PHYSICAL REVIEW C 77, 034304 (2008)

p-shell � hypernuclei with the Nijmegen ESC04 baryon-baryon potential

Dean HaldersonDepartment of Physics, Western Michigan University, Kalamazoo, Michigan 49008, USA

(Received 25 September 2007; revised manuscript received 7 December 2007; published 13 March 2008)

The spectra of seven p-shell � hypernuclei are calculated with effective interactions derived from the newNijmegen ESC04 baryon-baryon interaction. The results of interactions ESC04A and ESC04B compare wellwith the known experimental levels, whereas those of ESC04C and ESC04D do not. The primary differencebetween the two pairs of potentials is the ratio of triplet to singlet strength in the central potential. The 16

�O resultsindicate that the tensor interaction is too weak and/or the spin-orbit odd interaction is too strong for all potentials.In addition the �N -�N coupling potentials have an unusual radial behavior. The calculations provide a naturalexplanation for the reduction of the spin-spin interaction of phenomenological interactions from the beginningto the end of the shell.

DOI: 10.1103/PhysRevC.77.034304 PACS number(s): 21.80.+a, 21.30.−x, 21.60.Cs, 21.10.−k

I. INTRODUCTION

Beginning in 1998 a series of experiments using theHyperball deterctor [1] attempted to measure gamma raysfrom p-shell �-hypernuclei. The experiments ended in 2005with KEK 566. The energy and yield of the measured gammarays from all experiments have provided valuable informationon the structure of seven nuclei. These experiments weremotivated by the work of Refs. [2] and [3]. It was shown inRef. [2] that the five pNs� two-body matrix elementsdepend on the radial integrals associated with each compo-nent of the standard interaction, V�N (r) = V0(r) + Vσ (r)sN ·s� + V�(rlN� · s� + VN (r)lN� · sN + VT (r)S12. The radialintegrals are conventionally denoted by the parametersV ,�, S�, SN , and T . The energy splitting of spin partnerscan be related to these parameters and, therefore, to the spin-dependent components of the �N interaction. Calculationsfor the p-shell nuclei, therefore, provide a test of thesecomponents. The new Nijmegen ESC04 potentials [4,5] havebeen shown to provide reasonable binding energies for the 0s-shell nuclei, but overbind heavier systems [6]. The followingcalculations for the p-shell nuclei test the spin-dependentcomponents of the Nijmegen ESC04.

II. THE EFFECTIVE INTERACTION

Effective interactions were generated for the four ESC04

potentials via the Brueckner-Hartree procedure with 4Heand 16O cores. The procedure is as follows. The fullBrueckner calculation with � intermediate states wasrun as described in Refs. [6–8]. The g-matrix elements〈0s1/2(�)n2l2j2(N )J |g|0s1/2(�)n2l2j2(N )J 〉, where n2l2j2 isbelow the Fermi sea, are identified. These two-body matrixelements have self-consistent starting energies. The Bruecknercalculation is repeated, but with an average-angle (Emery-Eden) Q,Q = 0 if 2n + 2N + l + L � Nm, where (n, l),(N,L) are the relative, center of mass quantum numbers. Nm

is taken as (1,2) for the (4He,16O) cores. In this last calculationonly one starting energy is allowed (which is correct for the4He core), and that energy is varied to best reproduce the

self-consistent two-body matrix elements above. The relativematrix elements with 2N + L = 0 are then fit to a sum of threeYukawas (r2 times Yukawas for the tensor force) as in Refs. [9]and [10]. The ranges for the Yukawas are 0.3, 0.4, and 0.6 fmfor �N and �N → �N and 0.25, 0.4, 1.414 fm for �N .This procedure differs from that in Ref. [8] in that the startingenergies for the calculations with an angle-average Q wereadjusted to fit the self-consistent, single-particle energies andnot the set of two-body matrix elements above. Therefore, theeffective interactions in this paper are intended for structurecalculations and not binding energy calculations.

III. STRUCTURE CALCULATIONS

Structure calculations were then performed for the sevennuclei in the recoil corrected continuum (RCCSM) [11]formalism and with the core states generated from the Cohenand Kurath (6–16) [12] NN interaction, however empiricalenergies are used. The oscillator constants, υ = mω/h, forA = 7, 9, 10, 11, 12, 15, and 16 are 0.270, 0.312, 0.320, 0.342,0.330, 0.320, and 0.342 fm−2, respectively. The RCCSMformalism allows expansion of the � wave function in aseries of oscillators as measured from the center of massof the core. This allows use of the � mass while stillproducing non-spurious states. The � states are also includedin the calculation by adding the �� mass difference to theappropriate core states. The � mass must be used to calculatethe � kinetic energy, but this error should be small.

