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Kinematically complete measurement of the absorption of stopped pions in 3He
Volume 112B, number 2
PHYSICS LFTTERS
6 May 1982
KINEMATICALLY COMPLETE MEASUREMENT
OF THE ABSORPTION OF STOPPED PIONS IN 3He ~a D. GOTTA, M. DI3RR, W. FETSCHER ~, G. SCI-IMIDT, H. ULLRICH Kernforschungszentrum Karlsruhe, lnstTtut fur Kernphystk und Unlverxltat Karlsruhe, Germany and G. BACKENSTOSS, W. KOWALD, 1. SCHWANNER and ll.-J. WEYER InstltUt fur Physik der Untversitat Basel, Basel, Switzerland Received 6 January 1982
The reacuons rr-atle -+ nnp and rr-3He ~ nd have been measured m a kinematlcally complete experiment with stopped plons. The results show strong preference/or colhnear events. About three quarters of all absorpUon processes occur as quasifree absorptmn on np- or pp-pmrs with a raUo ot R = 10.1 +- 1.5. The remaining quarter shows energy sharing among all three nucleons and ~slocated m the final-state mteractmn regmns. Branching ratms for all observed final states are given
Early studies of 7r absorption m nuclm showed emission of correlated rip- and nn-pairs [1 ]. Together with the fact that for kinematical reasons a pion cannot be absorbed by a single nucleon, but needs at least a nucleon pmr. this observation led to the suggestion that absorption takes place on a deuteron or, more generally, on a nucleon pair reside the nucleus. Pion absorption on a real deuteron, on the other hand, has been studied in some detad and is in fact the only absorption process reasonably well understood within the scope of a microscopic theory [2]. On this background theoretical calculations have been made to explain plon absorption m complex nuelm as a quaslfree process on two nucleons [ 3 - 5 ] representing a more general case of the absorption on the deuteron. However, there are some experimental observations in more complex nuclei which are not easily explmned by this model, for example the emission of high-energy deuterons and tritons, though the situation Is obscured
by possible occurrence of sequential processes like pickup and knock-out. Furthermore, total s-absorption rates as determined from the level widths in p~onic atoms are difficult to explain by the deuteron absorption alone [4]. On the theoretical site also models have been discussed considering direct pion interactions with larger groups of nucleons like e.g. alpha clusters [6]. In summary one can say that the importance of absorption processes involving more than two nucleons in the initial step has never become clear. Therefore, we felt that a crucial experunent would be p~on absorption on the three-nucleon system 3He, where a comprehensive investigation is justified. In a first step we measured total absorption rates from plonic atom level widths with the s-level width already reported [7] and the p-level width to be published shortly. As only three nucleons appear in the final state, a klnematlcally complete experiment is possible without prohibitive experimental effort, i.e. via the coincident measurement of two emitted nucleons. Also, the chances for two-step reactions should be small. To us Work supported m part by the German Bundesmmlstermm such an experiment seemed to be the natural continuafur Forschung und Technologm and the Swiss National tion of the numerous investigations on the deuteron Scmnce Foundatmn and an important step towards the understanding of 1 Present address: Laboratonum fur tlochenergmphyslk der plon absorption m general. Specml points of interest ETH-Zunch, CII-5234 Vllhgen, Switzerland. are: 0 0 3 1 - 9 1 6 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 North-Holland 129
Volume 112B. number 2
PIIYSICS LETTERS
(1) In 3He the quasifree two-nucleon absorption can be studied simultaneously on three nucleon oairs with different quantum numbers and relative wave functions different from that of the free deuteron. So far, only the process on the latter could be studied kinematically complete. Ttus process seems, however, not to be sensitive to details in the theoretical description. (2) In a three-nucleon system the influence of a thild nucleon on the two-nucleon absorption or even the existence of a genume three-nucleon absorption process can be searched for. So far, the absorption of stopped plons in 3He has only been studied in single-arm experiments. There, the