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UBC Theses and Dissertations

The dressing transformation and scattering of dressed particles in a model of fermions and bosons James, Howard Nelson 1982

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T H E D R E S S I N G T R A N S F O R M A T I O N A N D T H E S C A T T E R I N G O F D R E S S E D P A R T I C L E S I N A M O D E L O F F E R M I O N S A N D B O S O N S b y HOWARD N E L S O N J A M E S B . S c , T h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , 1 9 8 0 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T O F T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F M A S T E R O F S C I E N C E i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S D E P A R T M E N T O F P H Y S I C S We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A A u g u s t 1 9 8 2 © H o w a r d N e l s o n J a m e s , 1 9 8 2 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference and study. I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of ftd^s/ds  The University of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 D a t e Sy,/- 7 f??^ i i Abstract This thesis reviews a method in quantum f i e l d theory to y i e l d models dealing e x p l i c i t l y with physical p a r t i c l e operators. The method involves a unitary transformation known as a dressing transformation which transforms the o r i g i n a l p a r t i c l e creators into dressed physical p a r t i c l e creators. A dressing transformation i s applied to a system of fermions and. bosons interacting v i a a t r i l i n e a r Hamiltonian similar to the one found in The Cloudy Bag Model. The model i s then specialized to describe systems of physical N-particles ('nucleons' without spin and isospin) and physical / T-particles ('pions' without is o s p i n ) . Our model i s shown to describe the set of possible interactions involving N and 77-particles including T T-particle production and a n n i h i l a t i o n . In considering e l a s t i c N<7 scattering, an expression for the T-operator i s derived when the incident free p a r t i c l e s have a t o t a l energy near the energy of a ..-particle. Equations for a two N-particle bound state, the d-pa r t i c l e ('deuteron') are given and using a set of modified Faddeev equations, we study the reactions occurring within the three channels NN, NN and ff&. i i i TABLE OF CONTENTS Abs t r a c t i i L i s t Of Figures • — v Acknowledgements v i Chapter 1 I n t r o d u c t i o n 1 Chapter 2 The Fundamental Dynamical V a r i a b l e s And Hamiltonian 4 2.1 The Fundamental Dynamical V a r i a b l e s 4 2.2 Transformation P r o p e r t i e s Of The Fundamental Dynamical V a r i a b l e s 6 2.3 The Hamiltonian 9 Chapter 3 The Dressing Transformation And I t s A p p l i c a t i o n to The Hamiltonian 13 3.1 The Dressing Transformation 13 3.2 A p p l i c a t i o n To The Hamiltonian 17 Chapter 4 The Hamiltonian In The P h y s i c a l H i l b e r t Space . 23 4.1 The P h y s i c a l H i l b e r t Space 23 4.2 The Hamiltonian In The P h y s i c a l H i l b e r t Space 25 Chapter 5 P h y s i c a l N- S c a t t e r i n g 31 5.1 The Hamiltonian For Ph y s i c a l . N ~ S c a t t e r i n g 31 5.2 The T-Operator 33 Chapter 6 Systems I n v o l v i n g Two N - P a r t i c l e s 36 6.1 The Hamiltonian For Systems Containing Two N - P a r t i c l e s 37 6.2 The Channel Hamiltonians 40 6.4 The M o d i f i e d F a d d e e v E q u a t i o n s ... C h a p t e r 7 Summary And C o n c l u s i o n s B i b l i o g r a p h y A p p e n d i x A C a l c u l a t i o n Of The D r e s s e d H a m i l t o n i a n A p p e n d i x B The F u n c t i o n V^W**%') Ap p e n d i x C B a s i c S c a t t e r i n g T h e o r y A p p e n d i x D The d - p a r t i c l e A p p e n d i x E The M o d i f i e d F a d d e e v E q u a t i o n s A p p e n d i x F Some M u l t i c h a n n e l S c a t t e r i n g T h e o r y .. V LIST OF FIGURES F i g u r e 1 F i g u r e 2 F i g u r e 3 F i g u r e 4 F i g u r e 5 F i g u r e 6 F i g u r e 7 F i g u r e 8 F i g u r e 9 F i g u r e 10 F i g u r e 11 F i g u r e 12 F i g u r e 13 F i g u r e 14 F i g u r e 15 F i g u r e 16 F i g u r e 17 F i g u r e 18 F i g u r e 19 F i g u r e 20 F i g u r e 21 F i g u r e 22 F i g u r e 23 F i g u r e 24 F i g u r e 25 The Hf I n t e r a c t i o n 10 The I n t e r a c t i o n 10 The P h y s i c a l N-N P o t e n t i a l 26 The I n t e r a c t i o n Term V„ 26 v,*d.«'g) 27 v;/*-*'/T; 27 The I n t e r a c t i o n Term VlB 27 The I n t e r a c t i o n Terms V//0 And V,?B 29 The I n t e r a c t i o n Terms V,'iB And V*s 29 The I n t e r a c t i o n Terms V,£ And 30 The I n t e r a c t i o n Terms V„|B><B| And Y 6|F><F| ... 40 The O p e r a t o r s T, , T A , T, And 1¥ 46 The O p e r a t o r %„ . 47 The Operator T^ 48 The Op e r a t o r % B 49 The Op e r a t o r Tu 49 The Operator % d 50 VfttcK'n-fc) 60 ) 6 1 6 2 Vf* *:i<",K) 6 3 64 VJk . 65 Vt 6 6 The d - p a r t i c l e 76 Acknowledgements I would l i k e to thank my supervisor Dr. Malcolm McMillan for his constant support, input and patience. I would also l i k e to thank the National Research Council of Canada for i t s generous f i n a n c i a l assistance. 