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A field theory approach to pion deuteron elastic scattering Alexander, James Harry 1975

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A FIELD THEORY APPROACH TO PION ..DEUTERON ELASTIC SCATTERING • . by JAMES HARRY ALEXANDER B.Sc., University of Victoria, 1972 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in the Department of Physics We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August, 1975 In presenting 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 of the requirements f o r an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree 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 reference and study. I f u r t h e r agree that permission for extensive copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the Head of my Department or by h i s re p r e s e n t a t i v e s . It i s understood that copying or p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l gain s h a l l not be allowed without my w r i t t e n permission. Department of P-Ttysir.s-The U n i v e r s i t y of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date August 1 1 . 1 9 7 5 i i A b s t r a c t Pion-deuteron e l a s t i c s c a t t e r i n g i s s t u d i e d u sing a f i e l d t heory of pions and nucleons. By t r e a t i n g the nucleons i n t h i s manner, the d o u b l e - c o u n t i n g problem u s u a l l y a s s o c i a t e d with pion m u l t i p l e - s e a t t e r i n g i s a v o i d e d . The pion-deuteron T-matrix i s w r i t t e n as a s e r i e s expansion i n terms of o p e r a t o r s between one-nucleon s t a t e s . The f i r s t two terms i n the s e r i e s are examined. The f i r s t term y i e l d s the u s u a l s i n g l e - s c a t t e r i n g c o n t r i b u t i o n t o the T-matrix. The second term i n the s e r i e s can be expressed as a sum of twenty terms. By making an o n - s h e l l a pproximation and a s t a t i c a p proximation where p h y s i c a l l y s e n s i b l e , the magnitudes of the twenty terms are compared.- The dominant term i s s i m i l a r t o the c o n v e n t i o n a l double s c a t t e r i n g term r e s u l t i n g from the g e n e r a l i z e d impulse approximation. There are a l s o f o u r o t h e r terms whose magnitude cannot be e v a l u a t e d without doing numerical s t u d i e s with a p a r t i c u l a r f i e l d t h e o r e t i c p o t e n t i a l . i i i TABLE OF CONTENTS 1 I n t r o d u c t i o n 1 2 D e f i n i t i o n of Operators and S t a t e s ......4 2.1 P h y s i c a l S t a t e s With Barycn Number Zero .• 5 2.2 P h y s i c a l S t a t e s With Baryon Number One .7 2.2.1 The P h y s i c a l Ona-Nucleon S t a t e ....7 2.2.2 The Pion-Nuclecn S c a t t e r i n g S t a t e .............9 2.3 S t a t e s With Baryon Number Two 11 2.3.1 The P h y s i c a l Two-Nucleon State ................11 2.3.2 Tha Dauteron S t a t e and the Pion-Deuteron S c a t t e r i n g S t a t e ............11 2.4 The S-matrix and the T-matrix ..........13 3 Meson Exchange S e r i e s ....16 3.1 Cutkosky S t a t s s with Baryon Number Two ....16 3.2 Meson Operator I d e n t i t i e s and Cutkcsky Matrix E l e me nts ...........21 3.2.1 Operator I d e n t i t i e s ...........21 3.2.2 Cutkosky Overlap Matrix Elements and Diagrams .22 3.2.3 Cutkosky M a t r i x Elements of the Vertex Operator and the Hamiltonian .............25 3.3 Meson Exchange S e r i e s f o r P i c n - D e u t e r c n S c a t t e r i n g .30 3.3.1 The Deutaron S t a t e Vector 30 3.3.2 The Pion-Deuteron S c a t t e r i n g S tate Vector .....34 3.3.3 The Pion-Deuteron T-raatrix 37 4 rhe Meson Exchange S e r i e s For The T-matrix 38 4.1 Zero-Meson Exchange C o n t r i b u t i o n to the T-matrix ...38 4.2 One-Meson Exchange C o n t r i b u t i o n tc the T-matrix ....40 <4.3 The S t a t i c O n - S h a l l A p p r o x i m a t i o n 48 5 C o n c l u s i o n s . . » 55 B i b l i o g r a p h y • . . . 5 7 APPENDIX A Mason O p e r a t o r I d e n t i t i e s 58 APPENDIX B C u t k o s k y M a t r i x E l e m e n t s . . . 6 3 B.1 U n e x c i t e d O v e r l a p M a t r i x E l e m e n t s . . . . . . . . . 6 3 B . 2 U n e x c i t e d I n t e r a c t i o n M a t r i x E l e m e n t s . . . . 6 8 B . 3 E x c i t e d O v e r l a p M a t r i x E l e m e n t s . . . . . 7 5 B . 4 E x c i t e d I n t e r a c t i o n M a t r i x E l e m e n t s . . 7 7 APPENDIX C The T - m a t r i x i n t h e O n e - M e s o n E x c h a n g e A p p r o x i m a t i o n . . . 8 0 A P P E N D I X D 85 D. 1 T h e D e u t e r o n w a v e f u n c t i o n 85 D . 2 S i n g l e - S c a t t e r i n g . . 8 6 D. 3 D o u b l e - S c a t t e r i n g . . . 8 8 Acknowledgements I would l i k e t o take t h i s o p p o r t u n i t y t o express my thanks to my r e s e a r c h s u p e r v i s o r , Dr. Malcolm McMillan, f o r h i s encouragement and a s s i s t a n c e . I would a l s o l i k e to thank my w i f e , Margaret, f o r her c o n s t a n t moral support and a s s i s t a n c e - . The f i n a n c i a l a s s i s t a n c e p r o v i d e d by the N a t i o n a l Research C o u n c i l over the past two years i s g r a t e f u l l y acknowledged. 1 1 IHt£2.duct ion In the next few years many experiments w i l l be performed at the meson f a c t o r i e s which w i l l use the pion as a probe of nuclear structure. The pion i s a p a r t i c u l a r l y useful tool to investigate nuclear structure. Since i t comes i n three charge states, i t can p a r t i c i p a t e in charge exchange and double charge exchange scattering which w i l l hopefully further the understanding of nuclear states." Also, since the pion can be absorbed or emitted by a nucleon, pion-nucleus absorption experiments can be useful probes cf higher momentum components in nuclear wa vaf unctions. Since the i n t e r a c t i o n of the pion with the nucleus i s rather weak for pion k i n e t i c energies below about 100 MeV, the e l a s t i c s c a t t e r i n g of low energy pions can be u t i l i s e d i n a manner sim i l a r to electron scattering. However, since pions and electrons i n t e r a c t with nucleons via d i f f e r e n t types of forc e s , e l a s t i c pion scattering and electron s c a t t e r i n g can perhaps be used as complementary methods. The simplest way to treat pion-nucleus s c a t t e r i n g i s to use the s i n g l e - s c a t t e r i n g approximation, i . e . , to assume that the pion scatters from only one nucleon in the nucleus. However, i f e l a s t i c pion s c a t t e r i n g i s ever to be useful i n extracting d e t a i l s of nuclear structure using the s i n g l e - s c a t t e r i n g approximation , i t i s necessary to obtain an estimate of the terms neglected by taking the s i n g l e - s c a t t e r i n g approximation. Ihe generalized impulse approximation [Chew and Goldberger (1952) ] expresses the e l a s t i c scattering of a p r o j e c t i l e on a •2 nucleus as a sum of terms, the f i r s t three of which are single-scattering, double-scattsring, and binding corrections. Double-scattering describes the process where the p r o j e c t i l e s c atters from one nucleon, then propogates to another nucleon and scatters again before leaving the nucleus. Binding corrections describe the effect of the nuclear potential on the p r o j a c t i l e -nucleon scattering. If t h i s approach i s used to describe pion-nucleus scattering, a problem arises. Since the pion i s thought of as mediating the nuclear force, the p r o j e c t i l e scattering from the nucleons i s i d e n t i c a l to the p a r t i c l e being exchanged between these nucleons as a part of the nuclear force. Thus i t i s not cle a r to what extent the binding" corrections are already included i n the double-scattering term of the generalized impulse approximation. This i s usually referred to as the "double-counting problem". This t h e s i s avoids the double-counting problem by constructing a f i e l d theory of picns and nucleons. The nucleons are assumed to be composed of a bare nucleon core surrounded by a cloud of physical pions. By tr e a t i n g the nucleons i n t h i s manner, no double-counting w i l l occur since a l l the pions involved in the process are accounted for e x p l i c i t l y . The approach use! i n t h i s thesis follows a method used by Pendleton (1963). The case chosen i s e l a s t i c picn-deuteron scattering. The leuteron was chosen as the target since i t i s the simplest nucleus in which double-scattering and binding corrections can be non-zero. Since i t contains only two nucleons, the properties of i t s wavefunction are perhaps the 3 b e s t known o f any n u c l e u s . I n c h a p t e r 2 the p i o n , n u c l e o n and p i s n - n u c l e o n s c a t t e r i n g s t a t e s a r e d e f i n e d a l o n g w i t h t h e b a s i c o p e r a t o r s u s e d i n d e s c r i b i n g t h e s c a t t e r i n g . I n c h a p t e r 3 a p p r o x i m a t e t w o - n u c l e o n s t a t e s w h i c h c a n be w r i t t e n i n t e r m s c f o n e - n u c l e o n o p e r a t o r s a r e i n t r o d u c e d . T h e s e s t a t e s a r e u s e d t o w r i t e t h e p i o n - d e u t e r o n T - n a t r i x as a s e r i e s e x p a n s i o n i n t e r m s o f o p e r a t o r s b e t w e e n o n e - n u c l e o n s t a t e s . I n c h a p t e r 4 t h e f i r s t two t e r m s o f t h e s e r i e s a r e c a l c u l a t e d and a f t e r m a k i n g some a p p r o x i m a t i o n s , a r e c o m p a r e d t o t h e s i n g l e - and d o u b l e - s c a t t e r i n g t e r m s r e s u l t i n g f r o m t h e g e n e r a l i z e d i m p u l s e a p p r o x i m a t i o n . C h a p t e r 5 c o n s i s t s o f a summary o f t h e r e s u l t s . J 2 D e f i n i t i o n of 0 £ B r a t o r s and S t a t e s In t h i s s e c t i o n some of the o p e r a t o r s and s t a t e s which w i l l be used i n d e s c r i b i n g pion-deuteron s c a t t e r i n g w i l l be i n t r o -duced. The z e r o - and one-nuclaon s t a t e s w i l l be developed i n a manner s i m i l a r to t h a t used by Chew and Low (1956) and Wick (1955) i n d e s c r i b i n g pion-nucleon s c a t t e r i n g . The two-nucleon s t a t e s w i l l be t r e a t e d u s i n g a method f i r s t developed by H e i t l e r and London (1927) to d e s c r i b e the hydrogen molecule and l a t e r a p p l i e d to nucleon-nucleon problems by Cutkosky(1958) and Pendleton (1 963) . The fundamental dynamical v a r i a b l e s f o r the system of pions and nucleons are assumed to be boson a n n i h i l a t i o n and c r e a t i o n t o p e r a t o r s k and k and 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 a and a . These o p e r a t o r s s a t i s f y the u s u a l commutation and anticommutation r u l e s . That i s , the boson o p e r a t o r s s a t i s f y (2. 1) and the fermion o p e r a t o r s s a t i s f y {a,b} = {a +,b +} = 0 (2.2) In a d d i t i o n the boson and fermion o p e r a t o r s w i l l commute. The d e l t a f u n c t i o n s above are a c t u a l l y products of Kroenecker d e l t a 5 functions of spin and is c s p i n with Dirac delta functions of aomentura. The boson f i e l d quanta w i l l ke i d e n t i f i e d with physical pions and the fernion f i e l d quanta k i l l be i d e n t i f i e d with bare nucleons and antinucleons. It w i l l be assumed that a H a a i l t c c i a r car be constructed using the fundamental dynamical variables and having eicenstates corresponding to physical nucleccs, pion-nuclecn scattering states, deuterons, e t c . These states w i l l also he eicenstates cf the baryon number operator B . 2 . 1 Physica1 States With Barton Kumber Zero The state vectors correspcndirg tc barycn runber zero w i l l be mescn states or the physical vacuum st a t e . The mescn state |k> represents a mescn with guantum numbers (spin, i s c s p i n , momentum, etc.) l a b e l l e d by k. I t s a t i s f i e s H|k> = E k|k> P|k> = kjk> B|k> = 0 E, = /|k| z+m z k '— 1 ir where P i s the momentum operator, B i s the baryon ruaber cperatcr and i s the mass of the pion. The speed cf l i g l t and * are taken to be one in the abcve equations. The physical vacuum state | 0 > i s defined ty H |0> = 0 i 2 ' 1 ) B|O> = 0 < 2 * 8 ) (2 .3 ) (2 .4) (2 .5) (2 .6) 6 It a l sc s a t i s f i e s k|0> = 0 {2. 9) a|0> = 0 (2. 10) As stated atove i t w i l l be assuned that the boscn f i e l d quanta are physical pions. That i s , the physical nescn state |k> can be thought of as r e s u l t i n g frcm the action cf the meson creation operator on the physical vacuum state |0> k+|0> = |k> (2. 11) The Hamiltonian w i l l therefore be written in the form H = Zk E k k+k + V ( 2 . 12) where V i s an operator describing the meson-meson and meson-nucleon i n t e r a c t i o n . The summation over k i n the above equation denotes a sum over a l l the meson quantum numbers as well as an in t e g r a l over the meson momentum. (In subsection 3.2,3 a d i f f e r e n t s p l i t t i n g of H w i l l be introduced. That s p l i t t i n g w i l l be more useful in separating the nucleon energy part of H from the nucleon i n t e r a c t i o n part of H.) 7 2.2 P h y s i c a l S t a t e s W i t h B a r t o n Number One T h e s t a t e v e c t o r s c o r r e s p o n d i n g t o b a r y o n number one w i l l be e i t h e r p h y s i c a l o n e - n u c l e o n s t a t e s o r p i o n - n u c l e o n s c a t t e r i n g s t a t e s . As b o t h a r e o f g r e a t i m p o r t a n c e i n d i s c u s s i n g p i o n -d e u t e r o n s c a t t e r i n g , t h e y w i l l be t r e a t e d i n some d e t a i l . 2 . 2 . 1 The P h v . s i c a l O n e - N u c l e o n S t a t e T h e p h y s i c a l o n e - n u c l e o n s t a t e |A> d e s c r i b e s a p h y s i c a l n u c l e o n w i t h q u a n t u m n u m b e r s ( s p i n , i s o s p i n , momentum e t c . ) l a b e l l e d by A . I t s a t i s f i e s H|A> = EA|A> - (2 . 13 ) P|A> = kA|A> (2. 1") B|A> = +l|A> (2 . 15 ) E . = / | k , | 2 + M 2 (2. 16) w h e r e M i s t h e o b s e r v e d n u c l e o n m a s s . A p h y s i c a l o n e - n u c l e o n c r e a t i o n o p e r a t o r A + i s d e f i n e d a s f o l l o w s . In t h e C h e w - L o w t h e o r y [Chew a n d L o w ( 1 9 5 6 ) ] t h e p h y s i c a l o n e - n u c l e o n c r e a t i o n o p e r a t o r i s w r i t t e n a s t h e p r o d u c t o f a b a r e n u c l e o n c r e a t i o n o p e r a t o r a n d an o p e r a t o r w h i c h p r o d u c e s a " c l o u d " o f p i o n s . T h e p i o n c l c u d c r e a t i o n o p e r a t o r c a n be w r i t t e n a s a sum o f p r o d u c t s o f p i o n c r e a t i o n o p e r a t o r s . In t h e p r e s e n t t r e a t m e n t t h e p h y s i c a l n u c l e o n c r e a t i o n o p e r a t o r i s c o n s t r u c t e d i n a more g e n e r a l m a n n e r . T h e o p e r a t o r i s w r i t t e n as a l i n e a r c o m b i n a t i o n o f t h e p r o d u c t s o f n u c l e o n c o r e c r e a t i o n o p e r a t o r s and meson c l o u d c r e a t i o n o p e r a t o r s : 8 Z a(AM) A + M + ( 2 . 