A single particle energies

£=>S S - I l s • o sis £ ;«|l|s-p SEP 2 7 t9S3 A SINGLE PARTICLE ENERGIES OSTi A. R. Bodmer ( - J 3 < C O o 3 Department of Physics. University of Illinois at Chicago. Chicago. IL 60680. USA and Physics Division. Argonne National Laboratory. Argonne. IL 60439-4843. USA and Q. N. Usmani and M. Sami Department of Physics. Jamia Millia Islamia. New Delhi 110025. India 1. INTRODUCTION We consider the binding energies of A hypernuclei (HN), in particular the singleparticle (s.p.) energy data, which have been obtained for a wide range of HN with mass numbers A < 89 and for orbital angular momenta £\£4 [1]. We briefly review some of the relevant properties of A hypernuclei. These are nuclei £Z with baryon number A in which a single A hyperon (baryon number = 1) is bound to an ordinary nucleus A Z consisting of A - 1 nucleons = Z protons + N neutrons. The A hyperon is neutral, has spin 1/2, strangeness S = - 1 , isospin 1 = 0 and a mass M A = 1116 MeV/c 2 . Although the A interacts with a nucleon, its interaction is only about half as strong as that between two nucleons, and thus very roughly VAN ~ 0-5 YNN- AS a result, the two-body AN system is unbound, and the lightest bound HN is the three-body hypertriton A H in which the A is bound to a deuteron with the A-d separation energy being only = 0.1 MeV corresponding to an exponential tail of radius « 15 fm! In strong interactions the strangeness S is of course conserved, and the A is distinct from the nucleons. In a HN strangeness changes only in the weak decays of the A which can decay either via "free" pionic decay A —> N + JC or via induced decay A + N —» N + N which is only possible in the presence of nucleons. Because of the small energy release the pionic decay is strongly suppressed in all but the lightest HN and the induced decay dominates. However, the weak decay lifetime = 10" lo s is in fact close to the lifetime of a free A. Since this is much longer than the strong interaction time = 10"22s we can ignore the weak interactions when considering the binding of HN, just as for ordinary nuclei. In our work we consider the A separation energies B A defined by -B A = £ E - A - 1 E , where ^E is the total energy of the HN and A - 1 E is the ground-state energy of the "core" nucleus. For orientation consider a medium to heavy HN: A > 20. The A-nuclear interactions generate a A-nucleus potential which roughly follows the density distribution p(r) of the core nucleus, with an approximately constant value DA in the interior. This well depth D A is identified with the A binding in nuclear matter. Then for the ground SA states for which the A is in an s state: B A = D A - T A where the A kinetic energy T A ~ A" 2/3 since the radius of the A-nucleus potential is approximately that of the core nucleus (RA ~ A 1 ' 3 ). Figure 1 shows the experimental BA VS. A"2/3, in particular the s.p. energies obtained from the Jt+ + A Z -» ^Z + K + reaction [1]. Extrapolation to A -» °°, i.e. A"2/3 -> 0, in particular also for the SA states gives D A = 30 ± 3 MeV, a value which has been known for a long time. Our aim in this and previous work [2,3] is to learn about the A-nuclear interactions as well as the structure of HN from the B A data, making appropriate and adequate few- and many-body calculations. Here the emphasis is on the s.p. B A data. ANL-P-2f,1« A 89 | 89y 28 51 51 AT 40 1 i 4 AV °Ca A 20 CD cc LU 1 A 1 1 A A _ -W — \ 401. SEPARA1 1 A Si 12 \ 16 < 16 28 A 24 i 28 _ 12 - - \) 8L 4 — 0 - \ > i 0.04 l — , x 1 1 0.06 0.12 0.16 »-2/3 020 Figure 1 The experimental B A are showr with errors. The curves depict the calculated BA. The solid curve is for the first interaction of Table 2; the second interaction of Table 2 gives a very similar fit. The dashed curve is for a purely dispersive ANN potential: V o = 6.2 MeV, W = 0.016 MeV, Cp = 0. 2. A-NUCLEAR INTERACTIONS Our interactions are in large part phenomenological but are generally consistent with and suggested by meson-exchange models, and are such that they can be used in fewand many-body calculations. 