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A microscopic model of hypernuclei Johnstone, John Allistair 1982

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r , U A MICROSCOPIC MODEL OF I HYPERNUCLEI by JOHN ALLISTAIR JOHNSTONE B.Sc, McMaster University, 1978 M.Sc, McMaster University, 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES " (Department of Physics) -We accept t h i s thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1982 © John A l l i s t a i r Johnstone, 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 I E -6 ( 3 / 8 1 ) ABSTRACT A s e p a r a b l e p o t e n t i a l model i s c o n s t r u c t e d t o d e s c r i b e the coupled EN-AN systems. From t h i s the E s i n g l e p a r t i c l e p o t e n t i a l i s developed i n c l u d i n g P a u l i e f f e c t s . The momentum space Schroedinger equation i s then s o l v e d s e l f - c o n s i s t e n t l y f o r the complex e i g e n v a l u e s of 1s and l p s t a t e E° h y p e r n u c l e i . A r i s i n g from two q u i t e d i s t i n c t mechanisms these s t a t e s are a l l found to be long l i v e d . In s - s t a t e s , P a u l i s u p p r e s s i o n of the EN-»AN c o n v e r s i o n reduces the widths by as much as 50% from c l a s s i c a l e s t i m a t e s i n heavy n u c l e i , and in l i g h t n u c l e i produces widths as small as 1.8 MeV i n ^ l H e . In p - s t a t e s , P a u l i e f f e c t s are r e l a t i v e l y unimportant and the s t r o n g a b s o r p t i o n of the p o t e n t i a l c r e a t e s extremely narrow quasi-bound s t a t e s i n the E continuum. i i i TABLE OF CONTENTS PAGE ABSTRACT . i i LIST OF TABLES . v LIST OF FIGURES v i ACKNOWLEDGEMENTS v i i i AN INTRODUCTION FOR NON-SPECIALISTS ix CHAPTER 1 INTRODUCTION AND SUMMARY 1 CHAPTER 2 TWO-BODY IN-AN SCATTERING 12 2 . 1 . The Physical Basis for Separable 13 Interactions 2.II. OBE Predictions of the EN->AN 17 Interactions 2.III. T-Matrix Normalization and IN 19 Cross-Sect ions 2.IV. The Coupled-Channel, Separable 24 EN-AN Potentials 2.V. Coulomb Modifications to the Two- 26 Body T-Matrix 2.VI. Results of the F i t to the ZN Data 33 CHAPTER 3 THE Z-NUCLEUS POTENTIAL 50 3 . 1 . The Need for Many-Body Eff e c t s 50 3.II. The Z Si n g l e - P a r t i c l e Potential 53 3.III. Approximations in the Single- 62 Pa r t i c l e Potential 3.IV. The Pauli Exclusion P r i n c i p l e 69 CHAPTER 4 TECHNICAL DETAILS AND NUMERICAL 74 METHODS 4 . 1 . Bound States and Resonance Poles 74 4.II. Analytic Continuation of the Pauli 80 Operator 4 . 1II. The Z-Nucleus T-Matrix 83 4.IV. Numerical Solution of the T-Matrix 87 CHAPTER 5 " RESULTS IN LIGHT Z° HYPERNUCLEI 97 5.1 . 5 .II . Z~ Atoms S-State Hypernuclei 97 100 iv PAGE 5.III. Separable Approximation of the Z- 107 Nucleus Potential 5.IV. P-State Hypernuclei 110 CHAPTER 6 DISCUSSION AND CONCLUSIONS 126 BIBLIOGRAPHY 131 APPENDIX I ANGULAR MOMENTUM GRAPHICS 134 APPENDIX II SOLUTION OF THE COUPLED-CHANNELS 141 EQUATIONS APPENDIX III PARTIAL WAVE EXPANSION OF V v(k',k) 144 LIST OF TABLES TABLE I E + p Coulomb-Corrected Phase S h i f t s TABLE II Potential Parameters from Best-Fit to EN Scattering TABLE III Scattering Lengths and E f f e c t i v e Ranges in E'p-^-E'p Scattering TABLE IV Convergence of Pade Approximants to Exact Binding Energy TABLE V Is Binding in Light E° Hypernuclei TABLE VI Variation of 1s Binding Energy with Fermi Momentum and Nucleon Binding TABLE VII Convergence of Binding Energy with Number of Terms in the Form-Factor Expansion TABLE VIII .... 1s Binding Energy with Parameters from Ref. 28 TABLE IX 1p Binding in Light E° Hypernuclei TABLE X 1p Binding Energy with Parameters from Ref. 28 v i LIST OF FIGURES PAGE FIGURE 1 (a) (b) (c) (a) (b) (c) (d) FIGURE (a) (b) (c) (d) FIGURE (a) (b) ir Energy Spectrum Obtained in 9Be(K" , t r " )^Be FIGURE 2 IN D i f f e r e n t i a l Cross-Sections: l - p - ^ l ' p E l a s t i c Reaction I_p-*-An Conversion Reaction I + p-*"I*p E l a s t i c Reaction FIGURE 3 IN Total Cross-Sections: I*p-»I*p E l a s t i c Reaction I"p->-An Conversion Reaction I"p->I"p E l a s t i c Reaction I"p->-I0n Charge-Exchange IN-»IN Absorption C o e f f i c i e n t s and Phase S h i f t s : 3S, 1=1/2 3S, 1=1/2 3 P j 1=1/2 1 S 0 , 3S, 1 = 3 / 2 Energy Variation of the 3S, 1=1/2 Scattering Amplitude: Real Component Imaginary Component FIGURE 6 Movement of S-Matrix Poles for , Changing Absorption FIGURE 7 Variation of Potential Volume . Integral with Fermi Momentum and Nucleon Binding FIGURE 8 Variation of « and r with Fermi Momentum and Nucleon Binding FIGURE 9 I°-Nucleus P-Wave Phase S h i f t s and Absorption C o e f f i c i e n t s : (a) (b) (c) (d) (e) E-ftHe E - 6 L i I- 8Be I- 1 'C I- 1 50 36 36 37 38 39 39 40 41 42 44 44 44 45 45 49 49 49 78 99 103 115 115 116 116 117 117 v i i PAGE FIGURE 10 E°-Nucleus S-Wave Phase S h i f t s 118 and Absorption C o e f f i c i e n t s : (a) E-"He 1 18 (b) E - 6 L i 1 19 (c) E-8Be 119 (d) E~ 1 1C 1 20 (e) E- 1 50 120 FIGURE 11 E°-Nucleus D-Wave Phase Sh i f t s 123 and Absorption C o e f f i c i e n t s : (a) E-'He 123 (b) E - 6 L i 124 (c) E- 8 Be 124 (d) E - 1 1C 125 (e) E~ 1 50 125 v i i i ACKNOWLEDGEMENTS From among the many c o l l e a g u e s and f r i e n d s whose h e l p and guidance were so v a l u a b l e t o me i n the course of t h i s work, I should l i k e i n p a r t i c u l a r to r e c o r d my indebtedness to Dr. Tony Thomas f o r the b e n e f i t of h i s experi e n c e and i n s i g h t , and to Dr. Doug Beder f o r h i s i n t e r e s t and c o n s t r u c t i v e comments. A l s o , I wish e s p e c i a l l y t o thank my wife L e s l i e f o r the support and encouragement t h a t was so i n s t r u m e n t a l i n the development of t h i s t h e s i s . To a l l these people I g l a d l y a c c o r d my a p p r e c i a t i o n and thanks. ix AN INTRODUCTION FOR THE NON-SPECIALIST This short section i s intended to introduce the non-specialist to the concepts and terminology used in t h i s work. F i r s t , a 'hypernucleus' refers in general to any nucleus in which a nucleon has been replaced by one of the strange baryons A,Z,Z, or A . As the l i g h t e s t of these the A i s unique, in that i t i s stable in the nucleus. The heavier Z,Z, and hyperons, on the other hand, can decay to a A upon interacting with a nucleon, releasing a large amount of energy in the process. This energy release makes the conversion so favourable that the hyperons are expected to be very short-lived in a nucleus, simply because of the large number of nucleons present. Naively, the l i f e t i m e wouldn't be expected to be much d i f f e r e n t from about 1 0 ~ 2 3 seconds, which i s roughly the length of time required for the p a r t i c l e to cross the nucleus. More careful analyses which estimate the mean free path of the p a r t i c l e in the nuclear 'gas' do not change th i s figure appreciably. It has been quite a surprise recently then to f i n d that Z l i v e s nearly 10 times longer than t h i s figure in the nucleus. The aim of t h i s work has been to understand t h i s large discrepancy. Some guidance i s available from standard X nuclear experience where we know that the interaction of nucleons in a nucleus i s not l i k e the interaction of free p a r t i c l e s . In p a r t i c u l a r , the Pauli exclusion p r i n c i p l e ensures that no two nucleons can occupy the same quantum state. The result i s that no interaction can occur i f i t w i l l leave a nucleon in a state which i s already occupied. This i s an important result of early nuclear theory and the main reason why a nucleus can be considered a gas of nucleons which, to a good f i r s t approximation, are non-interacting. In fact, one of the e a r l i e s t triumphs of the Pauli p r i n c i p l e was reducing the calculated widths of excited nuclear., states from 30-40 MeV to the keV range, in l i n e with experiment. For the E one doesn't expect the e f f e c t s to be as dramatic because the exclusion does not apply to the E i t s e l f . Nonetheless, the r e s t r i c t i o n s on the f i n a l state N in the EN->AN reaction can suppress the conversion s i g n i f i c a n t l y . The basic outline of the study has then been to , f i r s t , understand and describe the EN interaction in free space. We then modify t h i s interaction with the Pauli p r i n c i p l e to describe the EN interaction in the presence of other nucleons in the nucleus. By summing the interaction of the E with each separate nucleon we arrive at an o v e r a l l description of the E-nucleus in t e r a c t i o n . It i s found by t h i s procedure that, for a I bound in the ground state, Pauli suppresion of the conversion i s s u f f i c i e n t to produce a l i f e t i m e consistent with the experimental value. 1 CHAPTER 1 INTRODUCTION AND SUMMARY Although the f i e l d of hypernuclear p h y s i c s i s now r e l a t i v e l y o l d , i n t e r e s t has been renewed i n recent years by the o b s e r v a t i o n of l o n g - l i v e d Z s t a t e s i n the nucleus. U n l i k e the A hyperon, which i s s t a b l e i n n u c l e a r matter, the I decays v i a the s t r o n g i n t e r a c t i o n to a A, r e l e a s i n g roughly -80 MeV i n the p r o c e s s . For t h i s reason i t had long been expected that I s t a t e s would be very broad, e.g. r ~ 2 5 MeV [1,2], on the b a s i s of s e m i - c l a s s i c a l arguments. Such s h o r t - l i v e d s t a t e s would be extremely d i f f i c u l t to observe even under i d e a l c o n d i t i o n s . Thus, i t was very e x c i t i n g when i n 1979 the H e i d e l b e r g - S a c l a y group r e p o r t e d f i n d i n g s t r u c t u r e i n 2|Be with widths l e s s than 8 MeV [3,'4], a p p a r e n t l y c o r r e s p o n d i n g to unbound p - s t a t e s . Since t h i s d i s c o v e r y a v a r i e t y of l i g h t I h y p e r n u c l e i have been s t u d i e d [5,6], and at Brookhaven i t has been found that i n ^ f L i the l e v e l may be as narrow as 3 MeV [ 5 ] . U n f o r t u n a t e l y the p i c t u r e i s not as c l e a r f o r s - s t a t e h y p e r n u c l e i , w i t h the only suggestion of narrow s t a t e s t o date being i n the ^|C [ 3 ] , and p o s s i b l y ^ , & e [6] systems. H i s t o r i c a l l y though i t i s A h y p e r n u c l e i which have a t t r a c t e d a t t e n t i o n . As the l i g h t e s t of the hyperons the A 2 decays only through the weak interaction and therefore has a r e l a t i v e l y long l i f e t i m e in the nucleus. In addition, the s i m i l a r i t y of the A and N masses, and the fact that the AN potential i s only s l i g h t l y weaker than that of NN, suggested that in the nucleus the A would behave much l i k e a neutron. With strangeness -1 though, the A would be distinguishable and therefore make a nearly ideal nuclear probe. The advances in thi s area and interesting future prospects, such as the study of doubly strange AA and H" hypernuclei formed in (K",K*) reactions, w i l l not be discussed further here but recent reviews are given i n , for example, r e f s . [ 7 , 8 ] . It should be clear though that i f one hopes to successfully compare the Y-nucleus (where Y i s I or A) with the N- nucleus interaction i t i s esse n t i a l to produce hypernuclear states which have s u f f i c i e n t l y simple configurations to allow a unique t h e o r e t i c a l interpretation. Ideally one would l i k e the hyperon to simply replace one of the target nucleons, assuming the same state quantum numbers and without disturbing the nuclear core. Conceptually at l e a s t , t h i s i s not a d i f f i c u l t problem. A nuclear target i s exposed to a beam of low energy kaons and, through the strangeness-exchange reaction KN-»»Y, one of the target nucleons i s transformed into a hyperon. In a rough approximation the amplitude for t h i s process occurring 3 in the nucleus i s proportional to the product of the elementary two-body amplitude, with a form factor F(g) which depends on the i n i t i a l state N, and f i n a l state Y wavefunctions as with CJ the momentum transferred to the hyperon, i , f label the state quantum numbers, and the d i s t o r t i o n of the K and n wavefunctions has been ignored. For very small values of g the exponential i s of order unity across the nuclear volume and the overlap integral i s therefore maximized for i=f. In addition F(g) decreases rapidly with the momentum transferred and, for example, F ( g ) ~ e x p ( - a 2 q 2 ) for harmonic o s c i l l a t o r wavefunctions. Experimentally then, for ' r e c o i l l e s s ' production (q=0) one would detect the pions scattered in the forward d i r e c t i o n and adjust the incident kaon momentum to minimize the momentum transferred to the hyperon. The 'magic' momentum which leaves the A at rest in the nucleus i s about 500 MeV/c, whereas for I i t i s lower, at 300 MeV/c. In practice, of course, the sit u a t i o n i s not nearly as clean as that outlined above. F i r s t of a l l , the optimum K momentum i s not easy to achieve. The kabns decay in f l i g h t by the weak interaction with a mean l i f e t i m e of ~ 1 0 ~ 8 4 seconds. Consequently the intensity of the beam reaching the target decreases sharply with decreasing momentum. Present experiments t y p i c a l l y use a momentum of ~700 MeV/c to p a r t i a l l y overcome t h i s intensity problem, but even so 20% of the kaons decay with every meter of f l i g h t . Secondly, t h i s higher momentum creates i t s own d i f f i c u l t i e s . For r e c o i l l e s s A production, 700 MeV/c i s not r a d i c a l l y d i f f e r e n t from the optimum value and the momentum transfer i s s t i l l f a i r l y low,~40 MeV/c. For I production however the corresponding momentum transfer i s 130 MeV/c and, with a value t h i s large, quasifree production .competes with the r e c o i l l e s s process. In t h i s case the E i s kicked into another state and, since there i s no Pauli r e s t r i c t i o n for the E in the nucleus, the number of available states i s large. The result i s that the experimental spectrum degenerates from a sharp r e c o i l l e s s peak into a small bump embedded in a large quasifree background. Despite these, and many other, complications i t i s possible to determine the binding energy and"width of the E states by measuring the momentum of the f i n a l state pions. The difference between the v and K energies gives d i r e c t l y the transformation energy (M H y-M A), where M H y i s the mass of the hypernuclear system and M A i s the target mass. The hyperon binding energy i s then simply B =BN~(M^M^)*(my-mN), 5 with B N the b i n d i n g energy of the nucleon that was r e p l a c e d , and 1 ^ , 1 1 ^ are the obvious masses. In f i g . 1 the experimental n energy spectrum of r e f . [3] i s shown f o r 720 MeV/c kaons i n c i d e n t on 9Be. The energy s c a l e i s given both as a f u n c t i o n of the t r a n s f o r m a t i o n energy and a l s o the A , Z b i n d i n g e n e r g i e s . Because of the l a r g e momentum acceptance of the n spectrometer (600-»-850 MeV/c) they were ab l e to measure the A and I hypernuclear s p e c t r a s i m u l t a n e o u s l y . In ^Be the peak at B^=-S MeV i s a t t r i b u t e d to r e c o i l l e s s p r o d u c t i o n on the l o o s e l y bound 1p 3/ 2 neutron, whereas the peak at -17 MeV i s a mixture of r e c o i l l e s s s t r e n g t h from the 1 p 3 / i and 1 s ^ neutrons. One can a l s o see the A ground s t a t e at B/^ = 7 M e V [ 3 ] . J u s t separated from the A spectrum by the E - A mass d i f f e r e n c e i s s t r u c t u r e which i s very s i m i l a r to that i n ^Be, and so i t i s n a t u r a l to a s s i g n these peaks to t h e ^ B e hypernucleus. A f t e r c o r r e c t i n g f o r the E - A mass d i f f e r e n c e they are found to l i e s y s t e m a t i c a l l y ~ 3 MeV higher i n e x c i t a t i o n than the A peaks, i n d i c a t i n g that the E-nucleus i n t e r a c t i o n i s s l i g h t l y weaker than that of A - n u c l e u s . More important i s t h a t , i n p - s t a t e at l e a s t , the l e v e l i s s u r p r i s i n g l y narrow. In f a c t the upper l i m i t of 8 MeV f o r the 'Be l e v e l was assesed because the experimental width was 6 no greater than that of the corresponding peak in £Be [3]. Conceivably then the E state i s much longer l i v e d than i s implied by the 8 MeV figure. These findings have sparked a f l u r r y of t h e o r e t i c a l a c t i v i t y and a number of mechanisms have now been proposed for suppressing EN-»AN conversion in the nucleus. Gal and Dover [9,10] have emphasized that the conversion proceeds only through' the isospin 1=1/2 channel, with the implication being that in special cases where the 1=3/2 interaction dominates, long-lived states can r e s u l t . For example, in 1 2C(K" , ir* )^-Be the t o t a l isospin (T,T 3) of the i n i t i a l state is (1/2,-1/2). Conservation of isospin then requires that the E" hypernucleus be a pure (3/2,-3/2) state, and some suppression of the conversion width from nuclear matter estimates i s expected. In very l i g h t nuclei t h i s quenching can be substantial, but the drawback to t h i s explanation i s that i t i s operative mainly for the ( K " , i r + ) reaction and, therefore, not applicable to either of the o r i g i n a l 2?Be or ^iC r e s u l t s . The ^iC hypernucleus i s a mixture of isospin 3/2" and 1/2 states and no suppression i s expected. Worse, in^lBe, which i s pure isospin=1, Gal's c a l c u l a t i o n s [2] predict that the p-state should be 20-25% broader than the nuclear matter estimate. 7 M H Y " M A ( M e V ) 150 175 200 225 250 275 300 325 350 300 250 > 200 at CM w 150 h-z z> o u 100 50 0 50 25 0 -25 -50 -75 -100 -125 -150 -175 B A (MeV) i i i — i — i — i i 25 0 -25 -50 -75 -100 -125 B 2 (MeV) F i g . 1. Spectrum o b t a i n e d from the ( K ~ , T T ) r e a c t i o n on s B e at a kaon momentum of 720 M e V / c . The u energy spectrum i s p l o t t e d as the d i f f e r e n c e between the the t a r g e t and h y p e r n u c l e a r masses ( M H Y * " M A ). The I and A b i n d i n g energy s c a l e s are a l s o g i v e n . A l l r e s u l t s a r e taken from r e f . [ 3 ] . 8 A p a r t i c u l a r l y i n t r i g u i n g explanation has been proposed by Gal, Toker, and Alexander [11]. Rather than seeking a mechanism to suppress conversion, they have pointed out that in special circumstances strong absorption can produce very narrow bound states embedded in the continuum. Using a phenomenological potential consistent with E" atomic data [12], they demonstrated that the strong absorption could cause an S-matrix pole in the unphysical, t h i r d momentum quadrant to cross into the physical, second quadrant and thereby become indistinguishable from an unstable bound state pole. However, i t has been pointed out by Stepien-Rudzka and Wycech [13], that the E" atomic data are sensitive primarily to interactions at the nuclear surface, leaving the interactions in the high density central region almost completely unconstrained. As a re s u l t , predictions for hypernuclei based on phenomenological analyses of E" atoms are inconclusive. For the moment then, no more w i l l be said of these unusual states, but they w i l l be discussed again in d e t a i l in chapter 4 where they are found to arise in a coupled channels c a l c u l a t i o n . A l l of the above models are mainly applicable to p-levels though. For s-state hypernuclei they generally pr.ed.ict very broad widths in agreement with the c l a s s i c a l 9 estimates [1,2]. One c a l c u l a t i o n which does predict f a i r l y narrow s-states (and i s closest in philosophy to our approach) i s the recent work of Stepien-Rudzka and Wycech [13]. Stressing the importance of Pauli blocking in reducing the EN->AN conversion width, they constructed a simple, l o c a l I-nucleus potential that was consistent with Z" atomic data but which also incorporated exclusion e f f e c t s for high d e n s i t i e s . Their model indeed produced r<10 MeV s-states in l i g h t nuclei but some rather severe approximations used to eliminate the energy dependence and n o n - l o c a l i t i e s in the potent i a l made i t unclear whether the res u l t i n g potential remained consistent with two-body EN interactions. In addition, there was no attempt to calculate the l i f e t i m e of p-states, where one would expect that Pauli suppression would not be as s i g n i f i c a n t because of the smaller overlap of the E wavefunction with the nucleus. It i s impossible to decide then i f t h i s model can also reproduce the observed p-state widths. A l l of the above proposals have some v a l i d i t y for the narrow range of examples they consider. Nonetheless, no attempt has been made to decide whether E states are expected to be narrow in general or only in special i s o l a t e d cases. Our aim i s to answer t h i s question by ca l c u l a t i n g 10 the I single p a r t i c l e p o t e n t i a l as accurately as possible, using the elementary EN scattering information as input. A separable p o t e n t i a l model i s developed in the following chapter to describe the coupled EN-AN systems. It i s argued that a convincing reproduction of the low energy data requires the inclusion of tensor and P-wave interactions. Perhaps more importantly, i t i s found that the conversion amplitude i s strongly energy dependent, suggesting that great care must be exercised in defining the E-nucleus p o t e n t i a l . With t h i s r e l i a b l e description of EN scattering as the foundation, the E single p a r t i c l e potential i s developed in chapter 3. A f a i r l y d e t ailed derivation of the coupled channel G-matrix i s presented, with special attention given to the energy va r i a t i o n a r i s i n g from nucleon binding and centre of mass motion. Pauli exclusion e f f e c t s are also e x p l i c i t l y introduced since, in s-states at l e a s t , t h i s i s expected to produce a sizeable suppression of the conversion width. F i n a l l y , following from these purely formal developments, a minimum number of ph y s i c a l l y reasonable approximations are made to reduce the pot e n t i a l to a calculable form. This careful treatment of the energy dependence of the pote n t i a l creates i t s own complications. Chapter 4 i s devoted to a consideration of these technical d e t a i l s , which include the self-consistent solution of the Schroedinger equation, and the interpretation of the complex eigenvalues as poles in the multi-sheet domain of the S-matrix. This l a t t e r discussion then leads to the necessity of a n a l y t i c a l l y continuing the Pauli exclusion operator to complex momenta. With these d i f f i c u l t i e s resolved, the self-consistent (complex) eigenvalues for the L=0,1 states in l i g h t 1° hypernuclei are presented in chapter 5. Remarkably, we find that a r i s i n g from two quite d i s t i n c t mechanisms both the s and p states are long l i v e d . In s-state Pauli e f f e c t s are found to suppress conversion by as much as 50% from the semi-classical estimates. By contrast, in p-state where the exclusion p r i n c i p l e i s r e l a t i v e l y unimportant, the re s u l t i n g strong absorption can create extremely narrow quasi-bound states embedded in the continuum. F i n a l l y , chapter 6 i s quite b r i e f and the findings of the study are simply highlighted. Some possible improvements of the c a l c u l a t i o n and di r e c t i o n s for experiments are also suggested. 