite nuclei and infinite nuclear matterdisfavor the tensor force, i.e., the optimal fit is achieved for vanishing pion field.On the other hand, the tensor force has the charac-teristic property of spin dependence. It can significantlyaffect the shell structure of nuclei, especially those lo-cated far away from the stability line [4]. One of themost famous benchmarks is the evolution of the energydifference between the proton states 1 h / and 1 g / inthe Sn ( Z = 50) isotopes and that between the neu-tron states 1 i / and 1 h / in the N = 82 isotones [45].Based on the Skyrme Hartree-Fock (SHF) [46], GognyHartree-Fock (GHF) [6], and the relativistic Hartree-Fock (RHF) [9] theories, it has been found that thetensor force plays crucial roles in reproducing the em-pirical trend of the shell structure mentioned above. Ac-cordingly, by reproducing the shell-structure evolution,one can expect to calibrate the strength of the effectivetensor force.Nevertheless, the single-particle states observed exper-imentally are usually fragmented [47–50] due to, for ex-ample, the coupling with the low-lying vibrations, whichis related to the quenching of the spectroscopic factors.The distraction arising from the beyond-mean-field cor-relations makes it ambiguous to compare directly thesingle-particle energies calculated by DFT with the corre-sponding experimental data. A hopeful solution to elim-inate such kind of distraction is to take into account theparticle-vibration coupling (PVC) in the theoretical cal-culations [51–53]. By doing so, the descriptions of theenergies as well as the wave functions can be improved,although the fragmentation of the single-particle statesmay be still a problem.Another choice to avoid the distraction of the beyond-mean-filed correlations is to seek for the ab initio cal-culations which can serve the meta-data, instead of theexperimental data. In the last decade, the ab initio cal-culations have achieved great progress [54–57]. In partic- a r X i v : . [ nu c l - t h ] J a n ular, Shen et al. have established the self-consistent rel-ativistic Brueckner-Hartree-Fock (RBHF) theory for thefinite nuclei and achieved much better agreement withthe experimental data only with the two-body interac-tion [58–60], in contrast to the previous nonrelativisticBrueckner-Hartree-Fock calculations. Like other ab ini-tio calculations, the RBHF calculation is computation-ally consuming for the heavy and even medium-mass nu-clei.In a neutron drop, a collectivity of neutrons confinedby an external filed, there exists only neutron-neutroninteraction, and the equations are easier to solve com-pared with the real finite nuclei. It thus has drawn greatattentions [61–67] and provides an ideal platform to linkthe ab initio and DFT calculations [68–74]. More im-portantly, the single-particle energies calculated by theRBHF theory and those by the DFT are both quantitieson the pure mean-field level, which ensures that one canmake a fair comparison between them.Great successes have been achieved in nuclear physicswith the nuclear covariant density functional theory(CDFT) [39–42]. As a branch of CDFT, the RHF the-ory shares the common advantages of it [75–81]. In ad-dition, the RHF theory can take into account the ten-sor force via the Fock term without extra free param-eters [9, 13, 77, 82–87]. In particular, the quantitativeanalysis of the tensor-force effects in the RHF theory wasachieved recently [88]. According to the famous mecha-nism revealed by Otsuka et al. [4], the spin-orbit (SO)splittings shall be sensitive to the tensor force. Tak-ing the SO splittings calculated by the RBHF theoryas meta-data, the strength of the tensor force was ex-plored in the RHF theory [67, 73, 89, 90]. Nevertheless,the contributions of the tensor force were not extractedquantitatively in these works. In the present work, wewill first quantitatively verify the tensor-force effects onthe SO splittings in the neutron drops within the RHFtheory. In addition, the strength of the tensor force willbe further explored. Motivated by the idea of renormal-ization persistency of the tensor force [91, 92], particularattention will be paid to the density dependence of thetensor force in nuclear medium.This paper is organized as follows. In Section II, theRHF theory and the method to evaluate the tensor-forcecontributions will be briefly introduced. In Section III,we will clarify the tensor-force effects on the evolution ofthe SO splittings in the neutron drops and then furtherexplore the strength of the tensor force. A summary willbe given in Section IV. II. THEORETICAL FRAMEWORK
