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Folded-diagram effective interactions for hypernuclei
NUCLEAR PHYSICS A Nuclear Physics A639 (1998) 165c-168~
ELSEVIER
Folded-diagram
effective interactions
for hypernuclei*
Yiharn Tzenga, S.Y. Tsay Tzengb, T.T.S. KuoC, T.-S.H. Leed aInstitute of Physics, Academia Sinica, Nankang, Taiwan bDepartment of Physics, Taipei University of Technology, Taipei, Taiwan CDepartment of Physics, State University of New York at Stony Brook, Stony Brook, New York 11794, USA dPhysics Div‘ISio n, Argonne National Laboratory, Argonne, IL 60439, USA We have performed folded-diagram G-matrix calculations of the particle-hole interaction for the hypernucleus i60, using both the Nijmegen soft-core and the hyperon-nucleon potentials. Significant differences between the energy spectra these two potentials are noticed. The effects of the folded diagrams and the three-body force induced by A-C coupling are found to be important.
effective Jiilich-B given by effective
The nuclear shell model is still by far the most promising model for describing the structure of nuclei, including hypernuclei. In this approach, we deal with the shell-model secular equation of the form PH,*PP,
= P(H,, + Veff)P9,
= E,PQ,.
(1)
Here, the effective Hamiltonian is denoted as He@, Ho is the unperturbed shell-model Hamiltonian, and P is the model-space projection operator, defined in terms of the eigenfunctions of Ho. Note that the above equation is a P-space equation, and usually P is chosen to be a small shell-model space. Clearly in this approach, our calculation depends a lot on the effective interaction V,B and its determination is of much importance. There are basically two approaches for the determination of V,E. One is to determine it empirically, namely we allow it to have certain adjustable parameters and then determine them by requiring an optimum fit to certain selected experimental data. Another approach is to derive V,. starting from the free nucleon-nucleon (NN) and hyperon-nucleon (YN) interactions, using many-body methods. This latter approach has been widely applied to ordinary nuclei (zero strangeness), and has been rather sucessful [1,2]. It should be of interest to extend this approach to hypernuclei [3-51. Let us consider fi”O as an example. We start from the full-space many-body problem H9 = E@ with H = T + VNN + VYN. We have taken VAN as either the Nijmegen soft-core potential [6] or the Jiilich-B potential [7]. They will be referred to as Nij and JB, respectively. For the NN potential, we have used in the present work the Bonn-A ‘Work supported in part by the US NSF Grant INT9601361, the US DOE Grant DE-FG02-88ER0388, the US DOE Contract W-31-109.ENG-38, and the NSC(Taiwan) Grant NSC86-2112-M-001.006Y. 0375-9474/981$19.00 0 1998 Elsewer Science B.V. All PII SO375-9474(98)00266-8
rights reserved
166~
I! Tzeng et al. /Nuclear Physics A439 (1998) 16.5-168~
potential [8]. We first choose a model space P consisted of nucleons confined within the s, p, and (sd) oscillator orbits and hyperons (As) within the s and p oscillator orbits. Then, by way of a folded-diagram reduction method [9,10], the full-space many-body problem can be formally reduced to a P-space one, which is precisely of the form of Eq. (1). In this way, the model-space effective interaction V,s has folded diagrams, and it is given as the following Q-box folded-diagram series:
v,,=s-~Js~olJ~Js-oJsS~J&+...