12 CHAPTER 2 TWO-BODY EN-AN SCATTERING The foundation of our aim to calculate E-hypernuclear states i s the description of the underlying EN interactions. This description i s complicated considerably by the coupling of the EN and AN channels. in addition, despite the sca r c i t y of EN data, there are clear indications that contributions from L>0 p a r t i a l waves are important. Even at energies as low as 5 MeV the E"p-»E~p and E~p->An cross sections show marked forward/backward asymmetries [14,15], indicating the presence of P-waves. It can also be expected that, because of the large AN momentum, the E"p( 3S!)->An( 3D,) t r a n s i t i o n i s s i g n i f i c a n t . We remark that the low energies of the EN scattering are consistent with n o n - r e l a t i v i s t i c kinematics. Clearly 5 MeV of kinetic energy i s ne g l i g i b l e r e l a t i v e to the EN mass of 2135 MeV. Even the EN-AN mass difference of 80 MeV i s small compared to the AN mass. So a l l discussions to follow are in the Schrodinger picture with the energy scale chosen to be zero at the EN threshold. Our program i s to f i t a l l of the low-energy EN scattering data within a separable potential model and which accounts e x p l i c i t l y for the multi-channel and multi-partial-wave features of the interactions. For spin-zero p a r t i c l e s a single-channel potential which i s 1 3 separable in each p a r t i a l wave can be defined as V(k',k) =Z (Jiii-)Vg(k' ,k)P £(k'- k) (2) with V^k' ,k) = v^(k' )Xv^(k) and X i s -1(+1) for a t t r a c t i v e (repulsive) interactions. A more convenient notation i s to write the potential (operator) in bra-ket form as Vg= | v^ >X.<vg | , with |v£> defined by v^(k) in momentum space. The more complicated case with coupled channels and p a r t i c l e s with spin i s discussed l a t e r , but for now we mention that there are several advantages to using separable models. It w i l l be shown that in each p a r t i a l wave the scattering amplitude can be solved a l g e b r a i c a l l y . In addition, separable models can reproduce NN phase s h i f t s over several hundred MeV [16], The r e l a t i v e s i m p l i c i t y of the model w i l l also allow us to perform sophisticated nuclear calculations with a minimum of purely technical complications. 2.1. The Physical Basis for Separable Interactions The more important consideration, of course, i s whether there i s any physical motivation for the separable form. It w i l l be shown that i t may be a good approximation to the 'true' po t e n t i a l i f the scattering amplitude i s dominated by a bound state or resonance. F i r s t , we decompose the t-matrix into i t s spectral representation as 14 t(E) = V+Y V|*n><#n V + / dE' V *(E')><*(E') V (3) n h + B„ J FT^F 1  with V the potential operator (not necessarily separable), |*(E')> i s the scattering wavefunction, and where the summation extends over a l l the (assumed) bound states |*n> with the B n(>0) the corresponding binding energies. Clearly for energies E 'near' to -B n, t h i s pole term w i l l dominate in t ( E ) , and the residue i s separable. 1 Conversely, i t i s straightforward to show that the separable form factor i s simply related to the bound state wavefunction (when i t exists) [16]. The wavefunction |•> obeys the homogeneous Lipmann-Schwinger equation |*> = -(B+K)"1V|*> (4) where B (>0) i s the binding energy, and K i s the kinetic energy operator. If i t is assumed that the underlying potential i s separable, as in eq.(2), then i t follows that |*> = (B+K)- 1 |v><v|4» ( 5 ) or v(k)= (B+kV2m)*(k) [<v | •>] which defines a potential which w i l l reproduce the wavefunction exactly. 1 For degenerate bound states, the residue i s modified to become a sum of separable terms [17]. 1 5 For the case in which t i s dominated by resonance poles, rather than bound states, the demonstration i s a l i t t l e more involved but the re s u l t s w i l l not be s u r p r i s i n g . A simple derivation of Lovelace's result [17] i s presented here. Let us define a new potential v, related to the V in eq.(3) above by the r e l a t i o n w(p,q)=e i 6V(e" i 6p,e- i Sq) , (6) with 6 an a r b i t r a r y phase parameter. S i m i l a r l y , a modified t-matrix (T) i s introduced as the solution of the Lipmann-Schwinger equation T(p',q';E)^(p',q')W dk k 2 v (p' ,k ) r ( k, q' ; E) (7) J (E*-kz/2m) where p' i s related to p by p'=pe"x^, and s i m i l a r l y q* =qe-ifc . The integration path can be d i s t o r t e d through the substitution k-fr-ke'^k (|6|<n/2), which rotates the contour O^k^00 about the o r i g i n into the fourth quadrant. Of course the d i s t o r t e d contour must not now enclose the pole at k2=2mE. The integrand dies quickly enough for k-»oo " that when the rotated contour i s ' joined to the real axis at i n f i n i t y , the i n t e g r a l along the i n f i n i t e connecting-arc 1 6 gives zero contribution. Equation (6) has then become r(p',q';E)=w(p',q')+e-is/dk k 2 i/(p' ,e~iS> k) r (ei% k .q' ;E) (8) J ( e z x b E-kz/2m) If eq.(8) i s multi p l i e d to the l e f t by e" 1 & i t i s found, using eq.(6), that t h i s i s just the Lipmann-Schwinger equation for t at the new energy e 2 ^ E , and therefore t (p,q;e 2 i sE)=e" i 6 r ( e - i & p , e - i & q ; E ) (9) Equation (9) defines the analytic continuation of t to the unphysical energy sheet and c l e a r l y , i f t has a resonance pole at e 2 1 & E then T has a pole at an energy E. Further, the phase 6 can be chosen such that E i s a bound state pole of v so that the contribution of t h i s pole to T w i l l be separable, in analogy with eq.(3). 'Bound state' is used here in the sense that the poles l i e in the second quadrant of the momentum plane and the wavefunction therefore decays exponentially at large distances. However, the pole i s not on the imaginary axis and so i s unstable and decays exponentially in time as well. This kind of pole w i l l be encountered throughout our discussion of hypernuclei as being the correct bound state description. It has been shown that i f the scattering amplitude is 17 dominated by bound states and/or resonance poles i t i s reasonable to expect the separable model to be a good approximation to the t-matrix. This observation w i l l be reassuring l a t e r when i t i s found that the EN amplitude is strongly influenced by a pole near the EN threshold. 2.II. QBE Predictions of the EN-AN Interactions As mentioned e a r l i e r , the EN cross sections are not well-determined and, therefore, as in a l l model ca l c u l a t i o n s , we need some reasonable premise for constraining the parameters. The most extensive treatments of EN to date are the one-boson-exchange calc u l a t i o n s of Nagels e_t a_l [18-20] and the i r results should provide s o l i d guidance for our f i t . Their philosophy i s that NN data can be described well in an OBE picture and that, with the assistance of SU(3) and SU(6) to f i x the r e l a t i v e strengths of coupling constants, the model can be extended consistently to the YN interactions. A l l of the i r c a l c u l a t i o n s include the exchange of nonets of pseudoscalar and vecto.r mesons, and uncorrelated two-pion exchange. Recent works also consider the contributions from a nonet of heavy, scalar mesons. The only phenomenology entering their models i s hard-core repulsion, and one of the main differences between models i s 18 the assumptions made regarding the core r a d i i in d i f f e r e n t channels. These r a d i i , plus F / ( F + D ) r a t i o s for meson couplings are treated as free parameters for f i t t i n g a l l the EN and AN data. The r e s u l t i n g f i t s are impressive, with a x 2/data-point less than one for YN in a l l models. Although in d e t a i l s the model predictions d i f f e r , in general conclusions they do not. The conclusions of t h e i r best YN f i t (model D [ 1 9 ] ) , which are relevant to our present purposes are summarized below. (i) isospin 3 / 2 : (a) *S0: a t t r a c t i v e . (b) 3 S , : weak and repulsive, ( i i ) isospin 1/2: (a) ' SQ : repulsive. (b) 3 S , : a t t r a c t i v e , with the e l a s t i c E"p cross section dominated by 3 S 1 - > 3 S 1 t r a n s i t i o n s , and the E~p-*An conversion dominated by 3 S 1 - > , 3 S 1 and 3 S 1 - * - 3 D 1 t r a n s i t i o n s . (c) 3 P i : a t t r a c t i v e , with the forward/ backward asymmetry in the cross section due mainly to 3 S , - 3 P , interference. In fact, t h i s model also suggests that the 1 = 3 / 2 'P, interaction i s important to the E*p description, but to include t h i s term in our model would require at least one extra parameter and, since our primary concern i s the 19 coupled EN-AN systems, such an additional freedom does not seem warranted. These results w i l l be r e v i s i t e d shortly, after specifying our potential model, but for the moment we only remark that our i n i t i a l suspicion that L>0 EN-AN interactions are s i g n i f i c a n t i s supported by Nagels et a l . 2.111 . T-Matrix Normalization and EN Cross-Sections Before proceeding to the s p e c i f i c form of the two-body potentials used in t h i s work i t i s worthwhile to describe the normalizations and re s u l t i n g EN-YN cross-sections. For two spin-1/2 p a r t i c l e s t can be expanded in the basis of the spin spherical harmonics |LSJM> [21], which are the eigenfunctions of the spin, o r b i t a l , and t o t a l angular momenta (S,L, and J, res p e c t i v e l y ) , and also of the z-component of t o t a l J (M). The normalization of the basis states i s chosen such that t= ^ |L'S"JM><L'S,'JM|t |LS'JM><LS'JM| (10) S'S" It has been assumed i m p l i c i t l y of course that the t o t a l angular momentum J i s a good quantum number in the int e r a c t i o n . The spin and o r b i t a l angular momenta need not be conserved separately however. This means that, in 20 addition to the expected tensor interaction 3S 1- 3D 1, there are also spin-changing t r a n s i t i o n s such as 'P,- 3?,, 1D 1- 3D 1, et cetera. In th i s work these l a t t e r interactions are ignored. P-wave i s the lowest angular momentum state in which they occur but, according to the OBE ca l c u l a t i o n s , the 'P, amplitude i s expected to be small in the 1=1/2 channel, and the 3P, small in 1=3/2. It i s reasonable to conclude that the 1P,<—*3P1 t r a n s i t i o n s are unimportant r e l a t i v e to the e l a s t i c 1=1/2 3P, and 1=3/2 1P 1 amplitudes. To be more s p e c i f i c , l e t us project out the t-matrix element for scattering from an i n i t i a l state of momentum |k.>, and spin \Sv> (with v the z-projection of S), to a f i n a l state | k_' > and |Sv'>. Then i t follows from eq.OO) that JS' <k' ; S i / ' 11 t k ; Sv>= T" < n ' - S i / ' |L'S'JM>t,,, (k' ; k ) <LS ' JM I Sv; J) k > JMU (1D L'S' This result becomes a l i t t l e more familiar i f the spin spherical harmonics are expanded in the basis functions |Lm>|Sv> as, for example < n i ; S i / ' |L'S' JM>= £ < n i ; S i / ' |L,m,S ,i/"><Llm,S ,i/,,|JM> m'v" M< (12) where <LmSv|JM> i s a Clebsch-Gordan c o e f f i c i e n t and Y L i s a spherical harmonic. The t-matrix element of eq.(1l) in th i s 21 representation i s < k ' ;Si/' | t | k;Si/>= £ <LmSi/1 JM><L'm'Si/' | JM> (13) mm' Y^n' ) Y L ( n ) t L < ^ ( k ' ; k ) Independent of the angular momentum decomposition, t may also be expressed in an isospin basis, assuming t o t a l I-spin to be conserved. In analogy with the above derivation, for scattering from an i n i t i a l state of I-spins | Ij, i 2 > 11 1 i 1 > to a f i n a l state 11^ [ i»[ > 11' i ' > <I ( i ' l a i a l t |I ,i,I 4i a>« £ <I ( i ' I£ i£ | I i><I , i , I a i * | I i> I i (14) <Ii|t|Ii> The isospin wavefunctions of the relevant E,A, N pairs are |r-p>=| 1-1>| 1/2 l / 2 > = > / 2 / 3 ' | 1/2>+n/T73" |3/2> I r°n>= | 1 0> [ 1 / 2 - 1 / 2 > = -X / 1 / 3 ' | l / 2 > + i / 2 / 3 " | 3/2> * (15) |I +p>=|1+1>|l/2+l/2>= |3/2> |A n>=|0 0 > | l / 2 - l / 2 > = | l / 2> With these states, the t-matrix decomposes as <E-p|t|E-p>= 2 /3 t ( l / 2 ) + 1/3 t ( 3 / 2 ) <A n|t|E-p>= J 2 / T t ( 1 / 2 ) V (16) <E°n 111 E-p>=-yJT / 3 t (1 / 2 ) +yfT / 3 t (3 /2) <E*p|t|E*p>= t ( 3 / 2 ) . 22 We are in a position now to derive the d i f f e r e n t i a l cross-sections for the processes described in e q s . ( 1 4 ) . With our 6-function normalization of momentum states, the t-matrix has the same d e f i n i t i o n as that used by Goldberger and Watson [ 2 2 ] . Their re s u l t for the cross-section describing scattering from an i n i t i a l state |o> of spin S and z-projection v, to a f i n a l state |^ > of spin S and z-projection v' i s dor ( S i / ' ; S i / ) = ( 2 ! L ) a 6 o ( E 6 - E 0 C ) 6 3 U » + E a ) | t a - ( S i / ' ; S y ) | 'dk^dfi^ ( 1 7 ) ' where v a i s the incident flux = | k_ocl/ moc an& iip'Ep a r e t n e f i n a l state momenta. The 6 functions allow us to integrate immediately over p_£ and the magnitude of k^, with the result do- ( S i / ' ; S i / ) = ( 2 n ) "kpn^mp I t ^ S v ' ;Sv)\ 2 d 0 p ( 1 8 ) Experimentally, only the unpolarized cross-sections have been measured . Since the incident beam comprises a random mixture of spins, and orientations, e q . ( l 8 ) should be summed over a l l S and averaged over v - that i s , we sum over v and divide by the t o t a l possible spin projections ( 2 S ^ + 1 ) ( 2 S N + 1 ) . We should also sum over v' since in the f i n a l state a l l orientations are detected. The unpolarized cross-section i s then 23 d7 =(2ir)"k^m(Xm^(1/4) I U k ^ S v ' y ^ S i / ) 12dSl$ (19) p k^  Svy' p If the scattering amplitude f w ^ i s defined in terms of the t-matrix by f ^ =-4irf / kftnvmft t ^ A (20) then the familiar r e l a t i o n d * ^ = | f ^ | 2 d n r e s u l t s . The summation over v,v' can be performed to simplify eq.(19). For the moment we concentrate on just the | t | 2 term, which . i s 11 ( k ' , S w ' ;k , S i / ) | 2 = < L m S i / | JMXL'm 'Si/' | JM><L"m"Si' | J \M' > T T ' i l'\"\u' kA1 M ' " * aiim <L"'m"'sy' |j'M'>Yj(n')y|l(n') (21) Y^ (n)Y^ fn)tJ£(k';k )t^(*k';k ) Spe c i a l i z i n g to the case where the z-axis i s the incident d i r e c t i o n , m=m" = 0, M=M'=i/, and m' =m"'-v-v' , eq. (21) s i m p l i f i e s considerably since the spherical, harmonics become vi' tfi,1"* /—, Y.,(n' )Y.„,(n' ) = (-) v" v £<L'0L"'0|10><L'i/-i/,L" ,i/'- l/|10> 1 L.1 L" ' Pj^k-k') (22) with Y0(n)Y0„(n)= LL"/4n , and ( 2L+1 ) 1 ' 2 . LI It can be shown that the summation in eq.(21) plus the sum over just gives a product of Wigner 6J symbols 24 (appendix I ) , and the c r o s s s e c t i o n takes the convenient form d7 =ff2(kfl/kttme,mfl.) (1/4) H LL'L"L" ' 1 2 J 2 J 2 ' P, (k'k) (23) (L L" l \ f L ' L - ' A ( L " ' J ' S W J ' L" S ) t** t *L** \0 0 O^VO 0 0/ \ 1 L' J ) \ 1 J L j L L L L In our case, where no s p i n t r a n s i t i o n s are al l o w e d the above sep a r a t e s i n t o s i n g l e t , and t r i p l e t c r o s s - s e c t i o n s . 2.IV. The Coupled-Channel, Separable EN-AK P o t e n t i a l s We assume that the c h a n n e l - c o u p l i n g p o t e n t i a l s are a l l rank-one s e p a r a b l e . That i s , the p o t e n t i a l o perator c o u p l i n g any two channels (a,*) V«^ can be w r i t t e n as |vc<>x^<vp|, i n our e a r l i e r n o t a t i o n . For N coupled channels, the t r a n s i t i o n o perator d e s c r i b i n g s c a t t e r i n g from the channel |o> to the channel |fi> (where o,$ r e f e r to the set of a l l quantum numbers f o r the s t a t e , i . e . L,S,J,I, p l u s a channel l a b e l ) , i s i n g e n e r a l N t B a - Iv6>*6a<va| + |v6>X; A6Y<vY|Gj *y* • { 2 4 ) where X ^ — H + l ) f o r a t t r a c t i v e ( r e p u l s i v e ) i n t e r a c t i o n s , and Go i s the r-channel Green's f u n c t i o n , d e f i n e d by Gj(E) - (E + Emy - (mj+mN) -K 2/2u Y+ i e ) " 1 , (25) with Im r the sum of the masses of the p a r t i c l e s i n that 25 channel, and ny t h e i r reduced mass. F o l l o w i n g the procedure of Londergan et a l . [ 2 3 ] , we impose the r e s t r i c t i o n t h a t a l l c o n s t a n t s X are the same s i g n w i t h i n any set of coupled c h a n n e l s . In t h i s case the s o l u t i o n has the p a r t i c u l a r l y simple form (appendix I I ) K > x e a < v a l (26) Ha • N ' - Y I / Y<V Y | GJ|V Y> where the summation extends over the d i a g o n a l elements of a l l coupled channels. The form f a c t o r s are chosen to have the b e h a v i o u r 2 <p|vY> = v Y (Bo + P 2 ) " 1 f ° r 1 • 0 • = v ^ p B l ^ B i + P 2 ) " 1 for £ = 1 , (27) = V Y p 2 (82 +P 2 ) for £ = 2 , where the fi^ are the i n v e r s e ranges, and v r the s t r e n g t h s of the i n t e r a c t i o n s . With the n o r m a l i z a t i o n of the t-matrix given by e q . O O ) , the o v e r l a p i n t e g r a l s <v|G e|v> i n eq.(28) are found to be /d « 2 v | ( K 2 ) -nuvv 2 , r , „ „v 1 . E * * ™ ^ . . * ( i S ^ J f L«/2»(« 2 -«?) • - y ] . • « - 0 . 2 T h i s c h o i c e of momentum dependence ensures t h a t the phase s h i f t 6l~p2l*'i f o r p->0. 26 [ » « ( » ' • * ? ) • I K ? ] . [ I / 1 6 B ( B 6 + 5 B V 2 + 15B2KY-5KY)+!K5] ,£=2 , (28) e2(e2+KY) (e2+<Y 2.V. Coulomb Modifications to the Two-Body T-Matrix So far no mention has been made of the Coulomb inter a c t i o n and yet i t s inclusion i s c l e a r l y important to a complete description of E*p scatt e r i n g . It w i l l soon become apparent that another c a l c u l a t i o n a l advantage of separable potentials i s that i t i s s t i l l possible to solve for the strong interaction amplitude in closed form even in the presence of the Coulomb potential [24,25]. In analogy with eq.(l3), the partial-wave expansion of the scattering amplitude i s f (k' ,Sv' ;k,S»/)=4n £ <LmSi/1 JM><L' m' S i /' | JM> V n k ) Y L ( n i } f ^ ( k ; k ' ) (29) with f j * ( k ; k ' ) = [exp(2iA|£ )-6 L. L]/2ik (30) where A^f i s the phase s h i f t in that channel. If the underlying potential i s the sum of the Coulomb and a short-range interaction then i t i s p r o f i t a b l e to write A as the sum of the pure Coulomb phase s h i f t « L and a 27 Coulomb-modified strong phase s h i f t 6 ^ • The p a r t i a l amplitude can then be rearranged to read f ?f (k';k)=6l(, (e 2 l° L-1)/2ik+e 2 i ( 5 l-(e a i C & S-6.. l )/2ik (31) The f i r s t term i s just the 1-th partial-wave Coulomb amplitude f L(k;k') (apart from the Kronecker d e l t a ) , and the second term i s the perturbed strong amplitude f , L(k';k). The summation over a l l angular momenta variables for the Coulomb amplitude can be calculated in eq.