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Partial wave decomposition of finite-range effective tensor interaction D. Davesne,1 P. Becker,1 A. Pastore,2 and J. Navarro3 1Université de Lyon, F-69003 Lyon, France; Université Lyon 1, 43 Bd. du 11 Novembre 1918, F-69622 Villeurbanne cedex, France CNRS-IN2P3, UMR 5822, Institut de Physique Nucléaire de Lyon 2Department of Physics, University of York, Heslington, York, Y010 5DD, UK 3IFIC (CSIC-Universidad de Valencia), Apartado Postal 22085, E-46.071-Valencia, Spain (Dated: January 13, 2016) Weperformadetailedanalysisofthepropertiesofthefinite-rangetensortermassociatedwiththe Gogny and M3Y effective interactions. In particular, by using a partial wave decomposition of the equationofstateofsymmetricnuclearmatter,weshowhowwecanextracttheirtensorparameters 6 directlyfrommicroscopicresultsbasedonbarenucleon-nucleoninteractions. Furthermore,weshow 1 thatthezero-rangelimitofbothfinite-rangeinteractionshastheformoftheN3LOSkyrmepseudo- 0 potential, which thus constitutes a reliable approximation in the density range relevant for finite 2 nuclei. Finally, we use Brueckner-Hartree-Fock results to fix the tensor parameters for the three effectiveinteractions. n a PACSnumbers: 21.30.Fe21.65.Mn J 2 1 Oneofthekeyingredientsofthe barenucleon-nucleon effective interaction, even for the tensor part. This de- (NN) interactionis the tensor part,which representsthe compositioncanbeusedforaninitialguessoftheinterac- ] most distinct manifestation on meson exchange process. tion parameters. The N3LO pseudo-potential represents h t Amongthemostimportantpropertiesrelatedtotheten- the most general non-local zero-range pseudo-potential - sorinteraction,onecanmentionthequadrupolemoment that includes all possible terms up to the sixth order in l c ofthedeuteron,thepropertiesofexcitedstates[1–4],the derivatives,thatisuptothenext-to-next-to-nextleading u contribution to the spin-orbit splitting [5], and the shell order. It has been constructed several years ago [23, 24] n evolution along isotopic chains [6]. Within the context and has been recently written explicitly in the more fa- [ of phenomenological interactions, one should in partic- miliar Cartesian basis, constraining it to be gauge in- 1 ular keep in mind that the strong competition between variant [25, 26]. In this approach, the standard Skyrme v spin-orbit and tensor on the shell structure properties of interaction with tensor terms corresponds to N1LO plus 6 atomic nuclei [2, 7–9], prevents to freely explore the pa- the usual density-dependent term. 5 9 rameterspace,since a variationinthe tensorcanleadto Inthepresentarticle,weuseasaguidethepartialwave 2 a substantial change in the shell structure with the ap- decomposition of the symmetric nuclear matter EoS to 0 pearance of new magic numbers. A detailed discussion isolate the spin-orbit and tensor contributions of some . and list of references about these issues is given in the 1 finite-range interactions. Specifically, we consider the 0 recent review article of Sagawa and Coló [10]. finite-range interactions D1ST2a [27] and M3Y-P2 [28], 6 and the N3LO pseudo-potential as well. We show that 1 Despite their importance, only a few exploratory at- new constraints are obtained on the sign (even the sign : temptshavebeenmadeinthepast[11,12]toincludeex- v is notknownwith certainty)andthe orderofmagnitude plicit tensor terms within non-relativistic self-consistent Xi mean-field or density functional nuclear models. More of the tensor parameters. r recently,suchterms havebeenaddedtoexisting zero-or ThepotentialcontributiontotheSNMenergyperpar- a finite-rangeeffective interactions,asthe popularfamilies ticle can be decomposed by using a coupled spin-isospin