In the CDFT, the nucleons are considered to interactwith each other by the exchange of various mesons andphotons [39–42, 93–97]. Starting from the ansatz of astandard Lagrangian density, which contains the degreesof freedom associated with the nucleon field, the meson fields, and the photon field, one can derive the corre-sponding Hamiltonian as H = (cid:90) d x ¯ ψ ( x ) [ − i γ · ∇ + M ] ψ ( x )+ 12 (cid:88) φ (cid:90) (cid:90) d x d y ¯ ψ ( x ) ¯ ψ ( y ) Γ φ ( x, y ) D φ ( x, y ) × ψ ( y ) ψ ( x ) , (1)where ψ is the nucleon-field operator, and the φ denotethe meson-nucleon couplings, including the Lorentz σ -scalar ( σ -S), ω -vector ( ω -V), ρ -vector ( ρ -V), ρ -tensor ( ρ -T), ρ -vector-tensor ( ρ -VT), and π -pseudovector ( π -PV)couplings, as well as the photon-vector ( A -V) coupling.Here, Γ φ ( x, y ) and D φ ( x, y ) are the interaction verticesand the propagators of a given meson-nucleon coupling φ ,respectively; their explicit expressions can be referred toRefs. [77, 82, 83, 98–102]. Obviously, the photon field isnot considered in the case of neutron drop.The nucleon-field operators ψ ( x ) and ψ † ( x ) can beexpanded on the set of creation and annihilation opera-tors defined by a complete set of Dirac spinors { ϕ α ( r ) } ,where r denotes the spacial coordinate of x . In this work,the spherical symmetry is assumed. Then, the energydensity functional can be obtained through the expecta-tion value of the Hamiltonian on the trial Hartree-Fockstate under the no-sea approximation [93]. Variation ofthe energy density functional with respect to the single-particle wave functions gives the Dirac equations, (cid:90) d r (cid:48) ˆ h ( r , r (cid:48) ) ϕ ( r (cid:48) ) = εϕ ( r ) , (2)where ˆ h ( r , r (cid:48) ) is the single-particle Hamiltonian. In theRHF theory, ˆ h ( r , r (cid:48) ) contains the kinetic energy ˆ h K , thedirect local potential ˆ h D , and the exchange nonlocal po-tential ˆ h E ; see Refs. [78, 82, 101, 103] for the detailed ex-pressions. Notice that the tensor force contributes onlyto the nonlocal potentials.For the RHF theory with density-dependent effectiveinteractions, the meson-nucleon coupling strengths aretaken as functions of the baryonic density ρ b . For conve-nience, here we explicitly present the density dependenceof the π -PV coupling, which reads f π ( ρ b ) = f π (0) e − a π ξ , (3)where ξ = ρ b /ρ sat. with the saturation density of thenuclear matter ρ sat. , and f π (0) corresponds to the cou-pling strength at zero density. The density dependenceof the other meson-nucleon couplings can be referred toRefs. [77, 83].The external field to keep the neutron drops bound ischosen as a harmonic oscillator (HO) potential as U h.o. ( r ) = 12 M ω r , (4)with (cid:126) ω = 10 MeV. It is worth noticing that the choiceof the external field here is not completely arbitrary, butit is an optimal one, as extensively discussed in Ref. [90].In Ref. [88], the tensor forces in each meson-nucleoncoupling were identified through the nonrelativistic re-duction. They can be expressed uniformly asˆ V tφ = 1 m φ + q F φ S , (5)where m φ is the meson mass, q is the momentum trans-fer, and S is the operator of the tensor force in themomentum space, which reads S ≡ ( σ · q ) ( σ · q ) −
13 ( σ · σ ) q . (6)The coefficient F φ associated with a given meson-nucleoncoupling reflects the sign and the rough strength of thetensor force, as displayed in Table II of Ref. [88].The method to quantitatively evaluate the contribu-tions of the tensor force was also established in Ref. [88].Using this method, one can first calculate the tensor-force contributions to the two-body interaction matrixelements; the explicit formulae are referred to AppendixC in Ref. [88]. Then, the contributions of the tensor forceto the nonlocal potential can be obtained, and eventuallyits contributions to the single-particle energies are quan-titatively extracted. III. RESULTS AND DISCUSSION