1 7 ) The nucleon core creation operator A i s simply a jrccuct cf tare nucleon and antinuclecn creation operators with t c t a l baryon number one: Obviously i f A' i s to s a t i s f y the above equation i t must ccntair an odd number cf bare nucleon and antiruclecn c r e a t i c r + cperatcrs. The meson clcud operator H consists cf a product of meson creation operators. The summation i n eguaticn ( 2 . 1 7 ) i s ever a l l nucleon core creation operators and meson cloud creation operators subject tc the condition that the t o t a l charge ard mcffentuns cf each term must be equal to the charge and momentum of the state | A > . The c o e f f i c i e n t s a(AH) i n the l i n e a r ccttbiraticn w i l l be the wave function of the physical nucleon in the lock space cf the cores and physical mesons. From the form cf eguation ( 2 . 1 7 ) and the commutation and anticommutation r e l a t i o n s of eguatiens ( 2 . 1 ) ard ( 2 . 2 ) i t can be seen that physical nucleon operators A r E , e t c . obey the following anticommutation r e l a t i o n s : 8A |0> = +1 A |0> ( 2 . 1 8 ) { A , B } = { A ' , B } = 0 ( 2 . 1 9 ) However, the expression for ( f l , E T ) w i l l be verj complicated because of the necessity of anticommuting the products of the 9 annihilation operators with the products cf the creation 4. operators in the expressions for i and E • The calculations necessary to evaluate t h i s connutatcr are dene in an approximate nanner in Appendix B. 2 . 2 . 2 The P i o n - N u c l e o n S c a t t e r i n g S t a t e The pion-nucleon scattering state |Ak>+ represents a state composed of a physical nuclecn with qcantun numbers asymptotically l a b e l l e d by A and a mescn * i t h quantum numbers asymptotically l a b e l l e d by k, obeying either outgoing wave ( + sign) or incoming wave (- sign) boundary conditions. Since only outgoing wave states w i l l normally be dealt with, pion-nuclecr scattering states without + or - subscripts * i l l be assumed to sa t i s f y outgoing wave boundary conditions. The pion-nuclecn scattering state |Ak> s a t i s f i e s H|Ak> + = (E +E.)|Ak> A k ± (2.20) P|Ak> + = (k +k) |Ak> " ~~ ± (2.21) 8|Ak>. = +l|Ak> . . * * (2.22) where E and E. are as given previously. A K Hick (1955) has shewn that the pion-nucleon scattering state can be expressed s o l e l y in terms cf the Eaoiltcnian, meson creation operators, and physical nucleon creation operators. Sick wrote the pion-nucleon scattering state as a state consisting of a free mescn and a physical nuclecn plus a scattered state I Ak> = k + |A> + |Ak> s 1C (2.23) An e x p l i c i t expression for the scattered state |Ak> can le s obtained by substituting equation (2.23) i n t c the eigenvalue equation for H for the scattering state, equation (2.20) ( H - E A - E k ) | A k > s + Hk +|A> - (E A +E k )k + |A> = 0 (2.24) Equation (2.24) i s s i m p l i f i e d by writing Hk +|A> = k +H|A> + [H,k + ]|A> (2.25) Using the e x p l i c i t form cf the Hamiltonian, equation (2.12), the commutator in the above expression becomes [H,k + ]|A> = E k k + |A> + {V,k + ]|A> (2.26) By substitutinq equations (2.12), (2.25), and (2.26) intc equa-tion (2.24) the following equation i s obtained ( H - E A - E k ) | A k > s + [V,k +]lA> = 0 ( 2 > 2 ? ) An expression for the scattered state |Ak>s i s crtainec" ty inverting the operator (H-E -E ) and imposing outgoing wave A k boundary conditions |Ak> s = ( E A + E k - H + i e ) ' 1 [ V , k + ] | A > (2.28) 11 using Pendleton's notation for the vertex operator [Vk] = [V,k f] [kV] = [Vk] + (2.29) the outgoing wave pion-nucleon scattering state i s written It should be noted that equation (2.30) i s writtec i n c c r r e c t l j ty Pendleton, the operator [Vk] being replaced by [kV] . 2.3 States With Barton Number Two 2.3.1 The Physical Two^Nuclaon State The physical two-nucleon state, denoted by |AB>, represents two physical nuclsons with quantum numbers asymptotically l a b e l l e d by A an! B. I t s a t i s f i e s 2.3.2 The Deuteron State and the Pion-Deutergn Scattering State The physical deuteron state i s represented ly the state vector \V> . It s a t i s f i e s |Ak> = k+|A> + (E A+E k-H+ie) _ 1[Vk]|A> (2.30) H|AB> = CEA+EB)|AB> P|AB> = (kA+kfi)|AB> B|AB> = +2|AB> (2.31) ( 2 . 33 ) (2.32) H|P> = E J P > (2.34) 12 P|P> = kjp> ^ (2.35) B\V> = +2\V> (2.36) y y 0 (2.37) where ftp i s the observed deutercn rest mass. The pion-deuteron scattering state |Pk>+ represents a state composed of a physical deutercn with guartufc numbers asymptotically l a b e l l e d by V and a meson with quantum nunbers asymptotically l a b e l l e d by k. As with the pion-nucleon scattering state,the ± subscript refers to either outgoing cr incominq wave boundary conditions and a state «ith no subscript w i l l be assumed to s a t i s f y outgoing wave boundary conditions. The pion-deuteron scattering state |Pk> s a t i s f i e s H|Pk>± = (Ep+Ek)|t7k>± ( 2 < 3 8 ) P|Pk>± = Ckp-k) It?k>± ( 2 # 3 9 ) B|Pk>± = +2|t?k>± < 2 < U 0 ) where and Efc are as given previcusly. By writing the pion-deuteron scattering state as |Pk> = k + |P> + |t?k>s (2.41) and using exactly the same procedure as was used following equation (2.23), the analogous result i s obtained for the p i c r -deutercn scattering state |t?k> = k+|p> + (Ep+E^H+ic)" 1^]!^ (2.42) 13 2.4 The S-matrix ana the T-matrix The scattering operator S i s defined in g e E e r a l as (2.43) where | IK (0)> i s the tine independent factor cf the asymptotic in l i m i t cf the state vector a long time before s c a t t e r i n g , s a t i s f y i n g incoming wave boundary conditions and | ^  (0)> i s out the time independent factor cf the asynptctic l i i r i t cf the state vectcr a long time aft e r s c a t t e r i n g , s a t i s f y i n g outgoing wave boundary conditions. Fcr pion-nuclecn scattering eguation (2.43) can te writter a s (2.44) or, using the fact that S i s unitary <Bm S = <Bm (2.45) + The S-matrix for pion-nucleon scattering i s defi r e c as = <Bm|s|Ak>+ = <Bm|Ak>+ (2.46) by use of eguation (2.45). Using eguation (2.30) fcr | Enlace" the s i m i l a r eguation for |Bm>_ 14 |Bm>_ = m+|B> + ( E ^ - I l - i e ) ~ 1 [Vm] | B> (2.4 7) i t fellows that |Bm>_ = |Bm>+ + { ( E g + E ^ H - i e ) - 1 - ( E g + E ^ H + i e ) ' 1 } [Vm] | B> (2.48) Using the i d e n t i t y ( E B + E m - H - i e ) - 1 = (Eg+E^H+ie)" 1 + 2TT16 (E B+E m-H) (2.49) equation (2.48) can be written |Bm>_ = |Bm>+ + 27riS (Eg+E^-H) [Vm] | B> (2.50) Substituting t h i s into eguation (2.46), the expression f c r the £-matrix becomes SBm,Ak = +<Bn>|Ak>+ - 2 , i 6 O y E ^ E ^ ) < B | [mV] |Ak> + = 6Bm,Ak " 2iri«(E B +E m-E A-E k)<B|[mV]|Ak> + ( 2 . 5 1 ) It should be noted that the Kronecker delta function 6„ i s a Bm,Ak shorthand notation for the product cf Kronecker delta functions cf a l l the quantum numbers of the states |Em> and |flk> as well as a delta function of momentum. The general expression for the S-matrix i s S F I = 5 F I " 2 T t i 6 ( E F - E I ) T F I ( 2 < 5 2 ) 15 where I refe r s to the i n i t i a l state, F refers tc the f i n a l state li and 1 i s the T-matrix. Thus, comparing equations (2.51) arc (2.52), the T-matrix for pion-nuclecn s c a t t e r i r g can be written V A k = <B|[mV]|Ak>+ (2.53) S i m i l a r i l y , using equation (2.42) f c r |Pk>+ ace the corres-ponding equation for |Pk> , the T-uatrix f c r picn-derteron scattering can be written T t ? - m , t ? k = < t ? ' l t m V ] | P k > + (2.54) The objective now i s to calc u l a t e the r i g h t hand side cf eq. (2.54). To do t h i s , the deutercn state and the picn-dettercc scattering state w i l l be expanded in terms cf a p a r t i c u l a r set of two^nucleon states introduced i n the next chapter. 16 3 SS^QU Excharj2§ Esrias In order to proceed from t h i s pcint, a ccnplete set of states with baryon number two i s introduced. Bather than choosing physical two-nuclecn states which cannot readily be expressed in terms of cne-nuclecn creation operators, a set cf states i s constructed e x p l i c i t l y i n terms of cne-nuclecr creation operators and meson creation operators following a nethcd used by Cutkosky(1958) and based on the work cf F e i t l e r and London(1927). These states w i l l be c a l l e d Cutkosky states. Since Cutkosky states are written in terms cf physical nucleon and meson creation operators, matrix elements betveen these states w i l l be able to be reduced to matrix elements between one-nucleon st a t e s . Thus, by using these states tc expand the pion-deuteron T-matrix, i t w i l l te possible tc express the T-matrix e n t i r e l y in terms cf cne-nucleon matrix elements. 3.1 Cutkosky, States with Barren Nujfcer Two The simplest Cutkosky state i s that which consists cf twe nucleons with no mesons present. This state, c a l l e d an unexcited Cutkosky state and denoted by |AB}, i s defined by AB} = A B |0> (3.1) It s a t i s f i e s P|AB} = (k^+k^lAB} 8|AB} = +2|AB> (3.3) (3.2) 17 H o w e v e r , u n l i k e t h e p h y s i c a l t w o - n u c l e o n s t a t e | AB>, t h e C u t k o s k y s t a t e JAB} i s n o t an e i g e n s t a t e o f t h e H a m i l t o n i a n . T h i s i s shown e x p l i c i t l y i n e q u a t i o n ( 3 . 4 4 ) . U s i n g t h e d e f i n i t i o n o f t h e s t a t e |AB} a n d t h e f a c t t h a t p h y s i c a l n u c l e o n o p e r a t o r s a n t i c o m m u t e , i t i s o b v i o u s t h a t t h e s t a t e |AB } i s a n t i s y m m e t r i c |AB} = - | BA} ( 3 . 4 ) S i n g l y e x c i t e d C u t k o s k y s t a t e s , i . e . t h o s e i n w h i c h aE e x t r a meson i s p r e s e n t , a r e d e f i E e d by | ABk} = k + A + B f | 0 > + (Ak)V|0> + A + (Bk)*|0> ( 3 < 5 ) T h e s e s t a t e s s a t i s f y p|ABk> = (k^+k^+jylABk)- ( 3 . 6 ) B|ABk> = +2|ABk> <3.7) T h e s e s t a t e s a r e n o t e i g e n s t a t e s o f t h e H a m i l t o n i a n as i s s h o w n i n e q u a t i o n ( 3 . 4 6 ) . The g e n e r a l e x c i t e d C u t k o s k y s t a t e w i t h b a r y o n number t w o , |ABM }, w h e r e M r e p r e s e n t s a p r o d u c t c f mescn o p e r a t o r s , c a n be d e f i n e d i n a s i m i l a r m a n n e r b u t i t w i l l n o t be n e e d e d i n t h e s i n g l e - a n d d o u b l e - s c a t t e r i n g c a l c u l a t i o n s . G e n e r a l C u t k o s k y s t a t e s ( e i t h e r e x c i t e d o r u n e x c i t e d ) w i l l be d e n o t e d by |U}, |V} , |W } e t c . One o f t h e f u n d a m e n t a l a s s u m p t i o n s o f t h i s m e t h o d i s t h a t t h e s e t o f a l l C u t k o s k y s t a t e s f o r m a c o m p l e t e s e t . I n o t h e r 18 words, i t i s assuaed that any physical state with karyon cunter two can be expressed as a linear c c E b i n a t i c r , cf the states |U). In p a r t i c u l a r a physical two-nucleon state should te able to te written as where the c o e f f i c i e n t s Y« a r e * c be deternired. Icr the M calculations to be dene i t i s hoped that equation (3.8) «ill converge quickly. Since the deuteron i s rather weakly bound, a good approximation to the deuteron state should be able to be obtained just by taking the noninteracting term and the terms i t the summation i n equation (3.8) fcr which M contains only one mescn operator ( i . e . singly excited st a t e s ) . As the Cutkosky states defined abeve are ret orthogonal, crthcncrmal Cutkosky states, denoted ty |U), |V), |W) etc., are defined (following Pendleton (1963)) by the equation The summation in aquation (3.9) runs over a l l two-nucleon Cutkosky states. The orthonormal states are introduced for AB> = |AB} + Z XMIABM> (3.8) <3.9) computational convenience. Using the orthonormal states the unit operator can be written in the form 1 - Z |u)(u| (3. 10) which w i l l be used i n the c a l c u l a t i o n of matrix elements 19 i n v o l v e d i n s i n g l e - and d o u b l e - s c a t t e r i n g e x p r e s s i o n s . Eguation (3.10) i s not t r u e i f the complete set c f s t a t e s i s not orthonorma1. Using equation (3 .10) and the o r t h c n o r m a l i t y of the s t a t e s d e f i n e d i n equation ( 3 . 9 ) , an e x p r e s s i o n f o r the matrix f i s d e r i v e d as f o l l o w s . = EE F * F {X|U} O . 11) wv Defining the matrix G by the equation 6 + G = {X|U} (3 .12) xu xu equation (3 .11) can be written 6 = E F F + EE F G F (3 .13) wv uv uw xv xu uw I J . u j Writing the above equation i n matrix form y i e l d s the equation 1 = F + F + F +GF (3 . 14) where the adjoint of the aatrix F i s defined by F^v= Equation (3 .14) can be solved for F + F . The solu t i o n i s F + F = (1+G)" 1 < 3 * 1 5 > The abcve equation can be solved for F . By choosing F tc be Hermitian a solution can be written F = a+G)'h = i - ^  + jf2 + ••• 20 (3. 16) The expansion using tha binomial theorem w i l l be used to calculate approximations to . I t w i l l be shewn that for si n g l e -and double-scattering c a l c u l a t i o n s , no more than the f i r s t two terms i n the expansion w i l l be needed. For example, by taking the f i r s t two terms only, the state |AB) nay be written |AB) = | | A B } - | E|U){U|AB} ( 3 . 1 7 ) It w i l l be useful to define an operator K.^ as fcllcws K + |U) = |uk) ( 3 > 1 8 ) This operator i s not ea s i l y expressed in teras cf physical nucleon and meson creation operators. Taking the f i r s t two terms only i n eg. (3.16) i t follows that K+|AB) = |ABk) = ||ABk} - \ g|u}{u|ABk} (3.19) Because of the secend term i n the alove equation, the rela t i o n s h i p between K+ and physical nucleon and mescn c r e a t i c r operators i s very complicated. 21 3.2 Meson Operator I d e n t i t i e s and Cutkcskj Matrix Elements In order tc evaluate the pion-deuteron T-aatrix, the deuteron state and the pion-deutercc scattering state h i l l le expanded i n terms of the orthcncriral Cutkosky states. Thus the 1-matrix w i l l be written in terms of matrix elements letweet orthonormal Cutkosky states. Using eguaticns (3.8), (3.12), and (3.16) these matrix elements can be written in terms of natrix elements between nonorthonornal Cutkosky states. The purpose of this section i s t c evaluate Cutkosky matrix elements and presert meson operator i d e n t i t i e s which w i l l prove tc be useful i n evaluating and simplifying these matrix elements. 