of Skyrme [13], Gogny [14] or M3Y [15]. The parame- (S,T) basis as ters are usually fixed to reproduce some selected spin- orbit splittings, with either a partial or a complete fit of parameters. However, the parametrizations can lead to E = 3~2kF2 + V(ST)(k ), (1) F the appearance of unphysical instabilities of the Fermi A 5 2m XST surface both in the zero- and long-range regimes [16– 18]. To avoid them we have proposed [19, 20] to in- corporate into the fitting procedure the constraints ob- where k = (3π2ρ/2)1/3, ρ being the density, and V(ST) F tainedfromlinearresponsetheory. Furthermore,wehave isthepotentialenergyperparticleprojectedontothedif- also suggested [19, 21, 22] some new constraints from ferentchannels. Neither the tensor northe spin-orbitin- microscopic calculations based on the bare NN interac- teractionscontributetothe(S,T)components. Tostudy tion: specifically, a partial wave decomposition on the them one has to go one step back in the calculation and so-calledN3LOSkyrmepseudo-potentialofthe symmet- projectthepotentialenergytermsontotheJ,L,S,T sub- ric nuclear matter (SNM) equation of state (EoS) allows spaces, where J and L are the total and orbital angular onetoclearlyidentifythecontributionofeachtermofthe momenta,respectively. Usingthestandardspectroscopic 2 notation 2S+1L , the EoS is written as where P is the isospin exchange operator, S (ˆr) = J τ T 3(σ ·ˆr)(σ ·ˆr)−σ σ isthetensoroperator,andasingle E 3~2k2 1 2 1 2 = F + V(2S+1L ), (2) Gaussianhasbeenused. Duetothechosenisospinstruc- J A 5 2m ture of the interaction, V +V is the strength of the JXLS T1 T2 force acting between proton-proton or neutron-neutron where for simplicity the explicit kF-dependence in the pairs whereas VT2 is the strength of a neutron-proton potential energy terms has been suppressed. The value pair. of T is immediately deduced from the antisymmetry of The differences (4) we are interested in are written as thematrixelements. AddingupV(2S+1L )forallJ and J L values, one gets V(ST), with an exact cancellation of δ =−W0ρk2 − 27µ2G (V +V )G (k µ ) tensor and spin-orbit contributions. δP =−8207µ2GF(V 40−√πV )TG1 (kTµ2 )1 F G  (7) Our analysis is based on Brueckner-Hartree-Fock D 280√π T1 T2 2 F G  (BHF) calculations[29]ofthe partialwavecontributions δ =− 3µ2G (V +V )G (k µ ) F 10√π T1 T2 3 F G tothepotentialenergyperparticle,derivedfromthemi-  croscopic Argonne v14 nucleon-nucleon two-body inter- The functions G (k µ ) are given by the sum over mo- L F T action plus the Urbana model for the three-body term. menta on the SNM Fermi sphere of the radial multi- Although the SNM saturation point given by this calcu- pole L of the matrix elements of the tensor interaction. lation is slightly shifted to higher values of density and They are obtained after some tedious, but straightfor- energy per particle, these results provide us with a help- ward calculations, and are expressed as a polynomial in ful guide to assess the properties of effective interactions a = k µ plus a combination with polynomial coeffi- F G in a mean-field scheme. Other microscopic calculations cients of the exponential e a2 and the function S(a) = − exist, as low-k chiral effective field (χ-EFT) [30] or the γ−Ei(−a2)+log(a2), where γ is the Euler constantand many-bodyperturbationtheory(MBPT)[31,32],butall Eiis the exponentialintegralfunction. The firstofthese ofthem havebeenlimited to the (S,T)channels. Inthis functions is written as respect it is worth noticing that for these channels all these calculations agree below saturation density. G (a) = 1 6−6a2−a4−6e a2 +4a2S(a) .