energy spectra of charged particles have been measured m a diffusion chamber [81 or with NaJ detectors [9]. In the present experiment negative plons from the nEl-channel of SIN were degraded by graphite and stopped in a 80 mg/cm 2 gaseous 3He-target, which was cooled to about 5 K by liquid 4He. The emitted neutrons, protons and deuterons were detected in two large-area (2 m × 0.5 m) position sensitive TOF-counters capable of particle identification, each consisting of 48 bars (2 m × 6 cm × 1.5 cin) of plastic scintillator arranged m 8 horizontal layers and 6 layers totaling 9 cm in depth. They could be placed at various angles between each other or with respect to the charged-particle hodoscope such that the kinematical allowed region was covered. The TOF start signal was taken from the 50 MHz signal of the accelerator. With a distance of 4.20m between target and detector 8% energy resolution for 88 MeV neutrons could be actueved. In addition, a large total-absorption hodoscope with 12 plastic scinnllators (17 cm × 17 cm X 8 cm deep) in connection with two MWP-chambers was used for the charged particles. It covers a solid angle of ~2 = 0.5 sr with an energy resolution of 7% for 35 MeV protons. Neutron- neutron, n e u t r o n - p r o t o n and neutron---deuteron coincidences were recorded with two TOF-counters and a combination of T19F-counter and charged-particle counter, respectively. The resolution for the relative angle between the two pamcles was about 1.5 ° Ibr all cases. A more detailed description of the apparatus can be found elsewhere [10]. With the experimental set-up the complete momenta of two particles were measured. For each event the three independently determined quantities TI, T 2 and 191,2, i.e. the kinetic energies of the two particles and their relative angle were extracted. The kinematics of 130
6 May 1982
the reaction, however, Is completely determined by only two independent quantities, llence, our measurement contains one constraint, which can be used for background suppression and to improve the resolution tbr the presented quantities. Most of the background, however, has been eliminated through subtraction of empty-target measurements. All data have been corrected for neutron counter efficiency and 19-acceptance of the respective set-up. The two-particle final state nd is necessarily collinear and hence easy to measure The respective rates are presented in table 1. Results from the nnp channel are presented as a Dalitz plot in fig. 1. The complete kinematically allowed region is contained in the elhptical area. About three quarters of it have been explored by np-coincxdences. Only the quarter at left, which corresponds to protons below the experimental threshold of 20 MeV has been covered with nn-coIncldences. The remaining small gap in between wall soon be filled with data fiom an additional measurement. For the conclusions drawn in this paper, the missing information can, however, well be estimated by extrapolations from the adjacent regions and by symmetry considerations. The contour lines of fig. 1 show 6 density maxima, marked with (A) to (F). The kinematical condition for the peak (D) on the y-axis corresponds to zero proton energy and hence to the quaslfree absorption QFA (Tr-np ~ nn), the third nucleon (proton) being spectator. Similarly peak (F) on the x-axis originates from a zero-energy neutron coming from the quasifree process QFA ( ~ - pp ~ np). The latter process can also be seen in peak (B) with the only difference that the displayed neutron energy now belongs to the fast neutron and hence equals the proton energy. For the remaining three peaks none of the nucleons has an energy close to zero. The available energy is shared among all three nucleons which then must have been involved m the process. For peak (A) the proton energy Is at maximum. The kmemancal condition for this fact is that the two other nucleons (neutrons) both have a direction opposite to the proton and zero relative lnomentum. This is expected to be the classical nn-final-state interaction FSI region. With the same kind of arguments we identify peak (C), where the energy of one neutron is at maximum, as the np-FSI region. Since two neutrons are present in the final state all peaks with unequal neutron ener-
Volume 112B, number 2
PHYSICS LF,TTERS
6 May 1982
(c)
TnlMeV
.