3.2.1 Operator I d e n t i t i e s The meson operator i d e n t i t i e s w i l l only te stated here, the proofs being l e f t tc Appendix A. The f i r s t i d e n t i t y gives ar expression for the comoutator of a product cf nescn creation + operators H and a product of meson a n n i h i l a t i o n operators Q. It w i l l be written i n two forms [M + ,Q] - -4 [ [ r , M + J ] [ [ Q , r + ] ] / n ( R)I (3.20) QM+ = E R [ [ r , M + ] ] [ [ Q , r t ] ] / n ( R ) ' . (3.21) [ [ r , M + ] ] = [ r 1 , [ r 2 , [ . . . [ r n , M + ] . . . ] ] ] (3.22) [ [ M , r + ] ] = [ [ r , M f ] ] + (3.23) n (R)=n=numter of operators in R a product of meson ann i h i l a t i o n operators. The or where R represents p r i m e c n t h e s u m m a t i o n i n e q u a t i o n ( 3 . 2 0 ) i n d i c a t e s t h a t t h e u n i t o p e r a t o r i s t o be e x c l u d e d f r c m t h e s u a e v e r a l l R . When R=1 t h e n e s t e d c o m m u t a t o r o f e q u a t i o n (3 .22 ) i s d e f i n e d t o l e j u s t M+. The n e x t t h r e e i d e n t i t i e s w i l l be u s e f u l i n r e m o v i n g m e s c r o p e r a t o r s f r o m o n e n u c l e c n m a t r i x e l e m e n t s . T h e s e i d e n t i t i e s a r e r|A> = -(H+E - E - i e ) _ 1 [ r V ] |A> ( 3 . 2 4 ) L A r ( H - E ) " 1 = ( H + E r - E ) _ 1 r - ( H + E ^ E ) " 1 [ rVj ( H - E ) " 1 ( 3 . 2 5 ) r|Ak> = 6 k r|A> - ( H + E r - E A - E k-ie ) _ 1 [ r V ] | A > ( 3 . 2 6 ) where r i s a meson a n n i h i l a t i o n c p e r a t c r a n d %here E , E , and X* A E h a v e b e e n d e f i n e d p r e v i o u s l y . The p r o o f s o f t h e s e i d e n t i t i e s a r e g i v e n i n A p p e n d i x A . 3 . 2 . 2 C u t k o s k y , O v e r l a p M a t r i x E l e m e n t s a n d D i a g r a m s T h e m a t r i x e l e m e n t s b e t w e e n C u t k o s k y s t a t e s a r e e v a l u a t e d i n A p p e n d i x B . A s i d e f r o m t h e g e n e r a l r e s u l t f o r t h e m a t r i x e l e m e n t [ A B | C D } , o n l y t h o s e m a t r i x e l e m e n t s w h i c h w i l l be n e e d e d t o c a l c u l a t e s i n g l e - and d o u b l e - s c a t t e r i n g w i l l be p r e s e n t e d b e l o w . T h e s i m p l e s t m a t r i x e l e m e n t i s t h a t b e t w e e n two u n e x c i t e d C u t k o s k y s t a t e s , | AB} and | C D } . I t c a n be w r i t t e n {AB|CD} = _ Z n <A|Q +R|C><B|R tQ|D>/n(R):n(Q)*. -Zn <B|Q tR |c><A|R +Q|D>/n(R)!n(Q)! R,Q ^ 1 ' ( 3 . 2 7 ) where R and Q r e p r e s e n t p r o d u c t s o f meson a n n i h i l a t i o n 23 operators. The unit operator i s included it the above summations. Equation (3.27) i s not an exact r e s u l t . The terns neglected above are these in which the cere cperatcrs i n a pa r t i c u l a r term cf the product AE are egual tc the cere operators i n a p a r t i c u l a r terra cf the product CD but the core operator from A i s not equal to the core operator from C cr the core operator from C . This type cf tern can be thought cf as describing the exchange of one or more pairs of tare nuclects and antinucleons between nucleons in the i n i t i a l state tc form the nucleons i n the f i n a l state. A more complete explanation cf the above i s given i n Appendix B . The above expansion of a Cutkosky matrix element w i l l te ca l l e d a meson exchange s e r i e s . The series i s ordered by the number of mesons exchanged, i.e., by the t o t a l number of nescc + operators in the product Q R. The n nescn exchange contribution to a matrix element w i l l be denoted by a bracketed superscript n. Thus the zero meson exchange contribution t c the natrix element (AE|CD } i s { A B | C D } ( 0 ) = <A|C><B|D> - < B | O < A | D > = 6 (3.28) A B , CD This can be represented diagramatically as B D The one-meson exchange c o n t r i b u t i o n i s { A B | C D } ( 1 ) = E <A|[Vq]|c><B|[qV]|D>(E c +E q -E A )~ 1 (E B +E q -E D ) -1 (3.29) + ant isym. This i s r e p r e s e n t e d d i a g r a m a t i c a l l y as B -p- D A C The terms i n equation (3.29) re p r e s e n t e d by •antisym.' are those necessary to make the r i g h t hand s i d e o f the equation antisymmetric with r e s p e c t to exchange of A and B or C and D. The meson 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 have been removed from equation (3.29) ty use of equation (3.25). The s o l i d l i n e s r e p r e s e n t p h y s i c a l nucleons and the dotted l i n e s r e p r e s e n t mesons. The diagrams are drawn with the i n i t i a l s t a t e on the r i g h t and the f i n a l s t a t e on the l e f t . For example, the f i r s t diagram f o r {AB|CD} ( 1 J corresponds to a nucleon i n s t a t e D e m i t t i n g a meson i n s t a t e q and becoming a nucleon i n s t a t e B. The meson i s then absorbed by a nucleon i n s t a t e C becoming a nucleon i n s t a t e A. There i s no r u l e f o r o b t a i n i n g the energy denominators from i n s p e c t i o n of the diagram since the same diagram can r e p r e s e n t d i f f e r e n t matrix elements with d i f f e r e n t energy denominators. I f one or both of the Cutkosky s t a t e s are e x c i t e d the r e s u l t s are {AB|CDk} (0) = 0 (3.30) 25 { A B | C D k } ( 1 ) = E <A|[Vq]|ck><B|[qV ]|D>(E c +E k +E q -E A ) " 1 (E B +E q -E D ) " 1 + E < A|[qV]|ck > < B i[Vq ] | D > ( E c + E k - E q - E A+i e ) " 1 ( E B - E q - E D ) " 1 + ant isym. B D B q D \ Q N j . ' A c£ C . . . . \k \ <3'31> \k { A B P | C D k } ( 0 ) = « k p 5 A B , C D p -k B D {ABp|CDk} ( 1 ) = 6 k p { A B J C D } ( 1 ) + other terms. p -k ( 3 . 3 2 ) B JP D ( 3 . 3 3 ) A C C The o t h e r t e r m s i n e q u a t i o n ( 3 . 3 3 ) a r e t h o s e i n w h i c h t h r e e d i s t i n c t m e s o n s , p , k r and q , a r e p r e s e n t 3 . 2 . 3 C u t k o s k y M a t r i x E l e m e n t s o f t h e V e r t e x O p e r a t o r a n d t h e H i E l i l t o n i a n T h e m a t r i x e l e m e n t s o f t h e v e r t e x o p e r a t o r b e t w e e n C u t k o s k y s t a t e s a r e g i v e n b y {AB|[mV]|CD}(0) = <5AC<B|[mV]|D> + antisym. 'AC B \ D 0 . 3 4 ) A C {AB|[mV]|CDk}(0) = 6AC<B|[mV]|Dk>+ antisym. m , k ( 3 . 3 5 ) D {AB|[mV]|CDk} ( 1 ) = Z <A|[qV]|c><B|[Vq](H+Eq-EB)"1[mV]|Dk>(EA+Eq-Ec)"1 + \ <A|[qV]|ck><B|[Vq](H+Eq-EB)-l[mV]|D>(EA+Eq-Ec-Ek-iE)~1 + Z <A| [Vq] |C><B| [mVl(H+E q-E D-E k-i e) _ 1[qV] | Dk> ( E ^ - E ^ " 1 + 2j <A| [Vq] |Ck><B| [mV] (H+E q-E D) _ 1 [qV] | D> ( E ^ E ^ - E ^ " 1 + antisym. ^ ' • S v B c~-—*J D B qs«sA> D c A — C A • C n k B D + B - - W * D 3 t 2 6 ) A (/ C A <- C \ s m {ABp|[mV]|cDk}(0) = 6 V n {AB|[mV]|CU}(0) + a n t i s y m . kp p 1 Jc B D ( 3 . 3 7 ) The heavy s o l i d l i n e indicates that the intermediate state i s to be summed over a complete set of states with baryon number one. 27 That i s , to evaluate the matrix elements i n equation (3.36), a complete set of s t a t e s with baryon number one must be i n s e r t e d i n t o the seconl matrix element f o r each term. Note t h a t i n the diagrams drawn, the o r d e r i n g of v e r t i c e s has no s i g n i f i c a n c e i f the v e r t i c e s are not d i r e c t l y connected by e i t h e r a dotted l i n e or a s o l i d l i n e . For example, i n the f i r s t diagram f o r equation (3.36), the v e r t i c e s r e p r e s e n t i n g the emission of the meson q and the a b s o r p t i o n o f the meson k are not d i r e c t l y connected and thus they need not be drawn as o c c u r i n g at the same i n s t a n t . In w r i t i n g matrix elements of the H a m i l t o n i a n , i t w i l l prove to be u s e f u l to write the Hamiltonian as H = H +H' (3.38) o where H i s d e f i n e d by o HjAB} = CEA+EB)|AB} (3. 39) HjABk} = (EA+EB+Ek) |ABk} (3.40) with s i m i l a r r e l a t i o n s f o r m u l t i p l y e x c i t e d Cutkosky s t a t e s . The above eq u a t i o n s can be w r i t t e n g e n e r a l l y as H |u} = E |U} (3.41) for any Cutkosky s t a t e |U}. Usinq equation (3.41), i t f o l l o w s that {U|HQ|W} - \ (EU+EW) {U|W} (3.42) 28 As i s shown below, the remainder H' has the very important property of having no zero-meson exchange contributions to i t s matrix elements. This separation of H w i l l be useful when evaluating approximations to the pion-deuteron T-matrix. The matrix elements of the Hamiltonian between Cutkosky states are given below. Note that the diagrams are drawn for the matrix elements of H• only. {AB|H|CD}(0) - \ ( E A + E B + E C + E D ) 6 A B ) C D { A B | H | C D } ( 1 ) = \ ( E A + E B + E C + E D ) { A B | C D } ( 1 ) \ E <A|[qV]|C><B|[Vq]|D>(_ A _ Q +  I | [  |0  [Vq  | D> ( E ^ E ^ ) " 1 + \ E <A| [Vq] |C><B| [qV] |D> ( E ^ E ^ ) " * 1 + antisym. (3 .44 ) B Q: D B p D A ^ c A * — C {AB|H[CDk}(0) = 0 (3 . 45) / 29 {AB|H|CDk}(1) = | (E A+E B+E c+E D+E k){AB|CDk} ( 1 ) + Z <A|[Vq]|C><B|[qV]|Dk> a±(E) + Z <A| [Vq] |0<B| [Vk] (a 2(E) (H+E^-E^ie)" 1^.(E) (H+Ek Eg+ie)" 1) [qV] |D> + Z <A|[qV]|C><B|[Vq](H-E -E -i £) - 1(H+E -E +i £) - 1[Vk]|D> a (E) +. antisyra. / D B c*—-p A d C A D C / + q V A ^—^> C (3 .46) {ABp|H|CDk}(1) = 5 , {AB|CH+Ek)|CD}(1) + other terms pk P- k B D (3 .47 ) A —d—^ C The f a c t o r s a ± ( E ) i = 1 , . . . , 4 i n e q u a t i o n ( 3 . 4 6 ) r e p r e s e n t f u n c t i o n s o f t h e e n e r g i e s o f t h e p a r t i c l e s i n v o l v e d . T h e o t h e r t e r m s i n e q u a t i o n (3 .47) a r e t h o s e c o n t a i n i n g t h r e e mesons p , k r and t h e e x c h a n g e d meson q . M a t r i x e l e m e n t s b e t w e e n o r t h o n o r m a l C u t k o s k y s t a t e s c a n be c a l c u l a t e d i n t e r m s o f t h e a b o v e m a t r i x e l e m e n t s u s i n g e q u a t i o n s ( 3 . 9 ) , ( 3 . 1 1 ) , a n d ( 3 . 1 6 ) . One i m p o r t a n t r e s u l t i s t h e m a t r i x e l e m e n t o f t h e H a m i l t o n i a n b e t w e e n two o r t h o n o r m a l C u t k o s k y s t a t e s . From e q u a t i o n s (3 .38 ) a n d (3 .42 ) i t i s e v i d e n t t h a t t h e m a t r i x e l e m e n t s o f H b e t w e e n n o n o r t h o n o r m a l s t a t e s may be w r i t t e n 30 {U|H|W} = | ( E u + E w )(6 u w + G u w ) +• {U|H'|W> (3 .48) where G uw i s d e f i n e d by e g u a t i o n ( 3 . 1 1 ) . U s i n g e q u a t i o n s (3 .9 ) and (3 .16) i t f o l l o w s t h a t (V|H|X) = | u E w F * v il+G)w ( E u + E w ) (V|H'|X) = iu^w W)t < ^ u w (VV (V'H'|X) = E 6 + (VIH 1 |X) V vx 1 1 (3 .49) T h u s Hq i s d i a g o n a l i n t h e o r t h o n o r m a l C u t k o s k y s t a t e s a n d H' h a s no z e r o meson e x c h a n g e c o n t r i b u t i o n t o i t s m a t r i x e l e m e n t . 3 .3 M e s o n E x c h a n g e S e r i e s f o r P i o n - D e u t e r o n S c a t t e r i n g T h e p i o n - d e u t e r o n T - m a t r i x g i v e n b y e q u a t i o n ( 2 . 5 4 ) c a n b e e v a l u a t e d b y i n s e r t i n g c o m p l e t e s e t s o f o r t h o n o r m a l C u t k o s k y s t a t e s a n d e v a l u a t i n g t h e r e s u l t i n g m a t r i x e l e m e n t s u s i n g t h e e x p r e s s i o n s g i v e n i n s e c t i o n s 3 .1 a n d 3 . 2 . I n o r d e r t o p r o c e e d f r o m t h i s p o i n t i t i s n e c e s s a r y t o e x p r e s s (U | C> and (U|Pk>, w h e r e |U) i s a n o r t h o n o r m a l C u t k o s k y s t a t e , i n t e r m s o f known d e u t e r o n w a v e f u n c t i o n s . T h i s i s d o n e i n t h e n e x t two s u b s e c t i o n s . 3 . 3 . 1 The D e u t e r o n S t a t e V e c t o r The d e u t e r o n s t a t e v e c t o r \V> c a n be e x p a n d e d i n t e r m s o f the o r t h o n o r m a l C u t k o s k y s t a t e s u s i n g e q u a t i o n (3 .9 ) V> = E | U ) ( U | P > (3 .50) t 31 where (U|P> i s t h e d e u t e r o n w a v e f u n c t i o n i n t h e o r t h o n o r m a l C u t k o s k y r e p r e s e n t a t i o n . T h e c o m p l e t e d e u t e r o n w a v e f u n c t i o n (U|P> i n c o r p o r a t e s t h e meson d e g r e e s o f f r e e d o m i m p l i c i t l y . H o w e v e r , known d e u t e r o n w a v e f u n c t i o n s d c n e t i n c l u d e a l l t h e s e d e g r e e s o f f r e e d o m . T h u s a n e x p r e s s i o n f o r t h e e x a c t d e u t e r o n w a v e f u n c t i o n i s n e e d e d i n t e r m s o f a n a p p r o x i m a t e d e u t e r o n w a v e f u n c t i o n w h i c h h o p e f u l l y i s k n o w n . T h i s e x p r e s s i o n s h o u l d t a k e i n t o a c c o u n t e x p l i c i t l y t h e meson d e g r e e s c f f r e e d o m i n t h e c o m p l e t e d e u t e r o n w a v e f u n c t i o n . A t t h i s p o i n t , p r o j e c t i o n o p e r a t o r s a r e d e f i n e d f o r t h e u n e x c i t e d a n d e x c i t e d o r t h o n o r m a l C u t k o s k y s t a t e s . T h e s e p r o j e c t i o n o p e r a t o r s a r e d e n o t e d by P a n d P ' r e s p e c t i v e l y . A s t a t e v e c t o r |PQ> i s d e f i n e d w h i c h s a t i s f i e s t h e f o l l o w i n g • a q u a t i o n s P H P . |P 0 > = E P | P 0 > ( 3 . 5 1 ) P 1 1P Q > = 0 (3 . 52) By w r i t i n q t h e d e u t e r o n s t a t e v e c t o r a s \V> = | P Q > + \V±> ( 3 . 53) a n e x p r e s s i o n c a n be o b t a i n e d f o r t h e r e m a i n d e r \V^> . U s i n g t h e f a c t t h a t P + P ' = 1 , t h e e i g e n v a l u e e q u a t i o n f o r H f o r t h e d e u t e r o n s t a t e may be w r i t t e n (P+P')H(P+P')\V> = E J P > (3 .54 ) 32 U s i n g e q u a t i o n s ( 3 . 5 1 ) - (3 .53) t h e a b o v e e q u a t i o n c a n be w r i t t e n Il|t?1> + P'HP|PQ> = EV\VX> (3 .55 ) By i n v e r t i n g t h e o p e r a t o r (H-Ep ) an e x p r e s s i o n i s o b t a i n e d f o r f P ^ . T h u s f r o m e q u a t i o n (3 .53 ) t h e d e u t e r o n s t a t e v e c t o r may be w r i t t e n \V> = {l-(U-Ev)~h'KP}\VQ> ( 3 .56 ) U s i n g e q u a t i o n ( 3 . 3 8 ) t h e o p e r a t o r ( H - E )-* may be e x p a n d e d t o g i v e t h e f o l l o w i n g e x p r e s s i o n f o r \t> > \v> t i- nl 0 [ ( W" l H' ] n ( Ho - V " l p , H P } | V (3-57> T h e a b o v e e q u a t i o n e x p r e s s e s t h e e x a c t d e u t e r o n s t a t e v e c t o r i n t e r m s o f an a p p r o x i m a t e d e u t e r c n s t a t e v e c t o r s a t i s f y i n q e q u a t i o n s (3 .51) a n d . ( 3 . 5 2 ) . S i n c e m a t r i x e l e m e n t s o f H» do n o t h a v e any z e r o - m e s o n e x c h a n g e t e r m s , t h e a b o v e e x p r e s s i o n c a n be e v a l u a t e d t o any o r d e r o f meson e x c h a n g e . N o t e t h a t s i n c e H i s d i a g o n a l i n t h e o r t h o n o r m a l C u t k o s k y s t a t e s P'HP = P ' H ' P (3 . 58) One m i g h t w i s h t o i d e n t i f y (U|p > w i t h t h e c o n v e n t i o n a l o d e u t e r o n w a v e f u n c t i o n i n momentum s p a c e . A l t h o u g h t h i s i s a r a t h e r a r b i t r a r y a s s u m p t i o n , i t d o e s seem r e a s o n a b l e . The 3 3 c o n v e n t i o n a l d e u t e r o n w a v e f u n c t i o n d e s c r i b e s a s y s t e m c o m p o s e d o f o n l y two n u c l e o n s , t h e m e s o n s n o t t a k e n i n t o a c c o u n t e x p l i c i t l y . T h e u n e x c i t e d o r t h o n o r m a l C u t k o s k y s t a t e s c o r r e s p o n d t o o r t h o n o r m a l s t a t e s w h i c h , i n t h e z e r o - m e s o n e x c h a n g e a p p r o x i m a t i o n , c o n s i s t o f two n c n i n t e r a c t i n g n u c l e o n s w i t h no mesons e x p l i c i t l y p r e s e n t . T h e p r o p e r t i e s o f \VQ > s t a t e d i n a q u a t i o n s ( 3 . 