1 Chapter 1_ Introduction One of the present problems in the o r e t i c a l nuclear physics i s finding an accurate and e f f e c t i v e theory of interacting nucleons and pions. Current models involve the use of boson exchange potentials which imply that the underlying interaction Hamiltonian i s t r i l i n e a r . In t h i s thesis we use a t r i l i n e a r Hamiltonian and formulate a technique to find potentials for interactions betwen physical N-particles ('nucleons' without spin and isospin) and physical ^ - p a r t i c l e s ('pions' without i s o s p i n ) . We show that our model yi e l d s the set of possible interactions involving only physical N and r r - p a r t i c l e s including 7 T-particle production and an n i h i l a t i o n . The thesis begins with a description of systems of bosons and two types of fermions interacting v i a a t r i l i n e a r Hamiltonian. The fundamental dynamical variables of our system are the fermion creators F/ (p_) and F*(p_), the boson creator B*(g) and the i r adjoint operators. For s i m p l i c i t y we neglect spin and isospin. The t r i l i n e a r interaction i s given by This in t e r a c t i o n i s similar to the one used to describe systems of nucleons and pions in The Cloudy Bag Model by Theberge, Thomas and M i l l e r , 1980, who were able to re-express pion-quark interactions in terms of pion-baryon t r i l i n e a r i nteractions. One important feature of our model i s that the fermion kets |0> and F^|0> are not eigenkets of the Hamiltonian. We refer t o such fermions as ' u n p h y s i c a l ' . For s i n g l e p a r t i c l e kets that are eigenkets of the Hamiltonian, we c a l l the p a r t i c l e ' p h y s i c a l ' . In Chapter 3 we transform our model i n order to work with only p h y s i c a l p a r t i c l e operators. This i s done by f i n d i n g a u n i t a r y transformation known as a d r e s s i n g t r a n s f o r m a t i o n , which when a p p l i e d t o the o r i g i n a l p a r t i c l e c r e a t o r s F/ Fj 1 and Bf r e s u l t s i n a set of p h y s i c a l 'dressed' p a r t i c l e c r e a t o r s ?/, F/ and B*. The Hamiltonian i s re-expressed i n terms of the transformed operators as a p e r t u r b a t i o n s e r i e s i n the i n t e r a c t i n g c o u p l i n g constant. In Chapter 4 we s p e c i a l i z e our model to d e s c r i b e systems of 'nucleons' and 'pions' and d e s c r i b e the a c t u a l H i l b e r t space f o r such p a r t i c l e s . We r e f e r t o the operators F/, F/ and B1* as c r e a t i n g a bare N, _ and 77"-particle r e s p e c t i v e l y . We r e f e r to as c r e a t i n g a p h y s i c a l N - p a r t i c l e or 'nucleon' and fT* as c r e a t i n g a p h y s i c a l i V - p a r t i c l e or 'pion'. The Hamiltonian, w r i t t e n i n terms of the dressed operators t o second order i n the c o u p l i n g constant i s shown to i n v o l v e p h y s i c a l N-N and N- 7f~ i n t e r a c t i o n terms. T h i r d order items i n v o l v e 77*-particle production and a n n i h i l a t i o n (NN«-*tfNN and N<T«-»N7T77). P o t e n t i a l s f o r these i n t e r a c t i o n s are found i n terms of; the vertex f u n c t i o n s h- and h^ . In the next chapter we look at the e l a s t i c s c a t t e r i n g of one p h y s i c a l 7 T-particle and one p h y s i c a l N - p a r t i c l e i n the region where the i n c i d e n t f r e e p a r t i c l e s have a t o t a l energy near the energy of a bare ^ - p a r t i c l e . In Chapter 6 we introduce the d - p a r t i c l e Cdeuteron') which i s f u l l y described in Appendix D and use a set of modified Faddeev equations to study the T-operators for reactions occurring within the three channels NN, NN77" and /Td. The Appendices provide a description of some of the cal c u l a t i o n s done in the thesis and also review some basic scattering theory. 4 Chapter 2 The Fundamental Dynamical Variables and Hamiltonian The purpose of t h i s chapter i s to formulate a Hamiltonian to describe systems of bosons and two types of fermions intera c t i n g via a t r i l i n e a r i n t e r a c t i o n . For s i m p l i c i t y we consider a l l the p a r t i c l e s to be spinless, however, one could e a s i l y extend the system to contain spin and isospin. (The reader may refer to Hearn (1981) for systems including spin and isospin.) Our model w i l l also not contain any a n t i p a r t i c l e s . 2.1 The Fundamental Dynamical Variables The fundamental dynamical variables are the boson creator B^g) and the fermion creators F,£(rj) (=*=1,2) as well as their adjoints. The operator F^(p_) creates a type fermion of momentum g, while the adjoint operator F„(p_) annihilates such a fermion. S i m i l a r l y the operator B f(g) (B(q)) creates (annihilates) a boson of momentum g. Equivalently we can also use the f i e l d operators F^(x), F j x ) , B^(x) and B(x) as the fundamental dynamical variables. We define Fjf(x) and Fj(x) to respectively create and annihilate a type « fermion at the position S i m i l a r l y BMx) and B(x) respectively create and annihilate a boson at x. The p a r t i c l e creation operators are connected by a fourier transform The operators s a t i s f y the following equations (2.1.1) (2.1.2) (2.1.3) (2.1.4) where |0> i s the vacuum state. The operators also obey the following commutation and anticommutation r e l a t i o n s . (2.1.5) (2.1.6) (2.1.7) (2.1.8) where [A,B]=AB-BA, {ArB}=AB+BA and % represents either x or £. 6 2.2 T r a n s f o r m a t i o n P r o p e r t i e s o f t h e F u n d a m e n t a l D y n a m i c a l  V a r i a b l e s F o r s p a t i a l t r a n s l a t i o n s o f t h e s y s t e m by t h e v e c t o r a t h e r e i s a l i n e a r u n i t a r y o p e r a t o r D ( a ) w h i c h g i v e s (2.2.1) (2.2.2) Die) £%) = ^ (zrz) U s i n g (2.1.1) and (2.2.2) we g e t (2.2.3) D(g) tfMitf*)* eT*'a/k£ • ' a U (2.2.4) 0(a) 3%)D%) - £ " * ' 9 A 0ty) T h e r e i s a l s o a l i n e a r u n i t a r y o p e r a t o r ftl^^s) f o r a s p a t i a l r o t a t i o n of t h e s y s t e m by t h e E u l e r a n g l e s °$/B;tV • (See Rose (1957) f o r t h e p a r t i c u l a r c o n v e n t i o n u s e d ) . Under r o t a t i o n s t h e o p e r a t o r s t r a n s f o r m a s where a g a i n c a n be e i t h e r x o r £ and (2.2.5) (2.2.6) (2.2.7) (2.2.8) where (2.2.9) 7 with ( 2 . 