(29) using the orthogonality of the Clebsch-Gordan c o e f f i c i e n t s with the result [22] oo f C(6) = £ (21+1 )f J(k;k' JP^k-k' ) (32) 1=0 = - n e x p [ 2 i t f 0 - i n l n ( s i n 2 (9/2) ) ] 2ksin 2 (e/2) and n = am/k , g 0 = r(1+ in) , *, =c0 + rtan" 1(n/s) r(1-in) s Two remarks can be made about t h i s r e s u l t . F i r s t , i t should be re a l i z e d that when forming | f | 2 to calculate the cross-section, an o v e r a l l factor of exp(2itf 0) disappears c c T S since i t i s common to both f L and f L< Lin a l l p a r t i a l waves. C TS Second, the phase 6L/J_ i s not the same as i t would be without the' Coulomb interaction present. In t h i s work we assume 28 that the two are the same, and i t i s the error introduced by t h i s approximation we now wish to investigate. At t h i s point we consider the example in which there are two strongly coupled channels with the Coulomb potential present in only one of these, as in E~p-An (1=1/2) for example. The results can be e a s i l y generalized to the multi-channel s i t u a t i o n , or s p e c i a l i z e d to a single-channel case. The derivation which we follow i s an extension of the single-channel result of Haeringen and Wageningen [24]. The t o t a l t-matrix T for Coulomb plus short-range potentials i s the solution of the Lipmann-Schwinger equation T(E)=(V C+V S)+(V C+V S)G°(E)T(E) (33) where V,T, and G° are the matrices v s = T = with v i j , v c the strong, and Coulomb potentials, and C'x^ the free two-body propagator ( E J ^ - H 0 ) ~ 1 . A pure Coulomb t-matrix t c can be defined as the solution of t c= ( t T , (A =V C+V CG° ( t f i 0\ \0 0) \0 oJ (35) 29 and the t o t a l t-matrix written as the sum of t c and a c Coulomb-modified strong amplitude t s . After a l i t t l e c algebra i t follows that t<. i s expressable a s 3 t|(E) = [l+t c(E)G°(E) ]*t|(E)[ 1+G°(E)t c(E) ] (36) where t s ( E ) i s the solution of t|(E)=V s+V sG C(E)t5(E) (37) and G c i s a Coulomb propagator matrix, defined as G C =G°+G°t cG°= (Gf, f j \ \0 Gj (38) with G? 1 = (E"I"1-H0-vc ) - 1 The only difference between t s ( E ) and the purely strong t-matrix [eq.(24)] i s the replacement of the free propagator G° by one appropriate for interactions in the presence of the Coulomb p o t e n t i a l . If the v£4 are a l l rank-one i i * * ~ separable |v^ >X <vj I with a l l equal to X., then t^fE) can be obtained a l g e b r a i c a l l y : * t . ( E ) g 1V>X<V| = |V>D°(E)<V| (39) (1-X<V|GC(E)|V>) where |V> i s the column matrix of |v^>. From eq.(36), Coulomb modified form factors may also be defined as 3 The angular momentum projections of the operators in eqs.(36) and (37) obey the same equations. 30 |VC(E)>= (l+t c(E)G°(E))|V> = /(l+t 1 , (E)G, , (E) ) | v , >' (40) In the notation of eq.(39) the t-matrix becomes t|(E)=|V c(E)>D C(E)<V c(E)| (41 ) To compare the phase s h i f t predictions of eq.(41) with the corresponding results where the Coulomb potential has been turned off we s p e c i a l i z e considerably to the case of single-channel scattering with rank-one Yamaguchi form factors U 2 + p 2 ) - 1 . c The solution of D(E) in eq.(39) involves some n o n - t r i v i a l integration, leading to the hypergeometric functions. ' However, to lowest order in the fine-structure constant a, Haeringen [25] has shown that the Coulomb modified scattering length and e f f e c t i v e range are where v=am/p, r i s Euler's constant (0.5772...), and a s , r s are the corresponding values with no Coulomb in t e r a c t i o n . 1 =e-«v-2*i/(r+ln(4i/) )+0(i/ 2) a| as r$ =r* e-"v+2i//#(1+2/(3#as ))+0(i / 2 I 2 2 5 (42) 1 = - * ( * 3 / U i r m v 2 ) + l/2) 31 r s =(4/0as+3)/2* (43) 2 and v| i s the strength of the in t e r a c t i o n . At low energies the phase s h i f t s are related to the e f f e c t i v e range expansions by kcot6 s =l/a s+r s k2/2+0(k*) : No Coulomb (44) 2 i r n kcot6§+2knh(n) = 1/a|+rf k 2/2+0(k«) : Coulomb (45) earn.-! with h(T,)=-r+ln n + n 2 t[ 1 (1 2 + n 2 ) ]" 1 1=1 and since a l l the IN data e x i s t s for energies less than 8 MeV and V=.0185, t h i s approximation should be adequate for our purposes. Ant i c i p a t i n g the results of th i s comparison of the phase s h i f t s , the I +p scattering parameters of section 2.VI. w i l l be used. These values are 0=1.053 f nr 1 , and in 1 S 0 as=4.338 fm, r s=3.68l fm, (a£=3.380 fm, r§=3.493 fm) , and in  3S, as=-.358 fm, rs=-7.232 fm, (a|=-.399 fm, r£=-6.766 fm). The phase s h i f t s in 1=3/2 'So, and  3S^ corresponding to the two expansions eqs.(42) and (43) are given, in table I for energies less than 10 MeV. In the energy region of the IN data (~3<E<~7 MeV), the Coulomb modifications to the 32 E (MeV) 'So 6S° °s 6S° 3s, 1 .0 30.4 22.2 -3.26 -2.44 2.0 35. 1 29.8 -4.45 -3.75 3.0 36.8 34. 1 -5.27 -4.85 4.0 37.4 35. 1 -5.90 -5.54 5.0 37.5 35.5 -6.40 -5.98 6.0 37.2 35.8 -6.80 -6.33 7.0 36.9 35.7 -7.13 -6.68 8.0 36.5 35.6 -7.41 -6.97 9.0 36.0 35.3 -7.65 -7.23 10.0 35.5 35.0 -7.85 -7.46 Table I. Comparison of the Coulomb-corrected phase-shifts 6% with the purely strong phase 6S in £ +p scattering 33 strong phases may be neglected, to the extent that the correction i s less than 5% of 6 S and yet the errors in the data t y p i c a l l y exceed 20%. In E~p interactions the influence i s expected to be even smaller since the Coulomb potent i a l i s not present in the coupled AN channel and the propagator i s unmodified. 2.VI. Results of the F i t to the EN Data With the t-matrix completely defined for EN scattering our problem now i s to minimize the number of parameters in the model without s a c r i f i c i n g any of i t s essential features. The d i f f i c u l t y i s e s p e c i a l l y clear since, regardless of the chronic shortage of EN data, i t was decided e a r l i e r that the description of the 1 = 1 / 2 channel required P and D waves to be included. Consequently we are obliged to make some rather severe assumptions about the interactions to arrive at a unique f i t . F i r s t , the range p" 1 i s a r b i t r a r i l y chosen to be the same in a l l channels. C l e a r l y , in a more r e a l i s t i c c a l c u l a t i o n the AN pote n t i a l i s expected to be of shorter range than the EN i n t e r a c t i o n . For example, the longest range contribution to the EN potential i s generated by one-pion exchange, whereas in the e l a s t i c AN interaction two piohs must be exchanged. On the other hand, each ir in 34 AN—>AN scattering i s emitted with roughly 80 MeV of energy corresponding to the E-A mass diffe r e n c e . To some extent t h i s compensates for the shorter range of the two tr process. Second, in S state the 1 S 0 strengths are related to 3S, in each isospin channel by the weighting with the 2* """"N signs the same as (t^,- £ N ) (g_^. e_^) , in agreement with the OBE c a l c u l a t i o n . We also include the 1=1/2 3 P j waves assuming the potential strengths to be a t t r a c t i v e , and equal in a l l t o t a l J states. F i n a l l y , an a t t r a c t i v e 3D, interaction i s included in the AN channel, but i s ignored in EN because of the large kinematic suppression of D-waves at low energies. Assuming the 1=1/2 strengths to be the same in both EN and AN channels, t h i s gives a t o t a l of fiv e free parameters; a single range 0 " 1 , and one strength parameter in each of the 1=1/2 3S,, 3 P j , 3D,, and 1=3/2 3S, channels. The f i v e parameters of the model were adjusted to f i t a l l the e x i s t i n g EN data, which comprises: (i) e l a s t i c E"p,E*p d i f f e r e n t i a l , and t o t a l cross sections [15], ( i i ) conversion E"p-»An d i f f e r e n t i a l , and t o t a l cross sections [14], and the ( i i i ) charge-exchange E~p->E°n t o t a l cross section [14]. 35 Because of the smal l angular r e g i o n e x p l o r e d by the experiments, the t o t a l c r o s s s e c t i o n s quoted i n those works are the d i f f e r e n t i a l c r o s s s e c t i o n s i n t e g r a t e d over the small angular i n t e r v a l and then m u l t i p l i e d by a f a c t o r a p p r o p r i a t e f o r an i s o t r o p i c d i s t r i b u t i o n . That i s , „ * " / - ' ° S 8 H A X D < , ( E L c o s e M ( N Since with t h i s d e f i n i t i o n the t o t a l c r o s s s e c t i o n s c o n t a i n Coulomb amplitudes, and n e g l e c t p o s s i b l e P-wave c o n t r i b u t i o n s i n e x t r a p o l a t i n g to a l l an g l e s , we have d e c i d e d to f o l l o w the experimental procedure i n comparing c a l c u l a t e d c r o s s s e c t i o n s with the da t a . The r e s u l t s of our f i t are shown i n f i g s . 2 and 3 and i t i s found t h a t , d e s p i t e the heavy c o n s t r a i n t s p l a c e d on the parameters, the f i t i s over-determined, with a x 2 / d a t a - p o i n t of 0.75 f o r the 45 p o i n t s , i n d i c a t i n g that the e r r o r s i n the data have been o v e r - e s t i m a t e d . The b e s t - f i t c o u p l i n g s t r e n g t h s are l i s t e d i n t a b l e I I , and i n t a b l e III the s c a t t e r i n g l e n g t h s and e f f e c t i v e ranges i n the e l a s t i c E'p channel are g i v e n . A l s o , the phase s h i f t s ( 6 ) , and a b s o r p t i o n c o e f f i c i e n t s (n) f o r e l a s t i c Z'p s c a t t e r i n g i n the separate LSJI channels are shown i n f i g s . 4. 36 F i g . 2. The EN d i f f e r e n t i a l c r o s s - s e c t i o n s at 160 MeV/c momentum, a) E l a s t i c I'p r e a c t i o n , b) E*p An c o n v e r s i o n , and c) e l a s t i c E*p r e a c t i o n . 37 39 F i g . 3. The low energy EN t o t a l c r o s s s e c t i o n s , a) E l a s t i c E*p r e a c t i o n , b) E"p An c o n v e r s i o n , c) e l a s t i c E"p r e a c t i o n , and d) E"p-*-E°n charge-exchange. I - 1/2 3 S i 3Pj 3Di I - 3/2 i«; 3 s , l $ 0 164.1*6 0 0 0 -38.31^ 0 3 S j 0 -5^.822 0 -37.581 0 12.771 3Pj 0 0 -14.357 0 3 D i 0 -37.581 0 -25.762 T a b l e l l . B e s t - f i t c o u p l i n g s t r e n g t h s V g X 6 c t v " a ( M e V / f m ) . S t r e n g t h s a r e i d e n t i c a l f o r IN and AN c h a n n e l s , e x c e p t f o r 3 D j w h i c h i s i g n o r e d f o r IN . $ = 1 .0563 f m " 1 i n a l l c h a n n e l s . I «= 1/2 fm I .- 3/2 fm 3 S •1.448 + 10.374 0.515 - 10.595 1.027 + 14.757 3.010 - J0.717 1.045 + 10.461 4.338 3.682 -0.358 -7.232 T a b l e l E . S c a t t e r i n g l e n g t h s a and e f f e c t i v e ranges r i n the e l a s t i c I"p c h a n n e l . F i g . 4 . Phase s h i f t 6 and a b s o r p t i o n c o e f f i c i e n t s n fo r EN-*IN s c a t t e r i n g , a) ' S o 1=1/2, b) 3S, 1=1/2, c) 3Pj 1=1/2, and d) ' S o , * S , 1=3/2. ENERGY (MeV) 46 C o n s i d e r i n g the poor q u a l i t y of EN data i t i s c l e a r t h a t the f i t s o b t a i n e d w i l l not be s e n s i t i v e to even l a r g e changes in the parameters. However, there i s an a c c u r a t e l y known datum, f r e e of P-wave c o n t r i b u t i o n s , t h a t i s s e n s i t i v e to the r a t i o of 3S, to 1 S 0 amplitudes. T h i s i s the c apture r a t i o at r e s t , R = y > (2s + Q ° 2 s + 1 ( Z - p + I ° n ) ( 4 7 ) 4 o- 2 s+i(E~p+An) .+ c 2 s + i ( I " p + I 0 n ) ' where 2S+1=1,3 f o r s i n g l e t , t r i p l e t r e s p e c t i v e l y . The c a l c u l a t e d value of R i s 0.485, i n e x c e l l e n t agreement with the measured value of 0.47±0.01 [ r e f . 18, and r e f e r e n c e s given t h e r e i n ] . With some c o n f i d e n c e we can look at the p r e d i c t i o n s of the model i n more d e t a i l . Most i m p o r t a n t l y i t i s found t h a t the i n c l u s i o n of the 1=1/2 P and D-wave terms i s e s s e n t i a l to the d e s c r i p t i o n of EN-*>AN c o n v e r s i o n . The P-waves comprise l e s s than 2% of the e l a s t i c E"p c r o s s s e c t i o n at 160 MeV/c l a b momentum, but y i e l d n e a r l y 16% of the c o n v e r s i o n c r o s s s e c t i o n . The 1=1/2 3S,-> 3D, t r a n s i t i o n i s even more important, c o n t r i b u t i n g 25% of the c o n v e r s i o n c r o s s s e c t i o n . In agreement with Nagels et a l . , we f i n d t h at the 1=1/2 3 S 1 - » 3 S 1 amplitude dominates the E'p r e a c t i o n s and i s r e s p o n s i b l e f o r 75% of the e l a s t i c c r o s s s e c t i o n and over 50% of the c o n v e r s i o n r e a c t i o n . The 1=1/2 3S, amplitude i s of p a r t i c u l a r i n t e r e s t . In 47 the reaction K~d—*ir~Ap a strong enhancement of the Ap invariant mass d i s t r i b u t i o n has been established near the E*n threshold [26]. It has often been suggested that t h i s enhancement cannot be explained by threshold e f f e c t s alone and that the cusp a r i s e s from either a nearby resonance pole or an unstable IN bound state in the 3S, channel [27]. 1 The status of t h i s pole has been addressed in several model c a l c u l a t i o n s . The OBE ca l c u l a t i o n s of Nagels e_t a l . do not support a bound state, but the i r results are f a i r l y sensitive to the parameters in t h i s respect. In some of t h e i r models no resonances are found [18,20] while others predict a resonance above the E*n threshold on the unphysical energy sheet [19]. Toker e_t a_l [28] have studied the K"d-*-n'Ap reaction in a Faddeev c a l c u l a t i o n . Using only S-wave separable i n t e r a c t i o n s , 2 they found that the data could be reproduced with or without a EN bound state but that the best reproduction of the shoulder in the Ap d i s t r i b u t i o n favoured a resonance pole rather than a bound state. This model (A) placed the pole on the second Riemann 1 This state occurs in the SU(3) |10| representation of the dibaryon system, another member of which i s the deuteron. At least according to exact SU(3) symmetry, one therefore expects a pole in t h i s EN amplitude. 2 Toker et a_l reproduced the EN t o t a l cross sections using only the 1=1/2 3S, interaction which probably r e s u l t s in an o v e r - s e n s i t i v i t y to the pole p o s i t i o n . 48 sheet at a momentum of k=0.162-iO.163 fm" 1. The results of our work support model A of Toker e_t a l , producing a pole above the Z~p threshold at k=0.163-iO.082 fm" 1. The close proximity of t h i s pole to the physical sheet creates very strong energy dependence in the scattering amplitude near threshold. This i s demonstrated in f i g . 5 where, with zero coupling to the 3D, AN channel, the v a r i a t i o n of the o f f - s h e l l amplitude -irmt ( k ' =k = 0 ;E) near threshold i s shown for d i f f e r e n t values of the 3S, coupling strength. This strong energy dependence of the (dominant) 3 S T amplitude w i l l be of p a r t i c u l a r importance to our study of hypernuclear states. Clearly the large enhancement of the E"p—>An conversion near threshold w i l l tend to decrease the l i f e t i m e of the Z-hyperon in the nuclear medium. 49 i 1 1 1 1 1 r i i i i i ' i I i I I -8 -6 - 4 -2 0 2 4 6 8 ENERGY (MeV) F i g . 5 . The energy dependence of the o f f - s h e l l 1 = 1 / 2 3S, amplitude near the EN t h r e s h o l d f o r d i f f e r e n t 3S, c o u p l i n g s t r e n g t h s as shown ( i n u n i t s of MeV/fm). a) The r e a l component of the amplitude f R , and b) the imaginary component fx . 50 CHAPTER 3 THE E-NUCLEUS POTENTIAL Very l i t t l e i s known about the nature of the E-nucleus i n t e r a c t i o n , and one of the few i n d i c a t i o n s we do have comes from B a t t y ' s a n a l y s i s of E" atoms [12], In that work the atomic energy s h i f t s were reproduced with a phenomenological p o t e n t i a l of the f a c t o r e d (tp) form with the medium-corrected EN s c a t t e r i n g l e n g t h a B t r e a t e d as a (complex) parameter. As p o i n t e d out by many authors though [1,9], the value of a B c o n s i s t e n t with the atomic data l e a d s to p r e d i c t i o n s f o r the widths of hypernuclear l e v e l s which are very broad. (For example, i n s - s t a t e ^|C, c a l c u l a t i o n s u s i n g eg.(48) g i v e r~22 MeV [ 9 ] ) . F u r t h e r , t h i s e f f e c t i v e s c a t t e r i n g l e n g t h bears l i t t l e resemblance to the f r e e v a l u e , d i f f e r i n g by as much as an order of magnitude from some EN a n a l y s e s . (e.g. 0.19i fm as compared with the 1.22i fm of the present work ). 3.1. The Need f o r Many-Body E f f e c t s I t i s c l e a r from the above r e s u l t s t h a t n u c l e a r many-body e f f e c t s must p l a y a c r i t i c a l r o l e i n the c o n n e c t i o n between f r e e EN, atomic EN, and n u c l e a r EN i n t e r a c t i o n d e s c r i p t i o n s . Amongst these e f f e c t s , P a u l i (48) 51 blocking of the f i n a l state nucleon in EN-AN conversion and the binding of the nucleons are expected to be most signi f i c a n t . In the atomic s i t u a t i o n , the E i s bound with e s s e n t i a l l y zero energy in a Coulomb o r b i t a l of high angular momentum and, therefore, absorption occurs primarily on the loosely bound valence nucleons at the nuclear surface. This i s a region of low density and correspondingly low Fermi momentum (within a l o c a l density approximation). For example, in 3 2 S the 4f o r b i t a l i s the lowest l e v e l reached by the E" [12], and absorption occurs mainly in a region of ~l / 5 the central density [29], The l o c a l Fermi momentum i s therefore ~60% of the central value, or ~150 MeV/c. Since the momentum of the f i n a l state N in EN—*AN decay i s roughly twice t h i s value Pauli exclusion should not have too large an ef f e c t on the absorptive strength of the potential in t h i s circumstance. A E bound in the ground state of a *hypernucleus, on the other hand, i s confined almost exclusively to the nuclear i n t e r i o r and absorption occurs mainly on the deeply bound s-state nucleons. In t h i s high density central region both nucleon binding and Pauli e f f e c t s make important modifications to E absorption. To appreciate the size of correction that nucleon 52 binding may introduce, consider a E bound with (say) 5 MeV which i s absorbed on an s-state nucleon bound with 35 MeV. The t o t a l energy of the pair i s -40 MeV and upon decay the AN pair emerges with +40 MeV. Phase space considerations alone require that the amplitude for t h i s conversion be reduced by ~30% r e l a t i v e to the atomic case where the EN pair have roughly zero energy. In addition, for the EN decaying at rest the f i n a l state nucleon emerges with 285 MeV/c momentum which i s comparable to the Fermi momentum at the central density. This implies that a large proportion of decays are forbidden because the f i n a l state nucleon i s produced with sub-sea momentum. ( i . e . The naive picture presented above predicts ~50% suppression for Kf=285 MeV/c). These simple estimates have been e s s e n t i a l l y confirmed by Dabrowski and Rozynek [30]. Using an OBE potential of the Nijmegan group [ 1 9 ] in Brueckner reaction matrix theory, they demonstrated that in nuclear matter Pauli exclusion suppressed EN decay by 50% from c l a s s i c a l estimates. On the basis of. these arguments Pauli blocking and nucleon binding are esse n t i a l features in understanding the li f e t i m e s of E states and a complete microscopic description of the hypernucleus must take them into account. 53 3.II. The I S i n g l e - P a r t i c l e P o t e n t i a l We t u r n t o the problem of c o n s t r u c t i n g the E-nucleus p o t e n t i a l with s p e c i a l a t t e n t i o n to many-body e f f e c t s . The coupled equations d e s c r i b i n g the E + A-nucleon system (assuming o n l y p a i r - w i s e i n t e r a c t i o n s ) are ( K z + ^ K i + ^ v ^ l g u , j - E ) | * ( l . . . A ; E ) > - - 2 v i A | * ( l . . . A ;A )> i ( K A + ^ K J + £ V A A + I U U _ (E + Am ZA))|4 ' ( l . . .A;A)> ( 4 9 ) - - J ] v| E|4'(1 . . . A ; r ) > . i The are s i n g l e - p a r t i c l e k i n e t i c energy o p e r a t o r s , V y y ' a n d u i j are two-body YN and NN p o t e n t i a l s , the * are the Y + A -nucleon wavefunctions, and Am^ i s the E-A mass d i f f e r e n c e . The d i v e r s e schemes f o r approximating s o l u t i o n s to many-body equations l i k e ( 4 9 ) form the fo u n d a t i o n of the whole f i e l d of n u c l e a r s t r u c t u r e p h y s i c s . These techniques stem from the o b s e r v a t i o n t h a t , as a good f i r s t a p proximation, the nucleus comprises p a r t i c l e s which move independently - each bound i n an .average p o t e n t i a l generated by a l l the other p a r t i c l e s . The success of the i n d e p e n d e n t - p a r t i c l e model i n p r e d i c t i n g many f e a t u r e s of n u c l e i suggests that the most important c o r r e c t i o n s are due to two-body c o r r e l a t i o n s . 