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A new hamiltonian for systems of nucleons and pions Hsieh, William Wei 1978

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A NEW HAMILTONIAN FOR SYSTEMS OF NUCLEONS AND PIONS by WILLIAM WEI^HSIEH B.Sc, University of B r i t i s h Columbia, 1976 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n 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 June, 1978 © William Wei Hsieh, 1978 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d that c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . ( W i l l i a m W. Hsieh) Department o f P h y s i c s  The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook P l a c e V a n c o u v e r , Canada V6T 1W5 Date June 17, 1978 A b s t r a c t Th is t h e s i s p resents new i n t e r a c t i o n p o t e n t i a l s f o r s tudy ing systems of nucleons and p ions at i n t e r m e d i a t e e n e r g i e s . Us ing a quantum f i e l d theory approach, the fundamental dynamical v a r i a b l e s are taken to be the Fermion and Boson c r e a t i o n o p e r a t o r s , f and B . F i r s t , c o n s i d e r i n g systems w i t h on ly n u c l e o n s , a 2-body nuc leon i n t e r a c t i o n p o t e n t i a l i s i n t r o d u c e d , from which the f a m i l i a r p a i r of coupled d i f f e r e n t i a l equat ions f o r the deuteron i s d e r i v e d . Next , t u r n i n g to systems of nucleons and p i o n s , f o c u s i n g p r i m a r i l y on the r e a c t i o n f>+p TT++ d , we i n t r o d u c e u n c o n v e n t i o n a l , p e n t a - l i n e a r i n t e r a c t i o n p o t e n t i a l s of the form " F F FFB + a d j o i n t " . Wi th these unconvent iona l p o t e n t i a l s , we can i d e n t i f y F+l°"> and B + l°> w i t h p h y s i c a l nucleons and p h y s i c a l p i o n s — q u i t e u n l i k e the c o n v e n t i o n a l s i t u a t i o n w i t h the Chew H a m i l t o n i a n , where F |o> cannot be i d e n t i f i e d w i t h a p h y s i c a l n u c l e o n . The d i f f e r e n t i a l s c a t t e r i n g c r o s s s e c t i o n f o r |'|'-^¥+c( (w i th p o l a r i z e d i n c i d e n t protons) i s then d e r i v e d i n terms of our p o t e n t i a l s . F i n a l l y , we i n c l u d e a s imple p e r t u r b a t i o n study of the deuteron s t a t e u s i n g our p o t e n t i a l s . i i i TABLE OF CONTENTS page Chapter 1 Introduction 1 Chapter 2 Mathematical Preliminaries: Operators and Their Transformation Properties 4 2 .1 Fermion and Boson Creation and Annihilation Operators . 4 2 .2 Singlet and Tr i p l e t Operators A ^ , SL •> -Sf/» ~C0 , T m / l . 8 2 .3 The 2-Fermion Operator y 14 2.4 The 2-Fermion Transfer Operator ($Zm 15 2.5 The 2-Fermion Transfer Operator 16 Chapter 3 Systems of Nucleons 17 3 . 1 The 2-Body Potential Vte 17 3 .2 Derivation of the Deuteron Equations 22 3 .3 The Singlet Even Scattering State 25 3.4 The T r i p l e t Odd Scattering State 27 Chapter 4 Systems of Nucleons and Pions 31 4 . 1 The Potential V for Singlet-Triplet Reactions 33 4 . 2 The Potential V for T r i p l e t - T r i p l e t Reactions . . . . 37 4 . 3 The D i f f e r e n t i a l Scattering Cross Section for pp->ifd . . 39 4.4 A Perturbation Study of the Deuteron State 47 Chapter 5 Summary and Conclusion 52 Bibliography 55 Appendix A The Motion Reversal Operator [J 56 Appendix B Some Properties of the Clebsch-Gordan Coefficients . . 57 Appendix C Some Properties of the Rotational Matrices D*M,("0*) • • 58 Appendix D Some Properties of the Spherical Harmonics Y M^ . . . 60 i v page Appendix E Wigner 3 - j , 6 - j , 9 - j Symbols and Racah C o e f f i c i e n t s . . 62 Appendix F D e r i v a t i o n of the Deuteron Equat ions 65 Appendix G The D e n s i t y Operator W 70 V Ac knowled gement s I would l i k e to take this opportunity to thank my supervisor, Dr. J. Malcolm McMillan, for having been, through the daily labyrinth of research, a superb guide— and at d i f f i c u l t times, a deus ex machina. I am also grateful to the National Research Council of Canada for f i n a n c i a l assistance throughout the l a s t two years i n the form of scholarships. 1 Chapter _1 I n t r o d u c t i o n For a long t i m e , r e s e a r c h e r s have been i n t e r e s t e d i n systems of i n t e r a c t i n g nucleons and p i o n s , ( i n p a r t i c u l a r , the r e a c t i o n J3+|>->n++c( and i t s r e v e r s e ) ; however, d e s p i t e the l a r g e number of t h e o r i e s proposed, none of them has been e n t i r e l y s u c c e s s f u l . Us ing quantum f i e l d theory i n which the fundamental dynamical v a r i a b l e s are the Fermion and Boson c r e a t i o n o p e r a t o r s , and B + , (see s e c t i o n 2 . 1 ) , t h i s t h e s i s t a c k l e s the o l d problem w i t h a new approach by u s i n g an unconvent iona l H a m i l t o n i a n . In the c o n v e n t i o n a l Chew(1954, 1956a, 1956b) H a m i l t o n i a n , the n u c l e o n - p i o n i n t e r a c t i o n p o t e n t i a l i s of a t r i - l i n e a r fo rm, ( F FB + a d j o i n t ) , whereas i n our unconvent iona l H a m i l t o n i a n , the n u c l e o n - p i o n i n t e r a c t i o n p o t e n t i a l i s taken to have a p e n t a - l i n e a r fo rm, ( F" F FFB + a d j o i n t ) . The c o n v e n t i o n a l approach has a s e r i o u s drawback, namely the f a c t that F l°> i s not an e igenket of the Chew H a m i l t o n i a n , and the cumbersome n o t i o n of " d r e s s e d " p a r t i c l e s has to be i n t r o d u c e d . No such c o m p l i c a t i o n s a r i s e i n our approach, s i n c e both F lo> a n d B > > are e igenkets of our unconvent iona l H a m i l t o n i a n . However, we have to pay a p r i c e f o r our p e n t a - l i n e a r p o t e n t i a l s . Being' l e s s fundamental than the t r i - l i n e a r ones , our p o t e n t i a l s cannot be u s e d , f o r i n s t a n c e , to p r e d i c t the j»^TT + c{ c ross s e c t i o n . But we can use the f>J5-»irfc( data to o b t a i n e x p l i c i t forms f o r our p o t e n t i a l s , and then apply them to more compl icated r e a c t i o n s — a process analogous to tha t adopted i n low energy n u c l e a r phys ics where nuc leon i n t e r a c t i o n p o t e n t i a l s were f i r s t determined from NN->NN data before be ing a p p l i e d to other problems. We believe our approach w i l l be simpler than the conventional one i n studying pion production with complex nuc l e i . Below i s an outline of this thesis: The Fermion and Boson creation operators are defined i n section 2.1. In section 2.2, the coupling of two Fermions leads to the singlet and t r i p l e t operators, A . By further combining these with appropriate Clebsch-Gordan coef f i c i e n t s and spherical harmonics, we obtain, i n sections 2 . 3 , 2 . 4 , and 2.5, the operators > (BI(T), and J4 „^a , which w i l l be deployed i n la t e r chapters. The Fermions and Bosons are to be i d e n t i f i e d with physical nucleons and physical pions. In Chapter 3 , we consider systems with only nucleons. A 2-body nucleon interaction potential i s introduced i n section 3.1, and i s used to derive a pair of coupled d i f f e r e n t i a l equations for the deuteron state. These equations, under appropriate r e s t r i c t i o n s , reduce to the fam i l i a r deuteron equations i n Bla t t and Weisskopf(1952, p.102). In sections 3 . 3 and 3 . 4 , the singlet even and t r i p l e t odd scattering states for two interacting nucleons are presented (for use i n the following chapter). In Chapter 4 , we study systems of nucleons and pions. Two of our A unconventional penta-linear potentials, V and V are introduced i n sections 4.1 and 4.2. In section 4 . 3 the d i f f e r e n t i a l scattering cross section JSL for (=f5^>TT+o( (with the incident proton p a r t i a l l y polarized perpendicular to i t s d i r e c t i o n of motion) i s derived i n f i r s t order perturbation theory using our unconventional Hamiltonian— with results i n agreement with those i n Mandl and Regge(1955). In section 4 . 4 , as an example i l l u s t r a t i n g other types of calculations one can perform with our penta-linear potentials 3 we g i v e a s imp le study of the deuteron s t a t e . The p e n t a - l i n e a r p o t e n t i a l s g i ve to the deuteron s t a t e a p i o n i c component which we assume to be s u f f i c i e n t l y s m a l l , so tha t we can apply f i r s t order p e r t u r b a t i o n theory i n our d i s c u s s i o n . A summary and a c o n c l u s i o n i s g i ven i n Chapter 5 . 4 Chapter 2^  Mathemat ica l P r e l i m i n a r i e s : Operators and The i r T ransformat ion In t h i s c h a p t e r , the b a s i c mathemat ica l t o o l s we need to b u i l d our theory a r e d i s c u s s e d . C r e a t i o n and a n n i h i l a t i o n o p e r a t o r s f o r Fermions and- Bosons a re f i r s t i n t r o d u c e d , and t h e i r behaviour- under v a r i o u s t r a n s f o r m a t i o n s , such as ro ta t ion , , , p a r i t y , e t c . , a re then i n v e s t i g a t e d . Next , we use these elementary o p e r a t o r s to c o n s t r u c t more compl i ca ted ones, and the t r a n s f o r m a t i o n p r o p e r t i e s of the l a t t e r a re a l s o examined. 