2 . 1 0 ) /Cos** Sin-* o j /Cosfi ° ~S/'/ip | MH~( Sto* Cos** O o i o \ O o I J [ S/'n O Cosja> J f Cos r S/n r 0 ) [ o o / J For space inversion there i s a l i n e a r unitary operator such that ( 2 . 2 . 1 1 ) ( 2 . 2 . 1 2 ) We refer to the fermions as having p o s i t i v e i n t r i n s i c p a r i t y and the bosons as having negative i n t r i n s i c p a r i t y . F i n a l l y for time reversal of the system we have an a n t i l i n e a r antiunitary operator with ( 2 . 2 . 1 3 ) ( 2 . 2 . 1 4 ) ( 2 . 2 . 1 5 ) ( 2 . 2 . 1 6 ) 8 The p o s i t i v e and n e g a t i v e c o e f f i c i e n t s i n e q u a t i o n s (2.2.13) t o (2.2.16) a r e d e t e r m i n e d by r e q u i r i n g o v e r a l l TCP ( t i m e r e v e r s a l , charge c o n j u g a t i o n and space i n v e r s i o n ) i n v a r i a n c e (see Schweber (1961, p. 2 6 8 ) ) . F u r t h e r i n f o r m a t i o n on t h e t r a n s f o r m a t i o n p r o p e r t i e s of o p e r a t o r s which i n c l u d e s p i n and i s o s p i n can be found i n Hearn (1981). 9 2.3 The Hamiltonian We take the Hamiltonian H of the system t o be where H 0 i s given by (2.3.2) fl9 - [4 Zjf) 3«) B(fi)} w i t h (2.3.3) I*-and (2.3.4) w i t h (2.3.5) and (2.3.6) Notice that both H* and H,fc are t r i l i n e a r i n t e r a c t i o n s . (See Hearn (1981) f o r g e n e r a l i z e d t r i l i n e a r i n t e r a c t i o n s . ) A l s o note tha t the f r e e p a r t i c l e energies _><> and f^, are given r e l a t i v i s t i c a l l y . A complete r e l a t i v i s t i c model would r e q u i r e ten operators s a t i s f y i n g the Poincare algebra (see Hearn page 100). Through a p r i v a t e conversation w i t h Malcolm McMillan i t has been discovered that t o s a t i s f y the Poincare algebra we must have H, =0. The p o s s i b l e r e a c t i o n s f o r the system are v — r. f, + B 10 The'interaction terms are i l l u s t r a t e d as follows: F i * - ' ^ i d ' l i n ^ f iS. < ^io.s; dashed l i n e s are bosons. F i g . 2 The H/- Interaction S o l i d l i n e s are type 1 fermions; dashed l i n e s are bosons; thick lines are type 2 fermions. The vertex functions hL and hj depend only upon the momentum of the boson and not upon any fermion momentum. In order that H be invariant under D ,R, J and (P (see Hearn 1981) these functions must be real and can depend only upon the magnitude of the boson momentum. The above Hamiltonian i s a combination of the scalar f i e l d model and the Lee model. The scalar f i e l d model (containing only type 1 fermions and bosons)results when h^(q)=0, while the 11 Lee.model r e s u l t s when h < y(q)=0. In the l i t e r a t u r e on the Lee model, the type 1 fermion i s u s u a l l y r e f e r r e d to as the N p a r t i c l e , the type 2 fermion as the V p a r t i c l e and the boson as the O p a r t i c l e . A s i m i l a r combination of the s c a l a r f i e l d and Lee models (with spin and i s o s p i n ) i s used to describe nucleons and d e l t a p a r t i c l e s i n t e r a c t i n g with pions i n The Cloudy Bag Model by Theberge, Thomas and M i l l e r (1980). In t h i s paper the bare nucleon and bare d e l t a p a r t i c l e each c o n s i s t of three quarks confined w i t h i n a s p h e r i c a l s h e l l or bag. Coupled to each bag surface i s a pion f i e l d . T r e a t i n g the pion f i e l d as small and using the lowest order bag model quark wave f u n c t i o n s , Theberge et a l . are able to re-express the pion-quark i n t e r a c t i o n s i n terms of pion-baryon t r i l i n e a r i n t e r a c t i o n s . In the process they o b t a i n e x p l i c i t expressions for the t r i l i n e a r vertex f u n c t i o n s . Using t h i s model, Hearn (1981) has worked out the NN vertex f u n c t i o n w r i t t e n as h ^ ^ q ) , to be where t0 i s the NN/^coupling constant, ny i s the pion r e s t mass, , (2.3.8) (2.3.7) a n d (2.3.9) where i s a s p h e r i c a l B e ssel f u n c t i o n of order one and 12 R i s the bag radius. The following operators commute with the Hamiltonian and are thus constants of the motion: the t o t a l momentum P (2 .3 .10) (2 .3 .11) (2 .3 .12) (2 .3 .13) the t o t a l fermion number Nf The Hamiltonian H has the property that however for any function £(_)). That i s , single fermion kets are not eigenkets of the Hamiltonian. We refer to such fermions as 'unphysical'. Hence one says that Y + creates an unphysical type fermion. For single p a r t i c l e kets that are eigenkets of the Hamiltonian, we c a l l the p a r t i c l e s 'physical'. In the next chapter we show a technique to y i e l d systems containing only physical p a r t i c l e s . We then apply t h i s technique to our model. 13 Chapter 3 The Dressing Transformation and i t s Application to  the Hamiltonian To avoid d i r e c t l y working with unphysical p a r t i c l e s we w i l l re-express the Hamiltonian of Chapter 2 in terms of only physical p a r t i c l e operators. F i r s t we w i l l show a technique to fi n d a unitary transformation known as the dressing transformation. 1 We then apply a dressing transformation to the o r i g i n a l fundamental dynamical variables. The resulting operators w i l l be physical 'dressed' operators. The Hamiltonian w i l l then be re-expressed in terms of the transformed variables. We w i l l c a l l this re-expressed Hamiltonian the 'dressed Hamiltonian' . 