(2)
In the above, the symbol 0, often referred to as the &box, denotes the irreducible vertex function, which is to be calculated from free YN and NN interactions Vyn and VNn. Since both have strong repulsive cores, we need first convert them to the respective G-matrices. Then we can calculate the Q-box from the G-matrix interaction. Methods for the NN G-matrix have been relatively well developed [ll]. For the YN G-matrix, we have solved the AN-EN coupled equation [4,5] G(w) =V+VQ
1
w-Q(miv+t~+my+t~)Q
QG(w) >
(3)
where V denotes the YN interaction, while m and t denote the baryon mass and kinetic energies. There have been two main difficulties in calculating the above G-matrix. First the G-matrix depends on the energy variable w, and there is uncertainty as to its choice when we perform a structure calculation of, for example, A’rO.This difficulty is overcome when one includes the folded diagrams, because the effective interaction V,s then becomes energy (w) independent [9,10]. The Pauli exclusion operator Q above imposes a restriction on the intermediate YN states for the G-matrix, to prevent double counting. The presence of Q in the G-matrix has also been a major computational difficulty. We have overcome this difficulty by employing a matrix inversion method [11,4,5], which provides an essentially exact treatment for the Pauli operator. In the past, one often used [3] a local-density nuclear matter Pauli exclusion operator Q(ICF) where k~ denotes the local Fermi momentum. This is an approximation, and how to determine the appropriate local density is rather uncertain. In the present work we use a Q defined in terms of shell-model orbits. In short, we are now able to calculate the YN G-matrix for finite nuclei with high precision. Let us now briefly summarize our folded-diagram calculation for i60: (i) We choose a model space P as mentioned earlier. (ii) We calculate the NN and YN G-matrix in strict accordance with this P space. (iii) For the $-box we include diagrams through 2nd order in G, including the core-polarization diagram and the A-C three-body force diagram. (iv) Finally, the particle-hole (A-particle nucleon-hole) V es is obtained by summing up the entire folded-diagram series using the Lee-Suzuki iteration method [12,13]. Before presenting our results, we note that natural-parity states (O+,l-,2+) from coupling SA or pi to the Op hole states are preferentially populated in (K-, C) and (r+,K+) reactions, giving rise to four st.rong peaks in the experimental spectra. The in-flight (K-,7r-) reaction near 0” populates l- and O+ states [14], the (n+,K+) reaction l- and 2+ states [15-171, and the (K-,7r-) reaction at rest O+, l-, and 2+ states (with a preference for 2+ over O+) [18,19]. The experimental binding energies (E = -Bh) in Figs. 1
K Tzeng et al. /Nuclear
1
-0 t
-14
JB
Nij
2
4
8
8
I-=
I
Expt
t 0
167c
Physics A639 (1998) 165c-168~
10
Figure 1. J- states of i60. The experimental levels [18] (see also [14,16,17,19]) should be compared to the l- predictions.
-411 0
1+
JB
Nij I 2
1
1+
I
I 4
’
Expt
I 6
’
I 8
I
’ 10
Figure 2. J+ states of i60. The experimental level [18] (see also [14,16,17,19]) should be compared to the 2+ prediction.
and 2 are from Ref. [18], but it should be noted that the data from different reactions are not completely consistent. In Fig. 1, we compare our calculated negative-parity states for i60, obtained with the Nij soft-core and JB potentials, with an experimental spectrum [18]. As shown, there is a good general agreement between the positions of the calculated and experimental states. We note, however, the relative orderings of the “spin-doublet” states (O-, l-) and (l-, 2-) given by Nij and JB are different (cf. Ref. [20]). Millener et al. [21] have pointed out that the splittings of these doublet states are of essential importance for the spin dependence of the AN effective interaction. Our results thus indicate that Nij and JB would lead to rather different spin dependence for the AN effective interaction. Turning to Fig. 2, we see that the low-lying positive-parity states given by the two potentials are significantly different, particularly for the O+ state. This state has essentially configuration. In fact for this state, the effects of the folded diagrams a pure API~ZNP$ and the A-C three-body force are both significant. We shall report our results with more details in a forthcoming paper. The actual separation of the O+ and 2+ states is subject to considerable uncertainty; a separation of 1.56 f 0.12 MeV in Ref. [19] is to be contrasted with 0.0 f 0.2 MeV in Ref. [17]. In summary, we can now rather accurately calculate the YN effective interactions starting from realistic YN potentials, using a folded-diagram G-matrix framework. There are significant differences between the predictions from the Nijmegen and Jiilich-B potentials. To further test these potentials, it would be very useful to have new hypernuclear experimental data. In particular, the 0 +, 2+ separation is very sensitive to the nature of the YN interaction, especially the spin-orbit components.