2.1 AN Potential A basic difference between the AN and NN potentials is that one-pion exchange (OPE) is not allowed between a A and a nucleon since the A has isospin 1 = 0 and hence there is no (strong) A Are vertex. However, the £ hyperon also has S = -1 but has 1 = 1 and there is thus a AXrc vertex. Since the E is only about 80 MeV heavier than the A, the twopion-exchange (TPE) potential is a dominant part of the AN potential being dominated by the strong tensor OPE component acting twice. There will also be K,K* exchange potentials which will in particular contribute to the space-exchange and the AN tensor potential. The latter is of quite short range because there is no long range OPE and also quite weak because the K and K* tensor contributions are of opposite sign [4]. Also there will be short-range contributions from co, quark-gluon exchange, etc. which we represent with a short-range Saxon-Wood repulsive potential which - somewhat arbitrarily - we take to be the same as for the MN potential [5]. We then use an Urbana-type central potential [5] with space exchange and a TPE attractive tail which is consistent with Ap scattering. For the present work we need only the spin-average potential which is , Vx=-eV(r)(l-Px). (1) P x is the AN space exchange operator, V x is the space-exchange potential with e determining its strength relative to the direct potential which is = W 0 /[l+exp{(r-R)/a}]-V 2jt , V2jc = V0T2(r), (2) where Wo = 2137 MeV, R = 0.5 fm, a = 0.2 fm and r is in fm. The strength of the spinaverage AN potential consistent with Ap scattering is Vo = 6.15 ± 0.05 MeV. (In terms of the singlet and triplet strengths Vo = (Vs + 3Vt/4). T^(r) is the one-pion exchange tensor potential shape modified with a cut off: T ^ r ) = (l + 3 / x + 3 / x2) (e-x / x) (l - e~<*2 f (3) with x = 0.7r and c = 2.0 fnr 2 . The space-exchange parameter, e = 0.1-0.38, is quite poorly determined from the Ap forward-backward asymmetry. For our fits to the s.p. data we take e to be a free parameter. This determines the odd-state potential, in particular the p-state potential to be Vp=(l-2e)V(r). (4) In Table 1 we show some results for the ground state BA calculated with our AN potential. Five-body variational Monte Carlo (VMC) calculations [6] were made for ^He in which a A is bound to an a particle [2]; for ABe for which a 2 a + A model was used implemented by appropriate VMC calculations [3], and for the well depth DA for which the Fermi hypernetted chain (FHNC) method was used [2,7]. The space-exchange contribution for the s-shell HN (A < 5) and for ^Be was recently obtained with the VMC method [8]. For A He for all our interactions (including ANN potentials, see below) we obtain E x = 0.5 MeV repulsion for 8 = 0.3. For A Be more limited calculations give E x = 1.3 MeV. Table 1. Experimental and calculated BA (in MeV) of selected hypernuclei. The errors in the calculated BA are due to the uncertainties in the strength VQ of VAN- HN Exp.BA Calculated Calculated B A for e=0 BAfore=0.3 A He 3.12 ±0.02 6.1 ± 1 5.6+1 A Be 6.71 ±0.04 = 12 ±2 = 10.7 ± 2 74 + 4 60 + 4 DA 30±3 It is clear that with only a AN potential fitted to Ap scattering, and even with rather large space exchange, the HN for A > 5 are strongly overbound relative to the experimental values. This is an ancient result which has been sharpened over time. Furthermore, we shall show that the s.p. data will not permit a fit with only a AN potential even if the requirement that this fit the scattering data is relaxed. These results imply that many-body effects are very large. 2.2 ANN Potential - Many-Body Effects Many-body effects can arise for a central VAN through changes in the AN correlation function gAN due to the presence of other nucleons. Related, are modifications (suppression) by other nucleons of an effective interaction due to e.g. a tensor force which must act at least twice. Such tensor-force suppression of the NN force is a very important contributor to nuclear saturation. (A AN tensor force is suppressed much less because of its short range and weakness). For the TPE AN potential V2n a closely related suppression effect arises from the modifications of the propagation of the intermediate 2 or N by other nucleons (Fig. 2). Such effects have been calculated in a coupled-channel reaction-matrix approach and can give a large repulsive contribution because of the large couplings together with the small S-A mass difference [9]. We represent such suppression effects by a phenomenological (repulsive) "dispersive" ANN potential of the form (5) where TJA are the A-nucleon separations. A strength W = 0.02 MeV gives a repulsive contribution which is roughly consistent with the suppression obtained in coupled-channel reaction matrix calculations. The other type of three-body ANN force (Fig. 2) arises from TPE, appropriate to a p-wave pion interaction of the A with two nucleons (1 and 2), and has the form [10] (6) where {A,B} = AB+BA, Y(x) = exp(-x)(l-exp(-cr2))/x and T(x) is given by Eq. (3). Sy is the tensor operator for particles i,j and dj and Tj, are the spin and isospin Pauli operators for particle i. Theoretical estimates give C p = 1-2 MeV [10]. ANL-P-21,143 A N, A M2 7t Figure 2 Diagrams for dispersive and TPE ANN potentials. Thus, finally our A-nuclear interactions are i=l ^ fcj-iV ^ J (7) ^i^J' where the strengths of VyysjN are chosen as described below. 