25 11' 13
80
(B) (D)
17
6O
40
9
i"A) 20
(E) ]7
"7, 13
20
40
60
(F)
80
rp .eV
Fig I. Contour hne representation of the Dalitz plot for the nnp final state produced by ~r- absorption m 3He. Tile kinetic energy of one neutron versus the proton Is displayed. The numbers at the contour hnes give the densities in arbitrary umts. The density maxima are as described in the text" (A) nn-FSl, (B) and (F) QFA (npp ~ np), (C) and (E) np-l'Sl, (D) QFA (nnp ~ nn) gles must appear twice m the Dahtz plot. Thus peak (C) is m i r r o r e d m peak (E), as it was the case for peaks (B) and (F). Table 1 Partial rates as obtained by integration of the dens]ty peaks m fig. 1 and the rate for the nd-channel in percents ol the sum of nnp and nd rates. nnp channel
QI'A (np ~ nn) QI'A ( p p ~ np) 1,SI nn FSInp
nd charmel
68 4 -+ 11 6 2 ± 0.4 l 6 ± 0.4 12.3± 1 I 1.5 +- 1.4
In a Dahtz plot all events with 180 ° relatwe angle between any two particles fall on the p e r i m e t e r o f the kinematically allowed region. Smaller angles correspond to regions in the interior o f the plot. The special case o f 120 ° b e t w e e n all nucleons and equal energy sharing a m o n g all three would be located in the center o f the ellipse. No such cases seem to occur. It is a remarkable result o f our e x p e r i m e n t that there is a strong preference for collinear events. The measured angular distributions dN/d~(O) show halfwidths o f ~ 15 ° and ~ 10 ° in the regions o f quaslfree absorption and final-state interaction, respectively. In the np-FSI region tile angular distribut]on has a steep and a flat c o m p o n e n t corresponding to the spin singlet and spin triplet stale, respectively. 131
Volume 112B, number 2
PHYSICS LETTERS
J50
5
(D)
dN d~
I I
4
~o
I I
(c)
3o
(a) 20
10
0
I
I
I
25 50 75 (1)-distribution
I
I
I
150 100 125 (0 95-~r-~10)
I
0
175 ®/*
Fig. 2. Density dlstrlbuuon along the perimeter of the kmematlcally allov~edregmn showing peaks (A), (B), (C) and (D). The scale at the x-axis corresponds to the polar angle (p from the center of the Dahtz plot m triangular coordinates. Data are for 0.95 ~
6 May 1982
theoretical models seem to predict [3,5] which were, however, not specifically done for 3He. (2) There is a non-negligible contribution of absorption processes mvolvmg three nucleons. If we add the rates for the nd-channel to those from final state interaction we find the ratio "'three-nucleon" absorption rate to "two-nucleon" absorption = (25.4 -+ 2.8)/(74.6 -+ 11.4) = 1/3. (3) The "three-nucleon" absorption process occurs in the final-state interaction region. This mechanism seems also to be responsible for the nd-channel. It should be mentioned that absorption of stopped pxons in 3He was theoretically treated by several authors [11[ which are, however, only of hmited use for direct comparison with the data. Efforts are being made to calculate double differential rates as function of twoparticle energies [ 12] where detailed comparisons are being prepared. It is a pleasure to acknowledge the valuable technical assistance of H. Krause, the help of the technical staff and the support of the Schweizerische lnstltUt ftir Nuklearforschung (SIN). [ 1] M.E. Nordberg, K.t Kmseyand R F. Burman, Phys. Rev. 165 (1968) 1096. [2] M Brack, D.O. Rlska and W. We]se, Nucl. Phys. A287 (1977) 425; O.V. Maxwell, W. Welse and M. Brack, Nucl. Phys A348 (1980) 388. [3] F. Hachenberg and ll.J Pirner, Ann. Phys. (NY) 112 (1978) 401. [4] R. ShlmlZU,A. Faessler and H. Muther, Nucl. Phys. A343 (1980) 468. [5] G F Bertsch and D.O. Rlska, Phys. Rev. C18 (1978) 317. [6] T.I. Kopalelshvili and I.Z. Machabeh, Soy J. Nucl. Phys. 12 (1971) 286, T.U. Kolybasov and V.A Tsepov, Sov. J. Nucl. Phys. 14 (1972) 418. [7] R. Abela el al., Phys. Lctt. 68B (1977) 429, I Schwanner el al., Phys. Lett. 96B (1980) 268. [8] O.A. Zalmldoroga et al, Sov Phys. JETP 21 (1975) 848, 24 (1967) 1111. [9] 1 McCarthy, T. Meyer, R.C Minehart, A.E. Wadhnger and K O.H Ziock, Phys. Rev. C l l (1975) 266. [ 10] D. Gotta, report KFK 3226 Kernforschun&~zentrum Karlsruhe (1981). [11] P.P. Dlvakaran, Phys. Rev. 139B (1965) 387, I.T. Cheon, Phys. Rev. 145 (1966) 794, A. Figureau and M Ericson, Nucl. Phys. B10 (1969) 349, C. Pajares and R. Pascu',d,Nucl. Phys. B35 (1971) 631, A C Phlll,ps and F. Rolg, Nucl. Phys. B60 (1973) 93, B.K. Jam, Nucl. Phys. A296 (1978) 479 [12] T. Schucan, pnvate communication.