Displayed in Fig. 1 are the SO splittings of the doublets1 p , 1 d , 1 f , and 2 p in the N -neutron drops, calculated byboth the RMF and RHF theories. The results of theRBHF [73] theory using the Bonn A interaction are alsoshown for comparison, serving as the meta-data. Onecan see that the meta-data present a nontrivial pattern:the SO splittings vary monotonously and even linearlybetween the neighboring (sub)shells N = 8, 14, 20, 28,40, and 50. Such a feature arises from the characteristicspin-dependent properties of the tensor force, as pointedout in Ref. [73].For the RMF calculation with PKDD [104], one cansee that the results are obviously far away from the meta-data. This is mainly because of the absence of the ex-plicit tensor force in the framework of RMF [9, 73], dueto the lack of the Fock term. As a typical representativeof the RHF effective interactions, PKO1 [77] reproducesthe pattern of meta-data qualitatively, which attributesto the tensor force in the π -PV coupling [73]. Interest-ingly, even though the tensor force is explicitly involvedin PKO2 [9] and PKA1 [83], their results obviously devi-ate from the meta-data. In fact, the results of PKO2 andPKA1 are similar to that of PKDD rather than PKO1.To understand this phenomenon, one needs to go intothe details of the tensor forces arising from each meson-nucleon coupling, which were identified in Ref. [88].Among all the meson-nucleon couplings involved inthe current RHF effective interactions, only the π -PVcoupling gives the tensor force that matches the prop-erties of spin dependence revealed in Ref. [4], i.e., it is DE s.o. (MeV) N h . o . FIG. 1. From top to bottom, the SO splittings of the doublets1 p , 1 d , 1 f , and 2 p in N -neutron drops in a harmonic oscilla-tor (HO) trap. The results are calculated by the RHF the-ory with the effective interactions PKO1 [77], PKO2 [9], andPKA1 [83], as well as by the RMF theory with PKDD [104].The results of RBHF [73] with the Boon A interaction areshown as meta-data. The contributions of the HO potentialin the RMF calculation with PKDD are also presented, de-noted as PKDD h.o. . repulsive (attractive) when the two interacting nucleonsare parallel (antiparallel) in their spin states. The ten-sor forces from the other couplings are opposite to thatfrom the π -PV coupling in sign, as shown in Table II ofRef. [88]. Meanwhile, it has been shown that the ten-sor force in the π -PV coupling dominates over those inthe other couplings. Thus, for PKO2, where the π -PVcoupling is absent, the tensor force mainly comes fromthe ω -V coupling. This means the sign of the net tensor-force contribution in PKO2 is opposite to that in PKO1.That is why PKO2 cannot reproduce the pattern givenby RBHF even qualitatively. PKA1 contains not onlyall the meson-nucleon couplings involved in the PKO se-ries but also the ρ -T and ρ -VT couplings, which givetensor force with considerable strength. As mentionedabove, the tensor-force contributions arising from thesetwo couplings partially cancel the corresponding contri-butions from the π -PV coupling. Therefore, PKA1 givesworse description of the meta-data than PKO1 does.It is noticeable that the RMF effective interactionPKDD also presents some kinks. To make it clearwhere these kinks arise from, we calculate the contribu-tions of the external HO potential, denoted as PKDD h.o. N DE s.o. (MeV) P K O 1 E t o t E e x c E s + w e x c E r e x c E p e x c E p t N FIG. 2. The contributions from the exchange terms of eachmeson-nucleon coupling to the SO splittings of the 1 p and 1 d doublets in the neutron drops, calculated by the RHF theorywith the effective interaction PKO1. For comparison, thetotal SO splittings and the contributions of the tensor forcefrom the π -PV coupling are also shown. See the text fordetails. in Fig. 1. By comparing the results of PKDD andPKDD h.o. , one can find that the kinks given by PKDDare determined by the external HO potential. This alsoexplains why the results given by different RMF effec-tive interactions are so similar to each other, as shown inFig. 