3. FITS TO THE s-SHELL HN, ESPECIALLY A He 3.1 Thes-ShellHN Previously we have made VMC calculations of the s-shell HN: A H , A H and A He (both J = 0 and 1), and A He. We used a Mafliet-Tjon central NN potential [11] which fits both the energies and rms radii of the core nuclei: 2 H, 3 H and 4 He. The calculations included AN, NN and ANN correlations: fyusj, fNN, fANN= f ANN^ANN • ^ n e m o s t P e r t i n e n t result for our present work is that V^5,N alone cannot provide the repulsion needed for A He to compensate the overbinding obtained with only a AN potential, and that consequently a strongly repulsive ANN dispersive potential V j ^ is required. This is because the ANN correlations fA^,N reduce die contribution of V ^ N from an appreciably repulsive one to one which is only slightly repulsive or even attractive, whereas the effect of correlations on the repulsive contribution of V ^ j ^ is much less. 3.2 AN + ANN Potentials Fitted to A He For our calculations of the s.p. energies we consider three families of AN + ANN potentials of the type discussed above. These are - mostly - constrained as follows: 1. The experimental value of B A (^He)=3.1 MeV is reproduced for an exchange parameter e = 0.3. This value of e is consistent with the s.p. data and also the Ap scattering and gives an exchange contribution = 0.5 MeV for A He. 2. The s-wave Ap scattering is fitted (i.e. Vo = 6.15 ± 0.5 MeV). Our interactions are then: I. A AN potential VAN only, with Vo = 5.98 MeV. This gives the experimental B A ( A He) for e = 0, but gives too little scattering. II. AN plus ANN dispersive potentials: V AN + V ? N N . The strength Vo of VAN covers the values allowed by AN scattering (6.1, 6.15, 6.2 MeV) and the strength W of V^ N N is adjusted accordingly to give the experimental B A ( A He), with more repulsion needed for more attractive VAN (W = 0.007,0.011,0.016 MeV). in. AN plus dispersive plus TPE ANN forces: V ^ + V ^ , N + V ^ , N with fixed V ^ N (C p = 2 MeV). Again Vo (6.1, 6.16, 6.2 MeV) covers the values allowed by scattering and W is adjusted accordingly (W = 0.006,0.01,0.013 MeV). 4. CALCULATIONS OF THE S.p. ENERGIES The s.p. energies B A are obtained from a Schrbdinger equation with a A-nucleus potential UA and an effective mass m A which are obtained in the local density approximation, using the FHNC method [2,7]. This is used to calculate the A binding D(p,kA) for nuclear matter of density p and for a A momentum of kA- Thus for A —> » : (4,(A)|H(A)|*p(A)) where H(A), »F(A) are the Hamiltonian and wave function of the HN and H ^ " 1 ^ , »P(A-1) those of the core nucleus. The (variational) FHNC wave functions are -e"°rAF*p(A-l) —t *trT>- wifh A-l A-l F - T T f Av,fr-» \T1 f . . J ? . , ' w i u i JT — r-. r - l (Q\ I I 1 A M i i jA 11 1 1 A N N ! '•IA ' iA» ii f \"/ and A=l (10) <I>(A-l) i s the uncorrelated Fermi gas wave function for nuclear matter of density p. The factor F includes both AN and ANN correlations. Details of the correlation factors fAN» fNN and fANN as well as of the calculational method are given in Ref. [2]. The effective mass m A (p) is obtained from a quadratic fit in ICA to D(p,kA)-D(p JCA = 0). We also allow for a "fringing field" (FF) due to the finite range of the AN and ANN potentials, but do not discuss the details here, in particular since our procedure is approximate and subject to uncertainties; also the effects of a FF are relatively small and vanish for A -»<*>. Without a FF one has U A (r) = D(p(A-l)(r),k A =0), (11) where the densities p( A-1 ) of the core nuclei are obtained from electron-scattering data. Similarly, the effective mass as a function of r is given by m A (r) = m A (pA-l(r)). (12) Finally, B A is obtained as the lowest eigenvalue of the appropriate radial Schrodinger equation for an orbital angular momentum £A, 5. THE A BINDING AND EFFECTIVE MASS IN NUCLEAR MATTER We summarize the expressions for D(p) and m A (p) obtained with the FHNC method. We define (always for a given density p of nuclear matter) where D^N and D A N N are the AN and ANN