54 S p e c i f i c a l l y , we need to e x t r a c t the i n t e r a c t i o n of a s i n g l e YN p a i r from each of the (A+1)-body equations (49). The sum over YN p o t e n t i a l s i n each channel can be w r i t t e n i n terms of the E + nucleon p a i r , A + nucleon p a i r , s i n g l e p a r t i c l e , and r e s i d u a l i n t e r a c t i o n s as X^/ • , • v * ( 5 0 ) ^ ( v h l Y d h + v y Y U ) ^ - v f z |4'(E)> + v§ A|4'(A)> + V Z | * ( E ) > + [S(vEEI4'(Z)> + V I A I Y ( A ) > ) ' VII*(Z) >1. £MAI^ A)> + VJ[J:|1'(Z)>) = vJj*(A)> + vJr|<r(E)> + VA|Y(A)> i + I"SWA I * U ) > + v A I | v( z ) >) - v A | y(A) >] . LMA -I The V Y are the hyperon s i n g l e - p a r t i c l e p o t e n t i a l s and the aim i s to d e f i n e them such that the c o n t r i b u t i o n from the r e s i d u a l i n t e r a c t i o n s (the terms i n square b r a c k e t s ) i s minimized. In other words, we wish to f i n d the that g i v e s the best p o s s i b l e estimate of the E energy i n an e f f e c t i v e one-body H a m i l t o n i a n . S i m i l a r l y f o r nucleon A we may w r i t e 1 E u ! j c 1 E u u + UA +(X> J " U A ) • i * j i^jVA \j*A / < 5 1' and U A i s the nucleon s i n g l e - p a r t i c l e p o t e n t i a l . N e g l e c t i n g 55 the r e s i d u a l i n t e r a c t i o n s , the hyperon s i n g l e - p a r t i c l e p o t e n t i a l o p e r a t o r s are d e f i n e d f o r m a l l y u s i n g eg. ( 49 ) as Vll*<*>> - X X V W A ( E + A r n Z A - K A - f i - U A - , / 2 5 ^ A U i j - V A - V A A ) " x 2 v A z ) l , | , ^ ) > » j V * l » ( « > - TfrL + 'k ( E - K t . y , - U A - . ) 2 . Z u , . - V I . » t r ) " J ( 5 2 ) j and where, f o r economy of space, i n the equations ( 5 0 ) - ( 5 2 ) the (A+1)-body * -wavefunctions |*(1...A;Y)> have been a b b r e v i a t e d to |*(Y)>. Because r e s i d u a l i n t e r a c t i o n s have been dropped i n the propagators of e q s . ( 5 2 ) , matrix elements of Vy taken between i n d e p e n d e n t - p a r t i c l e s t a t e s w i l l r e s t r i c t the sum over j to the one term j = i . The (A+l)-body equations themselves now become ( K Z + K A + K A . 1 + V A I + V Z + U a + 1 £ U I J - E ) | T ( E ) > = -vJA|»(A)> , < 5 3> ( K A + KA + KA-1 + v A A + V A + UA +7 £ u.J-<E+*"lA>) l * < A > > " "vJi • i^jVA The E K i has been d i v i d e d e x p l i c i t l y i n t o the nucleon-A and ( A - l ) - c o r e k i n e t i c energy o p e r a t o r s to emphasize the three-body nature of the i n t e r a c t i o n . That the equations 56 (53) are indeed three-body e q u a t i o n s , and that the r e c o i l of the core i s necessary i n guaranteeing e l a s t i c u n i t a r i t y has been s t r e s s e d by many authors [31,32]. The IA p a i r can now be i s o l a t e d from eq.(53). D e f i n i n g the YA p a i r wavef u n c t i o n s |*ff«.(IA)>, and |*x«(AA)> as the o v e r l a p i n t e g r a l s < * ( A - 1 ) | * ( ! ) > and <•(A-1)|•(A)>, with < * ( A - 1 ) | the .(A-l)-core wavefunction, eqs. (53) are m u l t i p l i e d to the l e f t by <#(A-1)|. F u r t h e r , s e p a r a t i n g K^.-j i n t o centre-of-mass and i n t e r n a l components, i t i s I N T found t h a t , s i n c e only the average of K ^ + 1 / 2 I u i j i s needed, t h i s i n t e r n a l H a m iltonian can be r e p l a c e d by E - E C K , where Eux i s the IA p a i r energy. F i n a l l y , w r i t i n g the Y and nucleon k i n e t i c energy o p e r a t o r s i n r e l a t i v e , and centre-of-mass terms they can be combined with the core k i n e t i c energy to gi v e K Y + K A + K J J , = K Y A + p 2/2yy , with (54) Uy = m A_j (m N +m Y ) / (m A +m Y ) , and where P i s the YA p a i r t o t a l momentum as measured i n the Y-nucleus r e s t frame. Assembling a l l of the above r e s u l t s we a r r i v e at the coupled YA p a i r equations i n r e l a t i v e c o - o r d i n a t e s 57 (K Z A +P 2 /2ii j:+v z i : +V z +U A -E a a ) | i j ; a o i (ZA)> = - v Z A |*Ao(AA) > , (K A A +p2/2y A +v A A +V A +U A -(E C J a +Amj ; A ) ) | * X a (AA)> = - v A Z | * o a ( E A ) > . (55) As p o i n t e d out e a r l i e r , P a u l i e x c l u s i o n i s expected to p l a y an important r o l e i n the E l i f e t i m e . These e f f e c t s can be i n c l u d e d e x p l i c i t l y by modifying the two-body p o t e n t i a l s Here the |x> are i n d e p e n d e n t - p a r t i c l e s t a t e s f o r the nucleons, and the second term i n eq.(56) p r o j e c t s only onto those s t a t e s unoccupied i n the n u c l e a r ground s t a t e . The presence of the o perator _1_ f o r p r o j e c t i o n on hyperon s t a t e s j u s t r e f l e c t s the f a c t that no P a u l i r e s t r i c t i o n s a pply to the hyperon i n the n u c l e u s . If the p o t e n t i a l Vj- has been chosen by a v a r i a t i o n a l procedure to y i e l d the best s i n g l e - p a r t i c l e e n e r g i e s t, then the d i f f e r e n c e between E j * o o t > - <<r o | v^J icx > | « o>-<<rc | v ^ l #Xot > |o"o> and (c^+e^) | i0oL> i s a second order c o r r e c t i o n , and t h e r e f o r e eq.(55) becomes vyy'to the form QotVyy ' , where i s a p r o j e c t i o n o perator f o r hyperon and nucleon s t a t e s d e f i n e d as Y > K F (56) 58 (K Z A+P 2 / 2u E +Qv L 3 : +V j :+U A-(e G+ e a))|^ 0(lA)> = - Q v Z A J*Xa(AA) > , ( 5 7 ) (KAA +P 2 /2vA+QvAA+VA+U A - (e a +e a +Am I A ) )|*Xo(AA)> = -Qv A Z|* o a(EA)> , where Q p r o j e c t s o n l y onto u n f i l l e d l e v e l s i n the ground s t a t e . In t h i s form i t i s convenient to e l i m i n a t e the AA wavefunction from eq.(57) to g i v e the EA p a i r e q uation (K Z A+P 2 /2u z+Qy Z A +V £ +U A ) 14>aa (ZA) > «= U0«-eo) |* o a(ZA)> , ( 58 ) with the e f f e c t i v e EA complex p o t e n t i a l v ^ d e f i n e d as ' 2 A 5 V Z I + V Z A ( E a ^ a + A m Z A - K A A - P 2 / 2 M A - U A - V A - Q v A A ) ^ ' ( 5 9 ) A s y m p t o t i c a l l y must reduce to the model wavefunctions \ca> as vZr\~^®' T h e r e f o r e , |*0-«.> i s t h e s o l u t i o n of the i n t e g r a l r e l a t i o n l*o«<*A>> - ka> + ^ , ^  . ^  . ^  ^ . ^  Qv Z A |* 0 o(iA)>. ( 6 0 ) I f we now i n t r o d u c e the Brueckner r e a c t i o n matrix which has the p r o p e r t y that r^\ea> = V £ / J * D-oc > t n e n » u s i n g eq.(60), T a " VZA + VZA (% " K Z A - P 2 / 2 p z - V Z - U A ) " 1 Q t a . Here i s the e f f e c t i v e EN i n t e r a c t i o n , i n the sense that a c t i n g on the model wavefunctions i s e q u i v a l e n t to v £ A 59 a c t i n g on the p a i r wavefunction. At t h i s p o i n t we are i n a p o s i t i o n t o d e f i n e the Z s i n g l e - p a r t i c l e p o t e n t i a l . S i n c e V j . ^ i s independent of p a r t i c l e l a b e l s the Z - n u c l e u s Schroedinger equation can be w r i t t e n as ( K L A + H N + A V I K ) l * ( E ) > = E | * ( E ) > , (62) where K^/^ i s the Z - n u c l e u s k i n e t i c energy o p e r a t o r i n r e l a t i v e c o - o r d i n a t e s and H N i s the i n t e r n a l n u c l e a r H a m i l t o n i a n . Provided only the ground s t a t e e x p e c t a t i o n v a l u e of eq.(62) i s needed, E-H N can be r e p l a c e d by M u l t i p l y i n g eq.(62) to the l e f t by the ground s t a t e n u c l e a r wavefunction <0| produces the Z s i n g l e p a r t i c l e equation «l*> » K Z ^ e > + A E < 0 I V £ A I W ( 6 3 ) = ( K £ K + A Z < o | T o t | o > ) |*> T h e r e f o r e , the s i n g l e - p a r t i c l e p o t e n t i a l i s c l e a r l y V£|#> = A Z < O | T < < \a«> (64) S p e c i f i c a l l y , i n t r o d u c i n g the Z s t a t e s *|k>,|.k'> the t o t a l I p o t e n t i a l i s <k'|vz|k> =S/dP •J<p-9><k'. P-q|T0U)|k.p> 4»a(p) , < " ) 60 where q i s the three-momentum t r a n s f e r , i s the nucleon s i n g l e - p a r t i c l e wavefunction, and u i s the complicated energy v a r i a b l e d e f i n e d by eq.(61). U n t i l now the development of Vj; has been p u r e l y formal and so i t i s worthwhile to c o n s i d e r what V £ means at the mi c r o s c o p i c l e v e l . F i r s t of a l l , l e t us expand the i n t e g r a l e quation (61) f o r T i n the Born s e r i e s <oa\r\ea>=<aa\ v £ ^ | «ra>+J2<ff c | v £ A \c'd> Q <e'o'\ Vj.^ \ca> ( 66) + T <c o I v y. I e'd> Q <ca I v T. I ca> Q <o"o"| v T /, I « c> + . . . } r 1 r\ **** I 1 n n~{ where, as u s u a l , the Q op e r a t o r s r e s t r i c t the summations over | o ' > , |o"> to nucleon s t a t e s above the Fermi sea. In a d d i t i o n to t h i s i n f i n i t e s e r i e s , the d e f i n i t i o n of v ^ i n eq.(59) shows that each term <oa|v^|co> i m p l i c i t l y comprises a l l p o s s i b l e AN int e r m e d i a t e s t a t e s and so can a l s o be expanded i n a i n f i n i t e s e r i e s . I f we s i m p l i f y the n o t a t i o n of the propagator G of eq.(59) to the form G = (€ff+AmZ(+€-H0-QvAA )" 1 (67) then G = G ° + G ° Q v A A G , and G° = ( c ^ + A m ^ ^ H o ) - 1with H 0= K A A+P 2/2m+U A+V A. The e f f e c t i v e p o t e n t i a l v ^ i s e q u i v a l e n t to the s e r i e s < o o | V j . ^ |-oa> = < o o | V g E | etx>+y.<oo I I <ro> Q <<ra 1 v A £ | o o > (68) 61 + 7> C O|v r J\V> Q <xV|vftA XoC' X*ec* *'c*> Q <X*c" € 0 + € o l - 6 x - ^ oo> + . . . T h i s s e r i e s f o r T i s shown g r a p h i c a l l y below. As i s standard i n Goldstone diagrams, arrows p o i n t i n g i n t o a vertex are s t a t e s occupied b e f o r e the i n t e r a c t i o n , and arrows p o i n t i n g away are s t a t e s occupied a f t e r w a r d s . <tfo T co> = c Q ' w v ^ o i + ol \a' OC'I OC + cr k // If e oc , li e' oc oc + ... (69) and each wiggly l i n e of the form J ( w w j ^ r e p r e s e n t s the i n f i n i t e s e r i e s <ea|v£^|ao> : (70) A sum over a l l i n t e r m e d i a t e s t a t e s i s to be understood i n these diagrams. So we f i n d that using t h i s d e f i n i t i o n of the p o t e n t i a l V~£ to c a l c u l a t e the I b i n d i n g energy i s e q u i v a l e n t to summing the c o n t r i b u t i o n s fromm a l l p o s s i b l e diagrams which i n v o l v e o n l y two p a r t i c l e s . In other words, hig h e r order 62 c o r r e c t i o n s to the E energy must i n v o l v e at l e a s t three body c l u s t e r s . In p r a c t i c e V £ i s d i f f i c u l t to e v a l u a t e , p a r t i c u l a r l y s i n c e u a l s o depends on Vj.. The best approach i s the Brueckner-Hartree i t e r a t i v e method in which T i s c a l c u l a t e d i n i t i a l l y u s i n g a reasonable spectrum of s i n g l e - p a r t i c l e e n e r g i e s and the s i n g l e - p a r t i c l e p o t e n t i a l s i n u are set equal to z e r o . V £ i s then determined by eq.(65) and a new value f o r the E energy i s f i x e d by the E-nucleus Schroedinger e q u a t i o n . T h i s c y c l e i s repeated u n t i l a s e l f - c o n s i s t e n t s o l u t i o n i s a c h i e v e d . A remark worth making at t h i s p o i n t i s that a complete, s e l f - c o n s i s t e n t s o l u t i o n would a l s o i n c l u d e o p t i m i z a t i o n of the nucleon s i n g l e p a r t i c l e s t a t e s s i n c e the c o n t r i b u t i o n of the EN p o t e n t i a l to U A p e r t u r b s the n u c l e a r d e n s i t y . F o r t u n a t e l y , t h i s i n f l u e n c e can be s a f e l y ignored s i n c e f i r s t , i t i s an 0(1/A) c o r r e c t i o n and, second, i t i s known e x p e r i m e n t a l l y that the E i s o n l y weakly bound ( i f at a l l ) so that the EN p o t e n t i a l must be weak r e l a t i v e to NN i n t e r a c t i o n s . 3.111. Approximations i n the S i n g l e - P a r t i c l e P o t e n t i a l In the present work we s h a l l not demand a f u l l y s e l f - c o n s i s t e n t s o l u t i o n . We wish i s to approximate eq.(65) 63 to a more manageable form while r e t a i n i n g i t s most important f e a t u r e s . E s s e n t i a l l y our aim i s t o apply the Brueckner-Hartree method on l y to f i r s t o r d e r . That i s , the s i n g l e - p a r t i c l e p o t e n t i a l s e n t e r i n g T w i l l be set equal to zero and then the momentum-space Schrodinger equation s o l v e d s e l f - c o n s i s t e n t l y f o r the E b i n d i n g energy. Higher-order c o r r e c t i o n s to would i n v o l v e the E and N p o t e n t i a l s but in t h i s work the primary i n t e r e s t i s the l i f e t i m e of the h y p e r n u c l e a r l e v e l s and, t h e r e f o r e , with the energy a v a i l a b l e i n asymptotic s t a t e s . The energy dependence d e f i n e d above should be a p p r o p r i a t e f o r t h i s purpose. In p a r t i a l compensation f o r t h i s ambiguity i n the energy dependence we invoke c l o s u r e approximation to r e p l a c e the nucleon e n e r g i e s by an average value -B which i s t r e a t e d as a parameter i n p r a c t i c e . Our f i n a l s i m p l i f i c a t i o n i s to r e p l a c e by i t s s p i n - , and i s o s p i n - a v e r a g e d v a l u e 7, which i s exact to 0(1/A). The e x p r e s s i o n f o r the E° s i n g l e - p a r t i c l e p o t e n t i a l i s then V z(k',k;E z) = y*dp T(<',K;U,)^ * * ( p - q ) * a ( p ) , ( 7 1 ) a where k.,k' are the i n i t i a l and f i n a l s t a t e E-nucleus r e l a t i v e momenta; jt, jc_' are the i n i t i a l and f i n a l s t a t e EN momenta i n the EN r e s t frame, and u i s the three-body energy w - E E "B - (k + p ) 2 / 2 p z . ( ? 2 > 64 D e s p i t e our s i m p l i f i c a t i o n s , the many-body e f f e c t s embedded i n T s t i l l make the i n t e g r a n d of eq.(71) a very c o m p l i c a t e d f u n c t i o n . I t can be seen from the i n t e g r a l r e l a t i o n (61) t h a t T resembles the f r e e IN t - m a t r i x except that the f r e e propagator has been r e p l a c e d by one a p p r o p r i a t e f o r a I i n t e r a c t i n g with a bound nucleon. T h i s analogy can be pushed f a r t h e r . I f p a r t i a l - w a v e mixing i n T i s i g n o r e d (which i s e q u i v a l e n t t o using the angle-averaged P a u l i o p e r a t o r [33,34]) i t can be shown t h a t to d e s c r i b e s c a t t e r i n g from the s t a t e |o> t o | , T s a t i s f i e s T B o » - | v B > X 6 a ( l A ^ l Q ^ ^ l v ^ ) " 1 ^ ! , Y (73) = | v 6 > D e a ( c o ) < v J , where Q° i s the angle-averaged P a u l i o p e r a t o r (to be d i s c u s s e d f u l l y i n s e c t . 3.IV.), the |v 7> are the form f a c t o r s of the sepa r a b l e p o t e n t i a l s of chapter 2, and the summation over r i n c l u d e s a l l c o u p l e d channels. The main d i f f i c u l t y p r e s e n t e d i n i n t e g r a t i n g e q . ( 7 l ) i s the angular dependence of the energy v a r i a b l e u , s i n c e a l l other f a c t o r s normally have a c l o s e d , a n a l y t i c form. A number of approximation schemes have been d e v i s e d t o circumvent t h i s d i f f i c u l t y . In the s i m p l e s t , f u l l y f a c t o r e d form T i s removed from the i n t e g r a n d and e v a l u a t e d a t some 65 'average' v a l u e of the nucleon momentum.. A much b e t t e r approach i s to leave T w i t h i n the i n t e g r a l but r e p l a c e a l l angles i n T by t h e i r average v a l u e [32], Although the l a t t e r i s much b e t t e r than the former, the angular dependence i s s t i l l t r e a t e d only approximately. The obvious successes of such models i n d e s c r i b i n g s c a t t e r i n g s i t u a t i o n s may be a t t r i b u t a b l e t o i n c i d e n t e n e r g i e s which are l a r g e i n comparison with nucleon e n e r g i e s , and a T which i s a slo w l y v a r y i n g f u n c t i o n of energy. A p r i o r i , n e i t h e r of the above f e a t u r e s can be expected to h o l d t r u e i n E h y p e r n u c l e i . The I and N e n e r g i e s are comparable i n the hypernucleus. Perhaps more i m p o r t a n t l y , a l l s e p a r a b l e p o t e n t i a l p r e d i c t i o n s of EN s c a t t e r i n g l e a d to stron g energy dependence i n the (dominant) 3S, 1=1/2 channel near t h r e s h o l d [ c f . s e c t i o n 2.VI.,and a l s o r e f . 28]. Consequently, we make no f u r t h e r approximations i n e v a l u a t i n g e q . ( 7 l ) . That i s , we e v a l u a t e the f u l l , t h r e e - d i m e n s i o n a l i n t e g r a l e x a c t l y . The p o t e n t i a l (71) can be put i n a more convenient form f o r angular i n t e g r a t i o n by changing v a r i a b l e s from the nucleon momentum 2 to the EN t o t a l momentum P=k+p_. In t h i s way u becomes a s c a l a r f u n c t i o n of P and a l l angular dependence r e s i d e s i n the simpl e r T form f a c t o r s and n u c l e a r wavefunction. With t h i s s u b s t i t u t i o n Vj. becomes 66 V z(k',k;E E) = JdP T ( k ' - e P , k - e P ; a ) ( P 2 ) ) F ( P - k ' ; P - k ) , (74) and e = mj/tmj+rriN) , and F ( P - k ' ; P - k ) = ^  • £ ( ? - ! $ ' ) < J > a ( P - k ) . a With harmonic o s c i l l a t o r s i n g l e - p a r t i c l e nucleon wavefunctions F may be expanded i n p a r t i a l waves as F ( P - k ' ; P - k ) - f 0 ( l + a ^ ( p 2 - p . ( k + k ' ) + k - k ' ) ) e " 3 2 ( p 2 + ' ' < k 2 + k ' 2 | - ? - | ^ t ' ) » , . , „ e - a 2 < p 2 t l ='^ ' 2 ' )y j ( K +, ) ( 2,, + 1 )( pO i t + c. C.JA Hi ' P o ( P - k ) P „ r ( P - k ' ) , * (75) with f 0 = 4 a 3 A 3 / 2 , i/=(A-4)/6, and a 2 chosen to f i t the r.m.s. matter r a d i u s [ 3 5 ] . T h i s r e s u l t f o l l o w s by expanding the e x p o n e n t i a l s s e p a r a t e l y i n p a r t i a l waves and usi n g the r e c u r s i o n r e l a t i o n s f o r Legendre p o l y n o m i a l s . The f u n c t i o n s p ° £ . and p \ v are given by pll> = ( l - v U + £ ' ) + a 2 v P 2 ) l A ( x ) l A , ( x ' ) - v ( x i A + 1 ( x ) l £ r ( x ' ) + x ' l £ ( x ) ( x ' ) ) , p £ £ , = a 2vkk'i A(x)! £,(x r) , (76) with x=a 2Pk, x'=a 2Pk', and the i£ are m o d i f i e d s p h e r i c a l b e s s e l f u n c t i o n s . S i m i l a r l y , the two-body T can be expanded as a f u n c t i o n of the thr e e angles k«k',P-k, and P«k'. The n o r m a l i z a t i o n of T i s chosen to be the same as was used f o r the two-body t - m a t r i x . The s p i n - , and i s o s p i n - a v e r a g e d T i s then 6 7 with (77) I L S J T ^ J ( K ' , K ; O , ) = v L ( < ' ) D f S J ( a , ) v L ( K ) K L K ' L In our case the v L are the Yamaguchi form f a c t o r s i n both S and P waves. Note t h a t eq.(77) d e f i n e s the expansion of 7 i n the IN centre-of-mass whereas the i n t e g r a l (74) r e q u i r e s T i n the I-nucleus frame. Using jc =k_~ cP, and =_k' - tP i t has been shown [36,37] that the g e n e r a l r e l a t i o n c o n n e c t i n g s p h e r i c a l harmonics i n the two frames i s be (^ ::-LX^ )^va(^ vr(p)vr:a<Mvr:r<^ > • The two-body form f a c t o r s expand as »(.) - (•Mk-.t)')-' ,2n+„Qn(^|^)pn(?.C), 5 23 "nd'iPlPntk-P) . (79) n where the Q n are Legendre p o l y n o m i a l s of the second k i n d . With equations (75)-(79) i n s e r t e d i n t o eq.(74) the 68 angular i n t e g r a l s can be performed. T h i s i s a f a i r l y s t r a i g h t f o r w a r d , but lengthy, e x e r c i s e producing the g e n e r a l r e s u l t (appendix I I I ) v z ( k ' , k ; E z ) = l/lhr Z (2£+1 )V £ ( k ' , k ;E z . )P £ ( k .k ' ) , where j •0 V £ ( k ' , k ; E z ) = 4 T r f 0 e - a 2 / 2 ( k 2 + k ' 2 ) / d p p 2 e - a 2 P 2 2 I ]£ (2i"+l) (2A'" + y 1' =0 ILSJ ab £ v n < k : P ) v m ( k ' : P ) ( ( 2 a ) , j ^ ! a ) ) , ( 2 b ) , i ^ ! b | ) , ) V 2 < 2 L * » 2 (2L' + l ) ( 2 L » + l ) ( 2 L " ^ O ( 2 L ' v + l ) ( J t 0 ' ^ 2 / n t " L " ^ l_f L"L'"L1V /m £ " ' L f " \ 2 / L ' L" L - a \ / L ' L ' " L - b \ / b L " L i v \ / a L"' L i v \ \0 0 0 / \ 0 0 0 A O 0 0 A O 0 0 A O 0 0 j (L-a L' L" ) { L L-b b > ( a L'" L I v J (80) T h i s i s not a very c o n v e n i e n t - l o o k i n g e x p r e s s i o n , but with o n l y S and P waves i n c l u d e d i n T i t s i m p l i f i e s c o n s i d e r a b l y . Then at l e a s t two of the e n t r i e s i n the 9J symbol are always zero, thereby r e d u c i n g i t to a 3J. The advantage of the above form i s that we have found i t p o s s i b l e t o reduce the Brueckner p o t e n t i a l t o an exact one-dimensional i n t e g r a l without r e l y i n g on untested 69 approximations a f f e c t i n g the energy dependence. 3.IV. The P a u l i E x c l u s i o n P r i n c i p l e So f a r the P a u l i p r i n c i p l e has only been i n t r o d u c e d f o r m a l l y i n t o the ZN i n t e r a c t i o n s v i a the ope r a t o r equation (61). I t has been r e p e a t e d l y s t r e s s e d though t h a t P a u l i e f f e c t s w i l l be a major f a c t o r i n f l u e n c i n g the l i f e t i m e of Z s t a t e s , and so i n t h i s s e c t i o n a c l o s e r look w i l l be taken at the Q o p e r a t o r . As d e f i n e d by eq.(56), Q p r o j e c t s only onto unoccupied nucleon l e v e l s i n the n u c l e a r ground s t a t e , and thereby excludes the propagation of the in t e r m e d i a t e s t a t e nucleon through f i l l e d s t a t e s . The formal s i m i l a r i t y of the Lipmann-Schwinger equations f o r T ( O ) and the f r e e t-matrix t ( o ) can be e x p l o i t e d to w r i t e the exact r e l a t i o n T ( u ) = t U ) - t U ) Q G 0 ( u ) T ( u ) (81) where the m a t r i c e s i n e q . ( 8 l ) couple the ZN and AN channels (but have not been expanded i n p a r t i a l waves). G 0(u) i s the propagator i n the n u c l e a r medium, with the energy dependence u d e f i n e d by eq.