2.1 Fermion and Boson C r e a t i o n and A n n i h i l a t i o n Operators Assuming the p h y s i c a l system to be composed of Fermions and Bosons o n l y , we choose our H i l b e r t space to be a d i r e c t product of Fock spaces f o r Fermions and f o r Bosons. We s h a l l i d e n t i f y the Fermions w i t h p h y s i c a l n u c l e o n s , and the Bosons w i t h p h y s i c a l p i o n s . The fundamental dynamical v a r i a b l e s a re taken to be the Fermion a n n i h i l a t i o n and c r e a t i o n o p e r a t o r s , r e s p e c t i v e l y , r m / t and rm^ , and the Boson a n n i h i l a t i o n and c r e a t i o n o p e r a t o r s , By, and • By d e f i n i t i o n , F m^(r) , o p e r a t i n g on the vacuum s t a t e |6)> , c r e a t e s a s i n g l e Fermion a t p o s i t i o n _r, w i t h s p i n m (p ro jec ted a long the z - a x i s ) , and i s o t o p i c s p i n jx (p ro jec ted a long the z - a x i s i n the i s o t o p i c - s p i n s p a c e ) . A n a l o g o u s l y , F,^ (Ik) c r e a t e s a Fermion w i t h momentum k, s p i n m, and i s o t o p i c s p i n ju. Fn^_(t) and F„ (fe) a re r e l a t e d by a F o u r i e r t r a n s f o r m : P r o p e r t i e s ( a . l . I ) 5 S i m i l a r l y , B^(r) c r e a t e s a Boson at p o s i t i o n r_, w i t h i s o t o p i c s p i n /i, w h i l e B^fk) c r e a t e s a Boson w i t h momentum k and i s o t o p i c s p i n These two operato rs a re a l s o r e l a t e d by a F o u r i e r t r a n s f o r m , Us ing the c o n v e n t i o u a l n o t a t i o n f o r commutators and ant i - commutators ( i . e . [A,B]= AB-BA, and {A,B)= AB+BA), the f o l l o w i n g r e l a t i o n s h o l d : (a.l •3) 0= B^(t» = EjuW|o>= <o|B^ (k)= <°IB*(r) (a.l •4-) - 0 = ( V - r ) . f^'(r,)} (a.i (a.i •7) (a.i .8 ) Ca.i [B^OO, B;C)] = S(t-t')V (a.i .10) (a.i .11) Next , we s h a l l d i s c u s s the behaviour of these Fermion and Boson operato rs under v a r i o u s t r a n s f o r m a t i o n s . F i r s t , we i n t r o d u c e a u n i t a r y d isp lacement operator ]>(!>) w i t h the f o l l o w i n g p r o p e r t i e s : 6 J)(k)B£(OD*(10- B^(r + k) (2.1.13) combining with (2.1.1) and (2.1.2), we also have: I>00F*rfc)D+(i) = e " - ' U F ^ ( _ 0 ( 5 J . / f ) D(t)6^k)J) + ft) - e - l i ' U B*(A) (a . l ./S) We note i n passing that JXk) can be written i n the form: D00 = e - ^ A (Z. I . I O where P - J A k [F^(k)Fm/1(k) + B > ) & > ) ] (2.1. 17) Next, we introduce a unitary G a l i l e a n boost operator with the following c h a r a c t e r i s t i c s : Gr(^) B*(A) = B j ^ + V ) ( 2 . M 7 ) where ™1F and *">B are the masses of the Fermion and the Boson, r e s p e c t i v e l y . Using (2.1.1) and (2.1.2), we have: < W r £ ( i ) G + ( * ) = e ^ - r / * F^(r ) (zi.zo) We now introduce a unitary r o t a t i o n operator (R(<xpn) with the following properties: 7 M«tr)Fv&&«?» = Z D ^ Y ) (7.111) ( R ^ ( 3 y ) B ^ ) ( R % ^ ) = Bjrr.^) (z. 1.23) where cV, ^tf" are the Euler angles specifying the rotation, and we adopt the d e f i n i t i o n s i n Rose(1957, pp.48-75) for these Euler angles and the transformation matrices I ^ / 0 ^ * ) • T n e Fermion ( i . e . nucleon) has spin h, and hence transforms by D^M , while the Boson ( i . e . pion) has spin 0, and hence transforms by D E O (=1). Under rotations, the s p a t i a l coordinates also transform. In Cartesian coordinates, with tr written as a column vector, we have , N where the transformation matrix M i s defined i n Rose(1957, p.65). With the help of (2.1.1) and (2.1.2), we can easily show that <ftftff»m*(i)ftfy/s*) = Bj(JL,0 (2.1. *O Let (P be the unitary, Hermitian, parity operator with the following cha r a c t e r i s t i c s : Then using (2.1.1) and (2.1.2), the following also hold: To consider motion reversal, an anti-unitary, a n t i - l i n e a r operator 8 w i t h the f o l l o w i n g p r o p e r t i e s , i s needed. (See Appendix A ) . 7 c - c * : r (2.1.30 where c i s any complex number and c*, i t s complex con jugate . ( 2 . 1 . 3 1 ) i s a m a n i f e s t a t i o n of the f a c t tha t J i s a n t i - l i n e a r . We can a l s o cons ide r r o t a t i o n s i n the i s o t o p i c - s p i n space by i n t r o d u c i n g the u n i t a r y operator which s a t i s f i e s the f o l l o w i n g : (R IC«jsv)Bj(S) tf^^/sr)= Z D ^ p r ) ^ ; ^ ) ( 2 . 1 . 3 3 ) where £ stands f o r e i t h e r the p o s i t i o n v e c t o r _r or the momentum v e c t o r k. These t r a n s f o r m a t i o n equat ions f o l l o w from the f a c t that n u c l e o n s , be ing a doublet i n the i s o t o p i c - s p i n space , have i s o t o p i c s p i n h; w h i l e p i o n s , be ing a t r i p l e t , have i s o t o p i c s p i n 1. As<r o ° c e ~T"£ "T° 2 .2 S i n g l e t and T r i p l e t Operators Anyo O o o > Q<y*> I mo > \mji The c o u p l i n g of two nucleons leads to s i n g l e t and t r i p l e t s p i n s t a t e s of even and odd p a r i t i e s . We use the n o t a t i o n ( j , 3213 m) to denote the Clebsch-Gordan c o e f f i c i e n t r e s u l t i n g from the c o u p l i n g of two p a r t i c l e s of angular momentum (jji, w,) and (]x,^z) to form a t o t a l angular momentum of ( 3 , ^ . These c o e f f i c i e n t s a re a l s o used to couple sp ins and 9 isotopic spins. For properties of these c o e f f i c i e n t s , see Appendix B and Rose(1957). l We define a new creation operator resulting from the coupling of two nucleons as follows: C^R)=Z ^ a i ^ ^ h - V i ^ A l ^ l i / R H r ^ (B-ir) (2.2.1) where R i s the coordinate of the centre of mass of the two nucleons, and r , the r e l a t i v e separation between them. The coupling produces a state of spin s and isotopic spinO", with m and ju , respectively, their components along the z-axis i n the spin and isotopic spin spaces. Correspondingly, we define It can be shown, with the transformation r e l a t i o n (2.1.1), that By (B.2) i n Appendix B, s can only take the values 0 or 1, and (T, also only the values 0 or 1. We introduce here four new symbols to denote A s < r the operator Am„ when s and <T take on these s p e c i f i c values. Let where S 0° 0, S 0 ^ , X! a n d , (and their abbreviations: S ° , S e , T e , T ° ) The Clebsch-Gordan co e f f i c i e n t (j, }z m i "^z| i "'») defined here i s equivalent t o C (a.a'a i j m , m 0 i n Rose(1957) , since (g.jj.m, r»2 (-j m) =0 unless rn, + m2=r>i are r e f e r r e d to a s , r e s p e c t i v e l y , the s i n g l e t odd o p e r a t o r , the s i n g l e t even o p e r a t o r , the t r i p l e t even o p e r a t o r , and the t r i p l e t odd o p e r a t o r . We now l o o k a t the or thogona l r e l a t i o n s between these o p e r a t o r s , 'cr' "t Consider A* y (r', R') A^* (r,R)|o> . By d e f i n i t i o n , ASv(r'Jg/)A!,l(r,R)|o>=E, ^ l i ( i i K " i | s W X i i / ^ | ( r y)(±i « l W l | s « X i i / ' . ^ l ^ x F^(8 ' - ir ' ) Fm i > ; (Mr')F^/RHD F ^ ( E - tr)|o> Using ( 2 . 1 . 8 ) r e p e a t e d l y , and w i t h ( 2 . 1 . 3 ) , we get = V ^  *(*' + *r'- 5 " ir)S(R'-ir'- B +" S/./i V ; S(R'+^ r'-B + ir)srR'-ir'-B-^0|o> X [ 2 L i(44 - . " . | « , » , X i i « 1 ^ | s n ) ] [ Z ( U / v M ^ X ± V . ^ l 7 0 > > With the use of (B.5b) and ( B . 6 a ) , t h i s s i m p l i f i e s to - (-i)S+<r5(R'+ir'-R + ir)s(R ' - ir'-R - i;r)}|o> A n a l o g o u s l y , one can show t h a t : 11 - ( - 0 5 + < r § ( i K ' + k ' - i K + k ) S ( i K ,-V - ^ - k ) 5 l o > (a.a.st) I n t e g r a t i n g over d3R, i n ( 2 . 2 . 5 a ) , and over o(3f< i n ( 2 . 2 . 5 b ) , we have where we have in t roduced the n o t a t i o n Z ) to denote e i t h e r or We emphasize, i n p a r t i c u l a r , the f o l l o w i n g o r thogona l r e l a t i o n s from ( 2 . 2 . 5 ) : o= s;s:>>= s:.s$o>= S;T;>>= T ; O S ; V - S;,T;V> = T;S;>> = S . - tH l°> = T0 S o |o> = S o 0 Tf 1 0 = T m / So0 |o> = T „ , o 7 ^ |o> = T v T , o I ° > (2-2.7) where the arguments of these operators have been omit ted f o r b r e v i t y . We now examine the t r a n s f o r m a t i o n p r o p e r t i e s of the s i n g l e t and t r i p l e t o p e r a t o r s . Under d i s p l a c e m e n t s , ( 2 . 1 . 1 2 ) and ( 2 . 2 . 1 ) g i ve With ( 2 . 2 . 3 ) , t h i s can be t ransformed i n t o Wk)A£ = As„;+a.K) (*•*••>) S i m i l a r l y , under G a l i l e a n b o o s t s , we have and the cor responding r e s u l t 12 ^A^V*)^) = e I a ^ s * A # r , £ ) (z-2.ll) Let us examine how these s i n g l e t and t r i p l e t o p e r a t o r s t rans fo rm under r o t a t i o n s . Us ing ( 2 . 1 . 2 2 ) , we have Us ing (C.5) from Appendix C, we have S i m i l a r l y , one can show that * W A £ M M ' V ) = 5 W ^ ) A S m > a . , , ( 2 . 2 . ^ ) For the s i n g l e t o p e r a t o r s , s=0 and i s j u s t the i d e n t i t y . Hence, the t r a n s f o r m a t i o n equat ions s i m p l i f y to Under p a r i t y t r a n s f o r m a t i o n , ( 2 . 1 . 2 7 ) and ( 2 . 1 . 2 8 ) r e a d i l y show that One can r e c a s t ( 2 . 2 . 1 4 ) i n t o a more u s e f u l fo rm. Take 13 Commuting the two Fermion c r e a t i o n opera to rs by ( 2 . 1 . 5 ) , then i n t e r c h a n g i n g the dummy i n d i c e s "i, and m2 , and u s i n g ( B . 5 b ) , we get and by s i m i l a r arguments, A:;+(-k,-K)=H) i 4 s*v:;+(i,-^ ^ . ^ ) w i t h the f a c t o r ("0" =+1 f o r /\ = S or T, and (-') = -I f o r Am/, = S° or T°- In o ther words, Under mot ion r e v e r s a l , from ( 2 . 1 . 2 9 ) , we have Changing the summation i n d i c e s (""N, i^O to ( -^i , -^) , and u s i n g ( B . 5 a ) , we o b t a i n S i m i l a r l y , from ( 2 . 1 . 3 0 ) , we can show that 14 QJm<r7* 2.3 The 2-Fermion Operator J J A S I t w i l l be u s e f u l to d e f i n e combinat ions of s p h e r i c a l harmonics w i t h s i n g l e t and t r i p l e t o p e r a t o r s , which t rans fo rm under r o t a t i o n s by the V m a t r i c e s . These new o p e r a t o r s , which w i l l be used i n l a t e r c h a p t e r s , a re d e f i n e d as f o l l o w s : where (t,%)=(^>^) o r (t<!S) > a n d Y?"