contributions respectively. Furthermore DAN=DAN + D where the direct contribution is S -top. (16) gAN is the AN correlation function, f2 is the AN correlation factor, and HAN is the AN reduced mass. Typically to(p=0) = 400 MeV fm3; and to(p) decreases somewhat with p. The exchange contribution is proportional to ek^pFj where ICF is the Fermi momentum and Fi is a form factor. Thus ^ | J b o P 5/3 F l , (17) where ^ J 2 3 (18) The effective mass m A is given by (19) Fl(p). F2(p) are form factors which represent finite range effects of VAN (relative to kp 1 ); Fi, F2 = 1 for p = 0, or equivalently for a zero-range VAN- Finite range effects are much more important for %, i.e. m A , than for D ^ . Note that exchange contributes both through D ^ and therefore through the A-nucleus potential U\ as well as through m A . For e > 0, D ^ 4 is repulsive (odd state potential less attractive than even state) and m A < m A thus giving a larger kinetic energy relative to that for mA and therefore also an effective repulsion. The ANN contribution to the A binding is (20) where FANN(P) is a form factor such that FANN = 1 for p = 0 and equivalently for a zerorange ANN potential. FANN depends on various correlation functions [2] and varies by roughly a factor of two or less, depending on the specific interaction, over the density range considered (p < 0.23 fnr 3 ). The dominant p dependence therefore comes from the p 2 factor. We define D o s D ( £ = 0) = D f + D A N N , (21) which is the sum of the direct AN and of the ANN contributions, i.e. the A binding without the exchange contribution. The total A binding at p is then (22) 6. FITS TO THE S.p. ENERGIES We attempt fits to the s.p. B A with our three families of interactions I-III. The exchange parameter e is the only parameter varied for a given interaction. The well depth is given by D A = D(p 0 ) where p 0 = 0.165 fnr 3 is the density of normal nuclear matter. We recall that DA = B A ( A = <») for all £\. Before we discuss details of our fits we emphasize the general requirements on D(p) and m A needed for a fit to the s.p. data. These requirements have been pointed out by Millener et al. Ref. [12], and will also be demonstrated in the following discussion. Thus, for a satisfactory fit to the SA (^A = 0) data, D(p) must have the following "saturation" properties: DA = D(p 0 ) = 30 MeV in order to allow a satisfactory fit for large A. On the other hand for a fit for small A, D{p)/p must be larger for small p < p 0 , implying a maximum in D(p) with pmax not very different from p 0 . Further, to give the separation between the B A for different iA requires quite generally that m A = 0.7 mA which in turn in our approach requires e = 0.3-0.35. Fits with interaction I: AN potential gives too little scattering. VAN only- This fits BA(AHC) for e = 0 but 1. With only a direct AN potential (e = 0 and thus m A = m A ): D A N = 0 ^ = 370 p MeV, with the nonlinear dependence on p only a few % for p 5 0.23 fnr 3 as a consequence of the slight dependence of gAN and f2 (Eq. (16)) on p. Thus D A = D^ N (p 0 ) = 60 MeV. All the s.p. states and in particular the SA states are then much too strongly bound, even for quite low A. If V o were adjusted (without any justification) to give D A = 30 MeV so as to fit the B A for the heaviest HN, then conversely the B A for even medium heavy HN would be much too small (and the AN scattering would be very much too small). Thus a direct AN potential cannot fit the s.p. energies. 2. With AN space exchange: D ^ = D£ N + D£ N . To obtain DAN(PO) = 30 MeV requires a large and repulsive exchange contribution D ^ ~ -30 MeV which is obtained for e « 0.88. This implies a correspondingly small value of m A / m A = 0.48 at p 0 . The A binding D(p), shown in Fig. 3, then has a maximum = 35 MeV at p m a x = 0.215 fnr 3 . The results for the SA states are then reasonable for large A as expected, but the large A kinetic energy (small m A ) gives too small B A for smaller A and also much too small BA for the £A > 0 states. In fact, no even tolerably adequate fit to the s.p. data can be obtained with a AN potential with space exchange. If purely phenomenologically, we use m A = m A together with D ^ for an appropriately chosen e, i.e. we relinquish the common origin of D£ N and m A in exchange forces, then a quite adequate fit to the SA states can be obtained for 8 = 0.98. However for £A > 0 although the fits are now much better, the calculated B A are somewhat too large (too small TA) and the fit is