3.1 The Dressing Transformation This section i s b a s i c a l l y a r e p e t i t i o n of Hearn, section 4.1 The General Dressing Transformation. F i r s t l e t the unitary operator U be given by (3.1.1) U-eD with (3.1.2) 'See Hearn (1981) Chapter 4. The Dressing Transformation and Some Simple Applications and references therein. 14 The operator D i s a function of the fundamental dynamical variables. That i s where for brevity F represents a l l types of fermions New creators F*(r_) and B^rj) are given by (3.1 .3) (3.1 .4) (3.1.5) The symbol ~ i s to be used for a l l transformed operators. The new operators F and "B obey the same commutation and anticommutation relations as the o r i g i n a l operators F and B. By requiring D to be invariant under translations, s p a t i a l rotations, space inversions and time reversal, F and B w i l l obey the same transformation rules as the o r i g i n a l operators. As we s h a l l show, the operator D w i l l be constructed in order that F Y and B* create physical p a r t i c l e s . Letting A=A(F,B) be any operator, then UA(rtd)UL+*A(ZB) In p a r t i c u l a r (3.1.6) (3.1.7) 15 (3.1.8) (3.1.9) (3.1.10) Using the above equations we define H(F,B) by The t o t a l momentum i s given by where since [P,D]=0. Equation (3.1.8) gives H in terms of H and D. Now we wish to choose D such that • (3.1.11) and (3.1.12) where £>(p) i s the energy of a free physical fermion of momentum £, and £ ^ ( 3 ) i s the energy of a free physical boson of momentum a -To s a t i s f y (3.1.11) and (3.1.12) we f i r s t expand equation (3.1.8) as 16 W r i t i n g the Hamiltonian as H=He+AH, and expanding the dr e s s i n g operator i n a power s e r i e s i n A as (3.1.14) enables us to w r i t e (3.1.13) as H-fL *Yft f[H.A]j^^([H,AMCHMAJAT} t- - '3''''5' In order to s a t i s f y equations (3.1.11) and (3.1.12), with the exception of the terms F*F and HTB, there must be no terms i n H(F,B) which have only one fermion or boson a n n i h i l a t o r . That i s , there can be no terms c o n t a i n i n g (3.1.16) however terms of the form (3.1.17) T+?+??t 3*8*88 andf+B + FB .. are p e r m i s s i b l e Thus we choose D„ such that t o the order of A", there are no terms i n equation (3.1.14) of the form (3.1.16). In the case that H i s t r i l i n e a r , D i s chosen so that (3.1.18) Knowing D, , we r e f e r back t o the 0 ( ^ ) term 1 from equation (3.1.14) t o solve f o r D,_. Repeating t h i s type of process gives D t o any order of / I . The u n i t a r y t ransformation d e s c r i b e d i n t h i s s e c t i o n i s to 17 be known as a dressing transformation. The operator D, chosen such that equations (3.1.10) and (3.1.11) hold, i s c a l l e d a dressing operator. We refer to the o r i g i n a l operators F + and as creating 'bare' p a r t i c l e s . The new creators F* and B y are said to create 'dressed' p a r t i c l e s . In the next section we apply a dressing transformation to the Hamiltonian of Chapter 2. 3.2 Application to the Hamiltonian Performing the dressing transformation on the Hamiltonian of Chapter 2 (see Appendix A) and summing up the dressed Hamiltonian H to second order gives where with where (3.2.1) (3.2.2) (3.2.3) (3.2.4) (3.2.5) (3.2.6) 18 (3.2.7) and £.0(p) and £B (p) are given by (2.3.3) and where (3 2 8) (3.2.9) (3.2.11) (3.2.12) and (3.2.13) A l l functions ^(kis\&), XJU'K) , V?(**'.£) and 0*X^V a r e determined in terms of h 5 , hL , , £,a and €B . Here H° i s the dressed free Hamiltonian with £/(p) and £jp) being renormalized energies, V w V i s the interaction term for F w +Fy-* F„+Fff and V„B i s the interaction term for % +B-» F w + B. The reaction F, +F, % +F]L i s described by V/;1' +VA" and the reaction F, +B^F, +B i s given by V,*a . Notice that to second order, there are no Fa +F„. F^  , F, +F, Fr or B+B-* §+B interaction terms in the dressed Hamiltonian. 19 The t h i r d order term in the dressed Hamiltonian i s given by a 7,4 7 y tffa. ^lo,]-t i[H,, QjtfaA] This contributes the following terms to H r _ (3.2.15) where (3.2.17) (3 2 18) and (3.2.19) The terms Vt_fB and V/^B correspond respectively to the an n i h i l a t i o n and production of one physical boson by two physical fermions, while the terms V^*B and V^BfiB correspond respectively to the a n n i h i l a t i o n and production of one physical boson by one physical fermion and one physical boson. For each physical two body interaction mentioned on page 18 there are higher order (0 (^* ) , 0 ( ; \ 6 ) , etc.) interaction terms corresponding to the same process. For example, in addition to the second order term Vu , there i s also a fourth order term corresponding to the process F, +Fj-» F, +Ff . 20 There are a l s o f o u r t h order terms of the form F*F^'Bf% F^  1, F^B fB^F t B IT and F^F^F^Fff F. which correspond to i n t r i n s i c p h y s i c a l three body i n t e r a c t i o n s . F i n a l l y , i n terms of the bare operators the dressed p a r t i c l e c r e a t o r s are given as (3.2.20) J A(W,r) (3.2.21) ^fc*') (3.2.22) Not i c e that (3.2.23) however (3.2.24) J and , , (3.2.25) In f a c t , i f we were t o continue (3.2.24) and (3.2.25) we would f i n d terms of the form F*B*B f, F y B V S * B y , e t c . We thus r e f e r t o the p h y s i c a l dressed fermion as c o n s i s t i n g of a bare fermion and a 'cloud' of bare bosons. 21 Using j u s t the s c a l a r f i e l d model, i f one assumes th a t the energy of a f r e e fermion i s independent of i t s momentum, that i s (3.2.26) then a corresponding d r e s s i n g operator f o r the model may be expressed e x a c t l y (Hearn 1 981 ) . W r i t i n g (3.2.27) 4 = f?fy one f i n d s (3.2.28) J W (3.2.29) DsO Choosing the vertex f u n c t i o n h 5(q) to be given by (3.2.30) then the second order p h y s i c a l fermion-fermion i n t e r a c t i o n term Vj! becomes Using equations (2.1.1) and (2.1.2) we can r e w r i t e as (3.2.32) That i s , two p h y s i c a l fermions i n t e r a c t v i a a Yukawa p o t e n t i a l . 