^  ^ s a s P n e r : L c a l harmonic . I t i s to be understood that the s p h e r i c a l harmonic Y^f]*) depends on ly on the o r i e n t a t i o n but not the magnitude of the v e c t o r "£ With the he lp of ( B . 6 b ) , we can e a s i l y o b t a i n the i n v e r s e r e l a t i o n , Under r o t a t i o n s , w i t h ( 2 . 2 . 1 2 ) , we have mr)i£(t.Z)ftW= 2 „, (" ». I J-)Y^(f ) D ; , ( ^ ) Z J Using ( D . l ) from Appendix D , we have Th is s i m p l i f i e s by (C.5) to 15 2.4 The 2-Fermion T rans fe r Operator (Bi i^ We now c o n s t r u c t an operator 63^ which w i l l l a t e r be used i n Chapter 3 . L e t w i t h t r e s t r i c t e d to the v a l u e s 0 , 1 and 2 by ( B . 2 ) . (Qjim which conserves the number of Fermions , w i l l be c a l l e d a 2-Fermion t r a n s f e r o p e r a t o r . Other t r a n s f e r opera to rs can be c o n s t r u c t e d s i m i l a r l y u s i n g the other s i n g l e t and t r i p l e t o p e r a t o r s . Us ing the r e s u l t s of S e c t i o n 2 . 2 , and some i d e n t i t i e s from Appendices B and C, i t i s s t r a i g h t f o r w a r d to prove the t r a n s f o r m a t i o n p r o p e r t i e s of the (8gm operator as l i s t e d below: ^ ^ ( - M ; R ) ^ • = (-i)^d5,.ffl(^r;R) ( m ) (8jm (r,r;R) = (H)"<8 (r'.r, R) 16 2 .5 The 2-Fermion T rans fe r Operator J^ im^  Analogous to the l a s t s e c t i o n , we now c o n s t r u c t another 2 -Fermion t r a n s f e r operator M~jimjn which w i l l be used i n Chapter 4 . L e t w i t h I r e s t r i c t e d to the v a l u e s 0 , 1 , and 2 . The reason f o r t h i s c h o i c e of momentum arguments w i l l become c l e a r i n Chapter 4 . A g a i n , i t i s s t r a i g h t f o r w a r d to d e r i v e the t r a n s f o r m a t i o n p r o p e r t i e s of t h i s operator as l i s t e d below: 17 Chapter 3^  Systems of Nucleons In this chapter, we s h a l l consider only systems of nucleons. We sh a l l write down a 2-body potential for the t r i p l e t even states and use i t to derive the familiar pair of coupled d i f f e r e n t i a l equations describing the deuteron wave function and energy. In the l a s t two sections of thi s chapter, we write down the singlet even and t r i p l e t odd scattering states which w i l l be needed i n Chapter 4. 3.1 The 2-Body Potential V t e We introduce the following 2-body potential Vt e(for the t r i p l e t even S13. t £ S ) * V t e = ^ A V R g ^ ( r r ' ( R ) ^ ( r C ; R ) (3.1-1) where (f$ (£,£.', S.) i s t n e 2-Fermion transfer operator defined i n (2.4.1), and V. (r r' f O i s a function to be determined. From the d e f i n i t i o n of (Bj[m» and the orthogonal properties (2.2.5) and (2.2.7), we see that Vte operating on a t r i p l e t even state recreates another t r i p l e t even state, while V t e operating on other states vanishes. We s h a l l demand this \4e be invariant under various transformations, and t h i s , i n turn, w i l l impose certain constraints on l£m(c,E',R). F i r s t l y , we demand \4 E to be invariant under s p a t i a l displacements. Using (2.4.2), we have DU) V,eD+G>) = jVnfV/R. Z V^frr'ftQ (r,r', R+b) with appropriate change of variable. The right hand side equals V^e i f Me» U, R_i)= W r ' £ ' 5) > t h a t i s , must be independent of R, Second ly , we demand V t e to be i n v a r i a n t under G a l i l e a n b o o s t s . Us ing ( 2 . 4 . 3 ) , we have a u t o m a t i c a l l y , w i t h no a d d i t i o n a l c o n s t a i n t on , T h i r d l y , we r e q u i r e V t e to be i n v a r i a n t under r o t a t i o n s . Us ing ( 2 . 4 . 4 ) , we have £f*m' which equals V e^ if 4m' where ( 3 . 1 . 3 b ) f o l l o w s from ( C . 3 ) . F o u r t h l y , we demand \4e to i n v a r i a n t under p a r i t y . Us ing ( 2 . 4 . 5 ) , we o b t a i n Z t£ (-r -r')(B | which equals V-te i f W - r - c ' ) = ^J-r,-i') (3. F i f t h l y , we r e q u i r e Vte to be i n v a r i a n t under motion r e v e r s a l . Us ing ( 2 . 4 . 6 ) and ( 2 . 1 . 3 1 ) , we have = J W . ( V ^  Z < . h ( ^ - r ' ) f - ' / + W ^ m ( r , r ; ^ which equals V^e i f < ^ - r ' ) r - O i + ^ ^ ^ r ' ) fs.,.*) S i x t h l y , V t e must c e r t a i n l y be unchanged i f we r e p l a c e the dummy i n t e g r a t i o n v a r i a b l e t' i n ( 3 . 1 . 1 ) by - £ . From ( 2 . 4 . 7 a ) , we see that t h i s c o n s t r a i n t i s s a t i s f i e d i f By making the same manoeuvre w i t h £ , we get the analogous c o n d i t i o n : V£„ (-£,£')- V ^ ( r r ' ) (3.1-U) F i n a l l y , we r e q u i r e V t e to be H e r m i t i a n . With ( 2 . 4 . 8 ) , we have Vt  » J j s r M 3 R Z < ( r / ) f - ' ) f l ^ . J n ; r , R ) = jSr'frfR Z V » w ( r ' j t X H r ( 0 ^ r r , £ ' J E ) VtH - i f rfjr'r) = ^(r,r') ( 3 . 1 . 7 ) Summarizing a l l the c o n s t r a i n t s on , we have: ^ ( r , r ; R ) = ^(r,r') = vlm(-r,t')= = ^ M(-r,-r') (3./.8«) ^ d ' . D r ( - 0^ m(r,r') ( 5 . I . H ) vfm(r,r') = (-O^^.Jr.r') (3.1.8c) 20 S ince l £ m t ransforms under r o t a t i o n l i k e a s c a l a r , a v e c t o r , and a t e n s o r , r e s p e c t i v e l y , f o r £ = 0 , 1 , and 2 , i t w i l l be c a l l e d , r e s p e c t i v e l y , the s c a l a r , the v e c t o r , and the tensor p o t e n t i a l f u n c t i o n . We now choose a form f o r that s a t i s f i e s these c o n s t r a i n t s . Le t ^ m<I,C<) = £ a ^ « . ^ | i « n ^ ) Y * ^ ^ ' ) V W i ^ r , r ' ) (3 .1 .7 ) where 1/^ ^ ( r , r ' ) a re f u n c t i o n s that depend on ly on the magnitudes r and I*'. We s h a l l show that t h i s form f o r Vgn s a t i s f i e s ( 3 . 1 . 8 ) i f we impose a few r e s t r i c t i o n s on the f u n c t i o n s Vj^^ . Using ( D . 2 ) , we have which equals i f (r , r ' ) = 0 , u n l e s s ^, i s even (3.l-|0«.) S i m i l a r l y , ^ - r ' ) = (V, c ') , i f Vjli{ljr, r') - 0 , u n l e s s ^ i s even (3 .1- lot ) Thus, w i t h the c o n s t r a i n t s ( 3 . 1 . 1 0 ) on ^ ^ > ( 3 . 1 . 8 a ) i s s a t i s f i e d . A f t e r i n t e r c h a n g i n g (h, w 0 w i t h (^,"'2) , we have by ( B . 5 b ) . For ( 3 . 1 , 8 a ) to be s a t i s f i e d , we impose, bes ides ( 3 . 1 . 1 0 ) , the a d d i t i o n a l c o n s t r a i n t : 21 Us ing (D.3) and ( B . l ) , we have a f t e r changing to ( - « , , - i > i , ) and u s i n g ( B . 5 a ) . For ( 3 . 1 . 8 c ) to be s a t i s f i e d , we impose, b e s i d e s ( 3 . 1 . 1 0 ) , the e x t r a c o n s t r a i n t Wi th ( D . l ) and ( G . 5 ) , ( 3 . 1 . 8 d ) i s a u t o m a t i c a l l y s a t i s f i e d w i t h no a d d i t i o n a l c o n s t r a i n t s on ^ Summarizing a l l the c o n s t r a i n t s on "^ni^ > w e have: l ^ ^ r ' ) i s r e a l ( 3 . 1 . 1 3 a ) Vjyj (r ,r ' )^0 on ly i f both ^ and j?2 a re even (3.1-13 0 ) We note i n p a s s i n g that we cou ld have cons t ruc ted V+ e out of the M operato rs d e f i n e d i n ( 2 . 3 . 1 ) and ( 2 . 3 . 2 ) . By r e a r r a n g i n g the C l e b s c h -Gordan c o e f f i c i e n t s and the s p h e r i c a l harmonics i n ( 3 . 1 . 9 ) and ( 2 . 4 . 1 ) , we can r e w r i t e \4e i n ( 3 . 1 . 1 ) a s : w i t h ^ ( r , r ' ) = Zh)™ (zl+i)VJ(ilx I T i ^ ^ r ' ) (3./.\s) where i s a Racah c o e f f i c i e n t . ( S e e Appendix E). With the he lp of ( E . l l ) , we can a l s o e s t a b l i s h the i n v e r s e r e l a t i o n , 22 3 .2 D e r i v a t i o n of the Deuteron Equat ions We w r i t e down the deuteron s t a t e | K , m)> as f o l l o w s : ' . _ _ t •» « — r \ / — t — c ( 3 . ? . | f t ) Th is can be w r i t t e n i n the f o l l o w i n g more compact fo rm: w i t h J=s=l , and UJSs ^ ^ f ° r r ' |^  0 otiieruiise Demanding < Is ,m | K', ^ '"> = S (£ _ K') , we have, by u s i n g ( 2 . 2 . 5 a ) , (D.5) and ( B . 6 a ) , the f o l l o w i n g n o r m a l i z a t i o n c o n d i t i o n : DO f dr [V(rO+ UT2(r)3 = | ( 3 - * - * ) Using ( 2 . 2 . 3 ) and ( D . 7 ) , we can a l s o w r i t e the deuteron s t a t e as 'TMs w i t h J=s=l , and OO where ^ i s a s p h e r i c a l B e s s e l f u n c t i o n . Us ing ( 2 . 3 . 1 ) , ( 3 . 2 . 3 ) becomes i K, m>=z/Au J < s (H ' y;;1k , K ) i o> (,.».*> w i t h J=s=l . With the momentum operator P de f ined i n ( 2 . 1 . 1 7 ) , i t i s s t r a i g h t f o r w a r d 23 to show by u s i n g ( 2 . 1 . 3 ) , ( 2 . 1 . 7 ) and ( 3 . 2 . 3 ) , that I t i s a l s o easy to show that the f o l l o w i n g ho lds under p a r i t y (by ( 2 . 2 . 1 4 ) , (2 .2 .15b) and ( 3 . 2 . 3 ) ) , and under r o t a t i o n (by ( 2 . 3 . 3 ) ) : ( P|K,W>= 1-K,«> (3 .2 . a ) m' m Thus, these equat ions c o n f i r m that our deuteron s t a t e lK,m^> , hav ing momentum K , does have even p a r i t y and t ransforms under r o t a t i o n w i t h t o t a l angular momentum J=l and z-component m. We w r i t e down the u s u a l t ime- independent Schrodinger e q u a t i o n : H j f c . ^ s (K.+ VNN)|k,«> = £|k">> O.a.7) where £ i s the energy e i g e n v a l u e . K0 i s the n o n - r e l a t i v i s t i c k i n e t i c energy operator d e f i n e d by w i t h 1% be ing the mass of a n u c l e o n . The n u c l e o n - n u c l e o n i n t e r a c t i o n p o t e n t i a l V N N i s d e f i n e d by V N N = V t e + V ^ 0 + V s e + V s o (3.2.,) where \ft0> Vse > ^so > r e s p e c t i v e l y , the p o t e n t i a l f o r the t r i p l e t odd s t a t e s , the s i n g l e t even s t a t e s , and the s i n g l e t odd s t a t e s , are d e f i n e d s i m i l a r l y to V ^ g . However, due to the o r thogona l r e l a t i o n s ( 2 . 2 . 