5 2 ) and ( 3 . 5 3 ) t h u s c o r r e s p o n d t o t h e d e s i r e d p r o p e r t i e s o f t h e c o n v e n t i o n a l d e u t e r o n w a v e f u n c t i o n . T h a t i s , | P Q > i s n o t c o u p l e d t o s t a t e s c o n s i s t i n g c f mere t h a n two n u c l e o n s ( i n the z e r o - t n e s o n e x c h a n g e a p p r o x i m a t i o n ) and | P Q > i s an e i g e n s t a t e w i t h t h e c o r r e c t e i g e n v a l u e o f t h e H a m i l t o n i a n i n the r e d u c e d s p a c e o f t h e u n e x c i t e d o r t h o n o r m a l C u t k o s k y s t a t e s . U s i n g e q u a t i o n ( 3 . 5 7 ) d i a g r a m s c a n be d r a w n r e l a t i n g t h e d e u t e r o n s t a t e \V> t o t h e a p p r o x i m a t e d e u t e r o n s t a t e \QQ> . F o r e x a m p l e ( C D | P > c a n be d r a w n a s V ~D C V, + ... The f i r s t t e r m - a b o v e r e p r e s e n t s t h e z e r o - m e s o n e x c h a n g e c o n t r i b u t i o n a n d t h e t e r m s n e g l e c t e d r e p r e s e n t m u l t i p l e - m e s o n e x c h a n g e c o n t r i b u t i o n s . T h e o n e - m e s o n e x c h a n g e c o n t r i b u t i o n v a n i s h e s . T h e o n e - m e s o n e x c h a n g e a p p r o x i m a t i o n t o (CDk |P> c a n be d r a w n a s k ^ D k C V D C V, D C V, D 7 I V, 34 3 . 3 . 2 The P i o n - D e ute r o n S c a t t e r i n g S t a t e V e c t o r The p i o n - d e u t e r o n s c a t t e r i n g s t a t e v e c t c r c a n ba e x p a n d e d i n t e r m s of t h e o r t h o n o r m a l C u t k o s k y s t a t e s u s i n g e q u a t i o n (3 .9 ) \Vk> = E|u)(u|Pk> < 3- 5 9 ) S i n c e t h e w a v e f u n c t i o n (u|Pk> i s n o t k n o w n , e q u a t i o n ( 3 . 5 9 ) i s w r i t t e n i n t h e f o r m of a s e r i e s , t h e f i r s t t e r m o f w h i c h i s o b t a i n e d f r o m t h e i m p u l s e a p p r o x i m a t i o n . The i m p u l s e a p p r o x i m a t i o n a s s u m e s t h a t t h e s c a t t e r i n g t a k e s p l a c e s o r a p i d l y t h a t i t c a n be a p p r o x i m a t e d by a p i o n s c a t t e r i n g f r o m a f r e e n u c l e o n . T h u s the a m p l i t u d e f o r f i n d i n g a p a r t i c u l a r p i o n - t w o -n u c l e o n s t a t e i n t h e d e c o m p o s i t i o n o f t h e p i o n - d e u t e r o n s c a t t e r i n g s t a t e i s a p p r o x i m a t e d by t h e a m p l i t u d e f o r f i n d i n g t h a t t w o - n u c l e o n s t a t e i n t h e d e c o m p c s i t i c n o f t h e d e u t e r o n s t a t e . T h u s e q u a t i o n ( 3 . 5 9 ) c a n be w r i t t e n \Vk> = E|Uk)(u|fl> + |S> (3 .60 ) where t h e f i r s t t e r m r e p r e s e n t s t h e i m p u l s e a p p r o x i m a t i o n t o t h e s t a t e |Pk> a n d |S> r e p r e s e n t s t h e r e m a i n d e r . U s i n g t h e o p e r a t o r + K d e f i n e d i n e g . ( 3 . 1 8 ) , e q u a t i o n (3 .60) c a n be w r i t t e n |Pk> = EK+|U)(U|P> + |S> (3.61) o r i n s h o r t e n e d n o t a t i o n |Pk> = Kf\V> + \S> 35 ( 3 . 6 2 ) where the presence of the unit operator has been assumed. An expression f c r |S> can be obtained from the eigenvalue equation f o r H for the pion-deuteron scattering state, equation ( 2 . 3 8 ) , by substituting for \Vk> from equation ( 3 , 6 2 ) giving the eqnation (H-Eg)|S> = (-EKf+EsKf)\V> ( 3 . 6 3 ) where E s = E P + Ek ( 3 . 6 4 ) By inverting (H-E ) with outgoing wave boundary conditions, the s expression f o r |s> becomes * |S> = (E G-H+ie)~ 1(HK t-E gK t)|t?> ( 3 . 6 5 ) Using equation ( 3 . 3 8 ) and writing (Eg-H+ie)"1 = JnUVV^'^'^VV1^"1 ( 3 . 6 6 ) the pion-deuteron scattering state may be written \Vk> = Kf\V> + ? n[(E -H +i e)" 1U ,] n(E -H +ie)' 1(HK t-E K +)|P> ( 3 . 6 7 ) i • n—U s o s o ° This may be put i n a more convenient form by noting that (HK+-EsKf)|P> = H'Kt|P> +llQKf\V> -EvKf\V> -Ek<+\V> = U'Kf\V> + E (Eu+Ek)K+|u)(u|P> -KfR\V> -EjK+\V> = H'K+|P> + E /C+HQ|U)(U|P> -K+H|fl> = (H'K+ - K+H')|P> Thus the pion-deuteron scattering state may be written |Pk> = Kf\V> + E [(Es-Ho+i£)"1H,]n(Es-Ho+ie)"1(H'Kt-KtH')|t?> ( 3 . 6 9 ) n=0 The state \V> i s known from section 3 . 3 . 1 and thus the pion-deuteron s c a t t e r i n g state can be evaluated i n terms of the approximate deuteron state |PQ> whose wavefunction i s assumed to be known. The above equation appears to resemble the expression f c r the pion-deuteron scattering state derived i n section 2 . 3 . 2 . However they are quite d i f f e r e n t . Equation ( 2 . 4 2 ) separates the pion-deuteron state into an asymptotically free state and a scatterinq state. That type of expansion i s net p a r t i c u l a r l y useful for obtaining approximations to the pion-deuteron state. On the other hand., equation ( 3 . 6 2 ) contains more than just the asymptotically free state. I t also contains the i n t e r a c t i o n of the meson k with the nucleons i n the Cutkosky state | U ) . For example, the state K^|AB> , i n the zero - meson exchanqe approximation, i s equal to the state |ABk} which according to equation ( 3 . 5 ) contains the i n t e r a c t i o n of the meson k with both physical nucleons A and B. Thus the expansion of equation ( 3 . 6 9 ) should be useful in approximating the pion-deuteron state. 3 6 ( 3 . 6 8 ) 37 3 . 3 . 3 The P i o n - D a u t a r o n T - m a t r i x H a v i n g o b t a i n e d e x p r e s s i o n s f c r t h e d e u t e r o n s t a t e a n d t h e p i o n - d e u t e r o n s c a t t e r i n g s t a t e i n t h e p r e v i o u s two s u b s e c t i o n s , t h e p i o n - d e u t e r o n T - m a t r i x c a n be w r i t t e n s o l e l y i n t e r m s o f m a t r i x e l e m e n t s b e t w e e n C u t k o s k y s t a t e s a n d a p p r o x i m a t e d e u t e r o n wavef u n c t i o n s . U s i n g e q u a t i o n s (2 .54 ) , ( 3 . 5 7 ) a n d (3 .69) t h e T -m a t r i x becomes • <»:ifra,',.iJ"".-v"1,',W1 X {Kf+ 1,[(E - H + i e ) - 1 H ' ] n C E - H + i £ ) - 1 ( H ' » : t - t C t H ' ) } K ( n = u s o s> <J i x K!o[(vv"lH']n(Ho-v"lp,H'p}iv <3-7o> The a b o v e e x p r e s s i o n i s e x a c t . No a p p r o x i m a t i o n s h a v e b e e n made i n w r i t i n g t h e a b o v e e q u a t i o n . I n o r d e r t o e v a l u a t e the T - m a t r i x many a s s u m p t i o n s w i l l be made. T h e i n f i n i t e s e r i e s w i l l a l l be t r u n c a t e d t o t a k e i n t o a c c o u n t no more t h a n c n e - m e s c n e x c h a n g e t e r m s f o r t h e T - m a t r i x . I n t h e n e x t c h a p t e r t h e z e r o - a n d o n e -meson e x c h a n g e t e r m s o f t h e T - m a t r i x w i l l be c a l c u l a t e d a n d d i s c u s s e d . 38 4 T h e M e s o n E x c h a n q e S e r i e s F o r T h e T - m a t r i x 4 • 1 2 e r o - M e s o n E x c h i ng_e C o n t r i b u t i o n to t h e T - m a t r i x The e x p r e s s i o n f o r t h e p i c n - d e u t e r o n T - m a t r i x i s e v a l u a t e d by i n s e r t i n g c o m p l e t e s e t s o f o r t h o n o r m a l C u t k o s k y s t a t e s i n t o e q u a t i o n ( 3 . 7 0 ) t h e n w r i t i n g t h e r e s u l t i n g m a t r i x e l e m e n t s a s meson e x c h a n g e s e r i e s . T h e r e i s o n l y one z e r o - m e s o n e x c h a n g e t e r m i n e q u a t i o n ( 3 . 7 0 ) . I t c a n be w r i t t e n $1.1* - £u,w <lmul[*)»<>)<°V|V «"-1> U s i n g e q u a t i o n ( 3 . 5 2 ) , t h e a b o v e e q u a t i o n c a n be w r i t t e n The n o r m a l i z a t i o n f a c t o r o f 1/4 t a k e s i n t o a c c o u n t t h e f a c t t h a t |AB} =- |BA} a n d | C D ] =- | D C } . U s i n q t h e e q u a t i o n s of s e c t i o n 3.1 a n d 3 . 2 . 2 , « ( A B | [ r a V ] | C D k ) ^ - ZU f M r-*v% { u | [ m V ] | W } < ° > = {AB| [mV] | C D k } ( 0 ) ( 4 ' 3 ) U s i n g e q u a t i o n s ( 3 . 3 5 ) a n d ( 4 . 3 ) , e q u a t i o n (4 .2 ) c a n be w r i t t e n T£i,(fc - ^ A , B , C < P o l A B ) <*\W]]^> (CB|V ( U . U ) 39 The a b o v e e q u a t i o n i s d r a w n b e l o w . °0 V, E q u a t i o n (4 .4 ) c a n be r e w r i t t e n i n t e r m s o f t h e c o n v e n t i o n a l d e u t e r o n momentum s p a c e w a v e f u n c t i o n a n d p i o n n u c l e o n T - m a t r i c e s . T h i s i s d o n e i n A p p e n d i x D . T h e r e s u l t i n t h e l a b o r a t o r y f r a m e i s e x p r e s s e d b e l o w T L 0 ) Q S p t , m ; 0 , k ; M' ,M) 6 (kp,-hn-k) (4 .5 ) +1 • I i The w a v e f u n c t i o n ^ OO i s j u s t t h e c o m p o n e n t o f t h e c o n v e n t i o n a l d e u t e r o n w a v a f u n c t i o n w i t h t o t a l s p i n p r e j e c t i o n M a n d t o t a l n u c l e o n s p i n p r o j e c t i o n I . T h e T - m a t r i c e s T ^ ^ a n d T ™ ^ r e p r e s e n t a v e r a g e s o v e r n u c l e o n s p i n o r i e n t a t i o n s o f p i o n - p r o t o n and p i o n - n e u t r o n T - m a t r i c e s , r e s p e c t i v e l y . T h e e x p l i c i t e x p r e s s i o n s f o r <f>^  (jO, T £ ? £ a n d a r e g i v e n i n A p p e n d i x D . T h u s t h e z e r o - m a s o n e x c h a n g e c o n t r i b u t i o n t o t h e T - m a t r i x c o r r e s p o n d s t o t h e u s u a l s i n g l e s c a t t e r i n g c c n t r i b u t i c n t o p i o n -d e u t e r o n e l a s t i c s c a t t e r i n g . A f a i r l y e x t e n s i v e n u m e r i c a l s t u d y o f t h i s c o n t r i b u t i o n has t e e n p e r f o r m e d by M c M i l l a n a n d L a n d a u (1974), uo U . 2 O n e - M e s o n E x c h a n g e C o n t r i b u t i o n t o t h e T - m a t r i x T h e o n e - m e s o n e x c h a n g e c c n t r i b u t i c n t o t h e p i c n - d e u t e r o n T -m a t r i x g i v e n i n a q u a t i o n (3 .70 ) c a n be w r i t t e n v",Pk = W> ) (u|[nV,Kt|w)a)(w|V - I L < t ' ' | U ) ( U | H ' | X ) < 1 > ( X | [ m V ] K t | w ) ( o ) ( M | I ' o > ( E x - E p , r 1 e x c i t e d + [ n I x<t/|u)(u | [ m V]|x ) ( ° ^ X ^ ^ " ^ u , w ^ x < t ? l | u ) ( u | t m V ^ + l x ) ( 0 ) ( X I H ' l w ) ( 1 ) (w| V^x-V1 e x c i t e d (4 .6 ) T h e f o u r t h t e r m a b o v e c a n be r e w r i t t e n by n o t i n g K + ( E X " V _ 1 | X ) = " ( E S " E X + i G > " l K + l X ) ( 4 .7 ) The t h i r d a n d f o u r t h t e r m s i n e q u a t i o n (4 .6 ) c a n now be c o m b i n e d t o g i v e t h e f o l l o w i n g e x p r e s s i o n f o r t h e T - m a t r i x vV-V0^"i["viKtiw>(1)(wiv - l„ u L VluXulH'IXJ^CxItmVl/C+lHj^CWlP >(E - B p , ) " 1 U j W A O e x c i t e d + I U ) W I xV|u)(u| [ m V]|x) ( o )(x| H ' K +|w) ( 1 )(w| t ? o > ( E s - E x + i c ) - ^U,W ^ X < 0>)(u| [ m V ] K +|x) ( o )(x| H'|w) ( 1 )( W|P o > ( E s - E x + i e ) unexci ted { ^ Q ) U s i n q e q u a t i o n (3.52) t h s sums o v e r t h e s t a t e s | U} a n d |w] may be r e w r i t t e n y i e l d i n g 4 1 T ^ , P k = T WO^ '^V - { ( A B | [ m V ] | C D k ) ( 1 ) - J x ( A B | H ' | x ) ( 1 ) ( x | [ m V ] | C D k ) ( 0 ) ( E x - E I ? , ) " 1 excited + J x ( A B | [ m V ] | x ) ( o ) ( x | H ' | C D k ) ( 1 ) ( E s - E x + i e ) _ 1 - Xx ( A B | [ m V ] | x k ) ( o ) ( x | H ' | C D ) ( 1 ) ( E s - E x + i e ) - 1 } unexcited (4 .9 ) As e x p l a i n e d i n t h e p r e v i o u s s e c t i o n , t h e f a c t o r o f 1/4 i s a n o r m a l i z a t i o n f a c t o r r e s u l t i n g f r o m t h e a n t i s y m m e t r y o f t h e C u t k o s k y s t a t e s . T o p r o c e e d , t h e m a t r i x e l e m e n t s b e t w e e n t h e o r t h o n o r m a l C u t k o s k y s t a t e s a r e e x p r e s s e d i n t e r m s c f m a t r i x e l e m e n t s b e t w e e n t h e n o n - o r t h o n o r m a l s t a t e s . T h e z e r o - m e s o n e x c h a n g e m a t r i x e l e m e n t s i n e q u a t i o n ( u . 9 ) a r e e a s i l y e x p r e s s e d i n t e r m s o f t h e n o n - o r t h o n o r m a l s t a t e s a s was d o n e i n e q u a t i o n ( 4 . 3 ) . T h e same t e c h n i q u e c a n be u s e d f o r t h e o n e - m e s c n e x c h a n g e m a t r i x e l e m e n t s o f H ' , s i n c e m a t r i x e l e m e n t s o f H* do n o t h a v e a n y z e r o - m e s o n e x c h a n g e c o n t r i b u t i o n s . H o w e v e r , s i n c e m a t r i x e l e m e n t s o f t h e v e r t e x o p e r a t o r do h a v e z e r o - m e s o n e x c h a n g e c o n t r i b u t i o n s , i t i s more c o m p l i c a t e d t c e x p r e s s t h e o n e - m e s o n e x c h a n g e m a t r i x e l e m e n t o f t h e v e r t e x o p e r a t o r b e t w e e n o r t h o n o r m a l s t a t e s i n t e r m s c f m a t r i x e l e m e n t s b e t w e e n n o n -o r t h o n o r m a l s t a t e s . U s i n g e q u a t i o n (3 .9 ) t h e m a t r i x e l e m e n t o f the v e r t e x o p e r a t o r may be w r i t t e n (AB| [mV] |CDk) - I U ) W F j ) A B {U| [mV] |.W) <li. 10) 42 The o n e - m e s o n e x c h a n g e c o n t r i b u t i o n t c t h e a b o v e m a t r i x e l e m e n t i s o b t a i n e d b y w r i t i n g e a c h o f t h e t h r e e m a t r i x e l e m e n t s o n t h e r i g h t h a n d s i d e as a sum o f z e r o - a n d c n e - m e s o n e x c h a n g e c o n t r i b u t i o n s t h e n t a k i n g t h o s e p r o d u c t s w h i c h y i e l d a n e t o n e -meson e x c h a n g e . T h a t i s . U s i n g e q u a t i o n s (3.12) a n d ( 3 . 1 6 ) , i t f o l l o w s t h a t ,CDk ,CDk T h u s e q u a t i o n (4.11) b e c o m e s (AB | [mV] | CDk) ( 1 ) = {AB | [mV] | CDk} ( 1 ) (4. 11) F ( o > = 6 U,W UW Fu\w = - | { u l w } ( 1 > ( 4 * 1 2 ) - \ I {AB|U} ( 1 ) {u|[mV]|CDk} ( 0 ) U (4.13) {AB|[mV]|w}(o) { W|CDk} ( 1 ) 2 W 43 From the above discussion, i t follows that T P V P k = I WPJAB)(CDIV {fAB|[mV]|CDk} ( 1 ) - • | l u { A B | u } ( 1 ) { u [ [ m V ] | C D k } ( 0 ) - | Z u { A B | [ m V ] | u } ( o ) { u | C D k } ( 1 ) - { A B | H 1 | u } ( 1 ) { u | [ m V ] | C D k } ( o ) ( E u - E I ? I ) - 1 e x c i t e d + ^ { A B l f m V l l u l ^ ^ u l H ' I C D k l ^ ^ E p + E ^ E y + i e ) " 1 - {AB|[mV]|uk} (° ){u|H'|CD} ( 1 )(E p+E k-E u+i S)" 1} un e x c i t e d (4 . IH ) In order to evaluate the T-matrix as expressed above, i t w i l l be necessary to truncate the summations over the intermediate states |U}. This w i l l be done by disregarding a l l terms i n which more than three mesons are present. That i s , the only terms retained i n the above summations w i l l be those which contain the i n i t i a l and f i n a l mesons, l a b e l l e d by k and m respectively, and one exchanged meson, l a b e l l e d by q. Whan truncatinq the above summations, one must be c a r e f u l to include a l l terms which contain three mesons. For example, i n the second and th i r d terms, the summation should include those states |U} which are sinqly excited as well as unexcited Cutkosky states. If | U }= | FGp } i n the second term, then by using equation (3.37) i t follows that {AB|FGp} ( 1 ){FGp|[mV]|CDk} ( o ) = {AB|FGk} ( 1 )(FG|[mV]|CDk} ( o ) (4. 