168c
Y Tzeng et al. /Nuclear Physics A639 (1998) 16Sc-168~
REFERENCES 1. A. Covello, F. Andreozzi, L. Coraggio, A. Gargano, T.T.S. Kuo and A. Porrino, Prog. Part. Nucl. Phys. 38 (1997) 165. 2. F. Andreozzi, L. Coraggio, A. Covello, A. Gargano, T.T.S. Kuo and A. Porrino, Phys. Rev. C 56 (1997) R16. 3. Y.Yamamoto and H. Bando, Prog. Theor. Phys. 73 (1985) 905; T. Yamada, T. Motoba, K. Ikeda and H. Bando, Prog. Theor. Phys. Suppl. 81 (1985) 104; H. Bando, T. Motoba and J. Zofka, Int. J. Mod. Phys. A 5 (1990) 4021. 4. J. Hao, T.T.S. Kuo, A Reuber, K. Holinde, J. Speth and D.J. Millener, Phys. Rev. Lett. 71 (1993) 1498. 5. T.T.S. Kuo and J. Hao, Prog. Theor. Phys. Suppl. 117 (1994) 351. 6. M.M. Nagel, T.A. Rijken and J.J. de Swart, Phys. Rev. D 15 (1977) 2547; D 20 (1979) 1633; P.M.M. Maessen, T.A. Rijken and J.J. de Swart, Phys. Rev. C 40 (1989) 2226. 7. B. Holzenkamp, K. Holinde and J. Speth, Nucl. Phys. A 500 (1989) 485; A. Reuber, K. Holinde and J. Speth, Czech. J. Phys. 42 (1992) 1115; Nucl. Phys. A 570 (1994) 543. 8. R. Machleidt, Adv. Nucl. Phys. 19 (1989) 189. 9. T.T.S. Kuo and E. Osnes, Springer Lecture Notes in Physics 364 (1990) 1. 10. M. Hjorth-Jensen, T.T.S. Kuo and E. Osnes, Phys. Rep. 261 (1995) 126. 11. E.M. Krenciglowa, C.L. Kung, T.T.S. Kuo and E. Osnes, Ann. Phys. (N.Y.) 101 (1976) 154. 12. S.Y. Lee and K. Suzuki, Phys. Lett. B 91 (1980) 173. 13. K. Suzuki and S.Y. Lee, Prog. Theor. Phys. 64 (1980) 2091. 14. W. Briickner et al., Phys. Lett. B 79 (1978) 157. 15. R.E. Chrien, Nucl. Phys. A 478 (1988) 705~. 16. P.H. Pile et al., Phys. Rev. Lett. 66 (1991) 2585. 17. 0. Hashimoto, in these proceedings. 18. H. Tamura, R.S. Hayano, H.Outa and T. Yamazaki, Prog. Theor. Phys. Suppl. 117 (1994) 1. 19. R.H. Dalitz, D.H. Davis, T. Motoba and D.N. Tovee, Nucl. Phys. A 625 (1997) 71. 20. Y. Yamamoto, A. Reuber, H. Himeno, S. Nagata and T. Motoba, Czech. J. Phys. 42 (1992) 1249. 21. D.J. Millener, A. Gal, C.B. Dover and R.H. Dalitz, Phys. Rev. C 31 (1985) 499.
ELSEVIER
Folded-diagram
effective interactions
for hypernuclei*
Yiharn Tzenga, S.Y. Tsay Tzengb, T.T.S. KuoC, T.-S.H. Leed aInstitute of Physics, Academia Sinica, Nankang, Taiwan bDepartment of Physics, Taipei University of Technology, Taipei, Taiwan CDepartment of Physics, State University of New York at Stony Brook, Stony Brook, New York 11794, USA dPhysics Div‘ISio n, Argonne National Laboratory, Argonne, IL 60439, USA We have performed folded-diagram G-matrix calculations of the particle-hole interaction for the hypernucleus i60, using both the Nijmegen soft-core and the hyperon-nucleon potentials. Significant differences between the energy spectra these two potentials are noticed. The effects of the folded diagrams and the three-body force induced by A-C coupling are found to be important.
effective Jiilich-B given by effective
The nuclear shell model is still by far the most promising model for describing the structure of nuclei, including hypernuclei. In this approach, we deal with the shell-model secular equation of the form PH,*PP,
= P(H,, + Veff)P9,
= E,PQ,.
(1)
Here, the effective Hamiltonian is denoted as He@, Ho is the unperturbed shell-model Hamiltonian, and P is the model-space projection operator, defined in terms of the eigenfunctions of Ho. Note that the above equation is a P-space equation, and usually P is chosen to be a small shell-model space. Clearly in this approach, our calculation depends a lot on the effective interaction V,B and its determination is of much importance. There are basically two approaches for the determination of V,E. One is to determine it empirically, namely we allow it to have certain adjustable parameters and then determine them by requiring an optimum fit to certain selected experimental data. Another approach is to derive V,. starting from the free nucleon-nucleon (NN) and hyperon-nucleon (YN) interactions, using many-body methods. This latter approach has been widely applied to ordinary nuclei (zero strangeness), and has been rather sucessful [1,2]. It should be of interest to extend this approach to hypernuclei [3-51. Let us consider fi”O as an example. We start from the full-space many-body problem H9 = E@ with H = T + VNN + VYN. We have taken VAN as either the Nijmegen soft-core potential [6] or the Jiilich-B potential [7]. They will be referred to as Nij and JB, respectively. For the NN potential, we have used in the present work the Bonn-A ‘Work supported in part by the US NSF Grant INT9601361, the US DOE Grant DE-FG02-88ER0388, the US DOE Contract W-31-109.ENG-38, and the NSC(Taiwan) Grant NSC86-2112-M-001.006Y. 0375-9474/981$19.00 0 1998 Elsewer Science B.V. All PII SO375-9474(98)00266-8
rights reserved
166~
I! Tzeng et al. /Nuclear Physics A439 (1998) 16.5-168~
potential [8]. We first choose a model space P consisted of nucleons confined within the s, p, and (sd) oscillator orbits and hyperons (As) within the s and p oscillator orbits. Then, by way of a folded-diagram reduction method [9,10], the full-space many-body problem can be formally reduced to a P-space one, which is precisely of the form of Eq. (1). In this way, the model-space effective interaction V,s has folded diagrams, and it is given as the following Q-box folded-diagram series:
v,,=s-~Js~olJ~Js-oJsS~J&+...