7 ) , Vt. | ^ > = V s e ) K , ™ > = V S 0 I M > = 0 (mo) Hence, we are l e f t w i t h 24 ( « 0 + V f c e)|K,^>= Elt^y 0-2.ll) Putting (3.1.1), (3.1.9), (3.2.1a) and (3.2.8) together, we obtain, a f t e r some computation, the following pair of coupled d i f f e r e n t i a l equations: (For d e t a i l s , see Appendix F ) . ~ f T r l r ' 0 , r ' u ( r ' ) ^ o o ( r ^ ' ) - f W ( 0 = i f T [ r V r V ( r ' ) U l o 2 ( r , r ' ) (3.a.|*«) - t J i r j r V r V ( r ' ) V a 2 1 ( n r ' ) ] ~ £ V ( r ) = J f - r J r W r ' « ( r ' ) V a a o ( 1 r ' ) (3.2./2b) Note that our p o t e n t i a l functions "^2g (J.r> r ) a r e > thus f a r , p e r f e c t l y general. The presence of terms with i = 0, 1, and 2 corresponds to the contribution from, r e s p e c t i v e l y , the sc a l a r , the vector, and the tensor force. I f we now take the p o t e n t i a l functions to be l o c a l , i . e . ^g} ( (r-r')— Vj-j q ( O S ( r - r ' ) 5 we can deduce from (3.2.12) the following f a m i l i a r deuteron equations i n B l a t t and Weisskopf(1952, p.102): A + V c(r ) U - £ U = - J T V T ( 0 ^ (z.X.ISa) ' 'U off + [ V c ( 0 - 2 V » > - £ i . = - j f V T ( r ) U (3.2.131,) where we have set - j ? = " i f r 2 ^ 2 0 = [l r* l£„fr) = /? V T (r) 25 3 .3 The S i n g l e t Even S c a t t e r i n g S t a t e To pave the way f o r the r e a c t i o n ]»f>->TT*e(— the main t o p i c of Chapter 4— we s h a l l , i n t h i s and the next s e c t i o n , w r i t e down the s c a t t e r i n g s t a t e s f o r two i n t e r a c t i n g n u c l e o n s . We d e f i n e the s i n g l e t even s c a t t e r i n g s t a t e as f o l l o w s : where we i d e n t i f y K w i t h the t o t a l momentum, and 2k, w i t h the r e l a t i v e momentum of the two nucleons when they a re a t i n f i n i t e s e p a r a t i o n . V^ (-)h) i s the s i n g l e t even s c a t t e r i n g wave f u n c t i o n obeying the Lippmann-Schwinger i n t e g r a l e q u a t i o n . (See Newton(1966, p.181 and p . 2 9 9 ) . ) But from ( 2 . 2 . 3 ) , » d h k f ^ ' R e " 8 " " ' 6 ' * = S C E - E ) , (3.3.3) t h e r e f o r e , ( 3 . 3 . 1 ) can be r e w r i t t e n as fcU,r>= (^ftffilk) s£(S,KM°> (3.3.**) where Note tha t f o r the s p e c i a l case t u , j O = eik'r/ti (3.3.5-*) then W , i ) = ( 2 K t f 8(hk) ( 3 . 3 . ^ ) lfe , M > = S^ fk,!<)l0> , (3.3.5-c) as expected . By n o t i n g that ( 3 . 3 . 1 ) must be i n v a r i a n t i f we change the dummy 26 i n t e g r a t i o n v a r i a b l e r_ to -r_, and u s i n g ( 2 . 2 . 1 6 ) , we have and s i m i l a r l y , , , „ . , . ^(-|,k)= ^ ( l , k ) (3.3.6 fc) Demanding < K', ^ k ,/*> = S ( K'~ £)Sft'-fc) S/y , we o b t a i n by s t r a i g h t f o r w a r d computat ion , u s i n g ( 2 . 2 . 5 a ) . Under r o t a t i o n s , we demand A p p l y i n g the r o t a t i o n operator to ( 3 . 3 . 1 ) , and s u b s t i t u t i n g i n ( 2 . 2 . 1 3 b ) , we o b t a i n Changing the v a r i a b l e s (E_(R > B.^ ) to (n , R ) , and equat ing ( 3 . 3 . 8 ) and ( 3 . 3 . 9 ) , we o b t a i n J / r / R e' • & / k O S.^(r,5) = J V r A B « * Wr«,k) S^ V, & ) which i s s a t i s f i e d i f That i s , uiffi 6 denot ing the angle between r_ and k , f(r , k ) = ^ ( r . k . e ) (3.3.(0) The angu lar dependence on 9 can be expanded out i n terms of the Legendre po l ynomia ls T^(cos0) , wh ich , by ( D . 6 ) , can be f u r t h e r expanded i n terms of the s p h e r i c a l harmonics , as f o l l o w s : 27 By ( 3 . 3 . 6 a ) , ^ (n k) must v a n i s h u n l e s s £ i s even. The terms w i t h i-0 , 2 , correspond r e s p e c t i v e l y to the 'S0 , 'D^, p a r t i a l waves. Wi th (D.7) and ( 3 . 3 . 4 b ) , we a l s o have where <* t e ^ O - j o/r i / ^ ) ^ ( r , fc) ( 3 . 3 . | i t ) Thus, ( 3 . 3 . 4 a ) can be r e w r i t t e n as Fur thermore , one can r e w r i t e ^ ( r , fe) i n the f o l l o w i n g fo rm: (See Newton(1966, p p . 3 0 1 - 3 0 3 ) . ) ^ (r> ) = el cos SA ^ \r, k ) (3. 3. If) where V ^ f r k ) i s a r e a l f u n c t i o n , and 8g(j0 i s the phase s h i f t f o r the I t h p a r t i a l wave. 3 .4 The T r i p l e t Odd S c a t t e r i n g S t a t e Analogous to the p r e v i o u s . - s e c t i o n , we d e f i n e the - t r i p l e t odd s c a t t e r i n g s t a t e as f o l l o w s : where "Tgg M ( r , i ) i s the t r i p l e t odd s c a t t e r i n g wave f u n c t i o n . One can a l s o d e f i n e 28 Analogous to the c o n s t r a i n t s ( 3 . 3 . 6 ) and ( 3 . 3 . 7 ) , we have S(l- k' )S v i = ( ^ ' r & ^„(r, „, (r, K ) ( 3.*.* O Since the t r i p l e t odd s c a t t e r i n g s t a t e has s p i n 1 , we demand i t to s a t i s f y the f o l l o w i n g t r a n s f o r m a t i o n r u l e under r o t a t i o n s : The l e f t hand s i d e can be r e w r i t t e n w i t h the he lp of (2 .2 .12b) as f o l l o w s : where we have changed the v a r i a b l e s ( ' r _ ( R ) £ ( R ) to The r i g h t hand s i d e of ( 3 . 4 . 5 ) can be r e w r i t t e n as f o l l o w s : X D ' S s « s < ^ V ) T : j + / 1 ( r,R )|o> (3.t.y) We observe tha t ( 3 . 4 . 6 ) and ( 3 . 4 . 7 ) are equal i f Le t us w r i t e 29 S u b s t i t u t i n g ( 3 . 4 . 9 ) i n t o ( 3 . 4 . 8 ) , and u s i n g the r e l a t i o n s ( D . l ) , (C.4) and ( C . 5 ) , we f i n d t h a t w i t h t h i s form f o r m , ( 3 . 4 . 8 ) i s i d e n t i c a l l y s a t i s f i e d , and hence, the t r a n s f o r m a t i o n r u l e ( 3 . 4 . 5 ) i s a l s o s a t i s f i e d . With (D.7) and ( 3 . 4 . 2 b ) , we a l s o have where • «> ( 3 . 4 . 2 a ) can now be r e w r i t t e n a s : We now cons ide r the p a r i t y of our s t a t e . By d e f i n i t i o n of the p a r i t y o p e r a t o r , We r e q u i r e the s t a t e to be of odd p a r i t y , i . e . = -!-!<, k , ( 3 . ^ . ( 3 ) S u b s t i t u t i n g ( 3 . 4 . 1 ) and ( 3 . 4 . 2 a ) s e p a r a t e l y i n t o ( 3 . 4 . 1 3 ) , we o b t a i n the c o n s t r a i n t s : %«(*rk) = , \»Jtri>-V,ft .k) (3-*- '«-) These, together w i t h ( 3 . 4 . 3 ) , when a p p l i e d to ( 3 . 4 . 9 ) and ( 3 . 4 . 1 0 a ) l ead to the f o l l o w i n g c o n d i t i o n s T^( r , l0 = S^(1i,l0== 0 u n l e s s both JL and i ' a re odd . IS") i i The f u n c t i o n s can be f u r t h e r expressed i n the f o l l o w i n g fo rm: (See Newton(1966, p . 4 5 7 ) . ) 30 where a re r e a l phase s h i f t s , ^Y^P a re r e a l f u n c t i o n s , and are o r thogona l r e a l m a t r i c e s i n v o l v i n g mix ing parameters . 31 Chapter _4 Systems of Nucleons and P ions To study systems of nucleons and p i o n s , we use the f o l l o w i n g H a m i l t o n i a n , H= H 0+ V u t c+.o.i) w i t h , , , . H0 = K0+ VN N ( i f . o . i ) where K0 i s the t o t a l energy of the f r e e nuc leons and p i o n s . i . e . K. = JA [ Z £N(k) F^ (k)Fv(kH Z ejk) B>)B,(fe)3 w i t h mN and 1% the masses of the nuc leon and the p i o n . V N N i s the i n t e r a c t i o n p o t e n t i a l f o r the n u c l e o n s , as d e f i n e d i n ( 3 . 2 . 9 ) . (See a l s o ( 3 . 1 . 1 ) and ( 3 . 1 . 9 ) . ) Hence V N N i s of the form JVT^PVFF , and the p o t e n t i a l f u n c t i o n s VNN can be t a k e n , f o r i n s t a n c e , to be the Reid(1968) p o t e n t i a l s . V^rt, the i n t e r a c t i o n p o t e n t i a l f o r the nucleons and p i o n s , can be subd iv ided as f o l l o w s : Vut ~- V f f N N + V T N + V f f f f (t-.o.t-) VTTNN i s a p i o n p r o d u c t i o n (and a n n i h i l a t i o n ) p o t e n t i a l c o n t r o l l i n g the ^rrNN^FFFB + a d j o i n t . Governing p i o n - n u c l e o n s c a t t e r i n g i s the p o t e n t i a l V„. N , chosen to have the form ]V^ NF8FB . The p o t e n t i a l f u n c t i o n s V"-^  can be taken to be the Landau-Tabakin(1972) p o t e n t i a l s . D e s c r i b i n g the i n t e r a c t i o n s v^B B B8 • Here, we emphasize the f a c t that a l l these p o t e n t i a l s ( V N N , VJTNN J V-tiH and ) have a t l e a s t two a n n i h i l a t i o n o p e r a t o r s each . When o p e r a t i n g on Fn^ \oy and B^l0^ > these p o t e n t i a l s c o n t r i b u t e n o t h i n g . Hence, 32 and B i , ' 0 ) * a r e e igenkets of the H a m i l t o n i a n H. We, t h e r e f o r e , s t r e s s the f a c t tha t F^ , |o> and B u l 0 ^ r e s p e c t i v e l y represent a p h y s i c a l nuc leon and a p h y s i c a l p i o n . In t h i s c h a p t e r , we s h a l l be f o c u s s i n g on the WNN p o t e n t i a l , and the r e a c t i o n f>+ F TT + + c( (f.o-s-; TTNN i s taken to be composed of two p a r t s , V and V, r e s p e c t i v e l y , the A s i n g l e t - t r i p l e t and t r i p l e t - t r i p l e t i n t e r a c t i o n p o t e n t i a l s . V and V w i l l be d i s c u s s e d i n d e t a i l i n the next two s e c t i o n s , w i t h f i n a l forms g i ven by ( 4 . 1 . 1 ) , ( 4 . 1 . 1 1 ) , ( 4 . 2 . 1 ) , ( 4 . 2 . 3 ) . We s h a l l denote the momenta of the two incoming protons by "k & + k and "iK -k . , the momentum of the p i o n by + K , and the momenta of the two nucleons i n the deuteron by and 7 p K ~ i k ' _ ^ , (so tha t the deuteron as a whole has momentum i£ ~ k )• F i g - 1 i l l u s t r a t e s the s i t u a t i o n : F i g . 1. Momentum diagram f o r the r e a c t i o n p+|=-?"rr++d • E v e n t u a l l y , we s h a l l go i n t o the cen t re of mass frame where K=0. The c o o r d i n a t e system w i l l be chosen such tha t the i n c i d e n t p r o t o n , p o l a r i z e d i n the y - d i r e c t i o n , t r a v e l s a long the + z - a x i s , (w i th the u n p o l a r i z e d t a r g e t p ro ton t r a v e l l i n g i n the - z d i r e c t i o n ) . The p i o n then e x i t s w i t h angles ( 9, $ ) i n the s p h e r i c a l p o l a r c o o r d i n a t e system. 