22 A dressing operator for the Lee model may also be determined exactly. Piskunov (1974) has shown that such an operator i s given by (assuming m^<m/+m^) 4- j ^ d ^ f ^ ^ - ^ m - ^ / J where (3.2.33) (3.2.34) with (3.2.35) Furthermore the exact solution for the ket Fj£(p_)|0> in terms of bare operators i s given by (3.2.36) where (3.2.37) 23 Chapter 4 The Hamiltonian in the Physical H i l b e r t Space We now wish to sp e c i a l i z e our Hamiltonian to describe systems of 'nucleons' and 'pions'. We write both nucleons and pions in quotation marks since we are not including spin and isospin. In Chapter 3, we applied a dressing transformation to the o r i g i n a l fundamental dynamical variables in order to work only with physical p a r t i c l e s . In t h i s chapter we s h a l l specify the meaning of a l l the p a r t i c l e creators and describe the actual physical H i l b e r t space for systems of such 'nucleons' and 'pions'. 4.1 The Physical H i l b e r t Space From t h i s stage on, we w i l l refer to as creating a bare N-particle, F a s creating a bare - p a r t i c l e and as creating a bare7T-particle. We w i l l refer to the dressed operator F/ as creating a physical N-particle Cnucleon' ) . The dressed operator B w i l l be said to create a physical 7f-particle ('pion'). However, the dressed operator cannot s i m i l a r l y correspond to a 'delta' p a r t i c l e since the actual delta p a r t i c l e i s unstable and cannot therefore be described by an operator which forms an exact eigenket of the t o t a l dressed Hamiltonian as we have in equation (3.1.11). That i s , F^_ cannot describe an unstable p a r t i c l e . We wish to describe systems with only one type of stable fermion: the physical N-particle Cnucleon' ) , To th i s we w i l l consider 24 a c t u a l p h y s i c a l s t a t e s to c o n s i s t only of products of the two operators F; and B' a c t i n g on the vacuum s t a t e . So instead of working w i t h a H i l b e r t space which i s the d i r e c t product of F/ Fock space ), F^ Fock space ( fyj and % Fock space (fYs ), that i s j^/^fV^.® #8 , a l l p h y s i c a l processes w i l l be described i n the smaller H i l b e r t space rVy®#g. Hence a l l p h y s i c a l s t a t e s jfy have the f o l l o w i n g property (4.1.1) In p a r t i c u l a r (4.1.2) (4.1.3). (4.1.4) V^t 1+7*0 « or r-x We s t r e s s here that while we have e l i m i n a t e d F* and F^ from a l l p h y s i c a l q u a n t i t i e s , our model s t i l l c o n t a i n s the bare _ ^ - p a r t i c l e . For example kets c o n t a i n i n g one p h y s i c a l N - p a r t i c l e and one p h y s i c a l s - p a r t i c l e can be w r i t t e n r-. „ (4.1.5) Thus the bare A ~ p a r t i c l e (F-f) appears as a component of the p h y s i c a l N-ff system. Our approach i s s i m i l a r to that used i n quantum 25 electrodynamics where in addition to photons that have either p o s i t i v e (B 1^) or negative (B 2 , t) h e l i c i t y , f i c t i t i o u s 'timelike' (B 0^) and 'longitudinal' (B 3^) photons are introduced although physical states (f/ have (4.1.7) for^=0,3. 4.2 The Hamiltonian in the Physical H i l b e r t Space H', the projection of the dressed Hamiltonian onto the reduced Hilbert space fl,®H&is given to second order as ~ . \ (4.2.1) where (4.2.2) w i th (4.2.5) %<txMy L + I 7 26 F i g . ; 3 T h e P h y s i c a l N - N P o t e n t i a l T h e s o l i d l i n e s r e p r e s e n t t h e f e r m i o n s a n d t h e b r o k e n l i n e r e p r e s e n t s t h e b o s o n . F i g . - * : 4 T h e I n t e r a c t i o n T e r m a n d > ( 4 . 2 . 6 ) w h e r e ( 4 . 2 . 7 ) / F i g . 7 T h e I n t e r a c t i o n T e r m V / f l 28 The term V,, corresponds t o the i n t e r a c t i o n of two p h y s i c a l N - p a r t i c l e s . The term VIB corresponds to the i n t e r a c t i o n of one p h y s i c a l N - p a r t i c l e with one p h y s i c a l T T - p a r t i c l e . Note that i n ' Yil(t,*',£) i s t n e c o n t r i b u t i o n from the s c a l a r f i e l d model whereas VJC*,*', g) i s the c o n t r i b u t i o n from the Lee model. A l s o , we emphasize the importance of Figure 3 where the bare & -p a r t i c l e manifests i t s e l f as an intermediate s t a t e i n the p h y s i c a l N-7T i n t e r a c t i o n . In the f u n c t i o n ( ) there can be i n f i n i t i e s when m A o > m / 0 + m s ' I n such cases we w i l l change v,£ by modifying the energy denominators *lo(i£+*') + Z&faZ-*') ~ a n d + - ^(k) . For example, i n the center of mass frame (£=0) we modify the denominators as f o l l o w s ; Let be such that (4.2.9) Note that VfD(t,k.',°) goes to i n f i n i t y as e i t h e r k or k1 approaches k*. Now we introduce a new parameter <f, with cf >0. We modify ^(^a'/0) by r e p l a c i n g with whenever r < <T. S i m i l a r y we w i l l replace ^ 0( A /)+ ^ (*0-/nu,cr with £/g( **-</") + K*-f)-*Xt£ whenever *'< **+<f. That i s , we change the p o t e n t i a l i n the region of the s i n g u l a r i t i e s by making the energy denominators f i n i t e and constant whenever t h e i r arguments are between t?and A**/. We 29 w r i t e the modified f u n c t i o n as • As / approaches zero we obt a i n a very la r g e resonance i n the p h y s i c a l N-7T s c a t t e r i n g . Conversely, as /becomes large the resonance behavior vanishes. T h i r d order terms i n the dressed Hamiltonian which have n o n - t r i v i a l matrix elements i n the H i l b e r t space /^ #rVe are (4.2.10) The above terms, which give r i s e to the production and a n n i h i l a t i o n of one p h y s i c a l - ^ - p a r t i c l e , are d e f i n e d on page 19. F i g . 