2 of Ref. [73]. Definitely, the external potential canalso affect the shell structure given by the RHF effectiveinteractions.It has been shown that the meta-data can be betterreproduced by PKO1 when the coupling strength of π -PV is enhanced properly [67, 73]. Actually, the RHFeffective interaction PKO3 [9], which contains the samekinds of meson-nucleon couplings as PKO1 but slightlystronger π -PV coupling strength, can also give compara-ble description of the meta-data. According to the previ-ous works and the discussion above, it can be shown thatthe π -PV coupling, essentially the tensor force embedded,plays a crucial role in determining the evolutionary trendof the SO splittings in the neutron drops. Nevertheless,all these analyses about the tensor force effects are intu-itive to some extent, while the quantitative investigationis still missing.Since the tensor forces arise from only the exchangeterms of the relevant meson-nucleon couplings, we firstcalculate the contributions of the exchange terms of eachmeson-nucleon coupling to the SO splittings of the 1 p and1 d doublets. The results are shown in Fig. 2. It can beseen that the evolution of the SO splittings calculated byPKO1 (black filled circles) is mainly determined by thecontributions of the exchange terms (red triangles withcross). Remarkably, the patten given by the exchangeterm of the π -PV coupling (blue open stars) is almost thesame as that by the total exchange term. The combinedcontributions of the exchange terms of the σ -S and ω -Vcouplings (wine-red open circles) are also considerable,but in general not so decisive for the kinks as those of the π -PV coupling. The contributions of the ρ -V coupling arenegligible because of the small coupling strength. It is well known that the π -PV coupling contains notonly the tensor force but also the central component.Thus, it is of particular significance to quantitativelyevaluate the contributions of the tensor force from the π -PV coupling. We calculate the tensor-force contribu-tions using the method developed in Ref. [88]. The resultsare also shown in Fig. 2, denoted by the blue filled stars,which are almost hidden behind the blue open stars. Onecan find that the contributions of the π -PV coupling isalmost totally determined by the tensor force, whereasthe role of the central force is negligible. In other words,the π -PV coupling affects the evolutionary trend almostfully through the tensor force embedded. We thus certifyquantitatively that it is reasonable to explore the tensor-force effects by varying the π -PV coupling in the previousworks.It is notable that in both the RHF and RBHF calcula-tions the filling approximation is adopted, i.e., the last oc-cupied levels are partially occupied with equal probabilityfor the degenerate states. Such an approximation may af-fect the levels which are not fully occupied, but does notaffect the neutron drops with closed (sub)shells. If weconsider only the neutron drops with closed (sub)shells,i.e., those with N = 8, 14, 20, 28, 40, and 50, we can avoidthe distraction from the filling approximation. Here, wedefine the slope of the SO splittings in the neutron dropswith respect to the neutron numbers as L N - N = ∆ E s.o. ( N ) − ∆ E s.o. ( N ) N − N , (7)where ∆ s.o. ( N ) is the SO splitting in the neutron dropwith the neutron number N . For the neutron drops withclosed (sub)shells, the combination of ( N - N ) has thefollowing choices: (8-14), (14-20), (20-28), (28-40), and(40-50). Following the strategy proposed in Ref. [73],i.e., multiplying the f π (0) in PKO1 by a factor λ ( λ ≥ . (cid:115) (cid:80) i (cid:0) L RHF i − L RBHF i (cid:1) , (8)where L RHF i ( L RBHF i ) is the slope calculated by the RHF(RBHF) theory, and i runs over all the possible combi-nations of ( N - N ) for different SO doublets. The resultsare shown in the upper half of Table I. It can be seenthat λ = 1 .
42 gives the smallest ∆ among the values of λ , which is 0 . λ = 1 .
0, 1 .
2, and 1 . λ = 1 . λ = 1 .
4, which is quite close to the optimal value
TABLE I. The rms deviations ∆ (in the unit of MeV) of the slope of the SO splittings between the neighboring (sub)shells. Theupper panel gives the results of PKO1 with f π (0) multiplied by a factor λ ( λ ≥ . a π multiplied by a factor η (0 < η < . λ and η as well as the corresponding ∆are shown with the bold font. See more details in the text. f π → λf π λ . . . . . . . .
6∆ 0 . . . . . . . . a π → ηa π η . . . . . . . .
1∆ 0 . . . . . . . . P K O 1 : f p fi l f p N DE s.o. (MeV) B o n n A
P K O 1 (l = 1 . 0 ) P K O 1 (l = 1 . 2 ) P K O 1 (l = 1 . 4 ) FIG. 3. From top to bottom, the SO splittings of the doublets1 p , 1 d , 1 f , and 2 p in the neutron drops. The calculations areperformed by the RHF theory with the effective interactionPKO1, of which the π -PV coupling strength f π (0) is multi-plied by the factor λ ( λ = 1 .
0, 1 .
2, and 1 . .