33 4.1 The Potential V for Singlet-Triplet Reactions We introduce the following potential for s i n g l e t - t r i p l e t reactions: where "adj." denotes the adjoint of the f i r s t term i n braces, and V„ are functions. Our d e f i n i t i o n of V ensures i t to be Hermitian. We s h a l l demand this V to be invariant under various transformations, and this w i l l impose constraints on the functions Vm (k, %,£). F i r s t l y , we demand V to be invariant under s p a t i a l displacements. From (2.2.9) and (2.1.15), we have - V (</:/. 2) i d e n t i c a l l y . Secondly, we try to require V to be invariant under Galilean boosts. Using (2.2.10) and (2.1.19), we obtain Grit) vefoo=j?u3k'4 ft z v„ouu,i0 ^(k^+z^nt^ik-k'+z^ Letting K'= K+iin^v Now approximate by neglecting terms of order ^ . Letting K = K + 2"'fV ; 34 and i = k'-™^ > w e get which equals V i f ^ ( f c , i+w^,*, K-i»vO= Vk,fe!*,K) ^•'•3) (4.1.3) holds i f and only i f i.e. V can only depend on k and K v i a the combination i "K+k'. Because of having neglected terms of order — , our potential V i s only approximately Galilean invariant. Thirdly, we demand V to be invariant under rotations. By (2.1.23), (2.2.12) and (2.2.13b), we have ft fepO V rffy * ) = J A A'/fr A Z. V* (k, h\ * ) S^T (JL«, S <"f y) by appropriate change of variables. If we require then i t follows by (C.3a) that <R(«W V ( R f e / 3 J r ) = V Fourthly, we demand V to be invariant under parity. By (2.2.14) and ( 2 . 1 . 2 8 ) , we have, a f t e r a p p r o p r i a t e change of v a r i a b l e s , which equals V i f F i f t h l y , we r e q u i r e V to be i n v a r i a n t under motion r e v e r s a l . Us ing ( 2 . 2 . 1 9 ) , ( 2 . 1 . 3 0 ) and ( 2 . 1 . 3 1 ) , we o b t a i n tri/ T ^ [A^'4 A Z ^(k,^fe)5^(-irS)^rT;.(-*,-4<s+k')5,(-i(c-fc') + Changing (m, k, k, K ) to ( - m , -fe., -k', , - fc ) , we see tha t V i s i n v a r i a n t under mot ion r e v e r s a l i f H r v ^ i - i ' r V ^ ^ f M ' , ^ ) (4 .1 -7 ) F i n a l l y , V must be unchanged i f we change the dummy i n t e g r a t i o n v a r i a b l e \ to - $ i n ( 4 . 1 . 1 ) . With ( 2 . 2 . 1 6 ) , t h i s i s s a t i s f i e d i f vw(kIk'l-*1JS> VU'.&JS) (4.1-8) By an analogous manoeuvre w i t h the v a r i a b l e k., we have tfJ-U',= ^ ( k , k ' 4 , K ) (4. M ) Summarizing a l l the c o n s t r a i n t s on V^, W e have: = - ^ ( k , % , - ± ! < - k ' ) = - ^ - k . - 4 , - t k - k ' ) W - . M ° « ) v - m ( k 4 ^ ^ ) = ( - 0 V * ( k,|^lS+ k ' ) (^.i.lob) ^ ^ V i V U = 2 . » J « p ) l U k , ^ H + k ' ) O M - H 36 We now choose a form for v m that s a t i s f i e s these constraints. Let X ^ i ' ^ f c ^ ' O (4.1. I I ) where H8,f tJ'i 3 a r e functions that depend only on the magnitudes , and |iK + k'/ . This form for Vw s a t i s f i e s (4.1.10) i f we impose some re s t r i c t i o n s on the functions ^ju^'J3 Using (D.2), (4.1.10a) i s s a t i s f i e d i f (k,%li^H'l) = 0 unless JI, , B2 are both even and i 3 i s odd. C 4 - . U 2 ) Substiuting i n the complex conjugate of (D.3) for the three spherical harmonics i n (4.1.11), changing the indices ( w'< w 1 ( » ) 3 ) to (-m1,-i"1,,-•"i,-"^') and using (B.5a), (4.1.10b) i s seen to be s a t i s f i e d i f H ) W V ' < ^ ^ ( K ^ , l i . ^ J ) = ^ i t i ' i,(fc . j,li!c +J }'J) which, with (4.1.12), yields the condition that the functions ^ ^  must be r e a l . Using (D.l) and (C.7), i t i s straightforward to show that (4.1.10c) i s s a t i s f i e d with no constraints on the functions V j ^ j ' / Analogous to section 3.1, we can rewrite our potential V i n terms of ni may) the operators y introduced i n section 2.3. With the help of (E.12) and (4.1.12), one can show that v = s s v/- (k, t,m^yyfjM<'\^»'^) where 37 (4.1.l3c() and i s a Racah c o e f f i c i e n t . By ( E . l l ) , we also have .i,+a-r(2i1+i)fai'+iX2j+0" ' 4.2 The Potential V for T r i p l e t - T r i p l e t Reactions We introduce the following potential for t r i p l e t - t r i p l e t reactions: where i&j.njt a r e the 2-Fermion transfer operators defined i n (2.5.1), and Vjin are functions. Our d e f i n i t i o n of V ensures i t to be Hermitian. A Exactly similar to the l a s t section, we require V to be invariant under various transformations. Using (2.5.2)-(2.5.7), we find that V f u l f i l l s these invariance requirements i f we impose the following constraints on ^a,k\u)- v^a^-u+k') = -<u-u, ±*+k') = %n-h iK+k') 38 A (Aga in , analogous to the l a s t s e c t i o n , V i s on ly approx imate ly G a l i l e a n i n v a r i a n t by n e g l e c t i n g terms of order 7 ^ ) . A We now choose a form f o r . L e t hi'm, ^ 2 ^ 3 where ^it^n'g a re f u n c t i o n s . I t i s s t r a i g h t f o r w a r d to show that t h i s form s a t i s f i e s ( 4 . 2 . 2 ) i f we impose the c o n s t r a i n t that ^iJlj^i'Jl3 be r e a l and that y^j ^ ' j p = 0 u n l e s s jt, i s odd and both Sx and #3 a re even. (4-.2.4-) A Analogous to the p rev ious s e c t i o n , we can r e w r i t e our p o t e n t i a l V i n terms of the opera to rs 3 T J ! s i n t roduced i n s e c t i o n 2 . 3 . With the he lp of (E . 6 ) , one can show that where " U .1 T J j «' ' ^3 j and i 1 I ^ j i s a Wigner 9-j symbol . (See Appendix E). By (E . 8 ) , we a l s o have the i n v e r s e r e l a t i o n VM^C^M+^D^ 2L-CaT+")yf««'+0<aj«) j ^ ' tl V ^ ^ l i . l . l ^ + k ' l ) (4.2.4) 39 4 .3 The D i f f e r e n t i a l Scattering Cross Section for pp-» Tr+d For the reaction pp-»TT+d, we take the interaction potential to be V , ^ * V + V (4.3.1) A where V i s the s i n g l e t - t r i p l e t p o tential, and V, the t r i p l e t - t r i p l e t potential, defined respectively by (4.1.1) and (4.2.1). From Goldberger and Watson(1964) the d i f f e r e n t i a l scattering cross section i n the centre of mass frame i s given by^ where from f i r s t order perturbation theory, The symbols i n these equations are to be interpretted as follows: W i s the density operator— for incident protons p a r t i a l l y polarized i n the y-d i r e c t i o n . (See Appendix G). In accordance with F i g . l , (but with K= since we are i n the centre of mass frame), k and k_ are, respectively, the momentum of the incident proton and that of the outgoing pion, while 2$. i s the r e l a t i v e momentum between the two nucleons i n the deuteron. and , respectively, the energy of the deuteron and that of the pion, are given by £*(k')=/n$c*+hV ^ V k c H k ' V (+.3.3) where and *% are the masses of the deuteron and of the pion. | l<', k', } From Goldberger and Watson(1964), we substituted their eqn. (5) on p.223 into their eqn. (4) on p.93 (with "h and c restored) and used the fact that the two protons have the same energy i n the centre of mass frame to arrive at our eqn. (4.3.2a). 40 the f i n a l state composed of the deuteron and the pion, i s taken to be where — i d e n t i c a l to \1,..A%) defined i n (3.2.3) and (3.2.4) — are r e a l functions. J22' = 0 and 2 respectively represent the contribution from the s and d-wave deuterons. I K', b, i s an eigenket of the Hamiltonian H0 , where H„ = K 0 4 V N f j Introducing the notation we have, from (4.3.1), T r ( V W N W V ™ N ) = T r ( V W V ) + T r ( V W V ) + T r ( V W V ) + Tr (vW\>) (4-3.^) Hence, to obtain an e x p l i c i t expression for the cross section, we must f i r s t evaluate the four traces i n (4.3.6)— a straightforward, but tedious task. We s h a l l start off by calculating Tr(VWV). The orthogonal relations (2.2.7) guarantee that only a single term from W, (W given by (G.10)), survives i n VWV, i . e . V W V = £v|!U,i><k,U| V 0--3-7) where |£,k,l)> > the singlet even scattering state for the two i n i t i a l protons, i s given by (3.3.4a). Substituting i n (4.1.1) for V, i t i s straightforward to show that V W V = £V><o|3£ ( 4 . 3 . 8 a ) where J » ms s s 41 We then have to eva luate Tr(VWV)= Z f fe'.k'.ilVWVU*. k » f 0-3.1) S u b s t i t u t i n g i n ( 4 . 3 . 4 ) f o r the f i n a l s t a t e | (<', t', ""Xf. , and u s i n g ( 4 . 1 . 1 1 ) and ( 3 . 3 . 1 2 a ) , one can show that where, i n v o k i n g ( E . 1 2 ) , J?! A' I 3 3 w i t h g i ven by ( 4 . 1 . 1 3 d ) . S i m i l a r l y , by u s i n g ( 4 . 2 . 1 ) f o r V, and ( 3 . 4 . 2 a ) f o r | !<-, k, , •> — the t r i p l e t odd s c a t t e r i n g s t a t e f o r the two i n i t i a l p r o t o n s , one can show that V W V = Z dCv><'&n + ^ 5 ( ^ | o X - l i , - ^ l o > < - l l l*s S S 1 where w i t h vL g i ven by ( 4 . 2 . 3 ) , , by ( 3 . 4 . 1 0 a ) , a n d ? , by XT*. I S where j> i s the p r o b a b i l i t y of the ( p a r t i a l l y ) p o l a r i z e d i n c i d e n t p roton hav ing i t s s p i n o r i e n t e d a long the +y d i r e c t i o n . Fu r ther computat ion y i e l d s : 42 where, i n v o k i n g ( E . 6 ) , L = Z ".-.ITMXJF, I ^^^^\iK^0Y*Jk)Y . ftfc+i') ^. 3.12b) ^ U ^ k 1 ) = i g ^ (<f.3.Rc) w i t h ~^zii\t3 d e f i n e d i n ( 4 . 2 . 5 d ) , and "r^ , i n ( 3 . 4 . 1 0 a ) . A n a l o g o u s l y , one can show that VWV = + A><o\^) (+3-13) ^- A-j- A Since V =V, V =V, and W =W, we have \/WV = (VWV')'1' (T.3.15") Thus Gather ing a l l the p i e c e s t o g e t h e r , we have, i n the cen t re of mass f rame, (!S=o), + ^ ^ I . [ X m ( X , n i * ) ] j (4,3.17) With the c o o r d i n a t e system chosen such that k. p o i n t s a long the +z A d i r e c t i o n , and Z w w s i m p l i f y to : 43 where the p i o n e x i t s w i t h angles (9 ,<f> ) i n the s p h e r i c a l p o l a r c o o r d i n a t e system. With the help of ( B . 5 ) , ( D . 3 ) , ( 4 . 1 . 1 2 ) and ( 4 . 2 . 4 ) , the f o l l o w i n g symmetry r e l a t i o n s can be proven : X . M = ( - i f X, x = (-o'-*s+m t A d i r e c t consequence of (4 .3 .20b) i s X o . = 0 ( 4 . 3 . 2 1 ) A We s h a l l approximate X„, and Xnj„ by t a k i n g on ly the f i r s t few terms i n the i n f i n i t e sums ( 4 . 