(72). The operator Q=1-Q p r o j e c t s onto occupied l e v e l s i n the ground s t a t e . Since T i s embedded i n the three-body space of the Z,N, and (A-1) s p e c t a t o r c o r e , matrix elements are taken between 70 s t a t e s | , P> and |k' ,P>,with k the EN p a i r r e l a t i v e momentum, and P the core momentum r e l a t i v e t o the p a i r . That P must be the same i n i n i t i a l and f i n a l s t a t e s i s simply a r e f l e c t i o n of the f a c t that the core i s n o n - i n t e r a c t i n g . S i m i l a r l y , the other o p e r a t o r s have the matrix elements <k' , P | t U ) |k,P> = 6(P-P')<k' | t [ o , P 2 ] |k> (82) <k',P|QG 0U)|k,P>=6(P-P*)6(k-k')Q(k,P)G 0[o,P 2] The n o t a t i o n used above i s intended to emphasize that t and G 0 depend on P only through the magnitude P 2 , whereas Q(Jt,P) depends on the angle k«P as w e l l . For example, i n nuc l e a r matter the oper a t o r Q i s §(k,P) = 1 f o r |k-i,P| < kPERMi (83) = 0 otherwise with n=mN/(mN+m£) , and nP -Ji i s the momentum of the int e r m e d i a t e s t a t e nucleon. The d i f f i c u l t i e s of i n c o r p o r a t i n g the P a u l i p r i n c i p l e e x a c t l y become apparent i n the i n t e g r a l r e l a t i o n <k' ,P| r | k,P>=<k' |t|k>~y*d2 <k' 11 |E>G 0 («)Q(p_r P) (84) <Er£ | T|k,P> I f Q was angle-independent then e g . ( 8 4 ) c o u l d be expanded i n 71 p a r t i a l waves with k-k' the o n l y a n g l e . However with Q A A given by eq.(82), T ( U ) c l e a r l y depends not only on k«k' but a l s o on the o r i e n t a t i o n of the r e l a t i v e momenta with r e s p e c t to P. I t i s then i m p o s s i b l e to expand eq.(83) i n p a r t i a l waves of d e f i n i t e angular momentum. It has a l r e a d y been seen in the p a r t i a l wave expansion of the E s i n g l e - p a r t i c l e p o t e n t i a l that p a r t i a l wave mixing of the EN i n t e r a c t i o n a r i s e s j u s t from the three-body k i n e m a t i c s . Now i t i s apparent that mixing a l s o a r i s e s from the complete d e s c r i p t i o n of the P a u l i e x c l u s i o n p r o c e s s . In p r a c t i c e t h i s l a t t e r c o m p l i c a t i o n can be overcome by u s i n g the angle-averaged value of Q. That i s , Q i s expanded in p a r t i a l waves and only the 1=0 term i s r e t a i n e d . For the n u c l e a r matter Q of eq.(82) t h i s expansion i s Q(k,P) = ? Q„(k;P)P-(k-P) (85) with the c o e f f i c i e n t s Q j 2(k;P)= 6 0x f o r k+nP < k F (86) = 0 f o r |k-nP| > k F = i f ( xo )"P£ + 1(x 0 ) 3 otherwise and x 0 = ( k 2 + »? 2P 2-k F)/2j?Pk. The P^ are Legendre p o l y n o m i a l s . R e p l a c i n g Q(k,P) by ^f 0(k;P) c e r t a i n l y s i m p l i f i e s T c o n s i d e r a b l y , but i t must s t i l l be d e c i d e d whether the 72 approximation i s l i k e l y to be a good one. I t can be seen immediately t h a t f o r |x 0|>1, Q 0 i s the e n t i r e o perator Q. In the re g i o n |x0|<1 the on the average are a l l sm a l l e r than Q 0 with the maximum v a l u e s a t t a i n e d by the l=0->3 c o e f f i c i e n t s being 1,.75,.48,and .35. Of course f o r 1>0 the magnitude of P^tk-P) i s l e s s than one everywhere except at the end p o i n t s . In a d d i t i o n , the o s c i l l a t o r y nature of the Legendre polynomials causes some c a n c e l l a t i o n i n higher p a r t i a l waves. These c o n s i d e r a t i o n s are encouraging but we can be more p r e c i s e . The operator Q can be s p l i t i n t o the sum of the angle-independent term Q 0, and the angle-dependent terms, denoted by q^(k,P)=IQ£(k;P)P^(k«P). I t then f o l l o w s that T i s the s o l u t i o n of the i n t e g r a l equation T(U) = T ° ( u ) - T 0 ( O ) q £ G 0 ( « ) r ( u ) (87) A . A where T ° ( U ) i s the s o l u t i o n that i s independent of k«P, s a t i s f y i n g r ° U ) = t U ) - t U ) Q 0 G o U ) T ° ( u ) (88) I t i s s u f f i c i e n t f o r our purposes here to assume that the p a r t i c l e s are s p i n l e s s so that T°(k',jc) can be expanded i n p a r t i a l waves of the o r b i t a l angular momentum as 73 • A A T ° ( k \ k ) = 23 (_21+J_)T°(k' ;k)P„(k-k' ) (89) £=0 4 i r * * We estimate the most important c o r r e c t i o n to the approximation T = T ° by i t e r a t i n g eq.(86) once to form r ( k ' ,k;P) ^  T°(k* , k ) - £ ( 2 l + 1 ) ( 2 £ " + 1 ) / dpp 2r° (k' ;p) (90) £ £ ' £ " < 4 i r ) 2 y * Q £ ( p ; P ) G 0 U ) r ° / p ; k ) y* d n ^ P ^ d c l p ) P £.(P'P)^P-K) C o n s i d e r i n g only the l ' = 1 term, i t i s found that T(k',k;P) ^ T ° ( k ' , k ) - 1 / d p p 2 Q , ( p ; P ) G 0 U ) ( 9 1 ) [ r o ( k ' ; p ) T * ( p , k ) k . p + T ? ( k ' , p ) r 0 ( p , k ) k i P ] T h i s i s an important r e s u l t . For small momenta k.,k' the s-wave i n t e r a c t i o n s dominate, and yet the lowest order c o r r e c t i o n to T ° i n v o l v e s the p-wave T ? terms ( i n a d d i t i o n to the p-wave Q, term of c o u r s e ) . I t i s reasonable then to expect that t h i s c o r r e c t i o n i s suppressed at low momenta and t h e r e f o r e to a good approximation r(k',k;P) » T ° ( k ' , k ) (92) The angle-averaged P a u l i o perator Q 0 ( = 1-0*0) w i l l be d i s c u s s e d again i n s e c t i o n 4 . I I where we extend i t s d e f i n i t i o n t o encompass complex momenta. 74 CHAPTER 4 TECHNICAL DETAILS AND NUMERICAL METHODS With the development in the la s t chapter of a r e l i a b l e consider the search for E° bound states and resonances. These states appear as poles in the e l a s t i c S- or T-matrix and i t i s worthwhile to consider their movements in the IN channel as the i n e l a s t i c AN coupling is introduced. F i r s t we w i l l b r i e f l y describe the structure of the Riemann surface. 4 . 1 . Bound States and Resonance Poles For two coupled channels there are two square-root branch cuts along the po s i t i v e energy axis s t a r t i n g at the IN and AN thresholds. On crossing one of these cuts the imaginary component of the corresponding channel momentum changes sign, and another crossing returns i t to i t s o r i g i n a l value. As a result there are four sheets to the surface and i t i s convenient to label them by description of the I s i n g l e - p a r t i c l e potential we can sheet 1 sheet 2 (93) sheet 3 sheet 4 75 C r o s s i n g the energy a x i s above the E t h r e s h o l d changes the s i g n of both' Imp^ and Imp A and, t h e r e f o r e , sheets 1 and 3 are connected. S i m i l a r l y , f o r e n e r g i e s between the two t h r e s h o l d s j u s t Imp A changes s i g n and so sheets 2 and 4 are connected. In a simple, one-channel i n t e r a c t i o n bound s t a t e p o l e s are l o c a t e d on the p o s i t i v e , imaginary momentum a x i s , and the r e f l e c t i o n p r o p e r t y S(k)=S*(-k*) ensures that resonance p o l e s are symmetric about t h i s a x i s [on the u n p h y s i c a l (lmk<0) s h e e t ] . In a d d i t i o n , because of the u n i t a r i t y c o n d i t i o n S(k)S(-k)=1, a po l e of S at k on the u n p h y s i c a l sheet i s accompanied by a zero of S at -k on the p h y s i c a l sheet. In the e l a s t i c s c a t t e r i n g s i t u a t i o n the t h i r d quadrant po l e i s too f a r from the p h y s i c a l region to i n f l u e n c e the c r o s s s e c t i o n . The f o u r t h quadrant pole on the other hand can l i e very c l o s e t o the p h y s i c a l momentum a x i s and i s then r e s p o n s i b l e f o r the c h a r a c t e r i s t i c resonant bump i n the s c a t t e r i n g amplitude. With the i n t r o d u c t i o n of c o u p l i n g t o an i n e l a s t i c channel the u n i t a r i t y c o n d i t i o n s i n each p a r t i a l wave are m o d i f i e d to [21] S ^ 3 ( p A , p £ ) = S ^ ( - p X , - p ^ ) , and E S 1 S 3 " = 6 i y with the s u p e r s c r i p t s i , j r e f e r r i n g to. e i t h e r E or A 76 channels and the summation i n c l u d e s both c h a n n e l s . The s o l u t i o n of these equations can be w r i t t e n q u i t e g e n e r a l l y as S j 2(E) = 1 + 2 i P - e * ' l / a [ A £ ( E ) - i P 2 £ + 1 ] - 1 P i + 1 / a (94) with p " j = p ^ 6 i j . A^(E) i s a r e a l , symmetric matrix which depends on the channel e n e r g i e s E^, but not the momenta p^. The p o l e s of occur as zeros of the determinant | A £ ( E ) - i P 2 ^ 1 | = 0 = ( a Z E - i p | : ^ M ( a A A - i p X £ + 1 ) - ( a 2 . A ) 2 (95) S p e c i a l i z i n g to an S-wave i n t e r a c t i o n , the s o l u t i o n f o r the pole i n the IN channel i s pz = ~ i a z z + . < i a A A - P A > (96) Assuming that the c o u p l i n g i s s m a l l , we f i n d t h a t the e f f e c t of the open channel i s to s h i f t a bound s t a t e p o l e (a Z J.<0) i n t o the second quadrant of the p^. plane (on sheet 2). The small r e a l component of p £ produces the a b s o r p t i v e width of the s t a t e . S i m i l a r l y , a v i r t u a l s t a t e pole (a££>0) i s a l s o s h i f t e d to the l e f t i n the momentum plane with the pole on sheet 4. F o l l o w i n g the same arguments as above f o r P-wave i n t e r a c t i o n s , with a 2 £ > 0 and no i n e l a s t i c c o u p l i n g , i t i s simple to show that resonance p o l e s occur at p^ .=a £j. 3e 7 T"' 6, 77 and a ^ 3 e 1 1 - 1 X 1 / 6 on the u n p h y s i c a l sheet. As the channel c o u p l i n g i s g r a d u a l l y i n c r e a s e d from zero the f o u r t h quadrant p o l e moves away from the r e a l a x i s c a u s i n g the resonant width to i n c r e a s e with i n c r e a s i n g a b s o r p t i o n , as expected. The t h i r d quadrant pole however, s h i f t s c l o s e r to the negative r e a l a x i s . T h i s pole s t i l l does not a f f e c t the c r o s s - s e c t i o n but, as i n d i c a t e d e a r l i e r , i t i s accompanied by a zero of the S-matrix approaching the p o s i t i v e r e a l a x i s from the p h y s i c a l sheet. The presence of the nearby zero i s manifested by a l a r g e r e d u c t i o n i n the magnitude of the S-matrix at t h i s energy. In other words, f o r |S|~0 a l l the incoming waves are i n the E channel and a l l outgoing waves are i n the A channel. The p o s i t i o n s and movements of a l l these s i n g u l a r i t i e s are shown i n f i g . 6. There i s a p a r t i c u l a r l y i n t e r e s t i n g l i m i t of the above cases. If the channel c o u p l i n g i s s u f f i c i e n t l y s t r o n g , i t i s p o s s i b l e f o r the t h i r d quadrant p o l e ( e i t h e r resonant or v i r t u a l s t a t e ) to c r o s s from sheet 4 i n t o the p h y s i c a l r e g i o n of sheet 2. We then have the unique s i t u a t i o n that the s t r o n g a b s o r p t i o n of the p o t e n t i a l i s r e s p o n s i b l e f o r c r e a t i n g a quasi-bound s t a t e . In f a c t , i t can be seen from eq.(96) t h a t i f the c o u p l i n g i s so s t r o n g that the d i a g o n a l elements may be n e g l e c t e d by comparison, there i s always a s o l u t i o n with both p £ and p A p u r e l y imaginary. In t h i s ( r a t h e r u n r e a l i s t i c ) l i m i t a bound s t a t e appears below both t h r e s h o l d s . 78 ImK, A \ —S: • BOUND STATE POLE * RESONANCE POLE O ZERO \ • ReK. \ F i g . 6 . Movement of the S-matrix s i n g u l a r i t i e s i n the IN momentum plane as c o u p l i n g to the AN channel i s i n c r e a s e d from z e r o . For stro n g c o u p l i n g the pole i n the t h i r d quadrant can move i n t o the p h y s i c a l r e g i o n to become a bound s t a t e . 79 In the present study these extremely deep bound s t a t e s do not o c c u r . What i s of p a r t i c u l a r importance to us i s t h a t , f o r moderate s t r e n g t h c o u p l i n g , i n p r i n c i p l e the p o l e can l i e a r b i t r a r i l y c l o s e to the r e a l a x i s . I t i s p o s s i b l e then f o r the s t a t e to have a v a n i s h i n g l y s m a l l width, d e s p i t e the f a c t that the e f f e c t i v e E-nucleus p o t e n t i a l i s s t r o n g l y a b s o r p t i v e . We have l a b e l l e d t h i s s i t u a t i o n a bound s t a t e although i t i s r e c o g n i z e d to be an unusual one. The I wavefunction f o r t h i s energy (on the p h y s i c a l sheet) decays e x p o n e n t i a l l y at l a r g e d i s t a n c e s and the s c a t t e r i n g phase s h i f t has the c h a r a c t e r i s t i c bound s t a t e s i g n a t u r e 6 (0)-6 (°°) = n . C o n v e r s e l y , the r e a l component of the ' b i n d i n g ' energy i s p o s i t i v e ; embedded i n the E continuum s t a t e s . We a l s o s t r e s s again that the p o l e r e s p o n s i b l e f o r t h i s s t a t e i s not the usual bound s t a t e one, but r a t h e r has moved from a normally i n a c c e s s i b l e r e g i o n of the u n p h y s i c a l energy sheet. The o b s e r v a t i o n that a s t r o n g l y a b s o r p t i v e p o t e n t i a l can l e a d to narrow s t a t e s i s not new. T h e i r e x i s t e n c e was f i r s t r e a l i z e d by Fonda and Newton [38,21], and r e c e n t l y Gal et a l [11] have c o n s i d e r e d t h i s as the p o s s i b l e e x p l a n a t i o n i n E h y p e r n u c l e i . They found that with t h e i r one-channel phenomenological model narrow p - s t a t e s arose n a t u r a l l y from t h i s mechanism. I t should be noted though t h a t t h e i r 80 p o t e n t i a l p r e d i c t s very broad s - s t a t e s (e.g. r~23 MeV i n ^SC) because of the s t r o n g a b s o r p t i o n . 4.11. A n a l y t i c C o n t i n u a t i o n of the P a u l i Operator In the l a s t s e c t i o n i t was noted that at a I bound s t a t e the cor r e s p o n d i n g AN channel energy was per t u r b e d i n t o the lower h a l f of the momentum pl a n e . Because we search f o r the s e l f - c o n s i s t e n t e i g e n v a l u e s E =c - i r / 2 of the Schroedinger equation, T(O) [eq. (73)] must be e v a l u a t e d at complex e n e r g i e s . T h i s leads to d i f f i c u l t i e s i n e v a l u a t i n g the o v e r l a p i n t e g r a l s <v|Q°G|v> appearing i n T(O). The problem i s best c l a r i f i e d by an example. Consider the case i n which the AN energy k*,/2m i s r e a l and p o s i t i v e . The o v e r l a p i n t e g r a l s then have the gen e r a l form where 'P' denotes p r i n c i p a l value i n t e g r a t i o n , and the second (pole) term i s simply P r o v i d e d k Y i s chosen to be r e a l and p o s i t i v e then Q ° ( k T , P ) i s w e l l - d e f i n e d . If, however, k Y i s taken to be on the u n p h y s i c a l (negative) momentum a x i s , then Q° has no c l e a r -iirkyMk T)Q°(k T,P) (98) 81 i n t e r p r e t a t i o n with the d e f i n i t i o n used i n s e c t . 3 .IV. It i s r e a l i z e d then t h a t the c o r r e c t e v a l u a t i o n of the T o v e r l a p i n t e g r a l s r e q u i r e s the a n a l y t i c c o n t i n u a t i o n of the P a u l i o p e r a t o r Q° t o the second sheet. T h i s i s not a t r i v i a l problem s i n c e i n the us u a l Fermi gas approximation Q° i s only a piece-wise continuous f u n c t i o n of nucleon momentum. T h i s p o i n t was not c o n s i d e r e d i n the model of Stepien-Rudzka and Wycech [ 1 3 ] because they made no attempt to compute the I-nucleus p o t e n t i a l s e l f - c o n s i s t e n t l y . In n u c l e a r matter approximation Q i s d e f i n e d by the zero temperature l i m i t of the Fermi-Dirac d i s t r i b u t i o n : . q . i - |oxo| - i -•!:(• •.XP[»(«S-4)])"1. ( M ) where k N , k F are the nucleon, and Fermi momenta. E v e n t u a l l y the l i m i t of an i n f i n i t e 9 w i l l be taken but f o r now a f i n i t e value allows Q to be d e f i n e d f o r complex momenta. If p a r t i a l - w a v e mixing i n T v i a Q i s ignored then only the angle-averaged value of Q i s r e q u i r e d . With k T,P the r e l a t i v e , and t o t a l AN momenta t h i s means Q°(<y;P:<F) =J/2y"d(p.K Y )Q(|K Y -nP|) , (100) Y (1 • e x p [ * ( 4 - ( K Y + n P ) 2 ) ] ) ' 82 with Ti=m H/(m N+m A). In the l i m i t t h a t 6-><» i t i s found that f o r complex momentum k-y=k+ir Q° behaves as Q°(< Y;P:K F) = 0 f o r (<+nP) 2 , ( i c - n P ) 2 < ic^+Y2 , <101> = 1 f o r ( K + n P ) 2 , ( t c - n P ) 2 > K^+Y2 , = Kh(<ynP)2 (tc+nP)2<A f o r (ic+nP) 2 < K p + Y2 < (<-nP) 2 , =  iKyl%l 2 '  4 f o r ( K + n p ) 2 > K 2 F + Y 2 > ( K . n P ) 2 . In the l i m i t that y ->0 the above r e s u l t of course reduces to the p r e v i o u s d e f i n i t i o n of the angle-averaged P a u l i o p e r a t o r . With t h i s simple d e s c r i p t i o n of Q° f o r complex momentum i t i s s t r a i g h t f o r w a r d to e v a l u a t e the o v e r l a p i n t e g r a l s . The contour of i n t e g r a t i o n i s d i s t o r t e d from the r e a l a x i s to e n c l o s e the pole i n the lower h a l f - p l a n e at k T so that i n the AN channel, f o r complex k 2 the r e q u i r e d i n t e g r a l i s /d k k L v ! L k J j Q ° ( k , P ) = f d k k 2 v 2 ( k 2 ) Q . ° ( k r P ) ( 1 0 2 ) ( k 2 - k 2 ) I ( k 2 - k 2 ) - i » r k T v 2 ( k ^ ) Q 0 ( k T , P ) With t h i s t e c h n i c a l d e t a i l c l a r i f i e d we can c o n s i d e r s o l v i n g f o r the E s t a t e s . 83 4,111. The I-Nucleus T-Matrix The 1° s i n g l e - p a r t i c l e p o t e n t i a l has been completely d e f i n e d now and i t remains to determine the (complex) e i g e n v a l u e s € f f. By t h i s p o i n t i t has probably become apparent that the i n t e n t i o n i s to f i n d these v a l u e s by s e a r c h i n g f o r p o l e s i n the E-nucleus S-, or T-matrix. In the extreme s i n g l e - p a r t i c l e model there i s no d i f f i c u l t y i n d e f i n i n g the t r a n s i t i o n amplitude. The I obeys the e f f e c t i v e one-body Schroedinger equation (K+V s) | tf> = € ( r | c> ( 1 03) where V £ i s the o p e r a t o r <0|A7|0>, and |0> i s the n u c l e a r ground s t a t e wavefunction. By d e f i n i t i o n of the s i n g l e p a r t i c l e model, the nucleus remains i n the ground s t a t e throughout the i n t e r a c t i o n , and the T-matrix i s simply T = A<0 | T | 0>+A<0 | T | 0>( €<y-K) - 1T (104) While t h i s r e s u l t seems almost obvious, i t can not p o s s i b l y be c o r r e c t . For example, i n the l i m i t that there i s only one nucleon reduces to the f r e e IN t-matrix t , and T of course should reduce to the i d e n t i t y T=t. A c c o r d i n g t o e q . d 0 4 ) though we are l e f t with the s e r i e s T = t + t l ( G 0 t ) n f o r A=1 (105) 84 The d i f f i c u l t y i s apparent. Since t i s , by d e f i n i t i o n , the sum of a l l EN p o t e n t i a l ladder graphs, the terms with n>0 d e s c r i b e processes a l r e a d y i n c l u d e d i n t . The problem can be c o r r e c t e d i f i t i s p i c t u r e d that f o r the f i r s t s c a t t e r i n g event the E can i n t e r a c t with any of the A nucleons. The second event must be with a d i f f e r e n t nucleon though so t h a t there are o n l y A-1 c h o i c e s , and s i m i l a r l y , only A-1 c h o i c e s i n a l l higher order terms. R e p l a c i n g AT by ( A - 1 ) T i n the second term of eq.(l04) ensures the c o r r e c t c o u n t i n g to a l l orders i n the Born s e r i e s . To put t h i s another way, l e t us use Goldstone diagrams again f o r s i m p l i c i t y . The r e a c t i o n matrix 7 i s represented by a s o l i d l i n e | 1 , and the p o t e n t i a l v ^ by a wiggly l i n e . "We are saying t h a t , while i t i s q u i t e c o r r e c t to i n c l u d e the diagram i n eq.(l06) below [or eq.(69)] in the theory, second order (or higher) diagrams such as eq.(l07) should not be i n c l u d e d s i n c e they represent processes a l r e a d y i n c l u d e d in 7. That i s , we should count but not 85 T h i s should emphasize that T i s to be c o n s i d e r e d the e f f e c t i v e i n t e r a c t i o n only i n the sense given by the d i s c u s s i o n f o l l o w i n g equation (61). Although r e p l a c i n g A by (A-1) produces the r i g h t r e s u l t , i t i s not very s a t i s f a c t o r y to i n c l u d e p h y s i c a l behaviour i n such an ad hoc manner. Wi t h i n the extreme s i n g l e - p a r t i c l e model t h i s i s the best we can do, and to have c o r r e c t c ounting inherent i n the theory r e q u i r e s a more c a r e f u l treatment of the (A+1)-body eq u a t i o n . The I-nucleus Schroedinger equation again i s ( K + H N+V E)|*(E)> = E|*(E)> (108) where V s i s now the operator A v ^ t c . f . e q . ( l 0 3 ) ] , and H N i s the n u c l e a r H a m i l t o n i a n . I t f o l l o w s that the T-matrix i s T = A v £ A +Av z^ (E-Hn-Kj-'T (109) with A v Z A = A [ 1 + T ( E - H N - K ) - 1 ] " 1 T (110) S u b s t i t u t i n g e q . ( l l O ) i n t o (109), we f i n d t h a t T s a t i s f i e s the many-body equation T = AT+(A-1 )7(E -H N -K)-'T (111) The double-counting problem i n higher order terms has been e l i m i n a t e d , but not without e x a c t i n g i t s p r i c e . S i n c e we have e l e c t e d t o work i n the (A+1)-body H i l b e r t space the 86 int e r m e d i a t e s c a t t e r i n g s t a t e s i n eq .