9 The I n t e r a c t i o n Terms V,J*B and V (f B 30 Fourth order terms which give r i s e to p h y s i c a l i n t e r a c t i o n s are, (4.2.11 ) where and y£' have the form of V/f and V/Q r e s p e c t i v e l y while v//_f< a n d v/a_f correspond to i n t r i n s i c p h y s i c a l three body i n t e r a c t i o n s . F i g . 10 The I n t e r a c t i o n Terms Vff* and V^*3 I f we continue on to higher order, i n a d d i t i o n to terms that d e s c r i b e i n t r i n s i c i n t e r a c t i o n s between l a r g e r numbers of p h y s i c a l p a r t i c l e s we w i l l get higher order terms which a l s o describe the before-mentioned i n t e r a c t i o n s . In c o n c l u s i o n our p h y s i c a l dressed Hamiltonian contains terms that describe the set of p o s s i b l e i n t e r a c t i o n s i n v o l v i n g only p h y s i c a l N and 7 7 - p a r t i c l e s . In the next chapter we w i l l concentrate on the i n t e r a c t i o n between one p h y s i c a l N - p a r t i c l e and p h y s i c a l - f l - p a r t i c l e and study e l a s t i c N-77* s c a t t e r i n g . 31 Chapter 5 Physical N-77 Scattering In t h i s chapter we w i l l be looking at the e l a s t i c scattering of one physical T/f-part i c l e (B^) and one physical N-p a r t i c l e (F 1*). In the dressed Hamiltonian, the term gives the interaction between a physical N-particle and a physical 77*-p a r t i c l e . We show below that when the incident free physical p a r t i c l e s have a t o t a l energy near the energy of a free bare A " p a r t i c l e , the physical N-77' potential ysH,i'£) can be approximated as separable. Using t h i s condition we fi n d an expression for the T-operator (from which s c a t t e r i n g amplitudes, cross sections, etc. may be found). 5.1 The Hamiltonian for Physical N-tf Scattering We now wish to project the Hamiltonian H onto the subspace for one N-particle and one ^ - p a r t i c l e . The projected Hamiltonian H i s (5.1.1) where the projection operator //e i s given by One finds that (5.1.2) (5.1.3) 32 where (5.1.4) and with (5.1.6) as given on page 26. Note that H has non-zero matrix elements only between two kets each containing one physical N-particle and one physical fT-p a r t i c l e . In order to maintain a correspondence with actual systems of nucleons and pions we w i l l assume that (5.1.7) That i s , the mass of the bare .--particle exceeds the combined mass of the bare N-particle and 7T-particle. In the region where the energy of incident physical N and TT-particles, *t(t£+*)'fZa(4&'*) > i s near the energy of a bare_\-p a r t i c l e with momentum K (such a region e x i s t s when mae>m +m^), we may approximate " f a a s We rewrite v£ as 33 (5.1.9) where (5.1.10) 0 * /i is. ,\ .~ Thus in the energy region where equation (5.1.8) holds we may write (5.1.11) where (5.1.12) and (5.1.13) 5.2 The T-Operator Using the above expression for we were able to solve for the T-operator, which we define as (5.2.1) where G(Z) i s the Green's operator (5.2.2) From a knowledge of the T-operator one can solve for the scattering amplitude and cross section and hence describe any resonance behavior the system may have, Since the function 34 V/&(4,*s a separable p o t e n t i a l the T-operator defined i n equation (5.2.1) may be solved f o r e x p l i c i t y as f o l l o w s ; S t a r t i n g from the Lippmann-Schwinger equation f o r G(Z), (5.2.3) where "GjZ) i s the free Green's operator we f i n d that and with d and i2 3L M f<£f) (5.2.9) (5.2.5) (5.2.6) (5.2.7) (5.2.8) The above equations (5.2.5) and (5.2.6) are merely^two l i n e a r equations f o r the unknowns, G(Z)|g(K)> and G(Z)|h(K)>. S o l v i n g them, we f i n d fo)k&- 0-*(&»; J %L*)lkkp+ 6 ft, (5.2.10) 35 &)Mtf?~ (HJ(&))^/0>+<*M&M)> (5.2.11) Using equations (5.2.10) and (5.2.11) we are able to simplify the T-operator as (5.2.12) with (5.2.13) For center of mass on-shell (incident energy equal to f i n a l energy) reactions^=0 and (5.2.14) As mentioned in Chapter 4, one can show that by using the function Y^Ck&o) instead of Y£(&tf,o) and choosing a small p o s i t i v e value for the parameter cf the t-matrix tff/^Ojtftfyltjtit) leads to a resonance peak in the center of mass scattering cross section. 36 Chapter 6 Systems I n v o l v i n g Two N - P a r t i c l e s In t h i s s e c t i o n we w i l l be lo o k i n g at r e a c t i o n s i n v o l v i n g two p h y s i c a l N - p a r t i c l e s . We l i m i t ourselves to three p o s s i b l e channels: o channel: two fre e N - p a r t i c l e s f channel: two fre e N - p a r t i c l e s and one fre e r r - p a r t i c l e d channel: a bound s t a t e c o n s i s t i n g of two N - p a r t i c l e s and one free - r f - p a r t i c l e We w i l l c a l l the bound s t a t e the d - p a r t i c l e (see Appendix D). Negle c t i n g s p i n and i s o s p i n one can think of t h i s bound s t a t e as a deuteron. The d - p a r t i c l e s h a l l be described by the operators /3*V_) and &(*) where E*(*) i s defined i n Appendix/) . The operator _ 5 ? _ 7 creates a p h y s i c a l dressed d - p a r t i c l e of momentum g , while the operator £>(&) a n n i h i l a t e s such a p a r t i c l e . For the purposes of t h i s chapter we w i l l t r e a t the d - p a r t i c l e as c o n s i s t i n g of only two p h y s i c a l N - p a r t i c l e s and neglect any p r o b a b i l i t y of i t c o n t a i n i n g one or more 7 T - p a r t i c l e s . By using multichannel T-operators, we s h a l l describe i n t e r a c t i o n s which in v o l v e not only e l a s t i c s c a t t e r i n g , but a l s o p a r t i c l e production (N+N-*N+N+7r), the breakup of the d - p a r t i c l e (TH-d-* 77+N+N) and both p a r t i c l e absorption and breakup (77+d-^N+N). 