PHYSICS LFTTERS
6 May 1982
KINEMATICALLY COMPLETE MEASUREMENT
OF THE ABSORPTION OF STOPPED PIONS IN 3He ~a D. GOTTA, M. DI3RR, W. FETSCHER ~, G. SCI-IMIDT, H. ULLRICH Kernforschungszentrum Karlsruhe, lnstTtut fur Kernphystk und Unlverxltat Karlsruhe, Germany and G. BACKENSTOSS, W. KOWALD, 1. SCHWANNER and ll.-J. WEYER InstltUt fur Physik der Untversitat Basel, Basel, Switzerland Received 6 January 1982
The reacuons rr-atle -+ nnp and rr-3He ~ nd have been measured m a kinematlcally complete experiment with stopped plons. The results show strong preference/or colhnear events. About three quarters of all absorpUon processes occur as quasifree absorptmn on np- or pp-pmrs with a raUo ot R = 10.1 +- 1.5. The remaining quarter shows energy sharing among all three nucleons and ~slocated m the final-state mteractmn regmns. Branching ratms for all observed final states are given
Early studies of 7r absorption m nuclm showed emission of correlated rip- and nn-pairs [1 ]. Together with the fact that for kinematical reasons a pion cannot be absorbed by a single nucleon, but needs at least a nucleon pmr. this observation led to the suggestion that absorption takes place on a deuteron or, more generally, on a nucleon pair reside the nucleus. Pion absorption on a real deuteron, on the other hand, has been studied in some detad and is in fact the only absorption process reasonably well understood within the scope of a microscopic theory [2]. On this background theoretical calculations have been made to explain plon absorption m complex nuelm as a quaslfree process on two nucleons [ 3 - 5 ] representing a more general case of the absorption on the deuteron. However, there are some experimental observations in more complex nuclei which are not easily explmned by this model, for example the emission of high-energy deuterons and tritons, though the situation Is obscured
by possible occurrence of sequential processes like pickup and knock-out. Furthermore, total s-absorption rates as determined from the level widths in p~onic atoms are difficult to explain by the deuteron absorption alone [4]. On the theoretical site also models have been discussed considering direct pion interactions with larger groups of nucleons like e.g. alpha clusters [6]. In summary one can say that the importance of absorption processes involving more than two nucleons in the initial step has never become clear. Therefore, we felt that a crucial experunent would be p~on absorption on the three-nucleon system 3He, where a comprehensive investigation is justified. In a first step we measured total absorption rates from plonic atom level widths with the s-level width already reported [7] and the p-level width to be published shortly. As only three nucleons appear in the final state, a klnematlcally complete experiment is possible without prohibitive experimental effort, i.e. via the coincident measurement of two emitted nucleons. Also, the chances for two-step reactions should be small. To us Work supported m part by the German Bundesmmlstermm such an experiment seemed to be the natural continuafur Forschung und Technologm and the Swiss National tion of the numerous investigations on the deuteron Scmnce Foundatmn and an important step towards the understanding of 1 Present address: Laboratonum fur tlochenergmphyslk der plon absorption m general. Specml points of interest ETH-Zunch, CII-5234 Vllhgen, Switzerland. are: 0 0 3 1 - 9 1 6 3 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 North-Holland 129
Volume 112B. number 2
PIIYSICS LETTERS
(1) In 3He the quasifree two-nucleon absorption can be studied simultaneously on three nucleon oairs with different quantum numbers and relative wave functions different from that of the free deuteron. So far, only the process on the latter could be studied kinematically complete. Ttus process seems, however, not to be sensitive to details in the theoretical description. (2) In a three-nucleon system the influence of a thild nucleon on the two-nucleon absorption or even the existence of a genume three-nucleon absorption process can be searched for. So far, the absorption of stopped plons in 3He has only been studied in single-arm experiments. There, the energy spectra of charged particles have been measured m a diffusion