(111) must i n c l u d e n u c l e a r e x c i t a t i o n s . The c o n t r i b u t i o n of these s t a t e s to T can be i s o l a t e d by w r i t i n g eq.(111) as the p a i r of coupled equations T'=f'+T'|0> 1 <0|T' (112) T ^ K T and T ' = ( A - 1 ) J T + r|n> i <n|T' j (113) where |n> i s an e x c i t e d s t a t e , and € n the co r r e s p o n d i n g e x c i t a t i o n energy. T' i s r e l a t e d to the p h y s i c a l amplitude T by T=AT'/(A-1). ( o f course i f we are only i n t e r e s t e d i n poles of T, the n o r m a l i z a t i o n constant A/(A-1) i s i r r e l e v a n t ). The c o n t r i b u t i o n from e x c i t e d s t a t e s i n eq.(l13) i s expected to be 0(1/A) r e l a t i v e to the f i r s t term. T h i s i s because ground s t a t e t r a n s i t i o n s <0|E~|0> i n v o l v e a l l the nucleons, whereas i n <n|ET|0> only the one nucleon that i s e x c i t e d c o n t r i b u t e s . In the s p i r i t of the s i n g l e - p a r t i c l e model these t r a n s i t i o n s are n e g l e c t e d and T' reduces t o our e a r l i e r r e s u l t <0|T' |0>=(A-1)<0| ) T + T|0> 1 <0|T' I 10> (114) or, with e x p l i c i t E plane-wave i n i t i a l and f i n a l s t a t e s T' (k' ,k,t) •= V (k' ,k,£)+/*dp_ V (k* ,p,6)T' (p,k,t) (115) J Te-pV2m) 87 where V = (A-1)<0|T| 0>=(A-1)V£/A, and the ground state label has been suppressed. With the same normalization for T' as used elsewhere in t h i s work, e q . ( l l 5 ) decomposes into p a r t i a l waves as T^(k';k:t) = ( k ' ; k : e ) + /*dp p2V'(k' ;p: e (p; k : e ) (116) / (6-p z/2m) 4.IV. Numerical Solution of the T-Matrix Before searching for poles of the T-matrix we need to be able to solve the integral equation (116) for any (in general, complex) value of e. Unlike the true scattering s i t u a t i o n , the energies w i l l usually be located off the real axis in the complex k-plane and therefore there i s no s i n g u l a r i t y in the propagator along the contour of integration. We use the standard technique of approximating the integral equation by a system of linear equations. That i s , the continuous variable of integration p is replaced by a set of N (Gauss) quadrature points p m, and corresponding weights wm. In matrix form e q . ( l l 6 ) i s N T(pi,pj;€)=V(p i fpyc)+£ K(pi,ptr)',€)T(pm,py,e) (117) 88 with K(p i,p m;€)=w mp^V(p i,p m; €)/(€-p^/2m) (118) The s u b s c r i p t 1 and s u p e r s c r i p t ' have been dropped but t h i s should not cause c o n f u s i o n . The s o l u t i o n f o r T i ^ i s T ( P i , p ^ ; 6 ) = 1 J l ^ V ^ p . ; * ) (119) where C m ^ i s the mith c o - f a c t o r of |K'|, and |K'| has elements K i j = 6 i j " K ( P i , P j M) (120) While t h i s method of s o l u t i o n i s p e r f e c t l y adequate f o r bound s t a t e s where the r e a l component of e i s l e s s than zero, f o r resonance p o l e s i t becomes i n a c c u r a t e . I f there i s a pole at an energy «=u+ir with u>0, then f o r small r t h i s pole approaches the r e a l a x i s , c a using s t r o n g energy dependence i n the i n t e g r a n d of e q . ( l 1 6 ) . E i t h e r a l a r g e number of p o i n t s i s r e q u i r e d to achieve accuracy, or the i n t e g r a t i o n technique must be m o d i f i e d . One such m o d i f i c a t i o n i s based on the well-known r e s u l t t h a t i n the l i m i t T»-*0 (c + ii»-H 0)" 1 = P -iir6(c-H») (121) U-H 0) with P d e n o t i n g p r i n c i p a l v alue i n t e g r a t i o n . I f the energy 89 € i s allowed to become complex the two i n t e g r a l s above behave as fo r y=0 (122) otherwise, and k 2/2m=u+ir fo r y=0, and k2)/2m=o otherwise Returning to the i n t e g r a l e quation, i t i s found to be e q u i v a l e n t to T(k',k)=v(k',k)+ p/dp [ p 2 v ( k ' , p ) T ( p , k ) - k S v ( k ' , k 0 ) T ( k 0 f k ) ] J (o+ir-p 2/2m) -ijrk 2mV(k' , k 0 ) T ( k 0 , k ) (123) k t With u<0 we set k o=0 and t h i s reduces t o our e a r l i e r r e s u l t . For o>0 and r=0 the second term i n the in t e g r a n d e x a c t l y c a n c e l s the c o n t r i b u t i o n from the 6 f u n c t i o n . With r=0 the p r i n c i p a l value i n t e g r a t i o n of the second term i s zero, and eq.(123) then r e p r e s e n t s a t r u e s c a t t e r i n g s i t u a t i o n . The advantage of t h i s now technique i s that the in t e g r a n d i s a smooth f u n c t i o n of energy near the p o l e , and the p r i n c i p a l v alue 'P' can be removed. •f< dp = 0 u+ir-p 2/2m) = -inm - i i r Jdp6(o + i r - p 2/2m) =-irrm = 0 90 I f we are only i n t e r e s t e d i n the energy dependence of the T-matrix then a l i t t l e a l g e b r a shows t h a t with a new p o t e n t i a l d e f i n e d as u ^ - V ^ - V * n V f t j I (124) with V 1 0 = V ( p i , k 0 ; € ) , and I = 2mk?)[ E w m +iir ] m"Mk|-p») 2k e the s o l u t i o n f o r T(p^,p^;€) i s s t i l l given by eqs.(118)-(120) but with u ^ s u b s t i t u t e d everywhere f o r v i j • Once the values T ^ j have been determined, the o n - s h e l l , and h a l f o f f - s h e l l v a l u e s T 0 0 , and T 0 j are c a l c u l a t e d to be T 0;=u 0;+ E W m P ^ U g m T ^ ; T; 0=T 0; (125) and T 0 0 = u 0 0 + E w mp&u nmTmr>  m (u+iy-p 2/2m) The s o l u t i o n T ^ as given by e q . ( l l 9 ) shows c l e a r l y that a po l e i n T at an energy e i s e q u i v a l e n t to a zero of the Fredholm determinant |K*| at e. In p r i n c i p l e these zeros are simple to f i n d v i a a Newton Raphson a l g o r i t h m . With an i n i t i a l v a lue t 0 , the f i r s t i t e r a t i o n produces a value t , • • « « € 1 - £ 0 - | K , ( € 0 ) | / d e | K M € o ) | (126) 91 Of course i n ge n e r a l the d e r i v a t i v e of |K'| w i l l need to be c a l c u l a t e d n u m e r i c a l l y as w e l l . U s u a l l y the i n i t i a l value of e i s not a c r u c i a l f a c t o r in the i t e r a t i o n s i n c e i t i s found that any value of e 0 with the c o r r e c t s i g n and of the same order of magnitude as the root e w i l l l e a d to convergence. However with a co m p l i c a t e d e x p r e s s i o n f o r V ( c ) , as i s the case here, the time r e q u i r e d to c a l c u l a t e |K'(e)| a l s o becomes an important c o n s i d e r a t i o n . To e v a l u a t e |K'U)|, V(k',k;«) must be c a l c u l a t e d at N(N+l)/2 momentum p o i n t s f o r each value of €. The f i r s t d e r i v a t i v e of the determinant t h e r e f o r e r e q u i r e s V to be r e c a l c u l a t e d at (at l e a s t ) an a d d i t i o n a l N(N+l)/2 p o i n t s . With a poor i n i t i a l value of t 0 the number of i t e r a t i o n s r e q u i r e d f o r convergence may be l a r g e . In the event that the p o t e n t i a l i s energy independent or o n l y a s l o w l y - v a r y i n g f u n c t i o n of energy, t h i s does not pose any d i f f i c u l t y . To c a l c u l a t e the d e r i v a t i v e of |K'(e)| one can assume that a l l energy dependence a r i s e s from the propagators, which c e r t a i n l y w i l l reduce the computing time. However i t has been found that the p o t e n t i a l used i n t h i s work i s too s t r o n g l y energy dependent f o r t h i s technique t o converge. 92 I t i s p o s s i b l e to improve on the s t a r t i n g value € 0 by u s i n g the Born s e r i e s f o r T [21], In the immediate r e g i o n of the pole the Born expansion i s d i v e r g e n t , but t h i s f a c t can be used to advantage. W r i t i n g the T-matrix as the s e r i e s T = I V ( G 0 V ) N (127) then i n the r e g i o n s where the s e r i e s converges, the convergence t e s t c l a i m s that l i m n e V ( G Q V ) W ^ V ( G 0 V ) " < 1 ( 1 2 8 ) with the e q u a l i t y h o l d i n g o n l y at the pole i t s e l f . The s i m p l e s t approximation to t h i s r e s u l t i s o b v i o u s l y V » 1 ( 1 2 9 ) and while t h i s may seem to be a very crude approximation, i t i s exact f o r a coupled-channel separable model. To c a l c u l a t e T(0,0;«) to second Born approximation r e q u i r e s V at ( N + 1 ) p o i n t s . (There i s no p o i n t i n c a l c u l a t i n g T to t h i r d Born approximation because V w i l l be needed at another N(N+l)/2 p o i n t s ) . Again using a Newton-Raphson a l g o r i t h m , N/2 i t e r a t i o n s t o the root € 0 of eq.(129) can be performed i n the time r e q u i r e d to c a l c u l a t e |K'(«)| and i t s d e r i v a t i v e once. With t h i s v alue of c 0 as the 93 s t a r t i n g p o i n t we enter the exact i t e r a t i v e s o l u t i o n to the root e. For the p o t e n t i a l used i n t h i s work i t has been found t h a t e 0 was always w i t h i n 50% of the t r u e v a l u e . So f a r we have assumed i m p l i c i t l y t h a t the pole i s on the p h y s i c a l energy sheet. The Newton Raphson a l g o r i t h m w i l l f a i l i f the root i s l o c a t e d on the lower h a l f of the k-plane because |K'U)| i s i n v a r i a n t under a s i g n change of the momentum. One approach that would normally overcome t h i s problem i s to a n a l y t i c a l l y c o n t i n u e T by r o t a t i n g or d i s t o r t i n g the contour of i n t e g r a t i o n to e n c l o s e the pole on the second sheet. T h i s i s not p o s s i b l e with our p o t e n t i a l because the harmonic o s c i l l a t o r wavefunctions behave p a t h o l o g i c a l l y away from the r e a l a x i s . A d i f f e r e n t approach i s needed and we can now use the symmetries of the S-matrix to our advantage. As d i s c u s s e d i n s e c t . 4.1. , a pole of S a t -k on the second sheet i s accompanied by a zero of S at k on the p h y s i c a l sheet. F i n d i n g the p o s i t i o n of a zero i s then e q u i v a l e n t to f i n d i n g a p o l e . A l s o as d i s c u s s e d e a r l i e r , the only resonant p o l e i n S which may be of i n t e r e s t i s the normally i n a c c e s s i b l e one i n the t h i r d quadrant. The S-matrix can not be s o l v e d at complex momenta f o r the same reason that the i n t e g r a t i o n contour can not be r o t a t e d , so i n s t e a d an approximation method i s needed f o r c o n t i n u i n g S o f f the r e a l a x i s . 94 The S-matrix i s r e l a t e d to the phase s h i f t s by • -S, = k ' f ^ c o t & t + i k * ^ 1 (130) K k2Ji+ ' c o t ^ - i k 2 * * 1 and the c o n d i t i o n S(k)S(-k) = 1 r e s t r i c t s k 2-^* 1cot6£ to being an even f u n c t i o n of k. T h e r e f o r e i t can be approximated at low e n e r g i e s by the T a y l o r s e r i e s k 2^* 'cot6^ w1/a +rk 2/2+0(k a) (131) which of course i s j u s t the e f f e c t i v e range expansion. There are s e v e r a l c h o i c e s as to how to proceed. The obvious approach i s to determine the f i r s t few c o e f f i c i e n t s in the expansion by c a l c u l a t i n g 6^  at s e v e r a l e n e r g i e s and s u b s t i t u t i n g t h i s s e r i e s d i r e c t l y i n t o e q . ( l 3 0 ) . Our aim though i s to e x t r a p o l a t e r e l i a b l y to complex e n e r g i e s which would r e q u i r e a f a i r l y l a r g e number of terms f o r accuracy. On the other hand, high-order polynomial approximations are infamous f o r t h e i r i n s t a b i l i t y . A b e t t e r method i s to c o n s t r u c t an [L/M] Pade approximant of the e f f e c t i v e range expansion. T h i s i s d e f i n e d as the r a t i o of two p o l y n o m i a l s of orders L,M that e x a c t l y reproduces the f i r s t L+M+1 terms i n the T a y l o r s e r i e s . The g r e a t e s t advantage of the Pade approximant i s t h a t , s i n c e i t c o n t a i n s p o l e s , i t i s w e l l s u i t e d f o r 95 r e p r o d u c i n g the a n a l y t i c s t r u c t u r e of the f u n c t i o n . T h i s l a t t e r approach i s the usual Pade technique. However the step of determining the c o e f f i c i e n t s of the e f f e c t i v e range expansion seems redundant. Instead an [L,M] approximant can be c o n s t r u c t e d that reproduces k 2 ^ + , c o t 6 £ e x a c t l y , and not j u s t the T a y l o r s e r i e s , at L+M+1 p o i n t s . P r e c i s e l y t h i s problem has been examined a t le n g t h by Ha r t t [39] and he has found that the best r e p r o d u c t i o n of the s i n g u l a r i t i e s i n S i s given by the c h o i c e L=M+1.1 The Pade approximant i s d e f i n e d as P L I a 2 m k 2 m ' a ° = 1 — -,-ih ( 1 3 2> Qt-i i ^ 2 m and the (complex) c o e f f i c i e n t s are determined by s o l v i n g the set of l i n e a r equations P L U f )«k£ £* ,cot6|q_ 1(k|) , i = 1,2, ...2L (133) The convergence and a n a l y t i c c o n t i n u a t i o n p r o p e r t i e s of Pade approximants have been d i s c u s s e d by many authours ( f o r example, r e f . 4 0 ) , but the only c o n v i n c i n g argument f o r t h e i r 1 T h i s would not be true i n s p e c i a l cases, such as the sepa r a b l e p o t e n t i a l , f o r which the expansion t o k* i s exa c t . 96 use i s i f they work. Although t h i s i n f r i n g e s on the t o p i c of chapter 5, the bound s t a t e p r e d i c t i o n s f o r a few s - s t a t e h y p e r n u c l e i are r e p o r t e d i n t a b l e IV. The exact r e s u l t s l i s t e d are those c a l c u l a t e d using the Newton-Raphson method d i s c u s s e d e a r l i e r . The agreement between the two methods i s very good and the Pade s o l u t i o n converges q u i c k l y to the r o o t . As expected, the best r e s u l t i s o b t a i n e d f o r the pole nearest to the r e a l energy a x i s . In p - s t a t e , where t h i s method w i l l be used, the zeros are c l o s e to the p h y s i c a l a x i s , and so we have reason f o r c o n f i d e n c e . 9 B e 1 2 C 1 6 0 EXACT -.202 + .255i -.300 + .450i -.306 + .576i [2/1] -.208 + .251i -.286 + . 470i -.352 + .623i [3/2] -.204 + .253i -.296 + .460i -.338 + .602i [4/3] -.202 + .255i -.299 + .451i -.313 + .582i Table IV. P o s i t i o n s of the bound s - s t a t e pole i n the momentum plane p r e d i c t e d by [L/M] Pade approximants compared with the exact (Newton-Raphson) s o l u t i o n . U n i t s are i n fm* 1. 97 CHAPTER 5 RESULTS IN LIGHT E° HYPERNUCLEI A great d e a l of c a r e has been taken to c o n s t r u c t the E° s i n g l e - p a r t i c l e p o t e n t i a l c o n s i s t e n t l y with the two-body EN i n t e r a c t i o n s . Before proceeding to the c a l c u l a t i o n of bound s t a t e s we should a l s o check the c o n s i s t e n c y of the model with the E" atomic data. 5 . 1 . E* Atoms Although no attempt i s made to f i t the model to the data, a comparison with B a t t y ' s phenomenological r e s u l t can be made in some a p p r o p r i a t e l i m i t . In E" atoms the E i s bound with e s s e n t i a l l y zero energy and consequently the p o t e n t i a l s should be comparable f o r very s m a l l momenta. At zero momentum t r a n s f e r the volume i n t e g r a l of the model p o t e n t i a l i s p r o p o r t i o n a l to B a t t y ' s value a B , with the t h e o r e t i c a l value given by the e x p r e s s i o n i A J I L S J 3 0 ( 3 T ( P)2L ( 1 3 4 ) T L S J ( " ) g ( e 2 : e 2 p 2 ) 2 ^ S J (-B-PV2P) ., N o t i c e that even at zero i n c i d e n t momentum ( i n the E-nucleus frame) a t h r e c e i v e s c o n t r i b u t i o n s from L=1 and higher EN p a r t i a l waves. 98 For a.^ has been examined i n some d e t a i l . With the nucleon b i n d i n g parameter B chosen ( r a t h e r a r b i t r a r i l y ) to be 10 MeV and a Fermi momentum of 260 MeV/c i t i s found that a t h = .346+i.197 fm i n comparison with a 6=(.35±.04) + i(.19±.03) fm. The agreement of these r e s u l t s i s r e l a t i v e l y i n s e n s i t i v e to the ch o i c e of parameters, as shown i n d e t a i l in f i g . 7, where the e l l i p t i c a l r e g i o n d e f i n e s the valu e s allowed from Batty's a n a l y s i s . With a Fermi momentum of 260 MeV/c and any value of 0<B<20 MeV i s c o n s i s t e n t with a^. Conversely, with B=10 MeV any kp i n the range 250<k F<275 MeV/c i s c o n s i s t e n t ( i . e . an average n u c l e a r d e n s i t y of ~75-l00% of nuclear m a t t e r ) . In a d d i t i o n , with the c e n t r a l v a l u e s of B=10 MeV, and kp=260 MeV/c the I ° - 1 2 C s c a t t e r i n g l e n g t h s are s i m i l a r i n the two models. Batty's p o t e n t i a l g i v e s -2.94+i1.20 fm, as compared with the -3.41+i1.44 fm of our model. Although these agreements are su g g e s t i v e , they r e q u i r e f u r t h e r i n v e s t i g a t i o n f o r d e f i n i t e i n t e r p r e t a t i o n . It i s not c l e a r that e i t h e r the volume i n t e g r a l or the s c a t t e r i n g l e n g t h i s the r e l e v a n t q u a n t i t y f o r comparison. T h i s i s p a r t i c u l a r l y ambiguous s i n c e Batty's p o t e n t i a l i s l o c a l , whereas ours i s h i g h l y n o n - l o c a l . In a d d i t i o n , the Fermi momenta used above are probably l a r g e r than are a p p r o p r i a t e f o r L>3 atoms. 99 0.29 0.31 0.33 0.35 0.37 0.39 R e a t h ( f m ) F i g . 7 . V a r i a t i o n of the p o t e n t i a l volume i n t e g r a l a th with Fermi momentum ( k F i n MeV/c) and nucleon b i n d i n g energy (B i n MeV) i n 1£3.C. The e l l i p t i c a l r e g i o n i s the range allowed by the a n a l y s i s of Z" atoms [12] 100 However, i t i s worth n o t i n g that a decrease i n k F would le a d to a l a r g e r value of Ima^ . If the s i n g l e p a r t i c l e p o t e n t i a l was then s c a l e d i n some f a s h i o n to once again reproduce B a t t y ' s value of ImI B s m a l l e r widths f o r s, and p - s t a t e s than those r e p o r t e d here would r e s u l t . In a d d i t i o n , with kp reduced to 150 MeV/c and B i n c r e a s e d to 0 MeV, which are probably more r e a l i s t i c i n the atomic s i t u a t i o n , a"^ does not change d r a m a t i c a l l y , becoming 0.29r+i0.252 fm. T h i s i s s t i l l comparable to the phenomenological v a l u e . 5.11. S-State Hypernuclei In s p i t e of the a m b i g u i t i e s o u t l i n e d above we have been encouraged to proceed with the c a l c u l a t i o n of E nuclear s t a t e s . For a l l the l i g h t h y p e r n u c l e i c o n s i d e r e d i n t h i s work we have f i x e d the nucleon b i n d i n g B at 10 MeV f o r s i m p l i c i t y . Because in s - s t a t e the I i s l a r g e l y c o n f i n e d to the nuclear volume, f a i r l y high Fermi momenta have been chosen, ranging from 245 MeV/c in^lHe to 260 MeV/c i n (or, average d e n s i t i e s from 70-80% of nucl e a r m a t t e r ) . The s e l f - c o n s i s t e n t eigenvalues e - i r / 2 f o r s - s t a t e h y p e r n u c l e i are presented in t a b l e V. Se v e r a l c o n c l u s i o n s can be drawn from these r e s u l t s . Most i m p o r t a n t l y i t i s found that f a i r l y narrow bound s t a t e s 101 r e s u l t when P a u l i e x c l u s i o n and nucleon b i n d i n g are i n c o r p o r a t e d i n a m i c r o s c o p i c c a l c u l a t i o n . We a l s o n o t i c e that f o r A<9 the model p r e d i c t s that €>0. These are bound s t a t e s though and not resonances. That i s , they a r i s e from an S-matrix pole i n the second quadrant of the momentum plane. Nucleus (MeV/c) B (MeV) e - i r/2 (MeV) r (MeV) Z O L I  Zo 9Be 1 2 C 1 3 c 245 250 260 260 260 260 OOOOOO +2.49 - i0.88 +0.81 - i0.99 -0.46 - 11.95 -2.06 - i4.91 -2.59 - 15.31 -4.22 - i6.26 1.75 1.98 3.89 9.81 10.62 12.52 Table V. Is bin d i n g i n l i g h t Z° hypernuclei Although the e a r l i e r p r e d i c t i o n of a was encouraging we a l s o wish to t e s t the s e n s i t i v i t y of these bound s t a t e s (and the widths i n p a r t i c u l a r ) to small changes i n the parameters. Again f o r ^ i C , the v a r i a t i o n has been examined in d e t a i l . With v a l u e s of B and k F which approximately reproduce the extreme values p e r m i t t e d f o r a ^ the b i n d i n g has been c a l c u l a t e d and these r e s u l t s are presented i n 102 t a b l e VI. The r e a l b i n d i n g | t | v a r i e s between 2 and 5 MeV, but r seem f a i r l y i n s e n s i t i v e , changing on l y from 9-11 MeV. B a t h t-ir/ 2 r (MeV/c) (MeV) fm (MeV) (MeV) 250 22.5 0.39 + 10.19 -A .78 - iA .95 9.90 250 12.5 0.35 + i0.22 -2.22 - 15.63 11.26 260 10.0 0.35 + 10.19 -2.59 - 15.31 10.62 275 7.5 0.35 + iO.16 -3.00 - iA .65 9.30 275 -7.5 0.31 + 10.19 -2.00 - \k.Bk 9.68 T a b l e VI. V a r i a t i o n o f Is b i n d i n g i n l | c w i t h K p and B . Since P a u l i e x c l u s i o n and nucleon b i n d i n g e f f e c t s are major i n f l u e n c e s i n producing these narrow s t a t e s we have t r i e d to i s o l a t e t h e i r c o n t r i b u t i o n s as much as p o s s i b l e . The b i n d i n g has been c a l c u l a t e d with, f i r s t , B f i x e d and kp v a r i e d between 225-300 MeV/c and, second, kp f i x e d and B v a r i e d from 0-30 MeV. These r e s u l t s are d i s p l a y e d i n f i g . 6. I t can be seen that the i n c l u s i o n of P a u l i e f f e c t s i s e s s e n t i a l to the c o r r e c t d e s c r i p t i o n of the bound s t a t e , but, any reasonable c h o i c e of kp leads to a small width. 103 F i g . 