42. If λ becomes significantly larger, the results getworse in general. The results of larger λ will not be pre-sented here; they can be referred to Ref. [105].In addition to the strength of the tensor force, theproperties of the tensor force in nuclear medium is alsoof great interests and still under discussion [2]. It hasbeen argued that the bare tensor force does not undergosignificant renormalization in the medium, which is de-noted as the tensor renormalization persistency [91, 92].In other words, the effective tensor force would be similarwith the bare one. If this is true, one could also antici-pate to verify such a property in the framework of DFT.Within the RHF theory with density-dependent effectiveinteractions, the renormalization can be, to a large ex- tent, reflected by the density dependence of the couplingstrengths. Minor renormalization can naturally mani-fest as weak density dependence. For the π -PV coupling,which serves as the main carrier of the tensor force, tensorrenormalization persistency requires that the coefficientof density dependence a π should be small, as indicatedby Eq. (3). In our previous work [101], we have shownthat weakening the density dependence of the π -PV cou-pling, i.e., reducing a π , can improve the description of theshell-structure evolution along the N = 82 isotones andthe Z = 50 isotopes more efficiently, compared with en-larging f π (0) with a factor. Inevitably, the beyond-mean-filed correlations in the experimental single-particle ener-gies make the comparison with the results on the mean-filed level ambiguous. With the meta-data given by theRBHF theory, which are also on the pure mean-filed level,we can further explore the tensor renormalization persis-tency in a more convincing way.Aiming at this goal, we calculate again the SO split-tings in the neutron drops by the RHF theory usingPKO1, but the coefficient of density dependence a π ismultiplied by a factor η (0 < η < . η = 0 .
33, which is 0 . η is smallerthan that with λ , which means that the modification with η is more adequate. Thus, one can conclude that weak-ening the density dependence is an available and efficientway to improve the description of the evolution of theSO splittings. It should be stressed that this conclusionis reached by comparison between the CDFT calculationand ab initio one, both of which belong to the pure mean-field level. Even though we did not perform completerefitting procedure yet, we indeed provide a semiquanti-tative support for the renormalization persistency of thetensor force.The results with η = 0 .
3, 0 .
6, and 0 . a π , which meansweaker density dependence, gives a better description ofthe evolution of the SO splittings. When η = 0 .
3, whichis quite close to the optimal value 0 .
33, the meta-dataare reproduced quite well. If η is too small, the resultscould become worse in a visible manner; the details canalso be referred to Ref. [105]. P K O 1 : a p fi h a p N DE s.o. (MeV) B o n n A P K O 1 ( h = 0 . 3 ) P K O 1 ( h = 0 . 6 ) P K O 1 ( h = 0 . 9 ) FIG. 4. Similar with Fig. 3, but a π is multiplied by a factor η ( η = 0 .
9, 0 .
6, and 0 . IV. SUMMARY
In this work, we have investigated the tensor-force ef-fects on the evolution of the SO splittings in the neutrondrops within the framework of the RHF theory. Thecorresponding results of the RBHF calculation with theBonn A interaction were adopted as meta-data. Since theresults of the RHF theory and the meta-data calculatedby the RBHF theory are both within the pure mean-filedlevel, one can thus make fair comparisons between them.Through a qualitative analysis of the results by the RHF effective interactions PKO1, PKO2, and PKA1, we con-firmed that the tensor forces in the effective interactionsplay crucial roles in reproducing the meta-data. Mean-while, for the RMF effective interactions, it was foundthat the evolution of the SO splittings is mainly deter-mined by the external HO potential. Moreover, we foundthat the contributions from the exchange terms almostfully determine the evolution of the SO splittings. Amongall the meson-nucleons couplings, the exchange term of π -PV plays the dominant role. In particular, the tensor-force contribution was extracted, and it was found thatthe π -PV coupling affects the evolutionary trend throughthe tensor force embedded, while its central force hasalmost invisible effects. This conclusion verifies quanti-tatively that it is reasonable to explore the tensor-forceeffects by varying the π -PV coupling strength. In otherwords, the evolution of the SO splittings belongs to theobservables that can constrain the strength of the tensorforce.The strength of the tensor force was explored by en-larging the f π in two different ways: (i) multiplying afactor λ ( λ > .
0) as a whole, and (ii) weakening the den-sity dependence by multiplying a factor η (0 < η < . a π . To avoid the possible distractionsfrom the filling approximation adopted in the RHF andRBHF calculations, we took into account only the neu-tron drops with (sub)shell closure and calculated theslopes of the SO splittings between the neighbouring(sub)shells. Judging from the rms deviations of the se-lected slopes with respect to those calculated from themeta-data, we found that when λ (cid:39) . η (cid:39) .
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