3 . 1 8 ) and ( 4 . 3 . 1 9 ) , i . e . , we s h a l l n e g l e c t c o n t r i b u t i o n s from the h igher order p a r t i a l waves. Table 1 shows the nature of the terms we s h a l l r e t a i n . (4-. 3.20 a) (4.3. Zo£) I n i t i a l pp s t a t e (deuteron s t a t e - p i o n s t a t e ) T S i n g l e t s : 's. 0 1 z 1 T r i p l e t s : T 3 P , C3s,- *\ 1 1 o ' P , ( 3 s,<0, 1 1 z 7. 1 Z. Table 1 . P a r t i a l wave c o n t r i b u t i o n s i n c l u d e d i n our c a l c u l a t i o n of ^£ D i r e c t eva luat i o n of ( 4 . 3 . 1 8 ) and ( 4 . 3 . 1 9 ) y i e l d s the r e s u l t s g i ven i n Table 2 . 44 -1 O i ^(w-o , - i i^ , )Y, 0 jferte. + fc^OY,, \ m : -1 A 0 -t &(/£AW4*0Y„ 0 0 I ^ a / i ^ ^ ^ j Y , , Table 2 Table of X „ and „ c o e f f i c i e n t s . ( N o t a t i o n : Yjpw = Y ^ t M ) ) S u b s t i t u t i n g the v a l u e s from Table 2 i n t o ( 4 . 3 . 1 7 ) , we have vi = C(y°+ *>s*e) - x? s^ ecoscf) (x„ + x.cose + A^ e ) ] (4.3. where Yo = r(fc,k')L(^0o+2/rb0, + b „ V (^ M-/2t l 3 - / i 5 b 1 + + 4 b 3 3 - i ? b 3 1 f + f t ^ ) ] (f.3;£ib) ^ = r ( U T ( - ^ be i+ 3 b „ ) + (3^b 1 3 4 3/IS b,^- I b 3 3 + I b^.)] ( 4 . 3 . « c ) = T(kM')(Wl C.| - 4 C 0 3 - *C 1 S I - i JT C i S ) (if, 3. 22.4) X , = TU,k')(*fi5"c I H. + ? J s c 3 ( t . ) ( < f . 3 . « e ) ^ = TtkX) ( l / I C « 4 3jio C ^ ) (q-. 3. i* f ) w i t h the n o t a t i o n : b0= ^o, , L A 1 b 2 = wii , I A r 1 ( 4 . 3 . 2.3 a") b<f=A\ (4 . 3 . 2 3 b) 45 If we make the i d e n t i f i c a t i o n («.,%«*, « S L « * . 0 = [TY»')3* (b 0, U, k, b 3 j k,o, o) (4.3. i f ) where «ft are the amplitudes defined i n Mandl and Regge(1955), then our expression for ^  agrees with that of Mandl and Regge. <*s and A6 of Mandl and Regge are set to zero here because we have truncated the i n f i n i t e series i n (4.3.19) e a r l i e r than Mandl and Regge. The proper correspondence for « r and a(, should be: A 2 A 3 , — I f we had included ^ 3 7 L and u 3 2 i n our calculation of 5 ^ , we would expect our result to agree exactly with equations (17)-(22) of Mandl and Regge, thus yi e l d i n g a d i f f e r e n t i a l scattering cross section of the form: Since i s a function of ^» we s h a l l also denote i t by j£(?)• When the incident proton i s unpolarized, ? = 0. Using (4.3.22a) for we have lk<S° 0) = l f A + nc-s'e) = ± [ ^ S ) + &(-*)] (4.3.Z7) The analyzing power A, a parameter frequently used by experimentalists, i s defined as follows: A = ^ [ feft)" ©-t)3 / Cg (| ) + ( 4.3. .3) A = ( Xo + X t cos 8 + A 2 oos*e') sUx e cos 4> , v r e + r zc O S*e (4-. 3.2?) The t o t a l scattering cross section CT i s obtained by integrating' 46 (4 .3 .22a ) over JlSL-<r= ± (r. + i O (4.3.30) We can a l s o express the d i f f e r e n t i a l s c a t t e r i n g c r o s s s e c t i o n i n terms of Legendre f u n c t i o n s . (4 .3 .22a ) can be r e w r i t t e n as sj£= ( r „ + i O P 0 ( c » s e ) t f-fcP^cose) - a|«s<f> [(>„ + ^xjp '/cose) + i*,Pl(cose) + ^ A z P 3 ' ( « s e ) ] (4.3.31) where and a r e , r e s p e c t i v e l y , the Legendre po lynomia ls and the a s s o c i a t e d Legendre f u n c t i o n s . These are or thogona l f u n c t i o n s , obey ing : l ^ ) f M O - X ^ ' ^ X = ^ = U , S k m ( 3 3 ,c) Taking advantage of these o r t h o g o n a l i t y p r o p e r t i e s , an exper imenter can r e a d i l y o b t a i n the c o e f f i c i e n t s X0 , Yx , \ , X, and \ x from the e x p e r i -mental d i f f e r e n t i a l s c a t t e r i n g c r o s s s e c t i o n by an a p p r o p r i a t e i n t e g r a t i o n . For i n s t a n c e , ^ - T f «Ke* s e) (4.3.33a) X * = "If { _( ' j l ^ l P B ( " ^ «™*\/ I «** ' (/r. 3. 33b) e t c . 47 4.4 A Perturbation Study of the Deuteron State As an example i l l u s t r a t i n g the various types of calculations one can perform with our penta-linear potentials, we give, i n th i s section, a simple discussion of the deuteron state using f i r s t order perturbation theory. We require the deuteron state lot^C^)^ (with momentum K and spin m) to s a t i s f y the following eigenvalue equation: H I ^ ) > H ( H 0 + V ,^)I^ (<)>= eC0|*UG> (4.4.1) with Ho given by (4.0.2). Since t h i s section i s mainly of pedagogical value, we w i l l make a s i m p l i f i c a t i o n by taking V^t to be composed of only one penta-linear potential: v« = v where v = j A A ^ A ZI ( < 0 C z, (i i//|o o) s^ a,^ -k')B (^iK+k')]Tmeo^ Js) A (We could have included V and V i n V^- , but i t turns out that these potentials do not contribute i n f i r s t order perturbation theory. However, a t r i p l e t - t r i p l e t potential of the form " T ° B T E + adj.", which does contribute i n f i r s t order, has been neglected for sim p l i c i t y . ) Analogous to section 4.1, we can check the constraints imposed on from the various invariance requirements on V. I t turns out that the functions v m have to s a t i s f y the same constraints on , namely (4.1.10). Hence, we can l e t ^  have the same form as v», , i . e . (4-. 4. 3 ) 4 8 where Vjtj £'ji3 are r e a l functions. To lowest order, we take the deuteron state to be composed of two nucleons only, | = | E>M0<)> + higher order terms (4.4.4a) where . , , e^ |D,(k)>=^^l^^|lm)J^kU,(k)Y i m/k)TK,s0Ci^|o> (4.4.4k) analogous to (3.2.3) and ( 4 . 3 . 4 ) . 1 "D„(£S> s a t i s f i e s the equation Hoi TUfc)> = g^CWlDJ^ (4.4.*) For convenience, the following notations w i l l be used interchangably: The i d e n t i t y operator i i n the Fock space of Fermions and Bosons can be expressed i n terms of the eigenkets of H 0 : i - l°X°l+ F* (fc)|o><o|Fy(fc) + ZB*(fe»<o|B.(k)l+J^/< S |D mto><D m(^)l + z |D:;( j<,k)><p:;M] +P3ICA^^)B>IO><0|B/4)]).^) + P 3 U \ 4 C Z ^ a ^ ^ l ^ l B ^ ^ ^ L k ) ^ •••] + {-••} ( 4 . 4 . 7 ) where {•••} denotes terms with two or more Boson creation operators, and terms with more than two Fermion creation operators. D^tK,^), J-* CJS, k), •t 4 "to4 Dm are the creation operators f o r , respectively, the singlet even, the singlet odd, the t r i p l e t even and the t r i p l e t odd scattering state. Analogous to (3.3.4) and (3.4.2a), we l e t Dsok k » = &ZTJM r°(u) s;!a, * » (4.4. % b) 49 These s c a t t e r i n g s t a t e s are e igenkets of H0 , w i t h e igenva lues £S (i<,k) , £ fK/ £), e^CS,^, and e*°(K,k). From ( 4 . 4.1), we have i*UKY>= [eW - H.]"' V |<*Ul<» = [>(/<)- H.]~'l V|£L(K)> f 4.4.?) S u b s t i t u t i n g i n ( 4 . 4 . 7 ) , and remembering the o r t h o g o n a l i t y r e l a t i o n s ( 2 . 2 . 7 ) , we have *I^"(jc',k)><if ftk)\v\£JK)> + Z [ e ( » - I 0 K ; B > < D > ; b ) | V I ^ ) > } + W\ z [£( K) - e, (ic)- £T ft)]'1 D* )^B*tt)i*X»lB„($)Dm.(*•;v I W ) + JVicW > - ^ . k V E ^ t ) ] " ^ ^ ; k ) B ^ ^ ) | o X o | B A ^ I ) ^ ( K ; k ) v | c D ^ ) > } +{•••} (4.4. lo) I n v a r i a n c e of the m a t r i x elements under s p a t i a l d isp lacements and r o t a t i o n s g i ves the f o l l o w i n g requ i rements : <el^ (K')V|£L,(iC)>= ^-K'^, l(K) (4.4.11a) <o|DS;(fe;k)V|<SM(K)>= S((H<')f^(K.fc) (r.4.Hb) <^lD^(K ' ,k )V|<SD f f l (K )>= S d C - ^ S ^ f e O H . f . l l c ) <°!BA^ )D^(!C')i7|«qn(!c)>= S(K-|<-£) W (4,4. H o t ) <H^(#D* e(fa)Vf£^ C4.4. lie) 50 From ( 4 . 4 . 4 a ) and ( 4 . 4 . 1 0 ) , we have which i m p l i e s £((<) = £ OO , and £ ( K ) - €JK)+ Z(K) ( 4 . 4 . R ) Hence, |&(&> = l £ ( K » + JA {Z C^(K) + ?(K) -£ S e (K ,k) ] " ' f v (K , k )D^(<; k)l°> 4 fe(K)+ £(K) x D!(K')B^|C-J<')|O>+ / ^ { R W + ^ ^ - ^ K ^ - ^ O ^ O T ' Z , K r / , ( k ^ ; A ) X D*e(K', k)B* +{•••} (4 .4. 13) The equat ion i s so f a r e x a c t — g i ven our cho ice of . Let us now w r i t e l £ „ ( ( 0 > = \\(^?+ \£>l(£)> (4 .4 . (4) and M ^ H£-£)\m, l  }(K)= < ° ] ^ ^ ' ) V | £ ) m ( K ) > = : <0|Dm,^)v|cBl(g)> (+. 4 . /s-) To f i r s t order i n our p e r t u r b a t i o n theory , we have ( 4 . 4 . /6) where Cfa-ttXkW <°IB x^)D; e^;A)(/D^(i<» ( 4 . 4 . 1 7 ) 51 Th is m a t r i x element can be eva luated e x p l i c i t l y u s i n g ( 4 . 4 . 2 ) , ( 4 . 4 . 4 b ) , ( 4 . 4 . 8 a ) , and ( 2 . 2 . 5 b ) , y i e l d i n g : tflfM'.k)* J A A Z C ^ " ^ C44l8fc) S u b s t i t u t i n g i n ( 4 . 4 . 3 ) and ( 3 . 3 . 1 2 a ) , and u s i n g ( E . 1 2 ) , one can show that Given , one can use ( 4 . 4 . 1 6 ) to c a l c u l a t e the p r o b a b i l i t y of f i n d i n g a p i o n i n the deute ron . We now want to c a l c u l a t e the e igenva lue £ of ( 4 . 4 . 1 ) . S u b s t i t u t i n g ( 4 . 4 . 1 6 ) i n t o ( 4 . 4 . 1 5 ) , we get £(t K) « J A A lSd(K) - Bse(K\ k) - K-K'))]" Z, C/' ft ' x<o\T) B(lOv]^ki)Bj(K -K>> Invoking ( 4 . 4 . 1 7 ) and ( B . 6 a ) , we have e w (K )=J-IVA few- £ s%;k)-£.(i(<-i<'o]"'| h (>,/<; (4.4.20) I f we now, f o r s i m p l i c i t y , r e p l a c e £ SS(^k) by ZSe(*',k) , and approximate by some a p p r o p r i a t e average v a l u e Jfc-J^'l^ , we f i n a l l y o b t a i n , by (D.5) and ( B . 6 a ) : (4.4. 21) 52 Chapter 5^  Summary and Conc lus ion We s h a l l f i r s t g i v e a b r i e f summary of what we have done: In the b e g i n n i n g , there were the Fermion c r e a t i o n operator and' the Boson c r e a t i o n operator B^u. By c o u p l i n g two Fermions t o g e t h e r , we were l e d to the s i n g l e t and t r i p l e t opera to rs , S°0 , T„0 , ~Cja-By f u r t h e r combining these w i t h a p p r o p r i a t e C lebsch-Gordan c o e f f i c i e n t s and s p h e r i c a l harmonics , we obta ined s e v e r a l u s e f u l opera to rs i n s e c t i o n s 2 . 3 - 2 . 