of only moderate quality. There is of course no justification for taking m A = m A , especially since the effects of exchange on both D^ N and m A depend on quite basic many-body features and are already fully manifest in HF. Thus a central AN potential with and without exchange is ruled out by the s.p. data. Fits with interactions II: AN plus dispersive ANN forces. We recall that for a given the strength W of V } ^ is chosen to fit B A ( A H C ) for e = 0.3, and that for more attractive VAN the value of W is larger since more repulsion is then required. We find that no adequate fit to the s.p. energies can be obtained for these interactions, the "fit" being worst for the smallest V o = 6.1 MeV consistent with scattering, corresponding to the least VAN repulsive V^ N N . This inability to fit the s.p. energies is directly related to an insufficiently repulsive contribution D A N N (p 0 ) from V ^ N . Thus for V o = 6.2 MeV, which gives the most attractive VAN consistent with scattering and hence to the most repulsive VANN> we obtain DANN(PO) = -26 MeV. This together with the direct contribution D£ N « 74 MeV gives D 0 (p 0 ) = 48 MeV. To obtain D(p 0 ) = 30 MeV, needed to fit B A for large A, then requires a rather large exchange contribution D AN = -18 MeV which in turn requires e = 0.44. This implies a rather small m A (p 0 ) = 0.66 m A which although it allows an adequate fit to the SA states gives a mediocre fit for £A >0 as depicted in Fig. 1. The situation is much worse for smaller V o (< 6.2 MeV) for which V^ NN is correspondingly less repulsive. If we ignore the repulsive exchange contribution for A He (i.e. if e = 0) then a more repulsive V^ is needed to fit B A ( A Hej. Now V o = 6.2 MeV gives a V° N N (W = 0.02 MeV) just sufficiently repulsive for a satisfactory fit to the s.p. data. However, for smaller Vo a satisfactory fit is still not possible. The repulsive exchange contribution = 0.5 MeV in A He is thus necessary if purely dispersive ANN forces are to be completely ruled out. Our conclusions for a dispersive ANN force seem quite firm since the appropriate correlations have a relatively small effect in reducing the repulsive contribution of V^fN in A He. Table 2. The A binding D(p 0 ) and its components (in MeV), and m A = m^, all at p 0 = 0.165 fnr 3 for two interactions HI which fit the s.p. data for the stated values of e. Interaction DAN Vo W 6.16 0.01 0.34+015 70.5 30.5 6.20 0.013 0.31+015 71.9 34.5 6 ~ D ANN D mA/mA 12.3+.6 27.7+.6 0.71+.015 11.3+.6 26.1±.6 0.73+.015 -DAN Fits with interactions III: AN plus dispersive plus TPE ANN forces, with ^ , fixed. As for n, Vj^j N is again adjusted for any given VAN t o g i v e B A ^ H e ) . For these interactions we obtain excellent fits to all the s.p. data for Vo> 6.15 MeV (W 2 0.1 MeV) as shown in Fig. 1 for the 1st interaction in Table 2. This shows results for our best fits for two such interactions. All the A binding energies are for p 0 . Figure 3 shows D(p) and its components vs. p for the first interaction. D(p) has the characteristic saturation features needed for a fit to the s.p. data: DA = D(p o ) = 27 MeV required to fit B A for large A, and a maximum = 28 MeV at pmax = 0.14 fnr^. Since p m a x is quite close to p 0 the uncertainties in the predicted DA due to uncertainties in p 0 are quite small (= 0.5 MeV) for a given interaction. A combination V^ N N + VA^,N permits a fit to the s.p. data because the ANN correlations f Aj5IN in A He strongly reduce the repulsion due to V^J™ and can even give attraction, whereas this is not so for nuclear matter, i.e. for D. Thus a sizable V A J fN which gives a small repulsive or even an attractive contribution in A He can give a large repulsive contribution in nuclear matter. This together with the repulsion from V^ N N (which is required for A He and for which there is no such dramatic change between A = 5 to A = °°) provides sufficient overall repulsion DANN K -30 MeV needed for the s.p. data. More generally it seems clear that what is required for our family of interactions is that the effect of correlations for V^ NN does not change too much with A, whereas for V ^ j ^ the effect of f ^Jj N should depend quite strongly on A in such a way as to give relatively much more attraction for small A. ANL-P-21,145 1 35 i A BINDING ENE 1 1 ' / /° 30 - I 1 - 25 20 15 10 / / / 5 0 C) - Q I 0.04 0.08 0.12 0.1S 0.20 Figure 3 The A binding D(p) and its components vs. p for the first interaction of Table 2. Also shown is D(p) for interaction I (only VAN) with e = 0.88. 