For 7�Li the core states included are the 0+; T = 1, 1+

1 ; T =0, 1+

2 ; T = 0, 2+1 ; T = 0, 2+

2 ; T = 1, 3+; T = 0 states of 6Li,and the 0+, 2+ states of 6He and 6Be. The �+ and �− willcouple to these last states. The results with the 4He coreinteraction for ESC04A, B, C, D, and the refit NSC89 [13]are shown in Fig. 1. One sees that ESC04B gives excellentvalues for the 3/2+ − 1/2+ and 7/2+ − 5/2+ splittingsas compared to those extracted from experiment [14].The 3/2+ − 1/2+ and 7/2+ − 5/2+splittings, as givenin Ref. [15] in terms of the standard interaction pa-rameters, are 1.461� + 0.038S� + 0.011SN − 0.285T and1.294� + 2.165S� + 0.020SN − 2.380T , respectively. These

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DEAN HALDERSON PHYSICAL REVIEW C 77, 034304 (2008)

FIG. 1. The 7�Li spectra from ESC04A, B, C, D, and the refit

NSC89 compared to experiment [14].

expressions have some dependence on the core state wavefunctions and, hence, the NN interaction. But for reasonableinteractions the 3/2+–1/2+ splitting depends almost entirelyon the spin-spin parameter, �. However, one cannot alwaysextract a value of � for realistic interactions like ESC04because they have both even and odd components. This canbe seen from the following. With the Cohen and Kurath(6–16) interaction the 3/2+–1/2+ splitting due to the spin-spininteraction is given to first order by the LS coupled matrixelements,

0.448〈(0p0s)1(S = 1)1|Vσ �s1 · �s2|(0p0s)1(S = 1)1〉− 1.318〈(0p0s)1(S = 0)1 |Vσ �s1 · �s2| (0p0s)1(S = 0)1〉+ 0.870〈(0p0s)1(S = 1)2 |Vσ �s1 · �s2| (0p0s)1(S = 1)2〉.

If one approximates the � wave function by an oscillatorand uses equal-mass Brody-Moshinsky brackets, then 3/2+ −1/2+ splitting is 0.659(3S1 − 1S0) + 0.2243P1 − 0.6591P1 +0.4353P2, where [S]LJ are the relative matrix elements. Withonly a spin-spin interaction this becomes 0.659(3S1 − 1S0) +0.659(3P − 1P ). Depending on the signs of the even andodd contributions, it is possible to have a very strong spindependence, but no splitting. A sample set of effective, relativeg-matrix elements for 5

�He with only the triplet-even (TE),singlet even (SE), triplet odd (TO), and single odd (SO)interactions are shown in Table I.

TABLE I. Effective, relative g-matrix elements for the 5�He core

with only the triplet-even, singlet even, triplet odd, and single oddinteractions.

Interaction\[S]LJ3S1 (MeV) 1S0 (MeV) 3P (MeV) 1P (MeV)

A −3.500 −5.819 −0.522 0.983B −3.665 −5.448 −0.336 0.993C −4.421 −4.694 −0.736 2.738D −3.711 −4.690 −0.546 3.042Refit 89 −3.130 −4.611 0.113 2.311

If the odd state interaction is similar to the even, then onecan, of course, refer to the parameter �. This is not the case.However, if the odd state interaction is small, then one canalso refer to �. This criterion is reasonably well satisfied byESC04A and ESC04B. Since the standard interaction, withits �, does a very good job of fitting the known spectra inthe p-shell, a realistic interaction must somehow reflect itsassumptions.

One can then extract an effective � from the splitting withonly the triplet even, singlet even, triplet odd, and singleodd interactions. With the 3/2+ − 1/2+ splitting from theexpression of Ref. [15] given above, the 7

�Li calculationgives an effective � = 0.428. This values compares favorableto the 0.430 value obtained in Ref. [15] for 7

�Li with thestandard interaction. However, if, instead of using the effectiveinteraction with the 4He core, one uses that with the 16Ocore, the effective � becomes 0.381. This value comparesreasonably well with the 0.330 required in Ref. [15] from themiddle to the end of the p-shell. The effect on the spin-spineffective potentials, when going from the 4He core to the 16Ocore, is shown in Fig. 2. Here are plotted the triplet even minussinglet even and triplet odd minus singlet odd potentials forboth cases. One has a weakening of the spin-spin interactionfor the 16O core, especially for the odd states. This is mostlikely due to the nucleon 0p states dropping below the Fermisea. Therefore, one has a natural explanation for the reductionin � as one moves through the shell.