(8) To make a proper comparison of the tensor and spin- 1 a3(cid:26) − (cid:27) orbitcontentsofthe effectiveinteractionswerealisethat ThefunctionswithL>1involvehigherorderpolynomi- for a given L,S pair, the central part of the interaction als. gives the same contribution apart from a factor 2J +1. The M3Y interaction [28] is based on the so-called Therefore, it is possible to get rid of the central con- Michigan three-range Yukawa (M3Y) interaction [15], tributions –including the density-dependent terms– by which was derived from the bare NN interaction, by fit- considering weighted differences as ting the Yukawa functions to the G-matrix. The central 2J1+1V(2S+1LJ)− 2J1+1V(2S+1LJ′). (3) ptearrmt,iswchoimleptlehteedspwinit-horbaitzearnod-ratnegnesodrencosimtypodneepnetnsdeanret ′ written as In practice, we have considered the following three combinations e rµLS VY (r)= t(LSE)P +t(LSO)P − L ·S , δ =V(3P )/3−V(3P ) SO h TE TOi rµLS 12 12 P 1 0 (9) δD =V(3D2)/5−V(3D3)/7 (4) δF =V(3F3)/7−V(3F4)/9  e rµT VY(r)= t(TNE)P +t(TNO)P − r2S (rˆ), and neglected combinations involvingpartial waveswith T h TE TOi rµT T (10) L>3, which gives smaller contributions. where P and P are the triplet-even and triplet-odd We proceed now to give the explicit expressions for TE TO operators,andL =r×p isthe relativeorbitalangu- thesecombinationsascalculatedfromseveraleffectivein- 12 12 teractions. The Gogny interaction includes a zero-range larmomentumoperator,withp12 =i(∇1−∇2),andS12 is the total spin of the pair. Actually, both components spin orbit term contain a superposition of two radial functions with dif- VG(r)=iW (σ +σ )·[k ×δ(r)k], (5) ferent ranges, and a sum over them is to be understood LS 0 1 2 ′ in the following. where r is the relativecoordinate ofthe pair,andk′ and The differences given in Eq. (4) are written as k are the relative momentum operators acting on the left and right, respectively. For the tensor interaction δ =−3t(LSO)Y(LS) kF − 27t(TNO)Y(T) kF we follow the authors of Ref. [27], which on top of the P 128π 1 (cid:16)µLS(cid:17) 80πµ2T 1 (cid:16)µT(cid:17)  standardGognyinteraction[14]haveaddedafinite-range δD = t(1L2S8Eπ)Y2(LS)(cid:16)µkLFS(cid:17)− 2576t0(TπNµ2TE)Y2(T)(cid:16)µkFT(cid:17)  tensor effective interaction of the form δ = t(LSO)Y(LS) kF − 3t(TNO)Y(T) kF VTG(r)=[VT1+VT2Pτ]r2ST(ˆr)exp−r2/µ2G, (6) F 32π 3 (cid:16)µLS(cid:17) 40πµ2T 3 (cid:16)µT(cid:17) (11) 3 where the functions Y(LS)(k /µ ) and Y(T)(k /µ ) the BHF results [29] represented as black circles. Filled L F LS L F T are given by the sum over momenta on the SNM Fermi squaresandtrianglesaretheresultsobtainedwiththein- sphereoftheradialmultipoleLofthematrixelementsof teractions D1ST2a and M3Y-P2, respectively. One can the spin-orbit and tensor interactions, respectively. The seethatbothinteractionsleadtoresultswhichareindis- first ones are agreement with BHF ones. For D1ST2a we can roughly say the value of V +V has the right order of magni- 1 T1 T2 Y(LS)(a) = −4a2−88a4+128a3ArcTan(2a) tude, but should be changed of sign in order than the 1 a3(cid:26) calculated δ and δ be in agreement with BHF re- P F +(1−24a2+16a4)log(1+4a2) sults. Similarly,theabsolutevalueofV −V shouldbe T1 T2 slightly increased. As for M3Y-P2 interaction, the com- +16a2PolyLog(2,−4a2) , (12) parisonshows that while the interactioncoefficients pro- (cid:27) duce the rightsign for these combinations,the strengths 1 Y(T)(a) = 4a2(3−2a2)−3(1+4a2)log(1+4a2) should be increased in absolute value to get agreement 1 a3(cid:26) with BHF results. −8a2PolyLog(2,−4a2) .