15) + terms w i t h 4 mesons present S i m i l a r i l y , i f |U)=|FGpJ in the third term, then by using 44 equation (3.33) i t fol lows that {AB|[mV]|FGp} ( o ){FGp|CDk} ( 1 ) = {AB|[raV]|FGk} ( o ){FG|CD} ( 1 ) (4.16) + terms w i t h 4 mesons present When |U}=|FGp] in the fourth term using equation (3.37) w i l l give {AB|H»|FGp} ( 1 ){FGp|[mV]|CDk} ( o ) = {AB|H'|FGk} ( 1 ){FG|[mV]|CD} ( o ) ( ^ 1 ? ) + terms w i t h 4 mesons present The f i f t h term w i l l a lso have some terms coming from |U )=|FGp } s ince by using equation (3.47) {AB|[mV]|FGp} ( o ){FGp|H'|CDk} ( 1 ) = {AB|[raV]|FGk} ( o ) {FG|H'|CD} ( 1 ) (4. 18) + terms w i t h 4 mesons present Note that the part of the f i f t h term written above exact ly cancels the s ix th term of equation (4.14). Thus, keepinq only those terms contain ing three mesons, the T-matrix in equation (4.14) can be written T P X P k = \ WOAB)(C°l V { <AB| [ « V ] |CDk} ( 1 ) |FG} ( 1 ){FG|[mV]|CDk} ( o ) |FGk} ( 1 ){FG|[mV]|CD} ( o ) [raV]|FG} ( o ){FGJCDk} ( 1 ) [mV]|FGk} ( o ){FG|CD} ( 1 ) " J ^ F , G { A B - 1 £ F , G { A B + H , G { A B H' |FGk} ( 1 ){FG| [mV] | CD} ( ° } ( E p + E ^ - E ^ , ) _ 1 ( 4 ' 1 9 ) [ m V ] | F G } ( o ) { F G | H ' | C D k } ( 1 ) ( E p + E k - E F - E G + i £ ) " 1 | 45 T h a T - m a t r i x c a n now be e x p r e s s e d i n t e r m s o f o n e - n u c l e o n m a t r i x e l e m e n t s u s i n g th -3 e q u a t i o n s o f s u b s e c t i o n s 3 . 2 . 2 a n d 3 . 2 . 3 . The f i n a l r e s u l t c a n be w r i t t e n i n t h e f o r m TP'm,Pk " T d i f f + T d i f f + Ttame + Tsame ( 4 . 20) The t e r m s o n t h e r i g h t h a n d s i d e o f t h e a b o v e e q u a t i o n a r e e x p r e s s e d i n t e r m s o f o n e - n u c l e c n m a t r i x e l e m e n t s i n A p p e n d i x C . T h e t e r m s d e n o t e d by T ^ i f f r e p r e s e n t p r o c e s s e s where t h e i n i t i a l meson i s a b s o r b e d on one n u c l e o n o f t h e d e u t e r o n , t h e f i n a l meson i s e m i t t e d by t h e o t h e r n u c l e c n o f t h a d e u t e r o n , a n d t h e e x c h a n g e d meson i s e m i t t e d b y t h e f i r s t n u c l e o n and a b s o r b e d by t h e s e c o n d n u c l e o n . T h e s u b s c r i p t " d i f f " i n d i c a t e s t h a t t h e i n i t i a l a n d f i n a l mesons a r e a b s o r b e d and e m i t t e d by d i f f e r e n t n u c l e o n s . T h e r e a r e f o u r t e r m s r e p r e s e n t e d b y T ^ i f f a E t n e e x c h a n g e d meson c a n be e m i t t e d b e f o r e o r a f t e r t h e i n i t i a l meson i s a b s o r b e d and t h e n c a n be a b s o r b e d c n t h e s e c o n d n u c l e o n e i t h e r b e f o r e o r a f t e r t h e f i n a l meson i s e m i t t e d . One o f t h e f o u r t e r m s o f T ^ i f f i s d r a w n b e l o w . The s i n g l e s o l i d l i n e s d e n o t e p h y s i c a l n u c l e o n s a n d t h e d a s h e d l i n e s d e n o t e m e s o n s . The d o u b l e s o l i d l i n e s d e n o t e t h e d e u t e r o n . T h e d i a g r a m i s d r a w n w i t h t h e i n i t i a l s t a t e on t h e l e f t and t h e f i n a l s t a t e on t h e r i g h t . T h e d i a g r a m d r a w n a b o v e c o r r e s p o n d s t o t h e s e c o n d t e r m i n e q u a t i o n ( C . 2 ) . 46 T h e t e r m s d e n o t e d by T ^ f f a r e s i m i l a r t o t h e t e r m s d e n o t e d by T ^ ± f f e x c e p t t h a t t h e t e r m s o f T d i f f h a v e t h e e x c h a n g e d meson e m i t t e d by t h e s e c o n d n u c l e o n a n d a b s o r b e d b y t h e f i r s t n u c l e o n . A s a b o v e , t h e r e a r e f o u r t e r m s r e p r e s e n t e d by T <jiff a n ^ o n e o r " t h e s e t e r m s i s drawn b e l o w . \ V, /m The d i a g r a m c o r r e s p o n d s t o t h e s e c o n d t e r m o f e q u a t i o n ( C . 3 ) The t e r m s d e n o t e d b y T * r e p r e s e n t p r o c e s s e s w h e r e t h e f i n a l meson i s e m i t t e d by t h e same n u c l e c n w h i c h a b s o r b s t h e i n i t i a l meson and t h e e x c h a n g e d meson i s e m i t t e d by t h e f i r s t n u c l e o n a n d a b s o r b e d by t h e s e c o n d n u c l e c n . T h e r e a r e s i x t e r m s r e p r e s e n t e d b y T"*" c o r r e s p o n d i n g t o t h e s i x d i f f e r e n t S3.ni c o r d e r i n g s o f t h e a b s o r p t i o n o f t h e i n i t i a l meson a n d t h e e m i s s i o n o f t h e e x c h a n g e d and f i n a l m e s o n s . One o f t h e s i x t e r m s o f T i s drawn b e l o w , same » 0 M H k / — b - S - r O ^ — The d i a g r a m c o r r e s p o n d s t o t h e f o u r t h t e r m o f e q u a t i o n ( C . 4 ) . The t e r m s d e n o t e d by T~ a r e s i m i l a r to t h e t e r m s d e n o t e d same by T e x c e p t t h a t t h e t e r m s c f T c h a v e t h e e x c h a n g e d meson same * same e m i t t e d by t h e s e c o n d n u c l e o n and a b s o r b e d by t h e f i r s t n u c l e o n . a n d one o f As a b o v e , t h e r e a r e s i x t e r m s r e p r e s e n t e d b y T same t h e s e t e r m s i s d r a w n b e l o w . ' k ix-V, 47 The d i a g r a m c o r r e s p o n d s t o t h e f o u r t h t e r m o f e q u a t i o n ( C . 5 ) . I t i s w o r t h w h i l e a t t h i s p o i n t t o o u t l i n e t h e a p p r o x i m a t i o n s w h i c h h a v e b e e n made i n e x p r e s s i n g t h e p i o n -d e u t e r o n T - m a t r i x i n t h e f o r m g i v e n by e g u a t i o n ( 4 . 2 0 ) a n d A p p e n d i x C . When m a t r i x e l e m e n t s b e t w e e n C u t k o s k y s t a t e s were e v a l u a t e d i n s u b s e c t i o n s 3 . 2 . 2 and 3 . 2 . 3 , a l l c o r e e x c h a n g e t e r m s were n e g l e c t e d . T h a t i s , a l l t e r m s w h i c h c o u l d n e t be e x p r e s s e d i n t e r m s o f o n e - n u c l e o n m a t r i x e l e m e n t s were a s s u m e d t o be n e g l i g i b l e . A l s o , t h e e x p r e s s i o n s f o r t h e m a t r i x e l e m e n t s o f t h e H a m i l t o n i a n and t h e v e r t e x o p e r a t o r n e g l e c t e d t h e m e s o n - m e s o n i n t e r a c t i o n , V 1 , and any t e r m s i n V w h i c h were n o t l i n e a r i n t h e meson c r e a t i o n o r a n n i h i l a t i o n o p e r a t o r s , A m a j o r a s s u m p t i o n made i n w r i t i n g e q u a t i o n (4 ,20 ) was t h e n e q l e c t i n g o f a l l t e r m s i n w h i c h more t h a n t h r e e mesons were p r e s e n t . T h i s was d o n e f i r s t l y by t a k i n g t h e c n e - m e s o n e x c h a n g e a p p r o x i m a t i o n t o a l l t e r m s i n t h e T - r a a t r i x as g i v e n by e g u a t i o n (3 .67) a n d s e c o n d l y by t r u n c a t i n g a l l t h e s u m m a t i o n s o v e r i n t e r m e d i a t e s t a t e s s u c h t h a t o n l y t h r e e m e s o n s were p r e s e n t i n t h e f i n a l e x p r e s s i o n f o r t h e T - m a t r i x . T h e r e i s a n o t h e r m a j o r a s s u m p t i o n w h i c h must be made i f e q u a t i o n (4 .20) i s t o be i d e n t i f i e d w i t h t h e p i c n - d e u t e r o n T -m a t r i x . One must assume t h a t when t h e d e u t e r o n s t a t e v e c t o r i s w r i t t e n as i n e q u a t i o n ( 3 . 5 3 ) , t h e n t h e c o n v e n t i o n a l d e u t e r o n w a v e f u n c t i o n i n momentum s p a c e c o r r e s p o n d s t o I t i s v e r y d i f f i c u l t , w i t h o u t s p e c i f y i n g V , t o d raw a n y c o n c l u s i o n s a b o u t the o n e - m e s o n e x c h a n g e a p p r o x i m a t i o n t o t h e T -a i a t r i x when i t i s w r i t t e n a s i n e q u a t i o n ( 4 . 2 0 ) . I f one h a s some 48 p o t e n t i a l V t h e n , i n p r i n c i p l e , e q u a t i o n (4 .20) c o u l d be e v a l u a t e d n u m e r i c a l l y . H o w e v e r , t h i s w i l l n o t be d o n e h e r e . I n s t e a d , i n t h e n e x t s e c t i o n , some f u r t h e r a p p r o x i m a t i o n s w i l l be made w h i c h w i l l e n a b l e one t o draw some c o n c l u s i o n s w i t h o u t d o i n q any n u m e r i c a l c a l c u l a t i o n s . 4 . 3 The S t a t i c o n - S h e l l A £ 2 r o x i m a t i o n I n a c a l c u l a t i o n o f t h e t y p e d o n e i n t h e p r e v i o u s s e c t i o n s , i t i s u s e f u l t o g e t some i d e a o f t h e r e l a t i v e s i z e s o f t h e v a r i o u s t e r m s i n e q u a t i o n ( 4 . 2 0 ) . I n o r d e r t o do t h i s , some a p p r o x i m a t i o n s w i l l be made . The f i r s t a p p r o x i m a t i o n w i l l be t o a s s u m e t h a t t h e k i n e t i c e n e r g y o f t h e n u c l e o n s i n t h e d e u t e r o n i s s m a l l c o m p a r e d t o t h e t o t a l e n e r g y o f t h e i n c o m i n g p i o n . A l s o , a s s u m e t h a t t h e k i n e t i c e n e r g y o f t h e d e u t e r o n i s s m a l l c o m p a r e d t c t h e p i o n t o t a l e n e r g y . T h e s e a s s u m p t i o n s w i l l be c a l l e d t h e s t a t i c a p p r o x i m a t i o n . , I t i s a l s o n e c e s s a r y t o o b t a i n an a p p r o x i m a t i o n t o t h e p i o n - n u c l e o n T - m a t r i x . U s i n g e q u a t i o n s ( 2 . 4 2 ) a n d ( 3 . 2 4 ) , t h e p i o n - n u c l e o n T - m a t r i x a s g i v e n b y e q u a t i o n ( 2 . 5 3 ) c a n be w r i t t e n T B q , A k = " < B I [ q V ] C H - E A - E k - i e ) _ 1 [ V k ] | A > -1 , (4 .21) -<B|[Vk](H+E k -E B +ie) [qV]|A> E a c h m a t r i x e l e m e n t a b o v e c a n be s i m p l i f i e d by i n s e r t i n g a c o m p l e t e s e t o f o n e - n u c l e o n s t a t e s and p i o n - n u c l e o n s c a t t e r i n g s t a t e s a n d t r u n c a t i n g t h e s u m m a t i o n by n e g l e c t i n g a l l s c a t t e r i n g 49 states. With this assumption the T-matrix can be written TBq,Ak = -IG<Bl^lG><Gl™lA> ^ WV10"1 -IG<B|[Vk]|G><G|[qV]|A> ( E ^ - E ^ i e ) " 1 'his can be represented diagramatically as follows /k /k yk A = B :—p c* A + B ct— p^-( 4 . 22) One further assumption i s necessary to simplify the terms represented by T* ± f f and T* a m e > Most of the ten terms represented + + by T,.,.,. and T are proportional to a factor of dirt same (Eg+Eq-E^-E^ie)-1 (or a sim i l a r term with G interchanged with either B or D). The approximation made here i s to assume that the scattering involving the four p a r t i c l e s i n the above energy denominator takes place on-shell, i . e . assume that energy i s conserved i n the scattering. This i s equivalent to neglecting the p r i n c i p a l value i n t e g r a l when the energy denominator above i s s p l i t i n the following manner V V V V 1 ^ " 1 = * i 6 < E B + Y W + P ( EB + Eq-V Ek ) _ 1 ( 4* 2 3 ) Making use of these approximations the terms represented by T\ andT + become diff same 50 Tdiff + TLne 4 W o q ^ ^ l V * Ep-2M x | -TT i<A| [mV] |Cq><B| [qV] |Dk> 6 (E^+E^-Ep-E^) (2 - — ) V 2 M - A T T I I <A | [Vq] | G > < G | [mV] | C > < B | [qV] | Dk> 6 ( E ^ + E ^ - E ^ - E ^ ) (— ) x + ^ | XG<A| [Vq] | C > < B | [mV] | G > < G | [qV] |Dk> <$ ( V Y W ( — ) x 1c x ( E + E -E.+ie) C q A -1 E0-2M + IG<A| [Vq] | C > < B | [qV] [Gk><G] [mV] |D> 6 ( E j + Y W (— ) x x (E_+E - E . + i e ) " 1 C q A " I G<A|[Vq] |C><B|[qV]|G><G|[mV]|Dk> ( E c + E q - E A + i e ) " 1 ( E B + E q - E G - i e ) " 1 - IG<A| [Vq] |C><B1 [mV] |Gk><G| [qV] |D> (E c+E - E ^ i e ) " 1 (Eg+E - E ^ i e ) " 1 } ( 4 . 2 " ) T h e s e t e r m s c a n be r e p r e s e n t e d d i a g r a m a t i c a l l y a s b e l o w -P-v. • k >>~^— B 0,/ D A .'H r i>s /V. B G / D A C V o / k *0 B q, m m \ / k D r\ m \ • k B q / G D 51 The f i r s t two t e r m s r e s u l t f r o m T ^ i f f a n d t h e o t h e r f o u r r e s u l t f r o m T g a m e - In t h e s t a t i c a p p r o x i m a t i o n (Ep-2M) i s j u s t t h e b i n d i n g e n e r g y of the d e u t e r o n ( a b o u t 2 . 2 M e V ) . T h u s , m a k i n g u s e o f e q u a t i o n ( 4 . 2 2 ) and a s s u m i n g a l l m a t r i x e l e m e n t s o f t h e f o r m < \ | [ V g ] | 3 > , <G|[mV]|C> e t c . a r e o f t h e same o r d e r o f m a g n i t u d e , i t f o l l o w s t h a t t h e s e c o n d , t h i r d a n d f o u r t h t e r m s a r e a b o u t two o r d e r s o f m a g n i t u d e s m a l l e r t h a n t h e f i r s t t e r m ( a s s u m i n g a n i n c i d e n t p i o n k i n e t i c e n e r g y o f 50 Mev o r g r e a t e r ) . I t i s n o t o b v i o u s t h a t t h e f i f t h and s i x t h t e r m s a r e s m a l l c o m p a r e d t o t h e f i r s t t e r m . T h i s c o u l d o n l y be i n v e s t i g a t e d i f one d i d n u m e r i c a l c a l c u l a t i o n s w i t h some p o t e n t i a l V . When s i m p l i f y i n g t e r m s r e p r e s e n t e d b y T d i f f a n d T s a m e , i t i s n o t v a l i d t o make b o t h t h e s t a t i c a p p r o x i m a t i o n a n d t h e o n -s h e l l a p p r o x i m a t i o n . T h e e n e r g y d e n o m i n a t o r a s s o c i a t e d w i t h m o s t o f t h e s e t e r m s i s p r o p o r t i o n a l t o (E - E - E L - E , - i e ) - 1 (or a s i m i l a r Jj Q JJ K. t e r m w i t h G i n t e r c h a n g e d w i t h e i t h e r B o r D ) . S i n c e the e n e r g i e s i n t h e a b o v e e n e r g y d e n o m i n a t o r a r e t o t a l e n e r g i e s , n o t j u s t k i n e t i c e n e r g i e s , i t f o l l o w s t h a t t h e e n e r g y o f t h e n u c l e o n a f t e r a b s o r b i n g t h e m a s o n s k a n d q w i l l be a b o u t 300 MeV g r e a t e r t h a n t h e e n e r g y o f t h e n u c l e o n b e f o r e t h e i n t e r a c t i o n . T h u s t h e s t a t i c a p p r o x i m a t i o n w i l l n o t be made when d e a l i n g w i t h t h e s e t e r m s . M a k i n g o n l y t h e o n - s h e l l a p p r o x i m a t i o n , t h e t e r m s r e p r e s e n t e d by T~ c c a n d T ~ c a n be w r i t t e n c 1 d i f f same 52 T d i f f + T s a m e = 2 W P > ) ( C D l V * ' x <Aq|[mV]|C><B|[Vq]|Dk> 5(Eg-E^-E^-E^) - ^ I G <A| [qV] |C><G| [raV] |D > < B| [Vq] | Gk> 6 ( E ^ - E ^ ) (E^-E^-E^+ie) ~ ^2 U <Al [qV] |C><Bl [mV] |G><G| [V<1] |Dk> 6 (V Eq~ ED" Ek ) ( E C - E q " E A + i £ ) + lr, <A|[qV]|c><B|[Vq]|G><G|[mV]|Dk> (E^E^+ier^Eg-E^-ie) - 1 -1 -1 - lQ <A|[qV]|c><B|[mV]|Gk><G|[Vq]|D> (E^-E^-E^+ie) 1 ( E ( , - E q - E D - i e ) X} (4 .25 ) T h e s e t e r m s c a n be r e p r e s e n t e d d i a g r a m a t i c a l l y a s f o l l o w s k/ m qv » 0 B ^ N G A q N kx' *0 < B G x A q N D k' 4? sm k x s / \ • Q; B G s ' D A The f i r s t t h r e e t e r m s o f e q u a t i o n (4 .25 ) w i l l be s m a l l c o n p a r e d t o t h e c o r r e s p o n d i n q t e r m s i n e q u a t i o n (4 .24 ) s i n c e t h e n u c l e o n k i n e t i c e n e r g y o f o v e r 300 MeV w i l l y i e l d a r e l a t i v e momentum o f a b o u t 400 M e V / c f o r t h e n u c l e c n s i n t h e d e u t e r o n . s i n c e t h e d e u t e r o n w a v e f u n c t i o n i s v e r y s m a l l f o r s u c h l a r g e 53 r e l a t i v e momentum, the f i r s t three t e r e s cf e q u a t i c n (4.25) can te n eglected i n comparison to the f i r s t term o f eguation (4.24). As i n eguation (4.24) the s i z e c f the l a s t t%c t e r n s cf e g u a t i o n (4.25) can only be estimated n u m e r i c a l l y given some p o t e n t i a l V. Thus making an o n - s h e l l approximation and a s t a t i c approximation where p h y s i c a l l y s e n s i b l e , the dominant tern; c f T ^ P can be w r i t t e n P'm.Pk = " 1 7 1 l A B C D q < P o ' ^ ) ( C D I V < A l [ m V ] l C q > < B l [ q V ] l D k > 6 ( V V V V (4.26) In w r i t i n g t h i s eguation i t has been assuned that the b i n d i n g energy c f the deuteron i s n e g l i g i b l e compared t c the t o t a l energy of the incoming p i o n . The above equation can be represented d i a g r a m a t i c a l l y as below ' q V, / Asa In a d d i t i o n t o the term given i n e g u a t i c c (4.26) there * i l l te the l a s t two terms of both equations (4.24) a E d (4.25) whose c o n t r i b u t i o n s can only be e v a l u a t e d n u m e r i c a l l y . The above eguation can be w r i t t e n i r t e r n s cf the c o n v e n t i o n a l deuteron momentum space wavefuncticn and p i e r nucleon T - m a t r i c e s . T h i s i s dene i t Appendix D. The r e s u l t i n the l a b o r a t o r y frame i s expressed below .