7. CONCLUSIONS-WHAT HAVE WE LEARNED? The s.p. data require the A binding in nuclear matter D(p) to have the "saturation" features shown in Fig. 3 and which have been discussed by Millener et al. [12] and also obtained by us with our microscopic FHNC approach. With our interactions we obtain D A = 27 ± 1 MeV, consistent with earlier results, and required to fit B A for large A. The "saturation" of D(p) requires a large repulsive many-body contribution, depending nonlinearly on p, which is identified with ANN forces in our approach. To account for the s.p. data for all SLK further requires m^ / m A = 0.7 (at p 0 ) which implies an exchange parameter e = 0.32 ± .02. This value is expected to depend only slightly (through the form factor F2) on the details of VAN if this is of reasonable range and shape. The value of e is consistent with that obtained from scattering but is much more precisely determined by the s.p. data, and implies a p-state potential (Eq. (4)): V p = (0.35 + .05)Vs. These results depend only on the s.p. data and not on the s-shell B A and the Ap scattering. 2. Ap scattering fixes the spin-average VAN which then determines the direct contribution to D A to be D^ N (p 0 ) ~ 70 MeV. Many-body effects (nonlinear with p) are small. Even with a AN tensor potential such effects are quite small because of the short range and weakness of V^ N [4], and probably reduce D AN by not more than a few MeV [13]. The large value of D A N (p 0 ) and the approximate proportionality of D AN with p imply that a direct AN potential cannot fit the B \ data - even when a AN tensor force is included. With a space-exchange component, a very large e = 1 is required to fit the SA states for large A, and no reasonable fit to all the s.p. data is possible. Thus a AN potential with or without exchange seems excluded. Combining the values at p 0 of D^ N ~ 70 MeV, obtained by making use of scattering, with D = 27 MeV, D£ N = -12 MeV obtained from the s.p. data gives D A N N = D - D^ - D^ N = -32 ± 2 MeV. This large value is consistent with that required by just the s.p. data only and seems fairly well determined. Since many-body effects associated with VAN. in particular due to a tensor force, are expected to be small, D A N N must be identified almost entirely with other components of the interaction such as TPE ANN forces and suppression of the AN - LN coupling. It is a challenge to many-body theory to account for D*NN. 3. The results just discussed do not depend on the s-shell HN, in particular not on B A ( A Hej. If we use this to fix the strength of VANN then, with an exchange contribution = 0.5 MeV for A He, the s.p. data excludes a purely dispersive ANN potential since this is insufficiently repulsive to give D^NN _ .30 MeV. However, a combination of dispersive and TPE forces can give an excellent fit, since the effects of correlations makes the contribution of V j ^ , much less repulsive for A He than for nuclear matter. There are of course many outstanding questions although we believe that the results discussed above in 1 and 2 as well as the elimination of purely dispersive ANN forces discussed in 3 are reasonably secure. However, the detailed nature of the ANN forces, in particular the relative strengths of V^ N N and V j ^ is largely uncertain. Further elucidation requires, in particular, complete microscopic calculations of the s-shell and heavier HN which include 3-body tensor correlations for V A ^ N . Such calculations are in progress in particular for A He and J^O [Ref. 14] and will greatly help to clarify the character of the ANN forces. For heavier hypernuclei completely microscopic calculations are not yet feasible. However there are possible improvements in the local density approximation as used by us. Amongst these is the question of core distortion by the A. This is addressed by Professor Usmani in his talk where the will show that - for spherical nuclei - this distortion is very small even for quite light HN because the "forcing' action of the A on the core is much reduced because of the saturation features of D(p). Other questions relate to an adequate inclusion of the (nuclear matter) rearrangement energy and to the adequacy of the local density approximation, in particular to a better treatment of the fringing field. This work is supported in part by the U. S. Department of Energy, Nuclear Physics Division, under contract W-31-109-ENG-38. A. R. 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