37 6.1 The Hamiltonian for Systems Containing Two N-Particles The dressed Hamiltonian for our model was given in Chapter 4 as We wish to deal with incident and outgoing scattering states consisting either of two N-particles or two N-particles one one 77-particle. In the process we s h a l l not allow for intermediate states with two or more "//-particles. Henceforth we work with Tf, the projection of H' onto the subspace for the F^F*/ and F/F^B^ system. H is given by (6.1.2) where and We note that (6.1.3) (6.1.4) (6.1.5) The operator $i i s the projection operator onto the F+F~f 38 subspace and fya i s the p r o j e c t i o n operator onto the subspace. Summing up a l l i n t e r a c t i o n terms of the same form (e.g., a l l terms of the form ff F *\0><0 |FF). !. i s given by (6.1.6) where (6.1.7) f[44^ff(f>)t%(p) Pfa fa') Bfy /°y <°!(?) (6.1.8) " J (6.1.10) 1 8 (6.1.11) (6.1.9) (6.1.12) (6.1.13) '  w i t h . . (6.1.14) (6.1.15) 39 (6.1.16) (6.1.17) where % and ^  are given in Chapter 4 and y,"6 i s given in Appendix B. Notice that a l l matrix elements of the operator Vn except those of the form <0 |F(g)F(p^ V , , f/(ri)F / /(g /) | 0> w i l l vanish. The terms V^'BXB', V ^ ' F X F ' and V^f have non-zero matrix elements only between two kets each containing the two N-particles and one 7T-particle. S i m i l a r l y <0 | F(g)F (£)B( k ) V ^ f / ( p O F / ( g ) | 0> and <0|F(gOF-(^)V / /^F/(g)F/(ci)B' /(Ji) |0> are the forms of the only non-zero matrix elements of the respective terms V//8 and ^  . One can show that % /*? B %) Wig**) r,f(w- *'J/°> -fovMt<r)Pte-t')fr (6',',8) Thus the operator VR/|B><B| describes two N-particles interacting in the presence of one free T f - p a r t i c l e . One can also show that (6.1.19) 40 Hence V ^ l F x F l represents one T T-particle interacting separately with each of the two N-pa r t i c l e s . The terms V^JBxBl and V ^ j F x F l are shown below. Figure 11 The Interaction Terms V„|B><B| and V^|F><F|. 6.2 The Channel Hamiltonians For an interaction that began (finished) in channel 6*, the Hamiltonian that was in eff e c t long before (after) the c o l l i s i o n was not necessarily just the free Hamiltonian H° . To describe a system in channel —', we write the Hamiltonian as (6.2.1) where in t h i s equation V„ represents the interaction terms that vanish as the p a r t i c l e s in channel move asymptotically to or from the reaction s i t e . H-, w i l l consist of the free Hamiltonian plus any interaction terms that give use to bound states occurring in channel «*• . we refer to H., as the channel Hamiltonian. 41 For the d channel consisting of a free i t - p a r t i c l e and two bound N-particles one writes (6.2.2) P fL,f ti where (6.2.3) and (6.2.4) 7/ Notice that since channel d deals with a t o t a l of three p a r t i c l e s , we must use Vff|B><B| in equation (6.2.3) and not just For channels o and f there are no bound states and we write (6.2.5) with ^%+%'^%;^%k7<e>i+v,tmi (6'2'6> For an eigenket of the channel o Hamiltonian we s h a l l write (6.2.7) where K represents both R< and &. Sim i l a r l y for channel d (6.2.8) and for channel f 42 (6.2.9) 6.3 The Multichannel T-Qperators As in the one-channel case, multichannel scattering may be formulated in terms of T-operators. In order to solve for the scattering amplitudes and cross sections of our reactions i t i s again s u f f i c i e n t to solve for the on-shell t-matrix which describes a t r a n s i t i o n from channel «* with momentum £ to channel -*' with momentum (see Taylor 1972). The on-shell t-matrix i s given by (6.3.1) where T72) i s the multichannel T-operator given by (6.3.2) with _ . (6.3.3) and \6 i s defined on page 40. In equation (6.3.1) i s the energy of the incident p a r t i c l e s (for on-shell reactions rj. = _>' where i s the energy of the scattered p a r t i c l e s ) and implies taking / A ^ Ji-to . Hence to describe scattering that begins in either channel o or f and ends in either channel o or f ( i . e . , NN -* NN, NN-^ NNTT, NN7T^NN or NNTT-^ NNTT) we use 43 ^ (6.3.4) S i m i l a r l y for either channel o or f going to channel d ( i . e . , NN-* or NN77-*JTd) we use / ( 6 . 3 . 5 ) while for channel d going to either channel o or f ( i . e . , jfd-^NN or 7fd^»NN70 . . . ( 6 . 3 . 6 ) and for a reaction that stays in channel d ( i . e . , it d-» 7Td) (6.3.7) Notice that the operator tJ=t) contains several terms (ofXof) (e.g., V^'5 , V ^ ' B X B ' ) that always have zero matrix elements between two o-channel kets (F^Ff 10>). When describing NN e l a s t i c scattering we may avoid a l l these unnecessary terms by def ining ^ (6.3.8) 'oo V it l m g f ) // The operator 7^0 describes the reaction N+N-^ N+N and has non-zero matrix elements only between two o-channel kets. 44 We s i m i l a r l y d e f i n e t h e f o l l o w i n g o p e r a t o r s : 7fft) * if (ft f o r t h e r e a c t i o n N + N N + N + 77" ( 6 . 3 . 9 ) f0^-(%>(l)Y f o r N + N + ^ N + N ( 6 . 3 . 1 0 ) T(/*)*ttsf3$B f o r N + N + / r " N + N + ? ^ ( 6 . 3 . 1 1 ) t i ^ ^ W f i £ ° r N + N ^ + d ( 6 . 3 . 1 2 ) % t ( $ * (TdoM)* f o r 7T+d-*N+N ( 6 . 3 . 1 3 ) 9„if(Jfi (P„B f o r 7T+d-* N+N+/T ( 6 . 3 . 1 4 ) tifi^itfJ^Y f o r N + N + ^ ^ + d ( 6 . 3 . 1 5 ) Zt^YPnafiffifis tor r r + a . - f t r + a ( 6 . 3 . 1 6 ) 45 6 . 