chamber [81 or with NaJ detectors [9]. In the present experiment negative plons from the nEl-channel of SIN were degraded by graphite and stopped in a 80 mg/cm 2 gaseous 3He-target, which was cooled to about 5 K by liquid 4He. The emitted neutrons, protons and deuterons were detected in two large-area (2 m × 0.5 m) position sensitive TOF-counters capable of particle identification, each consisting of 48 bars (2 m × 6 cm × 1.5 cin) of plastic scintillator arranged m 8 horizontal layers and 6 layers totaling 9 cm in depth. They could be placed at various angles between each other or with respect to the charged-particle hodoscope such that the kinematical allowed region was covered. The TOF start signal was taken from the 50 MHz signal of the accelerator. With a distance of 4.20m between target and detector 8% energy resolution for 88 MeV neutrons could be actueved. In addition, a large total-absorption hodoscope with 12 plastic scinnllators (17 cm × 17 cm X 8 cm deep) in connection with two MWP-chambers was used for the charged particles. It covers a solid angle of ~2 = 0.5 sr with an energy resolution of 7% for 35 MeV protons. Neutron- neutron, n e u t r o n - p r o t o n and neutron---deuteron coincidences were recorded with two TOF-counters and a combination of T19F-counter and charged-particle counter, respectively. The resolution for the relative angle between the two pamcles was about 1.5 ° Ibr all cases. A more detailed description of the apparatus can be found elsewhere [10]. With the experimental set-up the complete momenta of two particles were measured. For each event the three independently determined quantities TI, T 2 and 191,2, i.e. the kinetic energies of the two particles and their relative angle were extracted. The kinematics of 130
6 May 1982
the reaction, however, Is completely determined by only two independent quantities, llence, our measurement contains one constraint, which can be used for background suppression and to improve the resolution tbr the presented quantities. Most of the background, however, has been eliminated through subtraction of empty-target measurements. All data have been corrected for neutron counter efficiency and 19-acceptance of the respective set-up. The two-particle final state nd is necessarily collinear and hence easy to measure The respective rates are presented in table 1. Results from the nnp channel are presented as a Dalitz plot in fig. 1. The complete kinematically allowed region is contained in the elhptical area. About three quarters of it have been explored by np-coincxdences. Only the quarter at left, which corresponds to protons below the experimental threshold of 20 MeV has been covered with nn-coIncldences. The remaining small gap in between wall soon be filled with data fiom an additional measurement. For the conclusions drawn in this paper, the missing information can, however, well be estimated by extrapolations from the adjacent regions and by symmetry considerations. The contour lines of fig. 1 show 6 density maxima, marked with (A) to (F). The kinematical condition for the peak (D) on the y-axis corresponds to zero proton energy and hence to the quaslfree absorption QFA (Tr-np ~ nn), the third nucleon (proton) being spectator. Similarly peak (F) on the x-axis originates from a zero-energy neutron coming from the quasifree process QFA ( ~ - pp ~ np). The latter process can also be seen in peak (B) with the only difference that the displayed neutron energy now belongs to the fast neutron and hence equals the proton energy. For the remaining three peaks none of the nucleons has an energy close to zero. The available energy is shared among all three nucleons which then must have been involved m the process. For peak (A) the proton energy Is at maximum. The kmemancal condition for this fact is that the two other nucleons (neutrons) both have a direction opposite to the proton and zero relative lnomentum. This is expected to be the classical nn-final-state interaction FSI region. With the same kind of arguments we identify peak (C), where the energy of one neutron is at maximum, as the np-FSI region. Since two neutrons are present in the final state all peaks with unequal neutron ener-
Volume 112B, number 2
PHYSICS LF,TTERS
6 May 1982
(c)
TnlMeV
.