(2)
In the above, the symbol 0, often referred to as the &box, denotes the irreducible vertex function, which is to be calculated from free YN and NN interactions Vyn and VNn. Since both have strong repulsive cores, we need first convert them to the respective G-matrices. Then we can calculate the Q-box from the G-matrix interaction. Methods for the NN G-matrix have been relatively well developed [ll]. For the YN G-matrix, we have solved the AN-EN coupled equation [4,5] G(w) =V+VQ
1
w-Q(miv+t~+my+t~)Q
QG(w) >
(3)
where V denotes the YN interaction, while m and t denote the baryon mass and kinetic energies. There have been two main difficulties in calculating the above G-matrix. First the G-matrix depends on the energy variable w, and there is uncertainty as to its choice when we perform a structure calculation of, for example, A’rO.This difficulty is overcome when one includes the folded diagrams, because the effective interaction V,s then becomes energy (w) independent [9,10]. The Pauli exclusion operator Q above imposes a restriction on the intermediate YN states for the G-matrix, to prevent double counting. The presence of Q in the G-matrix has also been a major computational difficulty. We have overcome this difficulty by employing a matrix inversion method [11,4,5], which provides an essentially exact treatment for the Pauli operator. In the past, one often used [3] a local-density nuclear matter Pauli exclusion operator Q(ICF) where k~ denotes the local Fermi momentum. This is an approximation, and how to determine the appropriate local density is rather uncertain. In the present work we use a Q defined in terms of shell-model orbits. In short, we are now able to calculate the YN G-matrix for finite nuclei with high precision. Let us now briefly summarize our folded-diagram calculation for i60: (i) We choose a model space P as mentioned earlier. (ii) We calculate the NN and YN G-matrix in strict accordance with this P space. (iii) For the $-box we include diagrams through 2nd order in G, including the core-polarization diagram and the A-C three-body force diagram. (iv) Finally, the particle-hole (A-particle nucleon-hole) V es is obtained by summing up the entire folded-diagram series using the Lee-Suzuki iteration method [12,13]. Before presenting our results, we note that natural-parity states (O+,l-,2+) from coupling SA or pi to the Op hole states are preferentially populated in (K-, C) and (r+,K+) reactions, giving rise to four st.rong peaks in the experimental spectra. The in-flight (K-,7r-) reaction near 0” populates l- and O+ states [14], the (n+,K+) reaction l- and 2+ states [15-171, and the (K-,7r-) reaction at rest O+, l-, and 2+ states (with a preference for 2+ over O+) [18,19]. The experimental binding energies (E = -Bh) in Figs. 1
K Tzeng et al. /Nuclear
1
-0 t
-14
JB
Nij
2
4
8
8
I-=
I
Expt
t 0
167c
Physics A639 (1998) 165c-168~
10
Figure 1. J- states of i60. The experimental levels [18] (see also [14,16,17,19]) should be compared to the l- predictions.