(13) (cid:27) V) 10 e Similarlytothe previousG functions, thosewithL>1 M 0 L involve higher order polynomials in the variables kF/µT δ (P-10 and k /µ . F LS For the Gogny interaction we have used the V) 0 parametrization D1ST2a [27], whose single tensor range e -0.5 D1ST2a M D1ST2a fit has been fixed to be equal to the longest range of the (D -1 MM33YY--PP22 fit Gogny D1S interaction, i.e. µG = 1.2 fm. The spin- δ -1.5 N3LO fit orbit and tensor strengths have been obtained by mak- 0.6 ing an adjustment on the neutron 1f spin-orbitsplitting V) 0.4 e for the nuclei 40Ca, 48Ca, and 56Ni. The resulting spin- M 0.2 orbitparameters is W =103MeV fm5, a value which is (F 0 0 δ smallerthantheoriginalD1STone(W =130MeVfm5). -0.2 0 0.8 1 1.2 1.4 1.6 The tensor parameters are given in Tab. I. For the M3Y -1 k (fm ) interaction we have used the parametrization M3Y-P2, F for which the ranges are µ−LS1 = 0.25 and 0.4 fm, and FIG. 1: (Colors online) Results for the combinations µ−T1 =0.4and0.7fm. The M3Y-P2setoftensorparam- δP,δD,δF defined in Eq. (4). Black circles are the BHF re- etersisfixedsoastoreproducethesingle-particleenergy sults[29],whilefilledredsquaresandbluetriangleshavebeen ordering for 208Pb. Its values are given in Tab. II. calculated with interactions D1ST2a [27] and M3Y-P2 [28], respectively. Open symbols and solid line are the result of a Ref. [27] Fitted fitofthetensorparameters,exceptforδF,asexplainedinthe VT1+VT2 -75 82.3 text. VT1−VT2 -195 -273.9 Similar drawbacks of D1ST2a and M3Y-P2 interac- TABLE I: Tensor parameters (in MeV fm5) of interaction tions have been signaledin Ref. [21], where tensor Lan- D1ST2a [27]. The last column displays the fitted values to dauparametersH(n) forpureneutronmatterwerecalcu- thecombinations δP and δD, as explained in thetext. ℓ latedwiththese interactionsandcomparedto the values obtained within the CorrelatedBasis Function (CBF) of Ref. [33] and the Chiral Effective Field Theory (CEFT) of Ref. [34]. Although these methods provides results Ref. [28] Fitted tT1NE -131.52 1299.3 which could differ up to a factor of ≃ 2, they all agree tT2NE -3.708 -299.50 onthe sign,whichisopposite towhathasbeenobtained tT1NO 29.28 1734.5 for the D1ST2a interaction. On the contrary, the re- tT2NO 1.872 29.709 sults obtained with M3Y-P2 interaction gives values of H(n) which are systematically much smaller than those ℓ TABLE II: Tensor parameters (in MeV fm−2) of interaction given by bare-interaction based methods, but with the M3Y-P2 [28]. The last column displays the fitted values to same sign. These points can be quantitatively assessed thecombinations δP and δD, as explained in thetext. by fitting the tensor parameters to the BHF results of partial waves. Consider first the corresponding Gogny interaction combinations (7). As we have realized that In Fig. 1 are displayed the three combinations δ , δ , the spin-orbit contribution to δ is relatively small, we P D P and δ as a function of the Fermi momentum k , with havekeptunchangedW ,andfitthestrengthsV ±V F F 0 T1 T2 4 to the BHF δ and δ , conserving the same value for be seen that a N3LO pseudo-potential contains the rel- P D the range µ . The combination δ is not included in evant degrees of freedom. In fact, the results for combi- G F the fit, and provides a control about its overall quality. nations δ and δ are of a quality comparable to those P D The fit has been limited to values of the Fermi momen- obtained from the fits with D1ST2a and M3Y-P2 finite- tum relevant for finite nuclei, that is k ≤ 1.7 fm 1. range expressions. As shown in (15), the predicted δ is F − F Similarly, for the M3Y-P2 interaction we keep the orig- proportionalto k9; it is not surprising to get a disagree- F inal spin-orbit parameters as well as the tensor ranges, mentwithBHFresultsforvaluesofk ≥1.5fm 1. One F − andhavefittedthetensorparametersenteringEqs.