5 . The Fermions and Bosons were i d e n t i f i e d w i t h p h y s i c a l nucleons and p h y s i c a l p i o n s . We then s t u d i e d systems of n u c l e o n s . A t r i p l e t even p o t e n t i a l \4 e was p r e s e n t e d , and used i n d e r i v i n g a p a i r of coupled d i f f e r e n t i a l equat ions d e s c r i b i n g the deuteron s t a t e . These e q u a t i o n s , under a p p r o p r i a t e r e s t r i c t i o n s , reduce to the f a m i l i a r deuteron equat ions i n B l a t t and Weisskopf (1952, p . 1 0 2 ) . The s i n g l e t even and t r i p l e t odd s c a t t e r i n g s t a t e s were in t roduced i n s e c t i o n s 3 . 3 and 3 . 4 , f o r use i n the f o l l o w i n g c h a p t e r . F i n a l l y , i n chapter 4 , we s t u d i e d systems of i n t e r a c t i n g nucleons A and p i o n s . The p e n t a - l i n e a r p o t e n t i a l s V and V, d i s c u s s e d i n s e c t i o n s .4.1 and 4 . 2 , were employed i n s e c t i o n 4 . 3 to o b t a i n the d i f f e r e n t i a l s c a t t e r i n g c r o s s s e c t i o n jj£ f o r the r e a c t i o n pp-»7T+d— w i t h r e s u l t s i n agreement w i t h Mandl and Regge(1955). As an example i l l u s t r a t i n g other types of c a l c u l a t i o n s one cou ld perform w i t h p e n t a - l i n e a r p o t e n t i a l s , we gave a f i r s t order p e r t u r b a t i o n study of the deuteron state i n s e c t i o n 4 . 4 . Having f i n i s h e d our summary, we s h a l l t r y to answer the f o l l o w i n g three q u e s t i o n s : "How should one proceed from h e r e ? " , "What are the uses of t h i s t h e o r y ? " , and "What are the advantages and d isadvantages of our theory compared to o t h e r s ? " 53 As po in ted out before i n Chapter 1 , our theory , by u s i n g a p e n t a -l i n e a r form (FFFFB+ a d j . ) f o r V and V, does not have as much p r e d i c t i n g power as the c o n v e n t i o n a l approach w i t h the Chew(1954, 1956a, 1956b) H a m i l t o n i a n , where the n u c l e o n - p i o n i n t e r a c t i o n p o t e n t i a l i s of a t r i -l i n e a r form ( F *FB + + a d j . ) . However, i n the c o n v e n t i o n a l c a s e , r"^Jo> i s not an e igenket of the H a m i l t o n i a n , (whereas By.|°> i s ) . Hence, w i t h the Chew H a m i l t o n i a n , l<>> cannot be i d e n t i f i e d w i t h a p h y s i c a l p a r t i c l e , and one has to i n t r o d u c e the n o t i o n of " d r e s s e d " p a r t i c l e s — l e a d i n g to very compl icated t h e o r i e s f o r processes i n v o l v i n g complex n u c l e i . In our theory , w i t h two or more a n n i h i l a t i o n opera to rs i n a l l our p o t e n t i a l s , both h^ 1°/ and B |o> are e igenkets of our unconvent iona l H a m i l t o n i a n . Due to the l o s s of p r e d i c t i n g power, our theory cannot make p r e d i c t i o n s f o r a r e a c t i o n as fundamental as pp - »n + d . However, we can work backwards; i . e . , we can use the exper imenta l data f o r pp-7>Tt+d to o b t a i n e x p l i c i t £ . A T f u n c t i o n a l forms f o r our p o t e n t i a l f u n c t i o n s and vSi . . (Th is i s analogous to u s i n g NN->NN data to o b t a i n e x p l i c i t forms of the n u c l e o n -nuc leon i n t e r a c t i o n p o t e n t i a l f u n c t i o n s i n low energy n u c l e a r p h y s i c s ) . A p o s s i b l e procedure i s o u t l i n e d below: Q One makes " i n s p i r e d guesses" f o r the f u n c t i o n a l forms f o r ( f e , ^ , ^ ) A T ) I \ and ~^itz\n3\k'lfi * / from prev ious work. That i s , one w r i t e s vl\h{K h O = * > « r , « „ O (S.O.U) A where f and f are s p e c i a l f u n c t i o n a l forms w i t h (X1^ . . . , 0 ^ parameters to be determined from d a t a - f i t t i n g . > ^ , % s (see ( 3 . 3 . 1 4 ) and ( 3 . 4 . 1 6 ) ) have been found by p rev ious exper imenta l works . S u b s t i t u t i n g these i n t o ( 4 . 3 . 1 0 c ) and ( 4 . 3 . 1 2 c ) , one can o b t a i n by n u m e r i c a l i n t e g r a t i o n : 54 A where ^ and ^ are aga in f u n c t i o n a l forms depending on the parameters , tf,,<Xa.; • - • ; K n . S u b s t i t u t i n g ( 5 . 0 . 2 ) i n t o ( 4 . 3 . 2 2 a ) , and f i t t i n g to the exper imenta l data f o r , one can get numer i ca l v a l u e s f o r the parameters , lt AT N\,-->°(r. • Wi th « , determined , | j f and v ^ z i J f j > and hence, V A and V are a l l known e x p l i c i t l y . A Having V and V determined from the p p - » n d d a t a , our theory can then proceed to make p r e d i c t i o n s f o r p i o n p r o d u c t i o n i n v o l v i n g an i n c i d e n t proton c o l l i d i n g w i t h a heav ie r t a r g e t n u c l e i , i . e . r e a c t i o n s of the type P + A -> i r + + ( A + 0 , A > 2 . (5r.o.3) Assuming the pr imary r e a c t i o n mechanism to be the NN-?TNN mechanism, A our V and V can be used to p r e d i c t the d i f f e r e n t i a l s c a t t e r i n g c r o s s s e c t i o n of ( 5 . 0 . 3 ) . A l s o of i n t e r e s t are i n t e r m e d i a t e s t a t e p i o n a b s o r p t i o n and p r o d u c t i o n e f f e c t s i n e l a s t i c s c a t t e r i n g of p ions o f f n u c l e i . That i s , p rocesses of the t y p e , TT++A A + 7T++A (S.O.if) A g a i n , our p o t e n t i a l s can be employed i n s tudy ing t h i s k i n d of r e a c t i o n . With these f u t u r e goa ls i n mind , we b r i n g t h i s t h e s i s to an end. Bibliography B l a t t , J.M. and Weisskopf, V.F. 1952. Theoretical Nuclear Physics (Wiley, New York). Brink, D.M. and Satchler, G.R. 1968. Angular Momentum, 2nd ed. (Clarendon, Oxford) . Chew, G.F. 1954. Phys. Rev., 94, 1748. Chew, G.F. and Low, F.E. 1956a. Phys. Rev., 101, 1570. Chew, G.F. and Low, F.E. 1956b. Phys. Rev., 101, 1579. deShalit, A. and Feshbach, H. 1974. Theoretical Nuclear Physics v . l (Wiley, New York). Goldberger, M.L. and Watson, K.M. 1964. C o l l i s i o n Theory (Wiley, New York). Landau, R.H. and Tabakin,F. 1972. Phys. Rev. D, _5, 2746. Mandl, F. and Regge, T. 1955. Phys. Rev., 99_, 1478. Messiah, A. 1958. Quantum Mechanics v . l (Wiley, New York). Newton, R.G. 1966. Scattering Theory of Waves and P a r t i c l e s (McGraw-H i l l , New York). Reid, R.V. J r . 1968. Annals of Physics, 50, 411. Rose, M.E. 1957. Elementary Theory of Angular, Momentum (Wiley, New York). 56 Appendix A The Mot ion R e v e r s a l Operator *J The mot ion r e v e r s a l operato r , a p p l i e d to a system, r e v e r s e s i t s momentum, but leaves i t s s p a t i a l c o o r d i n a t e s unchanged. By c l a s s i c a l ana logy , the angular momentum(=position v e c t o r x momentum) must a l s o be reversed under jT . S i m i l a r l y , we demand the s p i n to be reversed under motion r e v e r s a l , which i s e q u i v a l e n t to hav ing the observer ( i . e . c o o r d i n a t e system) r o t a t e by 180° a long some a r b i t r a r y a x i s . We choose a r o t a t i o n w i t h E u l e r angles (0,71,0") . The nucleons w i t h s p i n \ , thus t ransforms by Dmm,(o~n0) , as shown below: The p i o n s , on the other hand, hav ing no s p i n , t ransform, t r i v i a l l y by D°0(o"iro) • From Rose(1957 , p p . 6 2 - 7 3 ) , we have D* (OTTO) = ) - ( A - 3 ) S u b s t i t u t i n g (A.3) i n t o ( A . l ) and (A.2) y i e l d s ( 2 . 1 . 2 9 ) and ( 2 . 1 . 3 0 ) by s imple computat ion . Appendix 13 Some P r o p e r t i e s of the C lebsch-Gordan C o e f f i c i e n t s (j , tix w, "^ j In t h i s appendix , we l i s t some of the p r o p e r t i e s of the C l e b s c h -Gordan c o e f f i c i e n t s that are needed elsewhere i n t h i s t h e s i s . From Rose(1957, p p . 3 2 - 4 7 ) , we have (h iin'^z\iim^)-° u n l e s s + = w 3 ( B - l ) (ji h m*\hm3)-0 u n l e s s h',+ j z | * i 3 > Ik+islsj,, and | j 3 + 0',|< j 2 .(B - O C J i 1\ " i l " i l j , B 3 ) = ' 0 u n l e s s K | < j , . I »*| < g 2 ; and l " M < j , ( B - 3 ) ( 3 , 0 M , o | m3) = g • ^  S m i „ 3 and (o ^ o m z | j , * 3) = S . ^ S ^ ( B - 4-) From Rose(1957, p p . 3 8 - 3 9 ) , we have the f o l l o w i n g symmetry r e l a t i o n s : (j. U.n,.w*Us - ( L 0 4 , + 3* lB(l« 3 » - ^ l a " , - W 3 ) CB . S-ti) \ Z j, + t / 1 From B l a t t and Weisskopf (1952, p . 7 9 1 ) , we have the o r thogona l r e l a t i o n s Z ^ O . O z ^ ^ l ^ X i j ' z ^ ^ N ' « ' ) = S . . / 8Mffl, ( B . £ a ) £ ( 0 , 3 * *. " 1 I 3 « ) ( j , Oi I 0 » ) = S W ( n ; S ^ , ( B . 6 b) 58 i Appendix C Some Properties of the Rotational Matrices Dm m'(« In this appendix, we l i s t some of the properties of the matrices DrW^pY) > which are needed elsewhere i n t h i s thesis.. From Rose(1957, p.54), we have The orthonormal p r o p e r t i e s , from Rose(1957, p . 7 3 ) , a r e : From Rose(1957, p.58), we have where the common arguments (<*|3y) of the D-matrices have been omitted for brevity, and the summation over yu, and ^  i s such that = = constant. Correspondingly, we have Relabelling O\|,,>/0 by (i', »*<',/*') i n (C.5), multiplying the equation through by 2, (j'j3i*)'m3|jm)I^ 3m > and remembering (B.l^we have (j, j , i>, A i ) g . s (j- j , « ' »31 j »-> c i v D ; ; „ S Substituting (C.5) into the l e f t hand side, we obtain v>i rv>* TV's C C. 7 ) 59 where the summation i s such that m remains f i x e d . S i m i l a r l y , we have w i t h /*• f i x e d . 60 Appendix D_ Some P r o p e r t i e s of the S p h e r i c a l Harmonics Yim In t h i s appendix , we l i s t some of the p r o p e r t i e s of the s p h e r i c a l harmonics f r e q u e n t l y r e f e r r e d to i n t h i s t h e s i s . From Rose(1957, p . 6 0 ) , the s p h e r i c a l harmonic Yf„(r) t ransforms under r o t a t i o n by where i t i s understood tha t Y?