-411 0
1+
JB
Nij I 2
1
1+
I
I 4
’
Expt
I 6
’
I 8
I
’ 10
Figure 2. J+ states of i60. The experimental level [18] (see also [14,16,17,19]) should be compared to the 2+ prediction.
and 2 are from Ref. [18], but it should be noted that the data from different reactions are not completely consistent. In Fig. 1, we compare our calculated negative-parity states for i60, obtained with the Nij soft-core and JB potentials, with an experimental spectrum [18]. As shown, there is a good general agreement between the positions of the calculated and experimental states. We note, however, the relative orderings of the “spin-doublet” states (O-, l-) and (l-, 2-) given by Nij and JB are different (cf. Ref. [20]). Millener et al. [21] have pointed out that the splittings of these doublet states are of essential importance for the spin dependence of the AN effective interaction. Our results thus indicate that Nij and JB would lead to rather different spin dependence for the AN effective interaction. Turning to Fig. 2, we see that the low-lying positive-parity states given by the two potentials are significantly different, particularly for the O+ state. This state has essentially configuration. In fact for this state, the effects of the folded diagrams a pure API~ZNP$ and the A-C three-body force are both significant. We shall report our results with more details in a forthcoming paper. The actual separation of the O+ and 2+ states is subject to considerable uncertainty; a separation of 1.56 f 0.12 MeV in Ref. [19] is to be contrasted with 0.0 f 0.2 MeV in Ref. [17]. In summary, we can now rather accurately calculate the YN effective interactions starting from realistic YN potentials, using a folded-diagram G-matrix framework. There are significant differences between the predictions from the Nijmegen and Jiilich-B potentials. To further test these potentials, it would be very useful to have new hypernuclear experimental data. In particular, the 0 +, 2+ separation is very sensitive to the nature of the YN interaction, especially the spin-orbit components.
168c
Y Tzeng et al. /Nuclear Physics A639 (1998) 16Sc-168~
REFERENCES 1. A. Covello, F. Andreozzi, L. Coraggio, A. Gargano, T.T.S. Kuo and A. Porrino, Prog. Part. Nucl. Phys. 38 (1997) 165. 2. F. Andreozzi, L. Coraggio, A. Covello, A. Gargano, T.T.S. Kuo and A. Porrino, Phys. Rev. C 56 (1997) R16. 3. Y.Yamamoto and H. Bando, Prog. Theor. Phys. 73 (1985) 905; T. Yamada, T. Motoba, K. Ikeda and H. Bando, Prog. Theor. Phys. Suppl. 81 (1985) 104; H. Bando, T. Motoba and J. Zofka, Int. J. Mod. Phys. A 5 (1990) 4021. 4. J. Hao, T.T.S. Kuo, A Reuber, K. Holinde, J. Speth and D.J. Millener, Phys. Rev. Lett. 71 (1993) 1498. 5. T.T.S. Kuo and J. Hao, Prog. Theor. Phys. Suppl. 117 (1994) 351. 6. M.M. Nagel, T.A. Rijken and J.J. de Swart, Phys. Rev. D 15 (1977) 2547; D 20 (1979) 1633; P.M.M. Maessen, T.A. Rijken and J.J. de Swart, Phys. Rev. C 40 (1989) 2226. 7. B. Holzenkamp, K. Holinde and J. Speth, Nucl. Phys. A 500 (1989) 485; A. Reuber, K. Holinde and J. Speth, Czech. J. Phys. 42 (1992) 1115; Nucl. Phys. A 570 (1994) 543. 8. R. Machleidt, Adv. Nucl. Phys. 19 (1989) 189. 9. T.T.S. Kuo and E. Osnes, Springer Lecture Notes in Physics 364 (1990) 1. 10. M. Hjorth-Jensen, T.T.S. Kuo and E. Osnes, Phys. Rep. 261 (1995) 126. 11. E.M. Krenciglowa, C.L. Kung, T.T.S. Kuo and E. Osnes, Ann. Phys. (N.Y.) 101 (1976) 154. 12. S.Y. Lee and K. Suzuki, Phys. Lett. B 91 (1980) 173. 13. K. Suzuki and S.Y. Lee, Prog. Theor. Phys. 64 (1980) 2091. 14. W. Briickner et al., Phys. Lett. B 79 (1978) 157. 15. R.E. Chrien, Nucl. Phys. A 478 (1988) 705~. 16. P.H. Pile et al., Phys. Rev. Lett. 66 (1991) 2585. 17. 0. Hashimoto, in these proceedings. 18. H. Tamura, R.S. Hayano, H.Outa and T. Yamazaki, Prog. Theor. Phys. Suppl. 117 (1994) 1. 19. R.H. Dalitz, D.H. Davis, T. Motoba and D.N. Tovee, Nucl. Phys. A 625 (1997) 71. 20. Y. Yamamoto, A. Reuber, H. Himeno, S. Nagata and T. Motoba, Czech. J. Phys. 42 (1992) 1249. 21. D.J. Millener, A. Gal, C.B. Dover and R.H. Dalitz, Phys. Rev. C 31 (1985) 499.