(1) T L QSp.'.JS; 0 , k ; M ' . M ) <S(kp,+m-k) +1 = -rri I fc,Jt'=-l <«.27) , M » d ^ q $v (K+a-jQi-m))- ^ ( £ ) T £ T £ fiCkp.-hn-k) 54 The T - m a t r i x T ^ » £ r e p r e s e n t s an a v e r a g e o v e r n u c l e o n s p i n s a n d p i o n a n d n u c l e o n i s o s p i n s o f a p r o d u c t o f two p i o n n u c l e o n T -t n a t r i c e s . The e x p l i c i t e x p r e s s i o n f o r T ^ , ^ i s g i v e n i n A p p e n d i x E. E q u a t i o n ( 4 . 2 7 ) a g r e e s w i t h t h e d o u b l e - s c a t t e r i n g r e s u l t o b t a i n e d by P e n d l e t o n (1963 ) . 55 5 C o n c l u s i o n s U s i n g a f i e l d t h e o r y a p p r o a c h , t h e p i c n - d e u t e r o n e l a s t i c s c a t t e r i n g T - m a t r i x was e x p a n d e d i n a meson e x c h a n g e s e r i e s a n d t h e f i r s t two t e r m s o f t h i s s e r i e s were e x a m i n e d . T h e r e a s o n f o r u s i n g t h i s a p p r o a c h was t o a v o i d t h e d o u b l e - c o u n t i n g p r o b l e m s u s u a l l y a s s o c i a t e d w i t h m u l t i p l e s c a t t e r i n g c o r r e c t i o n s i n p i o n -d e u t e r o n s c a t t e r i n g . I n s e c t i o n 4 , 1 , t h e f i r s t t e r m i n t h e e x p a n s i o n o f t h e T -m a t r i x ( c a l l e d t h e z e r o - m e s o n e x c h a n g e term) was e v a l u a t e d . T h i s t e r m was shown t o be t h e u s u a l s i n g l e - s c a t t e r i n g a p p r o x i m a t i o n t o t h e T - m a t r i x ( e q u a t i o n ( 4 . 5 ) ) . I n s e c t i o n 4 . 2 , t h e n e x t o r d e r o f t e r m s i n t h e e x p a n s i o n ( c a l l e d t h e o n e - m e s o n e x c h a n g e term) was e v a l u a t e d . K e e p i n g o n l y t h o s e t e r m s c o n t a i n i n g one e x c h a n g e d meson a n d t h e i n i t i a l a n d f i n a l m e s o n s , t h e T - m a t r i x was e x p r e s s e d a s a sum o f t w e n t y t e r m s , e a c h w r i t t e n a s p r o d u c t s o f m a t r i x e l e m e n t s o f t h e v e r t e x o p e r a t o r b e t w e e n o n e - n u c l e o n s t a t e s ( e g u a t i o n ( 4 . 2 1 ) ) . I n o r d e r t o g a t a n i n d i c a t i o n a s t o w h i c h o f t h e s e t w e n t y t e r m s w e r e i m p o r t a n t , an o n - s h e l l a p p r o x i m a t i o n was made a s w e l l a s a s t a t i c a p p r o x i m a t i o n where p h y s i c a l l y s e n s i b l e . W i t h t h e s e a p p r o x i m a t i o n s , t h e o n e - m e s o n e x c h a n g e c o n t r i b u t i o n t o t h e T -m a t r i x was w r i t t e n i n a f o r m w h i c h f a c i l i t a t e d c o m p a r i s o n o f t h e m a g n i t u d e s o f t h e v a r i o u s t e r m s i n t h e e x p r e s s i o n f o r t h e T -m a t r i x ( e q u a t i o n s (4 .25 ) and ( 4 . 2 6 ) ) . A s s u m i n g an i n c i d e n t p i o n k i n e t i c e n e r g y c f 50 MeV, i t was shown t h a t one of t h e t e r m s was a b o u t two o r d e r s o f m a g n i t u d e g r e a t e r t h a n most o f t h e r e m a i n i n g t e r m s . T h i s t e r m was shown t o 56 be s i m i l a r t o t h e c o n v e n t i o n a l d o u b l e - s c a t t e r i n g t e r m r e s u l t i n g f r o m t h e g e n e r a l i z e d i m p u l s e a p p r o x i m a t i o n . T h e r e w e r e f o u r a d d i t i o n a l t e r m s whose m a g n i t u d e c o u l d n o t be e a s i l y c o m p a r e d t o t h e d o u b l e - s e a t t e r i n g t e r m . T h e s e f o u r t e r m s r e p r e s e n t e d s i n g l e -s c a t t e r i n g p r o c e s s e s w i t h a mescn e x c h a n g e d b e t w e e n t h e n u c l e o n s e i t h e r b e f o r e o r a f t e r t h e s c a t t e r i n g . T h e s i z e o f t h e s e t e r m s c o u l d o n l y be e v a l u a t e d n u m e r i c a l l y u s i n g a p a r t i c u l a r f i e l d t h e o r e t i c p o t e n t i a l V . P e n d l e t o n (1963) h a s c a l c u l a t e d t h e d o u b l e - s c a t t e r i n g c o n t r i b u t i o n ( e q u a t i o n ( 4 . 2 8 ) ) f o r a p i o n k i n e t i c e n e r g y o f 142 MeV. C a r l s o n ( 1 9 7 0 ) has c a l c u l a t e d t h e d o u b l e - s c a t t e r i n g c o n t r i -b u t i o n s a t p i o n k i n e t i c e n e r g i e s r a n g i n g f r o m 61 t o 300 M e V . I t w o u l d p e r h a p s be w o r t h w h i l e t o c a l c u l a t e o t h e r t e r m s c o n t a i n e d i n e i t h e r s e c t i o n 4 . 2 o r 4 . 3 . U s i n g a Chew-Low H a m i l t o n i a n [Chew and Low (1956) ] , n u m e r i c a l r e s u l t s c o u l d be o b t a i n e d w h i c h c o u l d be c o m p a r e d w i t h t h e d o u b l e - s c a t t e r i n g c o n t r i b u t i o n u s u a l l y c a l c u l a t e d . BIBLIOGRAPHY Blatt, J.M. and Weisskopf, V.F. 1952. Theoretical Nuelear Physios (John Wiley and Sons, New York). Carlson, C. 1970. Phys. Rev., C2, 1224. Chew, G.F. and Goldberger, M.L. 1952. Phys. Rev., 87_, 778. Chew, G.F. and Low, F.E. 1956. Phys. Rev., 101, 1570. Cutkosky, R.E. 1958. Phys. Rev., 112, 1027. Heitler, W. and London, F. 1927. Z. Physik, 4 4 ^ 455. McMillan, M. and Landau, R.H. 1974. TRIUMF Report, TRI-74-1. Pendleton, H.N. 1963. Phys..Rev., 131, 1833. Wick, G.C. 1955. Rev. Mod. Phys., 27, 339. 58 APPENDIX A Meson Operator I d e n t i t i e s The i d e n t i t y for the commutator of a product of mescr t creation operators M and a product cf nescii destruction operators Q i s [M+,Q] = -l'R [[r,M +J][[Q,r +]]/n(R).' { f l. 1) It can be proved by induction on the number of operators i r the product Q . + Let M be an arb i t r a r y product cf irescn creation operators and l e t Q=g. Then [q,M+] = E r [r,M +][q,r +] = ZR [[r,M +]][[q,r +]]/n(R): ( f l > 2 ) Thus eguation (A. 1) i s v a l i d for n (Q )=1. Assume the i d e n t i t y i s true f c r a r b i t r a r y C . It must now be shown that the i d e n t i t y i s true for Q k . The prccf cf th i s s i l l be done only f c r the case k ^ C . The proof for keQ w i l l be outlined only as i t i s conceptually s i m i l a r tc the case k^Q although i t i s more complicated a l g e b r a i c a l l y . lor k ^ C the prccf proceeds as follows. Using the r e l a t i o n [M+,Qk] = [M+,Q]k + [M+,k]Q + [[k,M+],Q] (A. 3) and assuming equation (A.1) i s true for Q , the conirutatcr 59 [M,Qk] oay be written - [ M + , Q k ] = E ' [ [ r , M + ] ] [ [ Q > r + ] ] k / n ( R ) . ' R t t + [ [ r , M T ] ] [ [ k , r T ] ] Q / n ( R ) . » R . . + E» [ [ r , [ k , M T ] ] ] [ [ Q , r T ] ] / n ( R ) : R Denoting by R q these products R which de net c e r t a i n defining S = R Q k , eguation (A.4) can be written - [ M f , Q k ] = E' [ [ r , M + ] ] [ [ Q k , r + ] ] / n ( R ) ! ° ° + [k,M ] [Qk,k] + E" [ [ S , M ] ] [ [ Q k,s + ] ] / n ( S 1 ) ! The double prime cn the summation indicates that the unit operator and single meson cperatcrs are ejclucec fron the summations. The factor of n ( S 1 ) ~ 1 comes from the fact that there are n C S ^ distinguishable ways of placing k in a given product R . o Relabelling the sums i n eguation (* .5) y i e l d s the desired r e s u l t . Thus eguation (A.1) has teen shewn, bj i r d u c t i c n , to be true for a set Q containing an a r b i t r a r y number of cperatcrs rc two of which are i d e n t i c a l and an arb i t r a r y set M. Tc prove eguation (A.1) i n general i t i s assumed that i t i s true for l M t , Q k J ] where keQ. Then i t i s shewn that i t i s trte f o r t i+1 [M ,Qk J ] . Using the following i d e n t i t y [ M + , Q k j + 1 ] = [ M + , Q k j ] k + [ M + , k ] Q k j + [ [ k , M + ] , Q k J ] (A.6) an eguation s i m i l a r to equation (A.1) can be written. Then the sums over R are separated intc S U B S ever B where F ccntairs n n 60 the operator k exactly n times. Denote by S the product cf the o operators not equal tc k in R^  . By writing the suns ever R^  as sums over £ with the n operators k e x p l i c i t l y written in the o nested commutators, i t i s a matter cf straightfcrvard manipula-tion tc prove that equation (A.1) i s true for iM^Qk^" 1" 1] i f i t i s true for [ M + , Q k ^ ] . Thus equation <A.1) i s true f c r any M and Q. The i d e n t i t i e s involving meson annihilation operators, equations (3. 24)- (3.26), can be proved as f c l l c v s . Since the one nucleon state i s an eigenstate of H i t follows that r(H-EA)|A> = 0 (A. 7) From equation (2. 12) [r,H] = E rr + [rV] ( J > 8 ) Thus eguation (A.7) can be written (H-E.)r|A> + E r|A> + [rV]|A> = 0 Inverting the operator (H-E^ +E^ ) the desired i d e c t i t j i s obtained r|A> = -(H+E r-E A-ie) _ 1 [rV]|A> (A•10) When two or more mescn operators are tc be removed, the commutator of a meson a n n i h i l a t i o n operator with (H-E)~ 1 w i l l be needed. Using equation (A.8), r (H-I) may be written r ( H - E ) = (H-E+E ) r + [ r V ] 61 (A. 11) Multiplying on the right by (H-E) - 1 and cr the l e f t kj (H-E+E^)-1, the desired r e s u l t i s obtained r ( H - E ) " 1 = ( H + E r - E ) - 1 - ( H + E ^ E ) " 1 [ r V ] ( H - E ) " 1 (A. 12) The t h i r d operator i d e n t i t y i s used i E relieving meson anni h i l a t i o n operators from matrix elements involving picn-nucleon scattering states. From equation (2.12) i t fellows that [ r k + , H ] = ( E r - E k ) r k + - r [ V k ] + [ r V ] k + (».13) Using t h i s i d e n t i t y the eguation rk+(H-Ec)|c> = 0 (A. 14) may be w r i t t e n ( H + E r - E c - E k ) r k + | c > - r [ V k ] | c > + [ r V ] k + | c > =0 (A.15) Now, s i n c e ( H + E r - E c - E k ) 6 k r | 0 = ( H - E C ) | C > = 0 ( j . 1 6 ) the following r e s u l t can be obtained freir equaticc (A.14) 62 rk+|c> - ( H + E ^ E g - E ^ i e ) " ^ ^ ] ! ^ ( A > 1 7 ) .= \ r|G> ~ (H+E r-E c-E k-i E) _ : L[rV]k + | c > O u t g o i n g wave b o u n d a r y c o n d i t i o n s a r e i m p o s e d when i n v e r t i r g (H+E-E-E). U s i n g t h e d e f i n i t i o n c f t h e p i o n n u c l e o n s c a t t e r i n g 10 \j lc s t a t e a n d e q u a t i o n ( A . 12) and ( A . 17) t h e d e s i r e d i d e n t i t y may be w r i t t e n r|ck> = 6 k r|c> " (H+E r-E c-E k-ie) 1[rV]|ck> (A. 18) 63 APPENDIX B C u t k o s k y M a t r i x E l e m e n t s E . 1 U n e x c i t e d C v e r l a £ M a t r i x E l e j r e c t s The s i m p l e s t C u t k o s k y m a t r i x e l e m e n t i s t h a t * h i c h g i v e s t h e o v e r l a p o f two u n e x c i t e d C u t k c s k y s t a t e s | A B } a n d |CD ]• D s i n g t h e d e f i n i t i o n o f C u t k o s k y s t a t e s t h i s m a t r i x e l e m e n t nay he w r i t t e n {AB|CD} = <o|BAC + D + |o> U s i n g e q u a t i o n (2 .16) t h e o p e r a t o r s A , E , C , and £ may t e w r i t t e n A = I a*(AK)AK A , K B = I b*(8L)8L B , L C + = I c ( C M ) C + M + C,M Dt = I d ( f l N ) P + N + t?,N U s i n g t h e s e e x p r e s s i o n s f o r t h e p h y s i c a l n u c l e c n o p e r a t o r s , t h e m a t r i x e l e m e n t c a n b e w r i t t e n { A B | C D } = I I I I a*(AK)b*(8L)c(CM)d( l7N )<o|8LAKC t M + P + N + |o> ( B # 3 ) A , K 8 , L C,M P ,N I n o r d e r t c e x p r e s s { A B | C D } as p r o d u c t s o f one n u c l e c n m a t r i x e l e m e n t s t h e t e r m s i n t h e s u m m a t i o n i n e g u a t i o n ( E . 3 ) a r e s e p a r a t e d i n t o f o u r t y p e s . T h e f i r s t t y p e c f t e r m c o n s i s t s o f t h o s e t e r m s i n t h e s u m m a t i o n i n w h i c h VC^BA ( i n a l l s t a t e n e n t s 64 in t h i s appendix regarding the equality or inequality cf products of tare nucleon cr antinuclecn operators, the ordering cf the operators w i l l be ignored. To be completely accurate the foregoing inequality should be written VC±±Bk but since the signs have no e f f e c t on the c l a s s i f i c a t i o n of terms, they w i l l be omitted.) These terms do net contribute tc the summation + + + + since at least one of the operators i n C V i s not in 8 A (cr vice versa) and t h u s w i l l anticcnnute w i t h a l l the operators i n 8A (cr ) making the term zero by an n i h i l a t i n g cn the vacuuir state. The second type of term consists of those terms in the summation in which C=A and V=B. The quadruple sunnaticn ever a l l these terms w i l l be abbreviated I^Q B D) • The t h i r d type cf tera consists of those terms in the summation in which C=8 and V=A, The quadruple summation over a l l these terms w i l l le abbreviated ^(BC AD)" I f a t € r i B i n €quation (B.3) i s to be ncr-2erc then i t can be shown that no two operators i n A and 8 or in C and V car be i d e n t i c a l . I f , for example, the operators aeA and beB are i d e n t i c a l then the matrix element on the ri g h t hand side cf eguation (E.3) can be written < o | . . . b a . . . j o > = < o | . . . a b . . . | o > ( E . 4 ) Eut since a and b anticoBmute <o| . . .ba . . . |o> = - < o | . . . a b . . . | o > ( E . 5 ) thus proving that the operators in A and 8 cr in C and V must be 65 d i s t i n c t i f the tern i s tc be ncn-zerc. Thus f c r rcr-zerc terms cf the second type C and 8 w i l l have no operators in common net w i l l A and V. S i m i l a r i l y , for ncn-zerc terms cf the third type A and C w i l l have no operators in common nor w i l l 8 and V, The fourth type cf term consists cf the reiiainder cf the terms in eguation (B . 