25 11' 13
80
(B) (D)
17
6O
40
9
i"A) 20
(E) ]7
"7, 13
20
40
60
(F)
80
rp .eV
Fig I. Contour hne representation of the Dalitz plot for the nnp final state produced by ~r- absorption m 3He. Tile kinetic energy of one neutron versus the proton Is displayed. The numbers at the contour hnes give the densities in arbitrary umts. The density maxima are as described in the text" (A) nn-FSl, (B) and (F) QFA (npp ~ np), (C) and (E) np-l'Sl, (D) QFA (nnp ~ nn) gles must appear twice m the Dahtz plot. Thus peak (C) is m i r r o r e d m peak (E), as it was the case for peaks (B) and (F). Table 1 Partial rates as obtained by integration of the dens]ty peaks m fig. 1 and the rate for the nd-channel in percents ol the sum of nnp and nd rates. nnp channel
QI'A (np ~ nn) QI'A ( p p ~ np) 1,SI nn FSInp
nd charmel
68 4 -+ 11 6 2 ± 0.4 l 6 ± 0.4 12.3± 1 I 1.5 +- 1.4
In a Dahtz plot all events with 180 ° relatwe angle between any two particles fall on the p e r i m e t e r o f the kinematically allowed region. Smaller angles correspond to regions in the interior o f the plot. The special case o f 120 ° b e t w e e n all nucleons and equal energy sharing a m o n g all three would be located in the center o f the ellipse. No such cases seem to occur. It is a remarkable result o f our e x p e r i m e n t that there is a strong preference for collinear events. The measured angular distributions dN/d~(O) show halfwidths o f ~ 15 ° and ~ 10 ° in the regions o f quaslfree absorption and final-state interaction, respectively. In the np-FSI region tile angular distribut]on has a steep and a flat c o m p o n e n t corresponding to the spin singlet and spin triplet stale, respectively. 131
Volume 112B, number 2
PHYSICS LETTERS
J50
5
(D)
dN d~
I I
4
~o
I I
(c)
3o
(a) 20
10
0
I
I
I
25 50 75 (1)-distribution
I
I
I
150 100 125 (0 95-~r-~10)
I
0
175 ®/*
Fig. 2. Density dlstrlbuuon along the perimeter of the kmematlcally allov~edregmn showing peaks (A), (B), (C) and (D). The scale at the x-axis corresponds to the polar angle (p from the center of the Dahtz plot m triangular coordinates. Data are for 0.95 ~
6 May 1982
theoretical models seem to predict [3,5] which were, however, not specifically done for 3He. (2) There is a non-negligible contribution of absorption processes mvolvmg three nucleons. If we add the rates for the nd-channel to those from final state interaction we find the ratio "'three-nucleon" absorption rate to "two-nucleon" absorption = (25.4 -+ 2.8)/(74.6 -+ 11.4) = 1/3. (3) The "three-nucleon" absorption process occurs in the final-state interaction region. This mechanism seems also to be responsible for the nd-channel. It should be mentioned that absorption of stopped pxons in 3He was theoretically treated by several authors [11[ which are, however, only of hmited use for direct comparison with the data. Efforts are being made to calculate double differential rates as function of twoparticle energies [ 12] where detailed comparisons are being prepared. It is a pleasure to acknowledge the valuable technical assistance of H. Krause, the help of the technical staff and the support of the Schweizerische lnstltUt ftir Nuklearforschung (SIN). [ 1] M.E. Nordberg, K.t Kmseyand R F. Burman, Phys. Rev. 165 (1968) 1096. [2] M Brack, D.O. Rlska and W. We]se, Nucl. Phys. A287 (1977) 425; O.V. Maxwell, W. Welse and M. Brack, Nucl. Phys A348 (1980) 388. [3] F. Hachenberg and ll.J Pirner, Ann. Phys. (NY) 112 (1978) 401. [4] R. ShlmlZU,A. Faessler and H. Muther, Nucl. Phys. A343 (1980) 468. [5] G F Bertsch and D.O. Rlska, Phys. Rev. C18 (1978) 317. [6] T.I. Kopalelshvili and I.Z. Machabeh, Soy J. Nucl. Phys. 12 (1971) 286, T.U. Kolybasov and V.A Tsepov, Sov. J. Nucl. Phys. 14 (1972) 418. [7] R. Abela el al., Phys. Lctt. 68B (1977) 429, I Schwanner el al., Phys. Lett. 96B (1980) 268. [8] O.A. Zalmldoroga et al, Sov Phys. JETP 21 (1975) 848, 24 (1967) 1111. [9] 1 McCarthy, T. Meyer, R.C Minehart, A.E. Wadhnger and K O.H Ziock, Phys. Rev. C l l (1975) 266. [ 10] D. Gotta, report KFK 3226 Kernforschun&~zentrum Karlsruhe (1981). [11] P.P. Dlvakaran, Phys. Rev. 139B (1965) 387, I.T. Cheon, Phys. Rev. 145 (1966) 794, A. Figureau and M Ericson, Nucl. Phys. B10 (1969) 349, C. Pajares and R. Pascu',d,Nucl. Phys. B35 (1971) 631, A C Phlll,ps and F. Rolg, Nucl. Phys. B60 (1973) 93, B.K. Jam, Nucl. Phys. A296 (1978) 479 [12] T. Schucan, pnvate communication.