Shown in Fig. 3 is the experimental spectrum for 9�Be [16]

and the results of calculations with ESC04A and B. The groundstate is set to zero energy in all cases. The calculations includethree 8Be states and two each of 8B and 8He. The 3/2+ − 5/2+splitting energy is shown in parentheses. This splitting hascontributions from several components of the potential. The�N -�N coupling contributes little to the splitting, but ifone shuts off the �N spin-orbit and antisymmetric spin-orbitinteractions the splitting is reduced from 103 keV to 5 keV.By adding back the spin-orbit contribution, one obtains183 keV. Then the return of both the spin-orbit and

FIG. 2. Triplet even minus singlet even and triplet odd minussinglet odd potentials from 4He and 16O core calculations.

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TABLE II. The 2−, 1− splitting in 12�C and 10

�B in keV.

Interaction,model space

A, 4He core A, 16O core B, 4He core B, 16O core 97f ,16O core

Exp. Standard interactionprediction

10�B, �N coupling 58 39 33 15 −34 <100 121

10�B, no �N coupling 173 146 177 101 −65 <100 135

12�C,�N coupling 426 384 281 238 −18 – 150

12�C, no �N coupling 80 51 38 14 −82 – 93

antisymmetric spin-orbit brings the splitting back to 103 keV,showing that the two spin-orbit components act in oppositedirections. The results for ESC04A and B are quite reasonable.Both potentials still prefer the interaction based on the 4Hecore.

Shown in Fig. 4 are the 11�B spectra with the 4He core

interactions. The calculation includes eight 10B states and twoeach of the 10C and 10Be states. Experimental levels [7] locatedwith dashes lines are those whose positions are less certain. Thetheoretical spectra are shifted so that the 5/2+ energy is zero.Again, ESC04A and ESC04B do a better job of reproducingthe experimental results. In all cases, the 3/2+

1 , 1/2+1 doublet

based on the 1+1 state of 10B is calculated to be much lower than

observed. This problem, but to a lesser extent, also occurs in thestandard interaction fit [15]. The experimental spread betweenthe centroid of the 5/2+

1 , 7/2+1 ground state doublet and the

centroid of the 3/2+1 , 1/2+

1 doublet is 1.60 MeV, which is veryfar from the unperturbed value of 0.718 MeV. Such a change isvery rare for the p-shell nuclei. This spread between centroidsdepends strongly on all the spin parameters, so a large changecould be obtained if all components add in phase. In Table IIseveral calculations of the ground state doublet splitting in10�B and 12

�C are presented. One sees the effect found in

FIG. 3. The 9�Be spectra from ESC04A and B with 4He and 16O

core interactions compared to experiment [16]. Splitting energy isindicated in parentheses in keV.

Ref [15]. The �N coupling increases the splitting for 12�C and

decreases it for 10�B. However, the magnitude of the change

is greater in both cases. In addition, the �N -�N couplingpotential and the resulting g-matrix elements look ratherpeculiar. The triplet-even, local, central �N -�N potential forESC04B is plotted in Fig. 5 along with that for NSC97f [18] asa comparison. The NSC97f appears much as one would expectin that it has a strongly repulsive core and goes negative in theintermediate range. The ESC04B potential changes sign withinthe region normally associated with a core and stays positivethrough the intermediate range.

The resulting, diagonal, 3S1,�N-�N , relative g-matrixelements at ω = −25 MeV, corrected with the average-angleQ, are shown in Table III. The NSC97f matrix elements behaveas one would expect, decreasing with principal quantumnumber. Although the NSC97f inverts the the 2−, 1−splitting,one sees in Table II that the �N -�N coupling contributionis small. In contrast, the ESC04B matrix elements are largerin magnitude, increase from n = 0 to 2, and are of oppositesign. Because the RCCSM calculation expands the wavefunctions in a series of oscillators, the matrix elements withn > 0 contribute to the calculation and hence, increase the�N -�N coupling contribution. Despite this behavior, it maybe premature to conclude that the �N -�N coupling istoo strong in the ESC04 potentials without more data. Inexperiment KEK 566 the investigators are searching for the2− → 1− γ -ray in 12

�C. The analysis is not yet complete.However, one should note that the discrepancy between theexperimental value and that calculated with ESC04A for the3/2+ − 1/2+ splitting in 7

�Li is largely due to the 183 keVcontribution from the �N -�N coupling.

The 15�N calculation includes six 14N states and one each

of 14O and 14C. The spectra are shown in Fig. 6. The positionof the first excited state is not known [1], and the dashedline represents a prediction of the standard interaction [15].Without a knowledge of its energy, it is difficult to choseamong the interactions. The agreement of ESC04A and Bwith experiment seems equally good.