3 ) , i . e . those terms for which CP=A8 tut which are neither of the second cr t h i r d type. These terms can be thought cf as describing the exchange of rare nuclecr-antinucleon pairs between the physical nuclecn ceres C and V • t t to form the physical nucleon cores A and 8 . These terms should have l i t t l e e f f e c t for two reasons. F i r s t l y , although the masses cf the bare nucleons are not known, i t w i l l be assumed that the bare nucleon-antinucleon pair w i l l have a s u f f i c i e n t l y large mass that these terms can be thought of as describing short range forces which w i l l not be s i g n i f i c a n t f c r medium energy scattering. Secondly, these terms w i l l only arise when at least two of the nucleon cores consist cf three cr acre bare nucleons and antinucleons. I t w i l l be assumed that the wavefuncticn cf the physical nucleon w i l l be small in that part cf the Fock space representing three or more bare nucleons and antinuclecrs. Thus i n a l l cal c u l a t i o n s of Cutkcsky matrix elements these 'core exchange' terms w i l l be neglected. Hith the above arguments, equation (B .3 ) car be written (ignoring cere exchange terms) as {AB|CD> - I <o|8LAKC +M +t? +N +|o> (AC.BD) ( E . 6 ) + I < O | B L A K C + M VN + | O > (BC.AD) where each term in the susmatiens i s tc be n u l t i p l i e d by a (AK) 66 e t c . S i n c e [ L , K ] = 0 a n d {A,8} = 0 t h i s c a n be w r i t t e n a s {AB|CD} = -I <o|AK8LC +M +t? tN +|o> + £ <o | 8LAKC V | o> (AC.BD) (BC,AD) = I <o|AKC+LM+8pV|o> - I <o|BLC + KM + AP + N + |o> (AC, BD) (BC,AD) ( B # 7 ) t h e s e c o n d e q u a l i t y r e s u l t i n g f r o m t h e f a c t t h a t f c r t e r n s c f t h e s e c o n d t y p e {8,C'}=0 a n d f c r t e r m s c f t h e t h i r d t y p e { A , C ' } = 0 . U s i n g e g u a t i o n ( 3 . 2 1 ) t h i s c a n be w r i t t e n as {AB|CD} = I I < o | A K C + [ [ r , M + ] ] [ [ L , r + ] ] 8 P + N + | o > / n ( R ) ! R (AC,BD) -11 < o | 8 L C + [ [ r , M + ] ] [ [ K , r + ] ] A P + N + | o > / n ( R ) ! R (BC,AD) ( £ . 8 ) I n o r d e r t c e x p r e s s t h e a b o v e m a t r i x e l e n e r t i n t e r m s c f o n e n u c l e o n m a t r i x e l e m e n t s , t h e u n i t o p e r a t o r i s i n s e r t e d b e t w e e n t h e n e s t e d c o m m u t a t o r s . T h e u n i t c p e r a t c r c a n be w r i t t e n 1 . ][ E f Q + |o><o|EQ/n(Q)!n(E) ! ( E g ) E,Q where Q i s a p r o d u c t c f mescn a n n i h i l a t i o n c p e r a t c r s and E i s a p r o d u c t o f b a r e n u c l e o n and a n t i n u c l e o n a n n i h i l a t i o n o p e r a t o r s s a t i s f y i n g t h e c o n d i t i o n 8E +|o> - + l E + | c > < B ' 1 0 > where 8 i s t h e b a r y o n number o p e r a t o r . Terms i n w h i c h E i s t h e 67 u n i t o p e r a t o r a r e a l s o t o be i n c l u d e d i n e q u a t i o n ( E . 9 ) . T h u s t h e m a t r i x e l e m e n t {AB|CD ] may be w r i t t e n {AB|CD} = I \l < o l A K C + [ [ r , M t ] ] E + Q t | o > < o | E Q [ [ L J r + ] ] B P + N t | o > / n ( R ) : E , Q , R (AC.BD) x n ( Q ) ! n ( E ) ! - I I < o | B L C + [ [ r , M + ] ] E + Q + | o > < o | E Q [ [ K , r + ] ] A P + N + | o > / n ( R ) : E , Q , R (BC,AD) x n ( Q ) ' . n ( E ) ! ( E . 11) Now, a l l t e r m s i n w h i c h E / l w i l l v a n i s h . F o r e x a m p l e , i n t h e f i r s t t e r m i f E + ^ l t h e n i t e i t h e r h a s a n c p e r a t c r i t common w i t h A ( i n w h i c h c a s e t h e o p e r a t o r i s i n w h i c h c a u s e s t h e m a t r i x e l e m e n t t o v a n i s h b e c a u s e c f t h e a n t i c o m m u t a t i c n c f i d e n t i c a l o p e r a t o r s ) o r i t d o e s n o t h a v e an o p e r a t o r i n common w i t h A ( i E w h i c h c a s e t h i s o p e r a t o r w i l l a n t i c c m m u t e w i t h A a n d a n n i h i l a t e c n t h e vacuum s t a t e ) . A s i m i l a r a r g u m e n t h o l d s f o r t h e s e c c r c t e r m o f e q u a t i o n ( B . 11). A l s o , Q + commutes w i t h [ [ r .M*] ] s i n c e [{r.M^]] c o n t a i n s c n l y meson c r e a t i o n o p e r a t o r s . S i m i l a r i l y Q commutes w i t h b o t h + + I [ L , r ]J a n d M K , r ]] . S i n c e r|0>=0 f o r any meson a n n i h a l a t i c r o p e r a t o r , [ [ r ,M + ] ]|o> RM | o> < o | [ [ L , r + ] ] <o|LR ( E . 1 2 ) < o | [ [ K , r + ] ] <o|KR T h u s e q u a t i o n (E.11) c a n be w r i t t e n 68 {AB|CD} = t I <o|AKQ +RC + M + |o><o|8LR +QP + N + |o>/n(R)!n(Q)! Q . R (ACBD) ( £ > 1 3 ) . -I I <o|BLQ+RC+M+|o><o|AKRtQP+N+|o>/n(R) ,.n(Q)I Q,R (BC.AD) The r e s t r i c t i o n s cn the sunmaticns in the above equation can now te removed without a f f e c t i n g the r e s u l t . That i s , tents in wticb CV AB w i l l be zero as w i l l a l l terns in which C/A acdP^B cr B#? and Aft). Also, terms of the t h i r d type can be added tc the f i r s t summation (since they w i l l be zero) and terms c f the second type can be added to the second summation (since they w i l l be zero). Terms of the second and t h i r d type in which A=8=C=P w i l l cancel between the f i r s t and second summation. Eerfcrning these summations, the overlap matrix element can be written {AB|CD} = I <A|Q+R|c><B|R+Q|D>/n(R),.n(Q)! R , Q ( E . 1 U ) - I <B|Q+R|c><A|R+Q|D>/n(R) !n(Q)'. R,Q Thus the matrix element between two urescitec Cutkosky states has been written as a sum of products cf natrix elements of physical one nucleon states. The only approximation in this r e s u l t i s that core exchange terns have been ignored. B .2 Unexcited Interaction Katrix Elements Interaction matrix elenents are natrix elenents cf the Hamiltonian, H, or the vertex operator, [mV], evaluated between Cutkosky states. The matrix element of the hamiltonian w i l l te evaluated f i r s t as i t uses a l l the ideas necessary tc evaluate matrix elements of the vertex operator. Using the d e f i n i t i o n of Cutkcsky states the natrix element 69 of the Hamiltonian between unexcited states car be written {AB|H|CD} = <o|BMICiD+|o> <E. 15) The general method used i s tc ccmoute H tc the right past eit h e r or and then separate the re s u l t i n g Cutkcsky aatrix elements into one nucleon matrix eleuents f c l l c w i r g the methods cf the preceding section. Since the f i n a l r e s u l t should te symmetric with respect tc i n i t i a l and f i n a l states the process should be repeated commuting H with either £ or E. The f i n a l r e s u l t w i l l be the average of the two results cbtained above. a. .j. Using the fact that C and E anticommute HC+D+|o> = (C+HD+-D+HC+)|o> - [H,D+]C+|o> + [H,C+]D+|o> - HC+D+|o> (E.16) t t Since C |0> and D |0> are eigenstates of the Hamiltonian (C+HD+-D+HC+)|o> = (Ec+ED)cV|o> < E* 1 7> Defining H u = E E k k k (E. 18) k the l a s t three terms of equation (B.16) can be written - [H,D+]C+|o> + [H,C+]D+|o> - HC+D+|o> = H C+D+|o> + D+H C+|o> - C+H D+|o> + VC +D +lo> + D+VC+Io> - C+VD+|o> Ihe terms involving H in equation (E.19) w i l l cancel as can te v e r i f i e d by writing c ' and D e x p l i c i t l y and evaluating those 70 terms. Thus the matrix element of the Hamiltonian can he written {AB|H|CD} = (EC+ED) {AB|CD} + <o|BAVCTD | O > t t t t ( E ' 2 0 ) + <o|BAD VC |o> - <o|BAC VD |o> Using the notation cf equations (B . 2 ) a t y p i c a l term cf the three matrix elements involving V i s < O | 8 L A K V C V P + N + | O > + <o | B L A K P + N + V e V | o> - <o| BLAKC + M + VP + N +|o> (E*21) where the c o e f f i c i e n t s a (AK) etc. have teen omitted. To proceed, V i s written as a sum cf f i v e terms. V can be written i n terms cf the fundamental dynamical variables as V= I v(l/,W,S,T)l/ +WS +T ( E . 2 2 ) l/,W,S,T + where I/ and W are eith e r unit operators or are products cf cdd numbers of bare nucleon and antinuclecn operators anc S and T are products of meson operators. The c c e f f i c i e r t s v(l/,W,S,T) w i l l depend upon the type of i n t e r a c t i o n chosen. New, the sum i t equation (E .22) i s broken up into five terms. The f i r s t term, V 1, consists of a l l those terms i n which l/=W=l . These w i l l be terms describing the meson-meson i n t e r a c t i o n . The ether four terms are denoted by V", V", V" and V". V" consists cf a l l these L D R o C terms i n which W=C and V" consists cf a l l these terms in which D W=V . V" consists cf those terms i n which Wv i s partly ccntainec R in C and partly contained in V while V" consists cf these terms o in which W contains some bare nucleon or antinuclecn operators 71 not i n C o r P . T h u s , w r i t i n g V = V + V C ' + V - + + V ' 1 < E . 23) some s i m p l i f i c a t i o n s c a n t e made i m m e d i a t e l y i n t h e m a t r i x e l e m e n t s i n v o l v i n g V . T h e f i r s t m a t r i x e l e m e n t i n v o l v i n g v i n e q u a t i o n ( B . 2 1 ) b e c o m e s <o |BLAKVC + M + I7 + N + |o> = < o | B L A K V ' C + M T P + N + | o > + < O | B L A K ( V ' , + v : , ) c T M V N + | o > + < O|BLAK V ; ' C + MVN + | O > ( E , 2 4 ) The m a t r i x e l e m e n t i n v o l v i n g V w i l l v a n i s h s i n c e a t l e a s t e r e o o f t h e a n n i h i l a t i o n o p e r a t o r s i n W w i l l a r t i c e m m u t e w i t h a l l t h e t t o p e r a t o r s i n C a n d V a n d a n n i h i l a t e on t h e v a c u u m s t a t e . S i n c e V ^ ' C + | o > = V ^ » P + | o > = o ( E . 2 5 ) t h e m a t r i x e l e m e n t i n v o l v i n g V " w i l l n e t be a b l e t c be e x p r e s s e d as p r o d u c t s o f o n e - n u c l e o n m a t r i x e l e m e n t s . S i n c e t h i s m a t r i x t t e l e m e n t o n l y a p p e a r s when e i t h e r C c r P ( c r b c t h ) c o n t a i n t h r e e c r more b a r e n u c l e o n a n d a n t i n u c l e o n o p e r a t o r s , t h i s m a t r i x e l e m e n t w i l l be n e g l e c t e d c n t h e b a s i s t h a t t h e n u c l e o n c o r e s a r e p r i m a r i l y c o m p o s e d o f s i n g l e b a r e n u c l e o n s . T h e s e c o n d m a t r i x e l e m e n t i n e q u a t i o n ( B . 2 1 ) b e c o m e s <O | 8LAKPVVC + M + |O> = < o | B L A K P + N + V C + M + | o > + < O | B L A K P + N + V ' * C + M + | O > < E ' 2 6 > The m a t r i x e l e m e n t i n v o l v i n g V " w i l l v a n i s h f c r t h e same r e a s o n o 72 as i n e q u a t i o n ( E . 2 U ) and t h e n a t r i x e l e m e n t i E v c l v i n q V " w i l l R v a n i s h a s a r e s u l t c f e q u a t i o n ( E . 2 5 ) . S i n c e anc c a n n o t h a v e any o p e r a t o r s i n c o a m c n ( o t h e r w i s e t h e n a t r i s e l e m e n t would v a n i s h a s a r e s u l t o f t h e a n t i c o m m u t a t i o n o f i d e n t i c a l t a r e n u c l e o n o p e r a t o r s ) V£ c a n n c t c o n t a i n a n y t e r n s i E w h i c h W=C . T h u s t h e m a t r i x e l e m e n t i n v o l v i n g V^' w i l l v a n i s h . By s i m i l a r a r g u m e n t s < O | B L A K C + M + V P + N + | O > = <O|8LAKCVV*PV|O> + <O|BLAKCVV^'P +N + |O> ( E . 2 7 ) T h u s t h e m a t r i x e l e m e n t s i n v o l v i n g V c a n be w r i t t e n <O|BLAK(V',+V')CTMVN+|o> + <O|8LAKP+N+(V'•+V,)C+Mt|o> + <O|BLAK(V^'+V')C+MVN+|O; - <o|8LAKC+M+(V^'+V*)P+N+|o> - <O|BLAKV-CWN +|O>. . ( E ' 2 8 ) The t e r m s i n t h e a b o v e e g u a t i o n c o n t a i n i n q VL c a n t e w r i t t e n - <o|8LAK (V C , +-V ' ) r ' T N T cV|o> + < O | 8 L A K P V ( V C , + V * ) C T M T | O > ( E . 29) = - < o | 8 L A K C + [ V ' ' + V ' , N + ] C + M + | o > C I n w r i t i n g t h e a b c v e e q u a t i o n u s e h a s b e e n made o f t h e f a c t t h a t s i n c e e a c h t e r m i n V " c o n t a i n s an e v e r number o f t a r e n u c l e c E and a n t i n u c l e o n c p e r a t c r s and s i n c e W c a n n c t c o n t a i n a n y c p e r a t c r s w h i c h a r e i n V, t h e n [P,v"]=0. Now s i n c e e a c h t e r m c f V '+V c o n t a i n s n e s c E a r n i h i l a t i o n C o p e r a t o r s , e q u a t i o n (3 .20 ) i s u s e d t o w r i t e [ N + , V " + V ] = - I , [ [ r , N + ] ] [ [ V ' » + V , r + ] ] / n ( R ) ! C R ° ( E . 3 0 ) 73 T h u s t h e t e r m s i n v o l v i n g v c ' fceccne - Z ,<olBLAK[[r,t?V]][[V c'+V ,,r +]]C +M +|o>/n(R)! <E.31) R U s i n g m e t h o d s i d e n t i c a l t o t h e s e u s e d i n e v a l u a t i o n c f t h e o v e r l a p m a t r i x e l e m e n t , t h e a t o v e e x p r e s s i o n c a n be w r i t t e n - E' E <o|AKP +QRP tN +lo><o|8LQ +P[(V''+V')R]C +M +|o> R Q,P G + E' E < o | 8 L P + Q R P + N + | o > < o | A K Q + p [ ( V c , + V ' ) R ] c t M + l 0 > <E- 3 2 > R Q,P e a c h t e r m b e i n g d i v i d e d by n ( Q ) ! n ( P ) ! . The G e n e r a l i z e d v e r t e x c p e r a t c r s [ V R ] a n d , [ R V ] a r e d e f i n e d by [VR] = [[V,r +]]/n(R)! [RV] - [VR] + ( E # 3 3 ) P e r f o r m i n g t h e sums o v e r A,8, e t c . a n d u s i n g t h e f a c t t h a t [(V' ,+V')R]cV|o>= [VR]C+M+|o> ( E . 3 H ) t h e a b o v e e x p r e s s i o n c a n be w r i t t e n - E* E <A|PfQR|D><B|Q+P[VR]|C>/n(Q)!n(P)! R Q,P + E* E <B|P+QR|D><A|Q+P[VR] |C>/n(Q)!n(P).» R Q,P ( E . 3 5 ) S i m i l a r i l y t h e t e r m s i n v o l v i n g v^' y i e l d t h e e x p r e s s i o n 74 E 1 E < A | P + Q R | C > < B | Q + P [ V R ] | D > / n ( Q ) ! n ( P ) ! R Q,P - E ' E < B | P + Q R | C > < A | Q + P [ V R ] | D > / n ( Q ) ! n ( P ) ! R Q.P ( E . 3 6 ) Ey d e f i n i n g H A C , B V D = Qy A l PV|c> < B | Q + P [ V R ] | D > / n ( Q ) ! n ( P ) : ( E # 3 7 ) t h e H a m i l t o n i a n m a t r i x e l e m e n t c a n be w r i t t e n { A B | H | C D } = (E +E ) { A B I C D } + H - H + H - H 1 1 C D / L 1 A C , B V D B C . A V D B D , A V C A D , B V C - { A B | V * | C D } + term i n v o l v i n g V " ( E . 3 8 ) R I n o r d e r t o make t h e d e r i v a t i o n s y m m e t r i c a l w i t h r e s p e c t t o i n i t i a l a n d f i n a l s t a t e s t h e w h o l e p r o c e d u r e i s r e p e a t e d s t a r t i n g w i t h t h e a n a l o g u e o f e q u a t i o n (B .16 ) < o | B A H = < o | ( B H A - A H B ) - < o | A [ B , H ] + < O | B [ A , H ] - <O|BAH ( E . 3 9 ) U s i n g t h e same m e t h o d t h e r e s u l t i s o b t a i n e d {AB | H | CD} = < E A + E B ) { A B | C D } + H A V C f B D - H ^ ^ + -- { A B | V | C D } + terra i n v o l v i n g V ^ 1 ( E . 4 0 ) where H A V C BD = V E < A l [ R V ] p + Q|c > < B | R + Q + P | D > / n ( P ) : n ( Q ) : ( E . 