6. Dependence of the s - s t a t e p o s i t i o n c and width r on Fermi momentum (k F i n MeV/c) and nucleon b i n d i n g (B i n MeV) i n 1 3.C. 104 The v a r i a t i o n of r with l a r g e v a l u e s of B i s dramatic. The width i s almost i n s e n s i t i v e to changes i n B from 0-20 MeV but decreases s h a r p l y f o r B>20 MeV. T h i s s t r o n g dependence of r on nucleon b i n d i n g f o r l a r g e B should not be s u r p r i s i n g . To take an extreme example, i f the E and nucleon b i n d i n g e n e r g i e s were .greater than the E-A mass d i f f e r e n c e then the EN-AN c o n v e r s i o n would never be e n e r g e t i c a l l y allowed and the decay width of the s t a t e would n e c e s s a r i l y be z e r o . For more r e a l i s t i c cases i t i s expected, from phase space c o n s i d e r a t i o n s alone, that the width must decrease r a p i d l y f o r l a r g e r e d u c t i o n s i n the AN channel energy. We have a l s o examined the s e n s i t i v i t y of the ^ i C s t a t e to the number of terms i n the two-body form f a c t o r expansion. A p r i o r i one a n t i c i p a t e s t hat the s e r i e s w i l l converge s l o w l y . For example, c o n s i d e r the simple T a y l o r s e r i e s expansion of the Yamaguchi form f a c t o r v ( k 2 ) v [ ( k - 6 P ) 2 ] = v ( k 2 + € 2 P 2 ) > €-S[-2v(k2 + € 2 P 2 ) P-k]^ ( 1 3 5 ) which i s v a l i d f o r a l l k,P. An i n d i c a t o r of the importance of the 1>0 terms i s given by the c o e f f i c i e n t €. For a l i g h t p a r t i c l e t& d i m i n i s h e s r a p i d l y with 1. For i n s t a n c e , f o r the pion € i s ~ 1 / 8 so that n e g l e c t i n g the angular dependence may be an e x c e l l e n t approximation i n many cases. By co 1 0 5 c o n t r a s t , f o r the sigma «>l/2 so that higher order terms are probably s i g n i f i c a n t . In t a b l e VII, we r e p o r t the v a r i a t i o n of the 1^iC b i n d i n g energy. Although the width may not be as s e n s i t i v e as a n t i c i p a t e d , t h r e e - , and f o u r - f i g u r e accuracy r e q u i r e s t h r e e , and four terms r e s p e c t i v e l y i n the expansion. C e r t a i n l y i t i s u n j u s t i f i e d to ignore the angular dependence e n t i r e l y . No. Of Terms E+ir/2 (MeV) r (MeV) % change i n r 1 -2. 264 - i 4 . 625 9.250 2 -2. 585 - i 5 . 253 10.506 1 1 .96 3 -2. 589 - i 5 . 304 10.608 0.96 4 -2. 587 - i 5 . 309 10.618 0.09 Table VII. Convergence of the 1s b i n d i n g energy i n Z o c with the number of terms i n the two-body f o r m - f a c t o r p a r t i a l wave expansion. The p r e d i c t i o n s f o r s - s t a t e seem f a i r l y i n s e n s i t i v e to moderate changes in the parameters of the model. However i t i s of i n t e r e s t to determine the s e n s i t i v i t y to the u n d e r l y i n g model of the ZN p o t e n t i a l s . We have r e c a l c u l a t e d 106 the s - s t a t e s using the model A parameters of Toker et a_l [28]. T h i s model, although s e p a r a b l e , d i f f e r s q u a l i t a t i v e l y from ours. In t h e i r work they f i t the I'p t o t a l c r o s s - s e c t i o n s assuming them to be determined completely by the 3S, 1 = 1 / 2 i n t e r a c t i o n , with no c o u p l i n g to the AN 3D, channel. The d i f f e r e n c e s are c l e a r l y r e f l e c t e d i n the r e s u l t s of the c a l c u l a t i o n , l i s t e d i n t a b l e V I I I . The r e a l b i n d i n g e n e r g i e s are s i m i l a r in the two models, but with Toker's parameters the s t a t e s are very narrow r e l a t i v e to ours. Although t h e i r two-body model i s r a t h e r l e s s a b s o r p t i v e than ours i n the channel alone, the major source of t h i s d i f f e r e n c e i n widths i s our i n c l u s i o n of the AN 3D! and 3 P j channels. These higher angular momentum s t a t e s are not a f f e c t e d by P a u l i s uppression to the same degree as the S-waves, and i n ^ 3C f o r example the P and D c o u p l i n g s are r e s p o n s i b l e f o r roughly 15 and 25% of the width r e s p e c t i v e l y . As mentioned e a r l i e r , our d e t e r m i n a t i o n of t i s expected to be l e s s r e l i a b l e than of r, mainly because of the n e g l e c t of s i n g l e - p a r t i c l e p o t e n t i a l s i n the energy v a r i a b l e of T ( U ) . Furthermore there i s no s t a t i s t i c a l l y s i g n i f i c a n t o b s e r v a t i o n of an s - s t a t e hypernucleus. 107 N e v e r t h e l e s s the p o s i t i o n and width c a l c u l a t e d f o r the ^>C s t a t e c o i n c i d e s with a small bump i n the spectrum of B e r t i n i et a l [ 3 ]. A l s o the weakly bound s t a t e i n ^.Be i s c o n s i s t e n t with a shoulder i n the data. C l e a r l y much b e t t e r experimental data are needed i n these cases. N u c l e u s <xp B c- ir/2 r (MeV/c ) (MeV) (MeV) (MeV) 2*5 10 +0.97 -- i 1.54 3.08 250 10 -0.03 -- iO.87 1 .74 ro 9Be 260 10 -0.67 -- i0.65 1 .30 1 2 C ro 260 10 -3 . 3 1 -- i 1.42 2.85 £ 0 ° 260 10 -4.96 -• i 1.53 3.06 TableVIIL S - s t a t e b i n d i n g e n e r g i e s i n l i g h t 1° h y p e r n u c l e i as c a l c u l a t e d with the model A parameters of Toker et a l [ 28]. 5.111. Separable Approximation of the E-Nucleus P o t e n t i a l I t i s an i n t e r e s t i n g f e a t u r e of the s i n g l e - p a r t i c l e p o t e n t i a l (80) t h a t , d e s p i t e the complexity of the e x p r e s s i o n , i n p r a c t i c e i t i s s e p a r a b l e . To see t h i s , l e t us r e w r i t e V L as V L ( k ' , k ) = / * d P y ^ g ^ k ' ; k : P ) £ v n ( k : P ) v m ( k ' : P ) C ^ P > (136) J JUT where C ( P ) i s a c o e f f i c i e n t t h a t depends on P but not on k 108 or k', and c o n t a i n s a l l the angular momenta summations as w e l l . The d e n s i t y term ^ can be s i m p l i f i e d i n n o t a t i o n as w e l l to l k' ; k: P) =^ Pc j,^ (P) k k a 2 kP) i ^ J a 2 k' P) (137) m,m*o with c(P) another P-dependent c o e f f i c i e n t (which can be z e r o ) . In t h i s form we n o t i c e that k and k' appear i n separate f u n c t i o n s i n a l l terms of the i n t e g r a n d of eg.(ASb). S t r i c t l y speaking, V L i s not s e p a r a b l e because of the P i n t e g r a t i o n but, i n p r a c t i c e , the i n t e g r a l i s always r e p l a c e d by a f i n i t e sum over N d i s c r e t e p o i n t s P^  . The r e s u l t i s that V L i s approximated by a f i n i t e rank separable p o t e n t i a l , and i n t h i s case the Lipmann-Schwinger equation can be s o l v e d a l g e b r a i c a l l y . In most i n s t a n c e s i t would simply not be p r a c t i c a l to s o l v e f o r T i n t h i s way s i n c e i t would i n v o l v e i n v e r t i n g a matrix of s i z e 102X102, (or l a r g e r , depending on the number of quadrature p o i n t s and terms i n the expansions of the i n t e g r a n d ) . For the s p e c i a l case of a "He core though a number of s i m p l i f i c a t i o n s become p o s s i b l e . A l l four nucleons are i n s - s t a t e so that the ^ term i n eq.03fe) i s r e s t r i c t e d to 1=0. For the same reason we can expect that the two-body p-wave i n t e r a c t i o n s are not as s i g n i f i c a n t i n |He as i n h e a v i e r n u c l e i . F i n a l l y , i f the Ev n can be 1 0 9 approximated by v 0 (and t h i s w i l l be t e s t e d ) , then the s - s t a t e p o t e n t i a l becomes N V 0(k';k) = £ <k' |g-, >x" <g5 | k> ( 1 3 8 ) with <k|g-,> = e - ^ ^ i o t a ^ i k J v o d c P ] ) and ),'" = 4nf nw, Pf e - ^ E (2S+1 ) (21 + 1 [ u ( P 2 ) 3 s r 4 6 The T-matrix i s s o l v e d , as i n the two-body case, t o be 6 T(o) = |g>[k- 1(u)-<g|G 0(«)|g>]- 1<g| ( 1 3 9 ) with | g> the row matrix of |g; >, and X . ( o ) the d i a g o n a l , energy dependent matrix of X . ". T h i s equation can be reasonably s o l v e d by matrix i n v e r s i o n s i n c e the dimension of X"1-<g|G|g> i s the same as the number of quadrature p o i n t s . We have s o l v e d the ^.He T-matrix by t h i s method and found that the b i n d i n g energy i s +2.72-0.76i MeV, as compared with the r e s u l t with both S and P-wave EN i n t e r a c t i o n s of +2.5l-0.80i MeV, and the exact r e s u l t with three f o r m - f a c t o r terms as w e l l of +2.49-0.88i MeV. The e f f e c t of i n c l u d i n g P-waves i s mainly to i n c r e a s e the r e a l b i n d i n g of the s t a t e , whereas the width i s more s e n s i t i v e to the number of terms i n the f o r m - f a c t o r expansion. T h i s i s 6 The term <g|G|g> must be m u l t i p l i e d by 3/4 because of the d o u b l e - c o u n t i n g problem. 110 simply a r e f l e c t i o n of the strong energy dependence i n the EN t - m a t r i x . The approximation works reasonably w e l l , and the s i m p l i c i t y of separable p o t e n t i a l s f o r c a l c u l a t i o n s suggests that t h i s approach may be u s e f u l i n systems where the i n t e r a c t i o n s are smoother f u n c t i o n s of energy than i n the EN case. 5.IV. P-State Hypernuclei E x p e r i m e n t a l l y , the c l e a r e s t evidence of narrow hypernuclear s t a t e s comes, not from the ground s t a t e s so f a r d i s c u s s e d , but rather from the r e c o i l l e s s l y - p r o d u c e d 1p l e v e l s . To date, the p o s i t i o n s and widths of these l e v e l s have not been f i x e d a c c u r a t e l y but some i n d i c a t i o n s are a v a i l a b l e . By comparing the E missing-mass spectrum with the c o r r e s p o n d i n g well-known A spectrum, Brueckner e_t a l . [ 4] have concluded that i n ^ B e the s t a t e i s unbound by roughly 9 MeV and have p l a c e d an upper l i m i t on the width of 8 MeV. They have a l s o found a narrow p - l e v e l i n l £ C at 5 MeV e x c i t a t i o n [ 3], and i t has been suggested r e c e n t l y that i n ^ i L i the width may be as small as 3 MeV [ 5], I t must be emphasized though that these are only i n d i c a t i o n s . The data are not yet p r e c i s e enough to p i n p o i n t the p o s i t i o n s to b e t t e r than a few MeV accuracy. 111 P a u l i e x c l u s i o n and nucleon b i n d i n g e f f e c t s are not expected to i n f l u e n c e the P-wave as much as i n the ground s t a t e c a l c u l a t i o n s . The c e n t r i f u g a l b a r r i e r o b l i g e s the I to i n t e r a c t p r i m a r i l y i n the low d e n s i t y r e g i o n of the valence nucleons so t h a t , f o r c o n s i s t e n c y , both the Fermi momentum k F and nucleon b i n d i n g B should be reduced from t h e i r s-wave v a l u e s . In a l l the l i g h t n u c l e i we c o n s i d e r , the nucleon b i n d i n g has been r a i s e d to zero MeV and the Fermi momenta have been reduced s u b s t a n t i a l l y , ranging from 75 MeV/c in ElHe to 175 MeV/c in ^ $ 0 . The method of s o l u t i o n i s f a i r l y s t r a i g h t f o r w a r d . We have d i s c u s s e d at l e n g t h the d i f f i c u l t i e s i n c a l c u l a t i n g the S-matrix at complex momenta, and so use the Pade technique d e s c r i b e d i n sect.4.IV. We search f o r p o l e s i n S by f i n d i n g the conjugate zeros on the p h y s i c a l sheet. A n t i c i p a t i n g the r e s u l t s somewhat, our a t t e n t i o n i s l i m i t e d to the pole in the t h i r d momentum quadrant. The corresponding zero l i e s c l o s e to the p h y s i c a l a x i s and, as we have seen, the Pade method should then give r e l i a b l e r e s u l t s . The p o s i t i o n of these p o l e s are l i s t e d i n t a b l e IX. The r e s u l t s are r i c h i n i n f o r m a t i o n and s e v e r a l important o b s e r v a t i o n s can be made. F i r s t , i t i s n o t i c e d that f o r A>9 the model p r e d i c t s the 1p l e v e l to be bound, which i s evid e n t from the p o s i t i o n of the pole i n the second 1 12 quadrant. That these s t a t e s do i n f a c t correspond to the resonance pole r a t h e r than the us u a l bound s t a t e pole i s confirmed by f o l l o w i n g t h e i r motion f o r an i n c r e a s e i n the Fermi momentum. I n c r e a s i n g the Fermi momentum e f f e c t i v e l y decreases a b s o r p t i o n by decr e a s i n g the c o u p l i n g to the AN channel. A b s o r p t i o n d e p l e t e s the I wavefunction at short d i s t a n c e s , thereby a c t i n g as a short-range r e p u l s i o n . By i n c r e a s i n g kp a bound s t a t e pole should then move towards the p o s i t i v e imaginary momentum a x i s , which i s e x a c t l y the behaviour encountered al r e a d y in s - s t a t e . A resonance p o l e , i n l i g h t of our e a r l i e r d i s c u s s i o n , w i l l move i n the opp o s i t e d i r e c t i o n , c r o s s i n g the negative r e a l a x i s i n t o the un p h y s i c a l t h i r d quadrant. Choosing joBe as r e p r e s e n t a t i v e , we in c r e a s e kp from 125 to 200 MeV/c. I t i s found t h a t the s t a t e becomes unbound, with the pole moving away to the t h i r d momentum quadrant at E =5.23+12.19 MeV, c l e a r l y demonstrating these p o l e s to be of the resonant v a r i e t y . The second f e a t u r e we n o t i c e are the remarkably long l i f e t i m e s of these s t a t e s , with r«>»4 f o r and as small as 0.5 f o r ^ l B e . T h i s i s t r u l y p a r a d o x i c a l : i t i s the very f a c t t h a t the p o t e n t i a l i s so s t r o n g l y a b s o r p t i v e that the l e v e l s are narrow. 113 N u c l e u s (MeV/c ) B (MeV) E p - c - i r / 2 (MeV) E l/2 « < + i  P ( M e V ) l / 2 r Z 0 H E 75 0 17.48 + 10.94 -4.18 - 10.112 -1 .88 100 0 11.13 + 10.26 -3 .34 - iO .039 -0.52 Z O B e 125 0 6.16 - 10.27 -2.48 + iO.054 0.54 1 2 c 150 0 4.58 - i l . 7 5 -2.18 + i0.402 3.50 16 0 175 0 3 .82 - 12.02 -2.02 + 10.501 4.04 T a b l e l X . 1p b i n d i n g e n e r g i e s i n l i g h t Z° h y p e r n u c l e i . S t a t e s with r>0 are quasi-bound The case of g>Li deserves s p e c i a l a t t e n t i o n because, although the pole i s very near to the r e a l a x i s , the s t a t e i s not q u i t e bound. However, s i n c e the o r i g i n a l c h o i c e of kp and B was to some extent a r b i t r a r y , the q u e s t i o n n a t u r a l l y a r i s e s whether f o r any reasonable c h o i c e of parameters the pole does c r o s s the r e a l a x i s to become a bound s t a t e . We f i n d i n f a c t t h a t i t does not. By reducing the Fermi momentum to zero and i n c r e a s i n g the nucleon energy to +20 Mev ( i . e B=-20 MeV) i t i s p o s s i b l e to decrease the width of the l e v e l to 0.06 MeV from 0.52 MeV, but the pole remains i n the t h i r d quadrant. 1 14 Larger changes i n k F and B than the above would not be warranted on any p h y s i c a l grounds, but the r e s u l t s of the model c a l c u l a t i o n should not be construed of course as meaning t h a t the £<>Li p - s t a t e i s n e c e s s a r i l y unbound. C l e a r l y a very small change i n the o v e r a l l s t r e n g t h of the c o u p l i n g to the AN channel would be s u f f i c i e n t to produce b i n d i n g i n L i , without q u a l i t a t i v e l y a f f e c t i n g the p - s t a t e r e s u l t s f o r the other n u c l e i . The e l a s t i c phase s h i f t s 6, and a b s o r p t i o n c o e f f i c i e n t s n, are a l s o shown i n f i g . 9. For the A>9 n u c l e i we n o t i c e the n phase change between k=0 and o° , c o n f i r m i n g our i n t e r p r e t a t i o n of these as bound s t a t e s . The deep minimum in the a b s o r p t i o n c o e f f i c i e n t i s a l s o c l e a r near the p o s i t i o n of the zero (or b i n d i n g energy). By c o n t r a s t , 6 and n f o r the s - s t a t e s are shown, in f i g . 10. For the he a v i e r (A>9)nuclei the curves are seen t o be n e a r l y d e v o i d of i n t e r e s t i n g s t r u c t u r e , as expected f o r bound s t a t e p o l e s at n egative e n e r g i e s . The l i g h t e r He and L i curves, however, show d i p near the ( p o s i t i v e ) bound s t a t e e n e r g i e s . Since i n these cases the pole s l i e c l o s e r to the negative, r e a l momentum a x i s than the imaginary a x i s , we know an S-matrix zero i s very c l o s e to the p o s i t i v e , r e a l a x i s . 115 8 I2 ENERGY (MeV) 16 F i g . 9. P-wave phases 6 and a b s o r p t i o n c o e f f i c i e n t s n f o r low energy e l a s t i c E-nucleus s c a t t e r i n g . For the l i g h t n u c l e i shown i n a ) , and b) no bound s t a t e s e x i s t , as seen by the zero-degree phase s h i f t s at zer o energy. The 180° phase s h i f t s f o r the n u c l e i w i t h A>8, shown i n c)->e), a r e evidence of bound s t a t e s . These s t a t e s are i n the continuum, about where n i s a minimum. ENERGY (MeV) 118 4 8 12 16 ENERGY (MeV) F i g . 10. S-wave phases 6 and a b s o r p t i o n c o e f f i c i e n t s n f o r low energy e l a s t i c I-nucleus s c a t t e r i n g . For the l i g h t n u c l e i shown i n a) and b) n has a d i p near the bound s t a t e e n e r g i e s . For the h e a v i e r n u c l e i t h e r e i s no i n t e r e s t i n g s t r u c t u r e , as expected f o r n e g a t i v e b i n d i n g e n e r g i e s . ENERGY (MeV) ENERGY (MeV) 121 Again, these s t a t e s have been r e c a l c u l a t e d u s i n g the parameters of Toker et a l , and these r e s u l t s are given i n t a b l e X. The p r e d i c t i o n s are s i m i l a r to those of our model but again there are some s i g n i f i c a n t d i f f e r e n c e s . F i r s t of a l l we remark that t h i s model a l s o p r e d i c t s a ^fO bound s t a t e . In l i g h t e r n u c l e i the r e a l component of the resonant e n e r g i e s are f a i r l y c l o s e to our v a l u e s , but i t i s important to a p p r e c i a t e that the bound s t a t e s and narrower widths of our model a r i s e from g r e a t e r a b s o r p t i o n , r a t h e r than g r e a t e r a t t r a c t i o n . Nucleus K F (MeV/c) 8 (MeV) E p - c-lT/2 (MeV) P(Mev)l/2 r I ° H e 75 0 9.01 + i 1 8 .7 -3.85 - i2.42 -37.4 100 0 5.43 + i 8 . 8 1 -2 . 8 1 - i 1.57 -17.6 i > 125 0 3.14 + i3.70 -2.00 - iO.93 -7.40 1 2C Z° 150 0 2.94 + i1.40 -1.76 - i0.40 -2.80 1 6 0 175 0 1.20 - i0.34 - 1 . 1 1 + iO. 15 .68 Table X. P o s i t i o n of the i p p o l e s c a l c u l a t e d with the model A parameters o f Toker et a l [28]. S t a t e s with y>0 are quasi-bound. 122 To complete t h i s study, 6 and n have been c a l c u l a t e d f o r D-wave E.-nucleus i n t e r a c t i o n s . As can be seen i n f i g . 11, a d i p i s beginning to develop near 12 MeV e x c i t a t i o n i n ^t^' ^ u t * n 9 e n e r a l there i s no i n d i c a t i o n of s t r u c t u r e . These f i n d i n g s are i n agreement with the phenomenological r e s u l t of Gal et. al. [11] where, although t h e i r p o t e n t i a l was more a b s o r p t i v e than ours, they' found no bound D - s t a t e s f o r n u c l e i l i g h t e r than s i l i c o n . In b r i e f summary then, we f i n d that when nucleon b i n d i n g and P a u l i e x c l u s i o n e f f e c t s are i n c l u d e d i n a m i c r o s c o p i c c a l c u l a t i o n narrow I s - s t a t e s r e s u l t . On the other hand we f i n d that i n p-wave, where these e f f e c t s are s m a l l , the consequent strong n u c l e a r a b s o r p t i o n produces remarkably l o n g - l i v e d s t a t e s i n the continuum, i n c l o s e agreement with experimental e n e r g i e s . 