TABLE III. Diagonal 3S1, �N -�N , relativeg-matrix elements at ω = −25 MeV, correctedwith the average-angle Q in MeV.

N 0 1 2 3

ESC04B 5.4 7.3 8.0 7.9NSC97f −3.9 −3.3 −2.3 −1.3

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FIG. 4. The 11�B spectra from ESC04A, B, C, D, and the refit

NSC89 from the 4He core interactions compared to experiment [17].Numbers on the ground state levels are the calculated � separationenergies in MeV.

The 16�O spectra are shown in Fig. 7. This is the only

case where ESC04A and B interchange the order of a spindoublet. The ordering of the 0−, 1−doublet was determinedby comparing the experimental [19] versus theoretical yieldsfor the 1+

2 → 0−, and 1+2 → 1−transitions. The two yields

differ by approximately a factor of two. ESC04B with the16O core interaction come the closest to reproducing theexperimental spectrum. One could ask whether the inversion

FIG. 5. The triplet even, central, local �N -�N potential forESC04B and NSC97f.

FIG. 6. The 15�N spectra from ESC04A and B with 4He and 16O

core interactions compared to experiment [1]. Splitting energy isindicated in parentheses in keV.

of the ground state doublet was somehow an artifact of thecalculated procedure, the NN interaction, or the simple coremodel space. A comparison can be made with a completelydifferent calculation, but the same YN interaction. In Ref. [20]the NN interactions starts with the Paris potential, theeffective interaction extracted with the unitary-model-operatorapproach [21], and the model space for the core is 2h-1p.The YN interaction for which a comparison is made is theoriginal NSC89 [22]. Figure 8 demonstrated that the splittingsof doublets are very similar, with the same level ordering. Thedifference in energy between the centroids of the two doubles

FIG. 7. The 16�O spectra from ESC04A and B with 4He and 16O

core interactions compared to experiment [16] Splitting energy isindicated in parentheses in keV.

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FIG. 8. A comparison of the results of Ref. [20] and the effectiveinteraction derived from the original NSC87 with an 16O core.

is due to the use of calculated core energies in Ref. [20] andempirical ones in this work.

The 0−, 1− splitting depends strongly on the tensor interac-tion. The standard interaction2 gives the splitting as −1/3� +4/3S� + 8T. In fact, the measurement of this doublet splittingwas motivated by the opportunity to measure the strength ofthe �N tensor interaction. The correct ordering and splitting ofthe 0−, 1− doublet can be obtained by increasing the ESC04B,16O core tensor interaction by 65%. This would be a veryreasonable alteration since the tensor interaction for ESC04Bhas about one-half the strength of NSC97e and NSC97f.The correct ordering and splitting can also be obtained bydividing the spin-orbit odd component by five. This wouldbe consistent with the conjecture in Ref. [6], where partof the overbinding was attributed to a large spin-orbit oddcontribution. Combinations of increasing the tensor interactionand reducing the spin-orbit interaction will give the correct

splitting. The two alterations were applied to 11�B to see

if one might improve that spectrum. Increasing the tensorinteraction by 65% made the spectrum slightly worse; dividingthe spin-orbit odd interaction by five made it slightly better,but the 1/2+

1 state still comes far too low.

IV. CONCLUSION

Calculation for the structure of seven, p-shell �-hypernuclei have been performed with effective interactionsderived from the Nijmegen ESC04 interaction. Bruecknercalculations for both 4He and 16O Fermi levels show a naturalreduction in the spin-spin interaction from the beginning to theend of the shell. Given the very sparse scattering data to whichit was fit, the ESC04 interactions A and B do a very good jobof reproducing the p-shell spectra. Interaction B is favoredover A because of the 16

�O results. Results for 16�O indicate

that the tensor interaction in ESC04A and ESC04B may betoo weak and/or the spin-orbit odd component too strong. Inaddition, the 12

�C results indicate that the �N -�N couplinginteraction may require revision. The refit NSC89 gives resultswhich are slightly worse than ESC04A. ESC04C and ESC04Ddo miserably. The primary difference between the pair A, Band the pair C, D is the spin-spin interaction. This can be seenin the ratio of the relative g-matrix elements 1S0/

3S1 in Table I.For A, B, C, D, and the refit 89, this value with the 4He core is1.66, 1.49, 1.06, 1.26, and 1.47, respectively. These numberswould be reflected in the effective spin-spin parameter, �.It would be desirable, and not prohibitively time consuming,to add the 1S0/

3S1 ratio to the parameter fitting routines thatdetermine the YN potential parameters.

ACKNOWLEDGMENTS

This work was supported by the National Science Founda-tion under grant PHY-0456943.

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