4 1 ) ' R Q,P The f i n a l e x p r e s s i o n f o r t h e H a m i l t o n i a n m a t r i x e l e m e n t i s 75 {AB|H|CD} = "|(E A+V EC + ED ) { A B L C D } + 2(HAC,BVD + HBVD,AC " V,AVD " HAVD,BC + HBD,AVC + HAVC,BD " HAD,BVC " HBVC,AD> " ^ ^ } ^ ^ + term I n v o l v i n g V ' f R The expression for the Cutkosky matrix elenent cf the vertex operator, [mV], fellows the above derivation s t a r t i r g at equation (E.20) with V replaced by [mv]. The s e c c E d and t h i r d natrix elements i n equation (E.20) w i l l not te present in t h i s derivation which w i l l r e s u l t in the unit operator being included in a l l the summations of the r e s u l t i n g expression (the unit operator i s included because equation (3.21) i s used rather than equation (3.20)). Since the energy terms come from manipul c ticts preceding equation (B.20) the r e s u l t f c r the vertex operator w i l l net include these terms. Thus the f i n a l expression f c r the natrix elenent cf the vertex operator [mV] i s {AB|[mV]|CD} = f(V A C ) B V D + VBVD.AC- VBC,AVD " VAVD,BC + VBD,AVC + VAVC,BD ~ V AD,BVC " V B V C , A D ) " [mV ] JCD} + term i n v o l v i n g [mV^] where v I <A|P +QR|o<B|Q +P[mVR] |D>/n(P) ! n ( Q ) ! ( E . U U ) AC.BVD R > ( ^ p and V A V C , B D = ^ < A | [ m R V ] P + Q | 0 < B | R V p | D > / n ( P ) : n ( Q ) ! ( £ ^ 5 ) R , Q , P E. 3 Ejjci ted Overlac Matrix B le ne n t s Matrix elements between excited Cutkosky states i r v c l v e some addit i o n a l complications. However the tasic method cf expanding these matrix elements i c t c matrix elenects cf one nuclecn states i s the same as for the unexcited Cutkosky states. The overlap matrix elenent calculated here w i l l be between 76 two singly excited Cutkosky states. Since t h i s c a l c u l a t i o n exhibits a l l the complicating features of overlap matrix elements between excited states, the cverlap matrix element between any Cutkcsky states can be calculated using methods based on the following c a l c u l a t i o n s . Using the d e f i n i t i o n cf singly excited Cutkosky states the matrix element can be written { ABm| CDk} = <o | B A m k V V | o> + <o |BAm(Ck) V " |o> + <o|BAmC+(Dk)*|o> IB.16) The f i r s t matrix element cn the right hand side cf the atove equation can be written and matrix elements involving only one meson operator can be written, for example. Thus every matrix element i n equation (E.46) can re written as <o|BAm(Ck) V | o > = <o|BA(Ck)^[m,D+] |o> + <o | BA [m, (Ck) + ] D + | o> (B.48) 4) can a l l be expressed in the form F, + = I f,(FM)F+M+ F,M and F + and W+ s a t i s f y equations (2. 18)-(2.18). (B.U9) 77 Thus a l l matrix elements r e s u l t i n g from equation (E.46) ry usinq equations (E.47) and (B.48) and similar eguaticns can be expanded in terms of one nucleon matrix elements using the tecniques of section B . 1 . The resu l t i n q expression can be written {ABm|CDk} = 6 {AB|CD} + £ < A | Q + R | c > { - 6 , < B | R + Q | D > + <Bm|R +Q|Dk> km R , Q ^ + <Bm|R +[k +,Q]|D> + <B|[R+,m]Q|Dk> + <B|[R +,m][k +,Q]|D>} + I <Am|Q+R|c> + <A|[Q +,m]R + |c>}{<B |R +Q|Dk> + <B |R +[k +,Q]|D>} R , Q + antisym. ( E . 50) The sign of the summations in the above equation i s determined by noting that the expressions w i l l be antisymmetric with respect to exchange cf A and B cr C and D. when the matrix elements are s i m p l i f i e d i n the above equation ty use cf equations (3. 24) - (3. 26) , the <5-functicn i n the summations w i l l vanish and a l l the terms i n equation (E.50) w i l l te able to re written as matrix elements involving one nuclecn states and/or pion-nuclecn scattering states and vertex operators. E.4 Excited Interaction Matrix Elements The int e r a c t i o n matrix element calculated here w i l l be between a si n g l y excited Cutkosky state and an unejcited Cutkosky state. The more complicated matrix elements w i l l not be given here since they are not used i n the calcu l a t i o n s cf the pion-deuteron T-matrix. However the methcd of evaluating them i s 78 s i m i l a r t o t h e t e c h n i q u e s u s e d b e l o w and i n s e c t i c r B . 3 . U s i n g t h e d e f i n i t i o n o f C u t k o s k y s t a t e s t h e m a t r i x e l e i r e t t o f t h e H a m i l t o n i a n b e t w e e n an e x c i t e d a n d a n u E e > c i t e d s t a t e c a n t e w r i t t e n {AB|H|CDk} = -<o|BAHC+D+k+|o> + <o|BAH(Ck)+D+|o> + <o|BAHCt(Dk)t|o> ( E . 5 1 ) T h e s e c o n d a n d t h i r d t e r m s a b o v e c a n be e v a l u a t e d u s i n g t h e m e t h o d s o f s e c t i o n B . 2 . The r e s u l t s a r e <o|BAH(Ck)+D+|o> = -|(EA+EB+Ec+ED+Ek)<o|BA(Ck)fD+|o> + I <A|M+QR|ck><B|Q+M[VR] |D>/n(Q)'.n(M) ! R Q,M I <A|R+Q+M|Ck><B|[RV]M+Q|D>/n(Q):n(M)! ( E . 5 2 ) R Q , M . + antisyra. <o|BAHC+(Dk)+|o> = ^ E A + E B + E C + E D + E k ) < 0 ' B A C + ( D k ) + ' 0 > + i J ' I <A|M+QR|C><B|Q+M[VR]|Dk>/n(Q)!n(M)! 2 R Q,M + Z <A|R+Q+M|C><B| [RV]M+Q|Dk>/n(Q):n(M)! R Q,M ( E . 5 3 ) + antisym. The f i r s t t e r m i n e g u a t i o n ( E . 5 1 ) c a n be w r i t t e n <o|BAHC+D+k+|o> = <o|B[A,k+]HC+D+|o> + <o|[B,k+]AHCfD+|o> + Ek<o|BAk+C+D+|o> + <o|BA[Vk]C+D+|o> ( E • 5") A l l t h e a b o v e t e r m s c a n be e v a l u a t e d u s i n g t h e n e t h c d s c f t h e p r e v i o u s s e c t i o n s . The o n l y d i f f e r e n c e s a r e t h a t t h e e n e r g y t e r n i n t h e f i r s t two m a t r i x e l e m e n t s a k c v e b e c c n e s i(E 4+E -E,+E„+E_) 2 A B k C D and there are extra matrix elements involving [Vk] since 79 <o|[A,k+]H = (E A-E f c) <o|[A,k+] - <o|A[Vk] ( E . 5 5 ) The f i n a l r e s u l t f c r the Hamiltonian matrix element i s {AB|H|CDk} = |(E A+E B+E c+E D+E k) {AB]CDk} + T I' I <A|M+QR|Ck><B|Q+M[VR]|D> + <A|R+Q+M|Ck><B|[RV]M+Q|D> R Q,M + <A|M+QR|C><B|Q+M[VR]|Dk> + <A|R+Q+M|C><B|[RV]M+Q|Dk> - <A|k+M+QR]C><B|Q+M[VR]|D> - <A|k+RfQ+M|C><B|[RV]M+Q|D> - <A|M+QR|c><B|k+Q+M[VR] |D> - <A|R+Q+M|o<B|k+[RV]M+QlD> - |<A|M+QR|C><B]Q+M[VkR]|D> - -|<A|R+Q+M]C><B|[RVk]M+Q|D> + jl I -<A|M+QR|C><B|QfM[VkR] |D> - <A|R+Q+M|C><B| [RVk]M+Q|D> 4 R Q,M ( E . 5 6 ) + ant isym. Although i t appears as though the l a s t two terms w i l l give zerc meson exchange terms, these terms w i l l cancel with seme cf the cne mescn exchange terms. The derivation for the matrix elements cf the vertex operator [mVJ between excited and unexcited or excitec Cutkcsky states w i l l not be given here as i t can be evaluated using the technigues of section B . 2 . 80 APPENDIX C T h e T - m a t r i x i n t h e Q n e ^ M e s o n Ejcchanqe A p p r o x i m a t i o n I n s e c t i o n 4 . 2 , t h e T - m a t r i x i n t h e o n e - m e s o n e x c h a n g e a p p r o x i m a t i o n was w r i t t e n as b e l o w (1) _ + - + Vm.flk " T d i f f + T d i f f + Tsame + Tsame (C. 1) The t e r m s o n t h e r i g h t h a n d s i d e o f t h e a b o v e e q u a t i o n a r e w r i t t e n i n t e r m s o f o n e - n u c l e o n m a t r i x e l e m e n t s i n t h e f o l l o w i n g e q u a t i o n s . " I I A B C D ^ l l A B X C D FGq x ( <A|[mV]|F><F +<A| [mV] [ F x F +<A|[Vq]|F><F +<A|[Vq]|F><F d i f f = 2 EABCD ^ J ^ ™ FGq x ( <A| [mV] | F x F +<A| [mV] | F x F +<A|[qV]|F><F +<A|[qV]|F><F V > x o [Vq] |C><B| [Vk] |G><G| [qV] |D> a ^ E ) [Vq]|C><B|[qV]|G><G|[Vk]|D> a 2 ( E ) [mV]|C><B|[Vk]|G><G|[qV]|D> a , (E) [mV]|C><B|[qV]|G><G|[Vk]|D> a 4<E) ) V > x o [ q V ] | C x B | [ V k ] | G x G | [ V q ] | D > a 5 ( E ) [qV]|C><B|[Vq]|G><G|[Vk]|D> a g ( E ) [mV]|C><B|[Vk]|G><G|[Vq]|D> a ? ( E ) [raV]|C><B|[Vq]|G><G|[Vk]|D> a g ( E ) ) ( C 2 ) (C .3) T + = i yA„™<p' |AB)(CD|P > x same 2 ^ ABCD o 1 1 o FGq ( <A|[Vq]|C><B| [Vk] | F x F | [qv] GxG| [mV] D> a 9 ( E ) +<A|[Vq]]C><B [qV]|F><F [Vk] G><G| [mV] D> a i o ( E ) +<A|[Vq]|C><B| [mV]|F><F| [Vk] G><G [qv] D> a n ( E ) +<A|[Vq]|C><B [mV] | F x F [qv] G x G [Vk] D> a 1 2 ( E ) +<A|[Vq]|C><B [qV]|F><F [mV] G><G [Vk] |D> a 1 3 ( E ) +<A|[Vq]|C><B [Vk]|F><F [mV] | G><G [qv] |D> a u ( E ) ) same F G q x (<A|[qV]|C><BJ[Vk] +<A|[qV]|C><B|[Vq] +<A|[qV]JC><B|[mV] +<A|[qV]|C><B|[mV] +<A|[qV]|C><B|[Vq] +<A|[qV]|C><BJ[Vk] F><F|[Vq] G><F|[mV] D> a 1 5 ( E ) F><F|[Vk] G><G|[mV] D> a l f i(E) F><F|[Vk] G><F|[Vq] D> a 1 ? ( E ) F><F|[Vq] G><G|[Vk] D> a l g ( E ) F><F|[mV] G><G|[Vk] |D> a 1 9 ( E ) F x F J [mV] |G><G|[Vq] |D> a 2 Q ( E ) ) where L 1 ( E ) ~ v ^ v ^ ) ^  v 2 " v 3 ( v 1 + v 2 + v 3 + v 5 ) ; (C.5) (C.6) a 2 ( E ) = a 3 ( E ) -2 v 1 + v 5 v l V 3 ( v 2 + V 3 ) ^ V i + V 2 + V 3 + V 5 ^ / 1 V l + V 2 + V 3 \ \ V 2 V V V V V / v 1 ( v 2 + v 3 ) (C.7) (C.8) 82 2v.4v. a (E) = + — 4 V 1 V 3 ^ V 2 + V 3 ^ V 1 + V 2 + V 3 + V 4 ^ (C9) a 5 ( E ) = V 1 V 2 ( V 2 + V 3 ) (CIO) l / 2v 1 +v 5 V E ) = v l ( v 2 + v 3 ) ( v 1 + v 2 + V 3 + v 5 ) V v 3 + . 2 v r v 2 - v 3 \ V 9 / ( C l l ) a 7 ( E ) = / . . 7 v 1 v 2 ( v 2 + v 3 ) (C.12) a 8 ( E ) = v l ^ 2 ^ 3 ^ ^ V l + V 2 + V 3 + V 4 _ ( ^ 1 * 4 + V r V 2 " V 3 \ CC13) a 9 ( E ) v 1( V2+v 3) ( I, V l + V 2 + V 3 2 V V V V V ) ( C 1 4 ) a i o ( E ) " — ( 1 v i v 3 V v : 2 v l * 4 2 (V 2 +v 3 ) ( v 1 - W 2 ^ 3 + v 4 0 ( C 1 5 ) a n ( E ) 1 f 2 V 1 ( V 2 + V 3 ) V v 2 2 v 2 + 2 v 3 - W 5 v 3 ( V l + v 2 + v 3 + v 5 ) ) (C.16) a 1 2 ( E ) = v l v 3 ^ v 2 + v 3 ^ ( V 1 + V 2 + V 3 + V 5 ^ ( C 1 7 ) a 1 3 ( E ) = 1 V1 V2 V3 (C.18) 83 a, . (E) = — 1 4 V 1 V 2 V 3 (C.19) a 1 5 ( E ) v 1 v 2 ( v 2 + v 3 > (C.20) a l f i ( E ) V l ( v 2 + V 3 ) ( v l + V 2 + V 3 + V 4 ) > V 3 3 V 1 + V 2+Y 3 + 2 V 4 2 V j + v V7 I (C.21) a 1 7 ( E ) V 1 V 3 ( V 2 + V 3 ) CC22) a 1 8 ( E ) 2v 2 +2v 3 +v 5 V 1 ( V 2 + V 3 ) ( V 1 " W 2 " W 3 + V 5 ) ^ V -V r V 2 ~ V 3 \ ( C 2 3 ) a 1 Q ( E ) = — i 19 V l v 2 v 3 (C.24) CC.25) . . , v 5 a r e o b t a i n e d a s o u t l i n e d The e n e r g y f u n c t i o n s b e l o w . - I f t h e v e r t e x i n v o l v i n g t h e mescn m ( t h e f i n a l e m i t t e d meson) i s r e m o v e ! f r o m e a c h d i a g r a m , t h e n a l l d i a g r a m s w i l l a p p e a r s i m i l a r t o t h e d i a g r a m b e l o w . B — a | c u 84 The energy function i s given by v i = E F i ~ E l i ( C ' 2 6 ) where the indices i are as spec i f i e d in the diagram above and where E^j, = sum of energy of p a r t i c l e s going i n t o vertex i and E . = sum of energy of p a r t i c l e s coming from ver tex i . r i -Note that since time i s assumed to " be increasing towards the l e f t i n the above diagram, E p i corresponds to the energy cf the pa r t i c l e s to the l e f t of the vertex i and E ^ corresponds tc the energy cf the p a r t i c l e s to tha right of the vertex i . Since the functions v^, v 2 , and v^ can vanish, a term -ie should be added tc each of these energy denominators. This w i l l insure that the meson at each of these vertices w i l l cbey the appropriate boundary conditions. 85 APPENDIX D D . 1 T h e D e u t e r o n Way_ef u n c t i o n T h e d e u t e r o n w a v e f u n c t i o n (AB|P > i s r e l a t e d t o t h e c o n v e n t i o n a l d e u t e r o n monientuni s p a c e w a v e f u n c t i o n o f M c M i l l a n and L a n d a u ( 1 9 7 4 ) i n t h e f o l l o w i n g way (AB|t>o> = ^ V V V 6 (%>~-} * (£*WM ) (C*1) where K = k A + ( C 2 ) j c - l ( k A - k B ) (D.3) and M i s t h e p r o j e c t i o n o f t h e d e u t e r o n s p i n . T h e f u n c t i o n i s t h e s i n g l e t i s o s p i n f u n c t i o n a s g i v e n by B l a t t a n d W e i s s k o p f ( 1 9 5 2 ) . The q u a n t i t i e s x a n d a r e p r e s e n t t h e i s o s p i n a n d s p i n , r e s p e c t i v e l y , o f t h e n u c l e o n s . I t w i l l p r o v e t o be c o n v e n i e n t t o s e p a r a t e t h e s p i n f u n c t i o n s f r o m t h e a b o v e d e u t e r o n w a v e f u n c t i o n . T h u s t h e w a v e f u n c t i o n w i l l be w r i t t e n * ( < > V V M ) = E * * t ( - X l * C ° A , a B ) < D ' 4 > where x 1 £ i r e t h e t r i p l e t s p i n f u n c t i o n s a s g i v e n by B l a t t a n d W e i s s k o p f (1 952) a n d • J . fe) - U GO Y O O <*) * m + W (<) Y 2 > M _ £ m * (D.5 ) 86 T h e f u n c t i o n s Y ( £ ) a r e s p h e r i c a l h a r m o n i c s o f the u n i t v e c t o r mn H a n d t h e c o e f f i c i e n t i n t h e l a s t t e r m i s t h e c o n v e n t i o n a l C l e b s c h - G o r d a n c o e f f i c i e n t . T h e f u n c t i o n s U a n d W a r e t h e S a n d D s t a t e momentum s p a c e w a v e f u n c t i o n s u s e d by M c M i l l a n a n d L a n d a u (1974) . D . 2 S i n g l e - S e a t t e r i n g _ In s e c t i o n 4 . 1 t h e p i o n - d e u t e r o n s i n g l e - s c a t t e r i n g T - m a t r i x i s w r i t t e n T£tek = ^A,B,C < P J a b ) < A I r m V H c k > <CBlV ( D * 6 ) D e l t a f u n c t i o n s o f momentum c a n be e x t r a c t e d f r o m t h e p i o n -d e u t e r o n and t h e p i o n - n u c l e o n T - m a t r i c e s . By e x p l i c i t l y w r i t i n g the a b o v e s u m m a t i o n as a p r o d u c t o f sums e v e r s p i n a n d i s o s p i n and a n i n t e g r a l o v e r momentum, t h e a b o v e e g u a t i o n c a n be w r i t t e n +4 -Ms d 3 k A d 3 k d 3 k „ <V »|AB)T(Am,Ck) A B t» o x (CB|P r t> 5 (kA+EQ.-kc-]S:) (D.7 ) • 5 ( k . + m - k „ - k ) o U s i n g e q u a t i o n s (D. 1) a n d (D.4 ) a n d u s i n g t h e r e s u l t i n g d e l t a f u n c t i o n s t o e l i m i n a t e two o f t h e i n t e g r a l s , e g u a t i o n (D.7 ) c a n be e x p r e s s e d i n t h e l a b o r a t o r y f r a m e a s 87 ,(0) +1 +4 d 3 K ^ ' (jc+|(k-m)) c^ OO 6(^,+m-k) X 2 X , V T A ' W V T B } W V " 1 T(kA,m; k^k; a ^ ; VVV^) ( C . 8 ) where k A = jc + k-m (C.9) kg --jc (D.10) The sums o v e r i s o s p i n c a n now be d o n e a n d t h e f i n a l r e s u l t w r i t t e n i n t h e f o r m T L 0 ) % ' » B ; 0,k; M ' , M ) 5 (k^.+m-k) (E. 11) +1 - I £ t A ' = - l where TTtl i and where T ^ , i s t h e same q u a n t i t y w i t h N T A = T C = ~ 2 I t s h o u l d be n o t e d t h a t t h e p i o n - n u c l e o n T - r a a t r i x i n e q u a t i o n (D. 12) r e f e r s t o t h e l a b o r a t o r y f r a m e . T h i s c a n be r e l a t e d t o t h e p i o n - n u c l e o n T - m a t r i x i n t h e p i o n - n u c l e o n CM f r a m e u s i n g e q u a t i o n (B.7) o f M c M i l l a n a n d L a n d a u (1974) . 88 D.3 D o u b l e t s c a t t e r i n g I n s e c t i o n 4 . 2 t h e p i o n - d e u t e r o n d o u b l e - s c a t t e r i n g T - m a t r i x i s w r i t t e n = — T T i <P ,|AB)<A|[mV]|Cq><B|[qV]|Dk>(CD|P >6(E +E - E D - E k ) P ' m ' t ? k A , B , C , D , q ° ( E . 1 3 ) U s i n g t h e same t e c h n i q u e s a s i n s e c t i o n D . 2 f t h e a b o v e e q u a t i o n can be w r i t t e n T T ( 1 ) (k^. 0 , k ; M' ,M) «S(k ,+m-k) L V V ( D . 1 4 ) +1 f ' * = -TTi I A,A»=-1 J d 3 K d 3 q (K+q-|(k - t n ) ) <^(K) T £ t £ 6(k p,-to-k) where T n = 2 I V V V V V V ^ X l i ' ( V f f B ) X U ( o C ' V T A , T B t ! A ' X - J , ( D . 1 5 ) +1 T =-1 q a n d k A = £ + £ - m ( E . 1 6 ) k B = - J i + k - a ( D . 1 7 ) ~~G = _ ~ D = ~— ( E . 18) A s s t a t e d i n t h e p r e v i o u s s e c t i o n , t h e p i o n - n u c l e o n T -m a t r i c e s i n t h e l a b o r a t o r y f r a m e c a n be r e l a t e d t o t h e T -t n a t r i c e s i n t h e p i o n - n u c l e o n CM f r a m e u s i n g e q u a t i o n ( E . 7 ) c f M c H i l l a n a n d L a n i a u ( 1 9 7 4 ) . 

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