11. D- Wave phases 6 and a b s o r p t i o n c o e f f i c i e n t s , f o r I L!?? rF e^ a s t l <r ^ n u c l e u s s c a t t e r i n g . Although a s m a l l d i p m „ i s d e v e l o p i n g i n 1 v 6 0, the curves are g e n e r a l l y d e v o i d of s t r u c t u r e 2 curves 124 4 8 12 16 ENERGY (MeV) ENERGY (MeV) 126 CHAPTER 6 DISCUSSION AND CONCLUSIONS Our aim i n t h i s work has been to c a l c u l a t e the widths of l i g h t E° hypernuclear s t a t e s as a c c u r a t e l y as p o s s i b l e . To t h i s end we c o n s t r u c t e d a separable p o t e n t i a l model to d e s c r i b e the EN s c a t t e r i n g where i t was found t h a t , i n a d d i t i o n the u s u a l S-wave i n t e r a c t i o n s , the 3S,(EN)-> 3D,(AN) t r a n s i t i o n was e s s e n t i a l to the d e s c r i p t i o n of the co n v e r s i o n r e a c t i o n . With these i n t e r a c t i o n s as fundamental input, the E nucleus s i n g l e p a r t i c l e p o t e n t i a l was developed with c a r e f u l a t t e n t i o n to c o r r e c t i o n s a r i s i n g from P a u l i e x c l u s i o n and nucleon b i n d i n g e f f e c t s . A f t e r making a minimum number of approximations we were able to ev a l u a t e a n a l y t i c a l l y the angular i n t e g r a l s i n the d e f i n i n g r e l a t i o n f o r t h i s p o t e n t i a l , and to show th a t i n the a p p r o p r i a t e l i m i t i t reduced to the phenomenological r e s u l t o b tained from the a n a l y s i s of E" atoms. Most im p o r t a n t l y , the s e l f - c o n s i s t e n t s o l u t i o n s of the momentum-space Schroedinger equation p r e d i c t e d narrow E s t a t e s i n both S and P waves, although the mechanisms r e s p o n s i b l e were q u i t e d i f f e r e n t . In s - s t a t e , P a u l i e x c l u s i o n and nucleon b i n d i n g e f f e c t s produced long l i v e d s t a t e s with widths ranging from 1.8 MeV in.jHe to 12.5 MeV 127 in ^fO. In p - s t a t e by c o n t r a s t , the P a u l i p r i n c i p l e had l i t t l e i n f l u e n c e on the p o t e n t i a l and the r e s u l t i n g s t r o n g a b s o r p t i o n produced narrow s t a t e s i n the E continuum with widths from 0.5 MeV i n ^ B e to 4.0 MeV i n 1 ziO. The mechanisms r e s p o n s i b l e f o r s u p p r e s s i n g EN-^ AN c o n v e r s i o n i n s - s t a t e s are w e l l understood. In p - s t a t e s though, the p h y s i c a l i n t e r p r e t a t i o n of s t r o n g a b s o r p t i o n producing narrow widths i s not at a l l c l e a r . There i s no denying that t h i s runs c o n t r a r y to one's p r e j u d i c e t h a t , with strong c o u p l i n g to an open channel, a bound s t a t e w i l l 'leak out' to the s c a t t e r i n g s t a t e of equal energy and hence can not e x i s t f o r any great l e n g t h of time. I t i s important that a p i c t u r e of these s t a t e s be developed and so l e t us f i r s t summarize t h e i r c h a r a c t e r i s t i c s . Without a weak c o u p l i n g h y p o t h e s i s , l o n g - l i v e d s t a t e s degenerate with the continuum can e x i s t i n the presence of an open channel. Such a s t a t e i s u n r e l a t e d to a bound s t a t e i n the i s o l a t e d channels and, indeed, would not e x i s t as a n o r m a l i z e a b l e s t a t e i n the absence of the strong c o u p l i n g . G a l , Toker, and Alexander [11] have t r i e d to i n t e r p r e t t h i s phenomena in a s i m i l a r framework to resonances i n ( f o r example) e-He* s c a t t e r i n g . i n the l a t t e r system i t i s w e l l known t h a t a s t a b l e s t a t e of He with both e l e c t r o n s i n the 128 n=2 l e v e l e x i s t s above the i o n i z a t i o n energy f o r one e l e c t r o n . Consequently, i n e-He + s c a t t e r i n g a sharp resonance r e s u l t s f o r the i n c i d e n t energy (~27 eV) at which t h i s s t a t e can be e x c i t e d . Gal e_t a_l have p i c t u r e d the narrow hypernuclear s t a t e s as resonances in A-nucleus s c a t t e r i n g . I t i s imagined t h a t . a t some p a r t i c u l a r energy the A c o n v e r t s on a nucleon with the f i n a l s t a t e N k i c k e d i n t o an e x c i t e d s t a t e and the E bound with negative b i n d i n g energy. The e x p l a n a t i o n i s tempting because of i t s f a m i l i a r i t y , and such s t a t e s may w e l l e x i s t , but the model i s q u i t e wrong f o r the l e v e l s we are c o n s i d e r i n g . F i r s t of a l l , t h e i r p o t e n t i a l which produced these s t a t e s was energy independent. I t i s not reasonable, t h e r e f o r e , to suggest that these s t a t e s depend on the m i c r o s c o p i c nucleon energy spectrum f o r t h e i r i n t e r p r e t a t i o n . Secondly, a resonance i n the e l a s t i c c r o s s s e c t i o n i s expected with t h i s p i c t u r e . As we have seen though, i n E-nucleus s c a t t e r i n g an S-matrix zero, r a t h e r than a p o l e , i s e x h i b i t e d and the c r o s s - s e c t i o n i s completely smooth. So the q u e s t i o n remains as ' to the meaning of these s t a t e s . To answer t h i s we w i l l c o n s i d e r the s i m p l i f i e d example of the E and A channels coupled by a square-well p o t e n t i a l matrix with elements Vyy' and range R. In the 129 scattering formalism the outgoing waves are related to the incoming ones via the S-matrix as in an obvious notation. A E bound state (of s t r i c t l y i n f i n i t e l ifetime) corresponds to S^-O. In other words, the two channels are completely decoupled so that no net flux i s lost from the E to A channels. It i s straightforward to show [21] that t h i s condition i s s a t i s f i e d for K LcotK zR = K^cotK^R (141) where the K$/2m are the channel eigenvalues. So we find that these states exist because of an interference e f f e c t , a r i s i n g from a fine balance between the channel potentials. More generally, the above i s related to Gal's result [11] that, for resonant or bound states above threshold, the width r (up to a normalization factor) i s • r = Re(k 2) | * Z(R) | 2 + Re(k A) |*A(R) | 2 (142) where, again, R is the range of the po t e n t i a l , and k^/2m, k^/2m are the channel eigenvalues. In t h i s case the influence of the potentials is disguised in the Wavefunctions, but i t should be clear that with a pole on 130 sheet 2 ( i n the n o t a t i o n of C h a p t e r 4) the r e a l components of k£,k A a r e o p p o s i t e i n s i g n and c a n c e l l a t i o n o c c u r s between the two terms of e g . ( 1 4 2 ) , becoming complete as the p o l e moves a c r o s s the r e a l a x i s from sheet 4 t o sheet 2. In view of the two d i s t i n c t c auses of narrow w i d t h s i n s and p - s t a t e s i t seems i m p o r t a n t t h a t measurements of p - s t a t e I h y p e r n u c l e i l e v e l s s h o u l d f i r s t be extended t o v e r y l i g h t n u c l e i where they are p r e d i c t e d t o be e x c e p t i o n a l l y l o n g - l i v e d . S e c o n d l y one would l i k e r e l i a b l e d a t a f o r the s - s t a t e s . 131 BIBLIOGRAPHY 1) C.J. Batty, Phys. Lett. 87B (1979) 324 2) A. Gal, Nukleonika 25 (1980) 447 3) R. B e r t i n i et a l , in Meson-Nuclear Physics, 1979 (Houston), ed. E.V. Hungerford (AIP, NY) p.703 4) W. Brueckner et a l , in Proceedings of the Kaon Factory Workshop, 1979 (TRIUMF), ed. M.K. Craddock, p.136 5) H. Piekarz et a l , Phys. Lett. 110B (1982) 428 6) R. B e r t i n i et a l , Phys. Lett. 90B (1980) 375 7) P.D. Barnes, invited talk at the 9th ICOHEPANS, V e r s a i l l e s ( 1 9 8 1 ) 8) A. Gal, in Proceedings of the Second Kaon Factory Workshop, 1981 (TRIUMF), eds. R. Woloshyn and A. Strathdee, p.148 9) A. Gal and C.B. Dover, Phys. Rev. Lett. 44 (1980) 379 10) C.B. Dover and A. Gal, Phys. Lett. 110B (1982) 433 11) A. Gal, G. Toker and Y. Alexander, Ann. Phys. 137 (1981) 12) C.J. Batty et a l , Phys. Lett. 74B (1978) 27 13) W. Stepien-Rudzka and S. Wycech, Nucl. Phys. A362 (1981) 14) R. Engelmann, H. F i l t h u t h , V. Hepp and G. Zech, Phys. Lett. 2_1_ (1966) 587 15) F. E i s e l e , H. F i l t h u t h , W. Fohlisch, V. Hepp and G. Zech, Phys. Lett. 37B (1971) 204 16) A.W. Thomas, in Modern Three-Hadron Physics, 1977 (Springer-Verlag), ed. A.W. Thomas, p.12 17) C. Lovelace, in Strong Interactions and High Energy Physics, 1963 (Scottish U n i v e r s i t i e s ' Summer School), ed. R.G. Moorhouse, p.437, and C. Lovelace, Phys. Rev. J_35 5B (1964) B1225 1 32 18 19: 20 26 29 30 31 M.M. Nagels, T.A. R i j k e n and J . J . DeSwart, Ann. Phys. 79 (1973) 338 M.M. Nagels, T.A. R i j k e n and J . J . DeSwart, Phys. Rev. D12 (1975) 744 ; Dj_5 (1977)2547 M.M. Nagels, T.A. R i j k e n and J . J . DeSwart, Phys. Rev. D20 (1979) 1633 21) R.G. Newton, S c a t t e r i n g Theory of Waves and P a r t i c l e s , McGraw-Hill (1966) 22) M.L. Goldberger and K.M. Watson, C o l l i s i o n Theory, John Wiley and Sons (1964) 23) J.T. Londergan, K.W. McVoy and E . J . Moniz, Ann. Phys. 8_6 (1974) 147 24) H. van Haeringen and R. van Wageningen, J o u r n a l of Math. Phys. j_6 ( 1 9 7 5 ) 1441 25) H. van Haeringen, N u c l . Phys. A253 (1975) 355 0. Braun et a l , N u c l . Phys. B124 (1977) 45 27) R. D a l i t z , in Meson-Nuclear P h y s i c s , 1979 (Houston) ed. E.V. Hungerford (AIP,NY) p.621 28) G. Toker, A. Gal and J.M. E i s e n b e r g , N u c l . Phys. A362 (1981) 405 J.A. Johnstone, EN Strong I n t e r a c t i o n s , (M.Sc. T h e s i s , unpublished) 1979 J . Dabrowski and J . Rozynek, Phys. Rev. C23 (1981) 1706 P.C. Tandy, E. Redish and D. B o l l e , Phys. Rev. L e t t . 35 (1975) 921 32) R,H. Landau and A.W. Thomas, N u c l . Phys. A302 (1978) 461 33) E. Lomon and M. Mc M i l l a n , Ann. Phys. 2_3 (19.63) 439 A.W. Thomas and R.H. Landau, Phys. Rep. 58 (1980) 121 35) R.C. B a r r e t t and D.F. Jackson, Nuclear S i z e s and S t r u c t u r e , Clarendon Press (1971) 36) M. Moshinsky, N u c l . Phys. j_3 (1959) 104 1 33 37) M. S t i n g l and A.S. R i n a t , N u c l . Phys. A154 (1970) 613 38) L. Fonda and R.G. Newton, Ann. Phys. j_0 (i960) 490 39) K. H a r t t , Phys. Rev. C 2 2 (1980) 1377 40) G.A. Baker, E s s e n t i a l s of Pade Approximants, Academic Press (1975) 41) J.N. Massot, E. El-Baz and J . L a F o u c r i e r e , Rev. of Mod. Phys. 39 (1967) 288 134 APPENDIX I ANGULAR MOMENTUM GRAPHICS The t e c h n i q u e o u t l i n e d below i s a g r a p h i c a l method f o r c a l c u l a t i n g sums of 3 J c o e f f i c i e n t s , adapted from the work of M a s s o t , E l - B a z , and L a F o u c r i e r e [ 4 1 ] . On ly those r e s u l t s which a r e needed i n the p r e s e n t work are g i v e n , and none of these a r e p r o v e n . F i r s t , a Wigner 3 J i s d e f i n e d by the p i c t u r e I The r u l e s f o r c o n s t r u c t i n g the p i c t u r e a r e : (1) The s i g n of the v e r t e x i s p o s i t i v e i f the j ' s are r e a d i n an a n t i - c l o c k w i s e d i r e c t i o n , and n e g a t i v e i f r e a d c l o c k w i s e . ( 2 ) The arrows p o i n t outward from the v e r t e x f o r p o s i t i v e m , and inward f o r n e g a t i v e m . ( 3 ) Changing e i t h e r the s i g n of the v e r t e x or the arrow d i r e c t i o n of a l l t h r e e l i n e s g i v e s a phase change of • • • ( - ) J ' * J i*3». (The d i r e c t i o n of one f r e e l i n e a l o n e must not be changed . ) (4 ) The 3 J r e p r e s e n t e d by the d iagram i s i n v a r i a n t under r o t a t i o n s and geometr i c d e f o r m a t i o n s . ( A I . 1 ) 135 To sum g r a p h i c a l l y o v e r one m which i s common t o two 3J c o e f f i c i e n t s the c o r r e s p o n d i n g f r e e j l i n e i n each d iagram a r e j o i n e d so t h a t the arrows p o i n t i n the same d i r e c t i o n . F o r example , -(-)3H<3S-«V"* £(->J«'-"W j i j» j , \ ( j i JM 3s\ m, ^m, m 2 m 3y \j-m , -m»4 - m 5 y and the sum i s r e p r e s e n t e d by The r u l e s f o r summing over the p r o j e c t i o n m of j a r e t h a t : (5) A f a c t o r of ( - ) J ' m m u s t be p r e s e n t . ( 6 ) Whenever two v e r t i c e s a r e j o i n e d by a l i n e j , the d i r e c t i o n of t h a t l i n e may be r e v e r s e d but t h i s i s accompanied by a phase change of ( - ) 2 3 . (7) I f t h e r e a r e t h r e e f r e e l i n e s , t h e i r l o o s e ends may may be p i n c h e d t o g e t h e r w i t h the arrows p o i n t i n g i n the same d i r e c t i o n . A 3J m u l t i p l i e s the d i a g r a m , . • r e p r e s e n t e d by the t h r e e l i n e s , w i t h t h e i r arrows p o i n t i n g i n the o r i g i n a l d i r e c t i o n , and the o p p o s i t e 136 s i g n at the v e r t e x . T h i s l a s t r u l e i s demonstrated much more c l e a r l y by a p i c t u r e CLOSED •3a • 3 * j i (AI.4) T h i s l a s t r u l e a l s o a l l o w s a l a r g e , c l o s e d diagram to be broken i n t o s m a l l e r components as (AI.5) w i t h < ^ = E <->*W-«"» / j , j a j s \ ( j . ji j s \ *»' ^m, m2 ma J -^m, -rrtj -mj y (AI.6) s 1 i f J i » J a ' J 3 obey the t r i a n g l e i n e q u a l i t y , a n d = 0 otherwise. For the purposes of t h i s work, the only other r e s u l t s needed are the diagrams f o r 6J and 9J symbols. These are given by 137 As an example of the u s e f u l n e s s of t h i s technique, equation (23) f o r the d i f f e r e n t i a l c r o s s s e c t i o n w i l l be d e r i v e d . C o n c e n t r a t i n g on j u s t the p i e c e which i n v o l v e s Zl vv' we have ^ < L 0 S i /1 J i / > < L " 0 S v i J ' y X L ' v - i / ' L " ' v ' -v | 1 0 > < L ' v~v' S i / ' \Jv> vv' , < L " ' i / - v ' S i / ' \J' v>(-)y'v (AI.9) F i r s t , t h i s i s c o n v e r t e d t o 3J's as A A A J 2 J 2 ' l (-)L*l?-v-v7s J L > \ f s J' L"VL' L"' L\ Vv -v OJ \v -v 0 Ay - v ' - v+v ' O / / 1 / S j V L"'S J ' ^ \v-v ' v' -w )\y-v' v ' -v / ( A l . 1 0 ) T h i s e x p r e s s i o n w i l l be m o d i f i e d somewhat so t h a t i t resembles the 3J's of ( A l . 2 ) more c l o s e l y f o r summation i n 138 d i a g r a m a t i c f o r m . We can i n t r o d u c e the dummy v a r i a b l e s M 3 ,Mj«,M , M ' , M " , M " ' , a n d m. There w i l l be no summation over M , M " , a n d m, and e v e n t u a l l y t h e i r v a l u e s w i l l be se t e q u a l t o z e r o , but f o r now n o n - z e r o v a l u e s w i l l a l l o w us t o a s s i g n d i r e c t i o n s and magnitudes t o the arrows i n the summation i n a m e a n i n g f u l way. The ( f o r m a l ) summation over the o t h e r M ' s w i l l not a l t e r 'the r e s u l t because t h e i r v a l u e s a r e a c t u a l l y f i x e d by M , M " , and m. A l s o S , i / ' w i l l be r e l a b e l l e d as S ' , v ' f o r the moment to d i s t i n g u i s h i t from S , ' . The summation ( A I . 9 ) can be e x p r e s s e d as J 2 J 2 » 1 ^ ( - ) 3 • « J / ( - ) J-wb ( . ) J ' - ^ ( _ ) 5 - v(-)s ' - v'(_)^. ,-M ( - ) l-"'-M• ,' (3 S L\/ S J ' L " Y L " ' L ' A \MJ v m ) \ r v - M ^ - M " / V M " ' M ' m) (AI.11) / L' S' J\ ( S' L"'J'N \~M' - v' -Mj/ V, v ' ' M j / We have a l l the phases ( - ) 3 _ f c A n e c e s s a r y to do t h i s i n d iagrammat ic f o r m . In the f o l l o w i n g , the s t e p s i n m a n i p u l a t i n g the graphs w i l l be shown w i t h the o v e r a l l phase A A A of the sum i n d i c a t e d . I g n o r i n g the J 2 J 2 ' 1 term f o r now, the sum i s r e p r e s e n t e d by ( i ) ( A I . 1 2 ) 139 (-) T - J ' - I : ( A I . 1 3 ) Rule 3: s i g n change of v e r t e x (-) T+y+L + C'-l (AI.14) Rule 7: s e p a r a t i o n - o f a 3J and p i n c h i n g the l i n e s . (-)J-J-il^'-£ (AI.15) Rule 8 : breaking of a l a r g e r diagram. ( ~ ) £ (AI.16) Rule 4: change of the arrow d i r e c t i o n s I t i s found by compar i son w i t h the p i c t u r e of a 6J symbol t h a t the whole e x p r e s s i o n has reduced t o J2J 2'1(-)LS L " ' J ' S ( J J ' L " S? (L L " l \ ( A I . 1 7 ) ) 1 L ' J J j l J L J \0 0 OJ I n s e r t i n g t h i s e x p r e s s i o n i n t o e q . ( 2 U g i v e s e q . ( 2 S ) f o r the d i f f e r e n t i a l c r o s s - s e c t i o n . 141 APPENDIX II SOLUTION OF THE COUPLED-CHANNEL EQUATIONS F o l l o w i n g the procedure of Londergan et a l . [ 2 3 ] , we w i l l o u t l i n e the steps i n v o l v e d i n s o l v i n g f o r the coupled-channels t - m a t r i x with rank-one sep a r a b l e potent i a l s . In g e n e r a l , f o r any p o t e n t i a l operator v, and N coupled channels, the t r a n s i t i o n o perator d e s c r i b i n g s c a t t e r i n g from channel |o<> to the channel |t3> i s N V G ° FCTK (AII.1) where G 0 i s the y-channel Green's f u n c t i o n d e f i n e d by eq.(25). I f the p o t e n t i a l s are a l l rank-one s e p a r a b l e , then can be w r i t t e n as \vp >x^  <v o t | i n D i r a c n o t a t i o n . Here, X. = -1( + 1) f o r a t t r a c t i v e ( r e p u l s i v e ) i n t e r a c t i o n s , and the s t r e n g t h of the p o t e n t i a l has been absorbed i n t o the form f a c t o r s |v>. In t h i s case ( A l l . 1) can be s o l v e d a l g e b r a i c a l l y . S e p a r a t i n g the |OC>-*|N> t r a n s i t i o n from (AII.1) g i v e s t40l *f*f p»y* Y r-i ( A l l . 2) + | v N > X H M < v M | G ? H-l 1 - i w < v N | G S | v N > ( A l l . 3 ) 142 The r e s u l t ( A l l . 3 ) i s then i n s e r t e d i n t o ( A I I . 1 ) t o e l i m i n a t e the | o > - » | N > term from the summation. I t i s found t h a t tpa becomes V - | v * > X * J ( E ) < v o t | + | v . > ^ X ^ ( E ) < v Y | G j t y o t ( A l l . 4 ) so t h a t the t - m a t r i x i s now e x p r e s s e d i n the form of N-1 fey c o u p l e d c h a n n e l s , w i t h the r e s c a l e d c o u p l i n g c o n s t a n t s x (E) energy-dependent and d e f i n e d as X?NCE)°= S Y m r 1 ^ - ^ y < V F,|G? Iv t l >i ( A H . 5 ) l -Xl^vjGo 1 1 v N > X*v By p l a c i n g some r a t h e r weak r e s t r i c t i o n s on the c o u p l i n g c o n s t a n t s X the second term i n p a r e n t h e s e s can be made to v a n i s h . ( i ) X^X 'S^-X" , ( A H . 6) and ( i i ) x'j =X^ The e f f e c t of these r e s t r i c t i o n s i s t o i n s i s t t h a t the X a r e i d e n t i c a l f o r a l l i n t e r a c t i o n s w i t h i n a g i v e n se t of c o u p l e d c h a n n e l s . The N-1 c h a n n e l can be removed from ( A I1 . 4 ) i n an i d e n t i c a l manner to the N - c h a n n e l . T h i s w i l l r e s u l t i n r e d e f i n e d c o u p l i n g c o n s t a n t s a g a i n . x£( E) - f r [ 1 -r < v j dt\ V>-X N M <v N |Go I v N >] - 1 (AI 1.7) 143 The r e m a i n i n g c h a n n e l s can a l l be removed i n t h i s way to l e a v e the e f f e c t i v e o n e - c h a n n e l p r o c e s s : V * I V x * * ( E ) < y * l + I V ^ * ( E ) < V ( A l l . 8 ) w i t h X ^ ( E ) = X ^ [ 1 - I) k r r < v y | G e | v y > ) - 1 ( A H . 9 ) N - 1 which has the s o l u t i o n V " l y ^ a l ( A l l . 1 0 ) 1- £ X r r < v r | G j | v r > The summation over r i n c l u d e s o n l y the d i a g o n a l e lements of a l l the c o u p l e d c h a n n e l s . 144 APPENDIX III PARTIAL WAVE EXPANSION OF Vj.(k' ,k) In t h i s s e c t i o n we w i l l b r i e f l y o u t l i n e the steps i n v o l v e d i n s o l v i n g f o r the s i n g l e p a r t i c l e p o t e n t i a l i n the 1th p a r t i a l wave. The p a r t i a l wave expansions of the nu c l e a r wavefunction and the EN form f a c t o r s f o l l o w simply enough from the r e c u r s i o n r e l a t i o n s f o r the Legendre p o l y n o m i a l s and so t h e i r forms w i l l j u s t be assumed here. The s i n g l e p a r t i c l e p o t e n t i a l was d e f i n e d i n terms of the n u c l e a r wavefunctions and r e a c t i o n m atrix T as [eq.( 7 ^ ) ] V(k' ,k) = J dP F(P-k';P-k)r(k',P-k';k,P-k:u) (AIII.1) oo = 1^(21+1) V £ ( k ' : k ) P^(k k') 4 " «.o With the expansions of F and T g i v e n by eqs.(75) and (77) the p o t e n t i a l Vg i s V k ' : K > - / a p 2 i ; ? 5 r E 5 ^ " ) S v " ( k , p , T ' " < k , , p > l L JH'Jt' L S J ^ i r ( 2 L + D I" \ r ( 2 a + 2 ) r ( 2 b + 2 ) r [ 2 ( L + 1 - b ) ] r [ 2 ( L + 1 - a ) ] J V/a (AIII.2) y ^ ( - € P ) a * b k u - < x k ' L'-b<aaL-aM-o|LM><bjL-bM-? |LM> 1 ydnKdn^dnp [ P x ( k . k f J P ^ . k ' J P ^ . p J P ^ j d c i t ? ) - P n ( ^ P ) P y k i P ) i ; ( P ) Y p P ) Y ^ ( i ) Y ^ » ) ] 146 Using the a d d i t i o n theorem f o r s p h e r i c a l harmonics the Legendre polynomials become P£ V i ^ n P m « ^ | <101' 01 L0> | 2 | <n01" 0 | L" 0> | 2 | <m01" ' 0 | L" ' 0> | 2 L ' L Y ' P L,(k.K' J P j . d t - P j P ^ k i P ) (AH1.3) and PL, P ^ L ' - 2 L " - 2 L " ' - 2 (4n ) ^ Y ^ ^ ) Y*fo' ) Y^?(^) (AIII.4) y^(P)Y^)Y5'(k') With the well-known r e s u l t f o r the i n t e g r a l of three s p h e r i c a l harmonics that (4n)1/yrdnyj^(n)y^n)Y^n) = i 4 i x i 3 U l^j i ^ i i j 1 * 5 ) the angular i n t e g r a l s of e q . ( A I I I . 2 ) become EA A A A A A — ^ , v (AIII.6) L ' W ' ^ ^ ( W l l L - t l / l 1' L'\ 2 /n 1" L B >\ 2 /m 1"' L"'\ 2 / V L" L -a \ ft,' L"'L-b\ o o ) \o o o ; \^ o o o J \o o o J fb L" L'\ /a L"' L'\ fl,' L" L-aN /L' L"'L-b\ \0 0 0 ,/ \0 0 0 J yM' M" M-oI U ' M"'M-*J A." b L'\ /L"' a L f >\ M^" » M'J IM"' O M'J 147 When t h i s r e s u l t i s combined w i t h the r e s t of the e x p r e s s i o n i n c u r l y b r a c k e t s o f the l a s t page , the sum over *m* g i v e s a term p r o p o r t i o n a l t o a 9 J c o e f f i c i e n t . In f a c t t h i s i s a p a r t i c u l a r l y s i m p l e sum t o do u s i n g the g r a p h i c a l t e c h n i q u e d e s c r i b e d i n appendix I . The columns can be i n t e r c h a n g e d w i t h i m p u n i t y i n s i d e any 3 J w i t h o u t a f f e c t i n g the o v e r a l l phase because the I j * s are r e s t r i c t e d to even v a l u e s . D i a g r a m m a t i c a l l y , the summation i s r e p r e s e n t e d by ( A I I I . 7 ) L' + b + w h i c h , w i t h the n o t a t i o n of appendix I , i s j u s t the 9 J symbol: ( A I I I . 8 ) Combining t h i s r e s u l t w i t h the r e s t of the 3 J symbols g i v e s 148 the f i n a l r e s u l t , e q . ( 8 0 ) f f o r the p o t e n t i a l i n the 1th p a r t i a l wave. With o n l y S and P wave IN c o n t r i b u t i o n s the 9J i s almost t r i v i a l because a t l e a s t two of a,b,L-a,and L-b are always z e r o . The 9J then has the v a l u e : ( i ) S-Wave; a,b,L,L-b,L-a 0 L' L' 0 0 0 0 L"' L 1 (2L' + 1 ) 6 L ' L ' L V (AIII.9) ( i i ) P-Wave; (a) a,b=0;L,L-a,L-b=l 1 L* L") l 3(2L"+1) 0 L"' L I (b) a,b=1;L-a,L-b=0 0 L' L" 1 0 1 1 L"' L 3(2L'+1) (c) a j L - b ^ O j b r L - a s I 1 L' L' 1 0 1 0 L"' L 1 3(2L'+1) (AIII.10) (AIII.11) (AIII.12) 149 (d) b , L - a = 0 ; a , L - b « l 0 L' L" ( A I I I . 1 3 ) 1 1 0 s 3(2L' + 0 1 L " ' L , 

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