(11). should keep in mind that a N3LO pseudo-potential can The fitted values of the parameters are given in Tables I be viewed as a low-momentum expansion (k,k ≤ 2k ) ′ F andII,andthepartialwaveresultsaredisplayedasopen ofafinite-rangeinteraction,aswasexplicitlyshownfora symbols with dotted lines in Fig. 1. Comparison of the central interaction [13, 35]. Considering the momentum originalandfittedparametersconfirmsthe drawbacksof matrix elements of the tensor interactions (6) and (10) these effective tensor interactions. In their present form and limiting ourselves to the first non-vanishing contri- and/or fitting procedure they lead to inconsistencies in bution for F-waves, it can be easily shown that in both the simultaneous description of finite nuclei and infinite cases the obtained structure is the same as the N3LO matter splittings (see [11, 27]). pseudo-potential. Actually, this result is general and it Let us now turn to the non-local zero-range pseudo- isstraightforwardtoseethatEq.(14)representsthecom- potentialN3LO[23,24]. Imposinglocalgaugeinvariance, monlow-momentumexpansionofanyfinite-rangetensor the tensor component is written as [22] interaction. The N3LO parameters are thus related to those of a finite-range interaction. 1 1 V(N3LO) = t T + t T T 2 e e 2 o o + t(4) (k2+k′2)T +2(k·k′)T e e o This relationbetween parameters can be alternatively h i + t(4) (k2+k′2)T +2(k·k′)T obtained by expanding in powers of kF the combina- o o e tions(4)ascalculatedwithafinite-rangeinteractionand h i 1 identifying the coefficients of k with the corresponding + t(6) (k2+k′2)2+(k·k′)2 T F e (cid:20)(cid:18)4 (cid:19) e powers of (15). Since the spin-orbit term of D1ST2a interaction coincides with the N3LO one, one has simply +(k·k′)(k2+k′2)To to Taylor expand the functions GL entering (7). One i canimmediatelyseethatthe resultingN3LOcoefficients 1 + t(6) (k2+k′2)2+(k·k′)2 T are not independent, since the following equalities hold: o (cid:20)(cid:18)4 (cid:19) o t(4) = −µ2t , and t(6) = −µ2t(4). That means that the +(k·k′)(k2+k′2)T , (14) toensorD1SGTo2ainteroa,cetionhaGsnoo,etthemostgeneralform, e i andatleastasecondGaussiantermshouldbeincludedin where T and T are the following even and odd tensor Eq. (6) to get independent N3LO parameters. This pro- e o operators cedureofrelatinginteractioncoefficientsismoreinvolved in the case of M3Y-P2 interaction due to its finite-range Te(k′,k) = 3(σ1·k′)(σ2·k′)+3(σ1·k)(σ2·k) spin-orbit term. Indeed, in that case the local-gauge in- − (k2+k2)(σ ·σ ), variance is not satisfied, and both spin-orbit and tensor ′ 1 2 termscontributetotheN3LOpartialwavecombinations T (k,k) = 3(σ ·k)(σ ·k)+3(σ ·k)(σ ·k) o ′ 1 ′ 2 1 2 ′ given in Eq.(4). In any case, the N3LO Skyrme pseudo- − 2(k′·k)(σ1·σ2). potentialprovidesareliableapproximationtoanyfinite- range effective interaction in the density range relevant Asforthespin-orbitpart,ithasbeenshown[25]thatthe to nuclear structure and astrophysical issues [36]. form given in Eq. (5) is the only possible one consistent withlocalgaugeinvariance. TheN3LOcombinations(4) then read In conclusion, we have shown that N3LO constitutes δ =− 1 k2W ρ P 80 F 0 a reliable substitute of any finite-range tensor interac- δD = 7+02070196k0F4ktF2(e4t)oρρ++3252900700kkF4F6tt(o(e46))ρρ+ 52070kF6t(o6)ρ  (15) ttieornacatinodn,tehiatthetrhefintietnes-roarnpgaeroarmneotenr-sloocfalazneyroe-ffraecntgive,eciann- δ = 1 k6t(6)ρ be obtained frommicroscopiccalculations based onbare F 875 F o  NN interactions. In particular, BHF results have been PresentlythereisnoparametrizationofaN3LOpseudo- used as an illustrative example, but other methods as χ- potential that incorporates finite-nuclei constraints. We EFT or MBPT could be used once these quantities will have therefore fitted the tensor parameters to the BHF be made available. These results canbe usedas astrong results, by keeping the value for W fixed in Ref. 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