™^ depends on ly on the o r i e n t a t i o n but not the magnitude of the v e c t o r jr; and where, i n C a r t e s i a n c o o r d i n a t e s , £<n-^-w i t h r_ w r i t t e n as a column v e c t o r , and the m a t r i x M as d e f i n e d i n Rose (1957, p . 6 5 ) . From Mess iah(1958, p p . 4 9 4 - 4 9 7 ) , we have the f o l l o w i n g r e l a t i o n s : Y^-rW-0^„(r) (j>.z) V ^ / ^ P , ( " s e ) ( J > ^ where !L-(r,0,<^) i n s p h e r i c a l p o l a r c o o r d i n a t e s , and ^ i s a Legendre p o l y n o m i a l . ^ P , ( c 0 5 . ) % £ Y>)Y,Jr') ( D - 0 where o( i s the angle between r_ and r'. where ^ i s a s p h e r i c a l B e s s e l f u n c t i o n op With the d i f f e r e n t i a l e ra to rs L T and L de f ined below, the s p h e r i c a l harmonics t u r n out to be e i g e n f u n c t i o n s of these o p e r a t o r s : 62 Appendix E Wigner 3 - j , 6 - j , 9 - j Symbols and Racah Coefficients This appendix l i s t s some of the properties of the Wigner 3 - j , 6 - j and 9 - j symbols and those of the Racah c o e f f i c i e n t s . For a more complete l i s t , see Brink and Satchler ( 1 9 6 8 , pp.136-145) or deShalit and Feshbach (1974, pp.925-932) . The Wigner 3 - j symbol i s related to the Clebsch-Gordan co e f f i c i e n t by From (B . 6 ) , we have the corresponding orthogonality relations: v /i,uu\(i< i * i i \ ~ J — s .« s , (E Za) From the symmetry relations (B . 5 ) , i t can be shown that the 3 - j symbol i s invariant under c y c l i c permutations of i t s columns, but i s multiplied by the extra factor ( - 1 ) + J i + 5,3 under non-cyclic permutations. The 6 - j and 9 - j symbols arise from the coupling of, respectively, three and four angular momenta. They can be expressed i n terms of the 3 - j symbols, as follows: K I + m ? mc ' Other useful relations.are: 63 The orthogonality relations for the 6-j and 9-j symbols are: ZL ( a ^ O O ^ - o f t V H P ' ' A = \y ( E . 7 ) The 9-j symbols can also be expressed i n terms of the 6-j symbols: \> M T J From the symmetries of the 3-j symbols, i t follows that: (1) The 6-j symbol i s invariant under interchange of columns, and also under interchange of the upper and lower arguments i n any two columns. (2) The 9-j symbol i s multiplied by the factor + + Jii+i 3q- +3 l 3 +U + T under an odd permutation of rows or columns, but i s invariant under an even permutation or a transposition. The Racah co e f f i c i e n t W(0ida^3 H > is i$) , frequently used i n place of the 6-j symbol, i s defined as follows: w(ii> u,,-j 5 ^ = ( - o i t l 2 + i , t i M i i ; t ) (E'lo) The orthogonality r e l a t i o n (E.7) becomes 05" (E .5) can be r e w r i t t e n i n terms of the Racah and C lebsch-Gordan c o e f f i c i e n t s : 65 Appendix F_ D e r i v a t i o n of the Deuteron Equat ions In t h i s appendix , we show how the deuteron equat ions ( 3 . 2 . 1 2 ) are der i ved from the e igenva lue equat ion ( 3 . 2 . 1 1 ) . S u b s t i t u t i n g ( 3 . 2 . 1 a ) (w i th K= 0) i n t o ( 3 . 2 . 1 1 ) y i e l d s : tXJ + D L > ( X l where, w i t h ( 3 . 2 . 8 ) , ( 3 . 1 . 1 ) and ( 3 . 1 . 2 ) , we have + ^ S %^«^|.«^r)r;^x JB)}io> ( F.ib) + ^ Z (*i ^  11m) Y2M,(r)Twet (r, E)\|o> (f. Ic) [1 ] = £ ( i k f I ^ R Y 0 0 ( x ) T ; > , S) + ^  Z/* I 11 *0 Y^dOT^ (r, )}|o> (F. W ) Let us focus on term C1] f i r s t . From ( 2 . 1 . 8 ) and ( 2 . 1 . 1 ) , we have { F „ f l O , F^ , (r)} = fef e ^ - ' r A S m „ , S / , ( F . ^ Us ing t h i s and ( 2 . 2 . 1 ) , we have Not ing tha t we can r e w r i t e ( 2 . 1 . 1 ) a s : (F-4) we can combine (F .3) and (F .4) to get 66 We now turn to term^E.^. Using (2.4.1) and (2.2.6), we have \i\ <8^r: r; g")Tm^ fc R » Z (li -s|X SX-CT' T*l Cr'; &"). Rs(r +r> s(r-r')>> Substituting this into term \X\ , integrating over Jfr , and writing r' as t , we get With (F.5) and (F.6) substituted back into ( E . l ) , we apply Z,TT0(E',&") to both sides of equation (F.la). We again focus on the f i r s t term. Using the anti-commutator relations (2.1.5) and (2.1.8), and (B.6a), i t i s straightforward to show that J Z ^(i;E')X A - fej { [ e ^ ( ^ + e^ ( t- r ' , A + e ^ ( - r - r ^ + e^(r+r')/fc]|o> Thus 67 S ince ^)-(4r^el^^\ and ^ = ^ [ f } e l x ^ , d=i,^ ( F . I ) the i n t e g r a t i o n s over J1^ and then over <^ r y i e l d : where the proper ty (D.2) has been invoked . In s p h e r i c a l p o l a r c o o r d i n a t e s , the L a p l a c i a n V* can be w r i t t e n a s : V* = + 2 2- - J - TZ (F- II) where the d i f f e r e n t i a l operator L i s de f ined i n (D .8b) . Us ing (D .9b ) , we o b t a i n Turning back to term , we have Z Teo(rU")En] = [Z , A ^ ( r V ) ( a k f { . K -« \ U0(-OT Y 0 0 + ^ >2L ^M^ s|i w)(n<- W s|r^)(-0' , S y2m(r)l|o> (F-13) a f t e r u s i n g ( F . 6 ) , ( 2 . 2 . 6 ) and ( 3 . 1 . 6 b ) . S u b s t i t u t i n g i n ( 3 . 1 . 9 ) f o r ^ (r',^) , u s i n g (D.5) and ( B . 4 ) , we o b t a i n + z z a 2 « , r v i r * x * i ^ ) 1«)(11 K-^ |Jrf)C-ons^ trr >»Y^ -r')V> (F.I4) Turning to term Tin:] , we have, w i t h (2.2.6) and ( D . 2 ) : Combining ( F . 1 2 ) , (F .14) and ( F . 1 5 ) , and i n t e r c h a n g i n g L and l ' , we get Apply J^Y,*!) t o ( F - 1 6 ) , where - IT Using the f a c t that Y 0 0 = Yo0 , and the orthonormal p roper t y ( D . 5 ) , we o b t a i n + K*' ( H w - " | oo)(-i? Vla0 (r> r') + jr<V £ Z (o 2 0 | J | ^ m s | | w ) Using the r e l a t i o n s i n Appendix B, the C lebsch-Gordan c o e f f i c i e n t s can be s i m p l i f i e d ; so that and Z Z ( 0 2 0 ^ I T « ) ^ i ^ ) i w X ' i w i - ^ I ^ X - 0 ^ V ( r ' r ' ^ = " i f ^ . * S u b s t i t u t i n g these i n t o ( F . 1 7 ) , we o b t a i n the f i r s t deuteron equat ion ( 3 . 2 . 1 2 a ) . To o b t a i n the second deuteron e q u a t i o n , we apply J^ - ^ r Y l w - f r ) t o ( F . 1 6 ) . Us ing (D.3) and ( D . 5 ) , we have + J r»Jr' Z { . Z Z (a Z | i inj, W41 I m)( 11 K | jr^-l)"5""*} ^ T £ „ ( V , r') = e ^ Z C * i ^ s | i » ) ( F . / 8 ) With the he lp of (B.5) and ( E . 1 2 ) , the terms i n s i d e the square b r a c k e t s 69 and the braces of (F.18) can be simplified to: = -{>2+0W(2£ II •, 21 ) Z (21 m j « . j | i»0 ( F - Itfci) where W(«bcc/;e-f) i s a Racah c o e f f i c i e n t . Thus, substituting (F.19) into (F.18), and cancelling out common factors, we have: -£S'*Z ~ C^)-J^\r'Jr'u(r,)^r' r < ) - f W+0W(aHI; *0/ r V r W ' ) v ^ ( r , r ' ) = ( F . 2 o ) From Rose(1957, pp.225-226), the Racah coefficients can be evaluated, yield i n g : W ( 2 0 l l ; a l ) = ^ f , W^lU;2l>^/i ^ W ( 2 2 H ; 2 l ) = ^ Substituting these values into (F.20), we obtain the second deuteron equation (3.2.12b). 70 Appendix (3 The Dens i t y Operator W In t h i s appendix , we i n t r o d u c e a d e n s i t y operator W f o r a two-p a r t i c l e s t a t e , w i t h one of the p a r t i c l e s p a r t i a l l y p o l a r i z e d i n the y d i r e c t i o n . F i r s t , cons ide r a s p i n \ system. We use X, ^ , and 2 to denote the three u n i t v e c t o r s a long the x , y , and z d i r e c t i o n s . We a l s o w r i t e l « > = |*t> , and ||8> a |n> where |$T}> and |£l)> a r e , r e s p e c t i v e l y , the s p i n up and s p i n down s t a t e s w i t h respec t to the z - a x i s . For an a r b i t r a r y d i r e c t i o n n , s p e c i f i e d by the angles © and <£> i n a c o n v e n t i o n a l s p h e r i c a l p o l a r c o o r d i n a t e system, we can w r i t e the s p i n up and s p i n down s t a t e s w i t h respect to the ri d i r e c t i o n , as |<U>= -e~^^||«> + cos||p> ($.a.l0 For the s p e c i a l case ft=ij , we have 6 ~\ , and $ = \ . Thus, I and I S O - H ^ + I P V ] ^ - 3 ) For a system c o n s i s t i n g of two s p i n h. p a r t i c l e s , we can form two-p a r t i c l e s t a t e s from ( G . l ) and (G.3) as f o l l o w s : iqtzi> = feO/j> + ^ > > A [ f H v > + ^ s > + l M > ] | ? i * T>=^[M^> + |(30(>>^[Mt,l>+jU|t ,o>- /t|s>] ($.<rc) | } A 2 ; > = jfe[i|«p>+|^>] = ^ [ ^ | t > o > + i | S > + | t J - i V ] 7 1 where it, r>?k*>, i^ °>=^Ci^ > + lr>] , K-i>=|(3/9> (Q.^b) a r e , r e s p e c t i v e l y , the s i n g l e t and t r i p l e t s t a t e s r e s u l t i n g from the c o u p l i n g of two s p i n % p a r t i c l e s . Suppose the f i r s t p a r t i c l e i s p a r t i a l l y p o l a r i z e d a long the y - a x i s , such that the p r o b a b i l i t y of i t hav ing s p i n up (a long the y - a x i s ) i s p . We can i n t r o d u c e a d e n s i t y operator W as f o l l o w s : S u b s t i t u t i n g (G.4) i n t o (G.6), and l e t t i n g f = \ > , we have V(5)«-!FI*,'><*.I| - jj|I*.•><*><>I + s a ^ ' X ^ I + ^K-f><Vl + V K - ' X V ' I + ^ K - X * I - ^ l * X * , ' | - U | s > < t ) H | + ^ | 5 > < s | ^ . 7 ) I f we l e t (T3 = |9t2t><3TST| + | S t ^ X 9 T l i | - l 9 i^><5iStl- l 9 i H X ^ 4 i \ and £ = Q-r'-r)]? = ^r-o5 = « 3 we can r e w r i t e (G.6) as = i d t P - o ; ) ($.«u) where 1 i s the i d e n t i t y o p e r a t o r . P i s commonly r e f e r r e d to as the p o l a r i z a t i o n v e c t o r , and i s the s p i n v e c t o r w i t h P a u l i s p i n m a t r i c e s 72 (for 2-particle states) as i t s components. We now turn to a r e a l i s t i c two-nucleon system, where we must take into account, besides spin, the momenta of the p a r t i c l e s and isotopic spin. In p a r t i c u l a r , l e t the two nucleons be two protons. Hence, the singlet and t r i p l e t states are respectively, |K,Js,/<,> i n (3.3.13), and [ K, fe, i n (3.4.11), with /< = I . Thus, we write the density operator W for a two-proton state with one of the protons p a r t i a l l y polarized i n the y di r e c t i o n , as follows: w= ^l ia.uXU.Ul - Uiu,u><a,°,i| +IIkiu><Ui| + ^|fe,k,o,i><ls,klu| + i o.iXis.k^il - ^lU^XUr1,'! 

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