4 T h e M o d i f i e d F a d d e e v E q u a t i o n s O n e c a n s h o w ( A p p e n d i x £ ) t h a t t h e m u l t i c h a n n e l T - o p e r a t o r s ' %f4^ a n d may b e w r i t t e n a s ± ( 6 . 4 . 1 ) w h e r e S S ' d T^i)--Tvtr(*)Qjt)ir.%) Tj*)=7:c*)&(*)l7:%) w i t h a n d " v. w h e r e i -/ V a n d ( 6 . 4 . 2 ) r - r " , ( 6 - 4 - 3 ) ( 6 . 4 . 4 ) ( 6 . 4 . 5 ) ( 6 . 4 . 6 ) ( 6 . 4 . 7 ) ( 6 . 4 . 8 ) ( 6 . 4 . 9 ) 46 (6.4.10) (6.4.11) The operators T^ity ( =1,2,3,4) are shown p i c t o r i a l l y as F i g . 12 The Operators T ^ T ^ , ^ and Ty Note that and (6.4.12) (6.4.13) (6.4.14) Equations (6.4.4), (6.4.5) and (6.4.6) are modified Faddeev equations. They are coupled i n t e g r a l equations involving the dr i v i n g terms T-. In the Faddeev equations the driving terms are known two-body T-operators which are used as input information to solve for three-body problems. While our 47 o p e r a t o r s T^, a r e n o t a l l t w o - b o d y o p e r a t o r s , t h e y d o d e s c r i b e a s e t o f b a s i c i n t e r a c t i o n s f r o m w h i c h t h e N N , NN7f a n d 7f d s y s t e m m a y b e d e s c r i b e d . T h e . o p e r a t o r T / ( Z ) d e s c r i b e s t h e e l a s t i c s c a t t e r i n g o f t w o p h y s i c a l N - p a r t i c l e s w h e n n o i n t e r m e d i a t e s t a t e s c o n t a i n i n g 77"-p a r t i c l e s a r e a l l o w e d . T ^ ( Z ) d e s c r i b e s t h e s a m e N - N s c a t t e r i n g i n t h e p r e s e n c e o f o n e n o n - i n t e r a c t i n g 7 T - p a r t i c l e . T h e o p e r a t o r T 3 ( Z ) i n v o l v e s V / f i | F > < F | w h i c h g i v e s t h e s c a t t e r i n g i n s y s t e m s w i t h t w o N - p a r t i c l e s a n d o n e 7 T - p a r t i c l e t h r o u g h i n t e r a c t i o n s b e t w e e n o n e ? T - p a r t i c l e a n d o n e N - p a r t i c l e . T y ( Z ) , i n v o l v i n g t h e f o u r t h o r d e r i m p l i c i t t h r e e - b o d y i n t e r a c t i o n v / ^ w i l l s e r v e a s a h i g h e r o r d e r c o r r e c t i o n i n t h e d e s c r i p t i o n o f s c a t t e r i n g i n v o l v i n g t h r e e p a r t i c l e s . T j . e q u a l t o V//B, a n d T f e e q u a l t o Vf'a ' g i v e r i s e t o T T - p a r t i c l e p r o d u c t i o n a n d a n n i h i l a t i o n r e s p e c t i v e l y . S u b s t i t u t i n g e q u a t i o n ( 6 . 4 . 1 ) i n t o e q u a t i o n ( 6 . 3 . 8 ) a n d u s i n g e q u a t i o n s ( 6 . 4 . 1 2 ) , ( 6 . 4 . 1 3 ) a n d ( 6 . 4 . 1 4 ) we f i n d t h a t ( 6 . 4 . 1 5 ) P i c t o r i a l l y F i g . 13 T h e O p e r a t o r T ( 48 For the production of one 7 7-particle by two N-particles the appropriate T-operator T^ Tfi) i s given by , (6.4.16) P i c t o r i a l l y F i g . 14 The Operator T^*J It i s worth noting that (6.4.17) The power series expansion of Tf + T Go does contain the same operators as the expansion of T , however with d i f f e r e n t c o e f f i c i e n t s . For example T00 contains Vtf'e G0 T3 Go V//3 while T, + T^G0 contains GgT3 G. V,;" . 4 9 F o r t h e r e a c t i o n N+N-97T+d ( 6 . 4 . 1 8 ) + o M F i g . 15 T h e O p e r a t o r F o r t h e r e a c t i o n 7 f + d - » N+N+7/ P i c t o r i a l l y ( 6 . 4 . 1 9 ) — — — — •f — -T " " ? — )— r, t Otf) F i g . 16 T h e O p e r a t o r T^, 50 F o r t h e e l a s t i c s c a t t e r i n g o f one d - p a r t i c l e and one fi~ p a r t i c l e ?7 +d-^7T+d _ (6.4.20) t Tim cu& & & ) n & r ) t ofty P i c t o r i a l l y i-9— >-> - . T3 - > F i g . 17 The O p e r a t o r T^, 51 B y a l l o w i n g f o r 7 f - p a r t i c l e a n d a n n i h i l a t i o n we o b t a i n e d e x p r e s s i o n s f o r t h e T - o p e r a t o r s d e s c r i b i n g t h e r e a c t i o n s N+N-^N+N+TT a n d N + N - » 7 T " + d . We a l s o o b t a i n e d a n 0(A6) c o r r e c t i o n t o t h e o p e r a t o r T, d e s c r i b i n g NN e l a s t i c s c a t t e r i n g . T h e m a i n c o n t r i b u t i o n t o rrd e l a s t i c s c a t t e r i n g (77+d ^77"+d) a n d t o t h e b r e a k u p o f t h e d - p a r t i c l e {rf+6 -^N+N+7T) o c c u r r e d t h r o u g h t h e o p e r a t o r T 3 i n v o l v i n g i n t e r a c t i o n t e r m V ^ S | F > < F | d e s c r i b e d o n p a g e 3 8 . F o u r t h o r d e r c o r r e c t i o n s t o t h e T -o p e r a t o r s d e s c r i b i n g t h e a b o v e t w o r e a c t i o n s w e r e g i v e n b y w h i c h i n v o l v e s V^'6, t h e i n t r i n s i c t h r e e - b o d y i n t e r a c t i o n . 52 Chapter 7 Summary and Conclusions We began our d e s c r i p t i o n of i n t e r a c t i n g N and /""-particles i n Chapter 2 with a model of bosons (Bf ) and two types of fermions (F/" and F/) i n t e r a c t i n g v i a a t r i l i n e a r Hamiltonian H s i m i l a r to the Hamiltonian found i n The Cloudy Bag Model. We noted that since the fermion kets F/" 10> and F/10> are not eigenkets of our Hamiltonian, F/ and F/ do not create p h y s i c a l p a r t i c l e s . In Chapter 3 we showed that by using a d r e s s i n g transformation the p a r t i c l e c r e a t o r s could be transformed i n t o p h y s i c a l p a r t i c l e c r e a t o r s , namely F/ , F/ and & . An expansion for the Hamiltonian i n terms- of p h y s i c a l p a r t i c l e operators (H(F/(F4, B)=H(f, B)) was given and we obtained renormalized fermion energies (equations (3.2.3) and (3.2.4)) and terms d e s c r i b i n g i n t e r a c t i o n s amongst the three types of p h y s i c a l p a r t i c l e s . To t h i r d order i n the c o u p l i n g constant the i n t e r a c t i o n terms i n H(F, fXfB) described two-body i n t e r a c t i o n s (e.g. f, +S-»F, +B) and boson production and a n n i h i l a t i o n (e.g. f, +B^ F, +B+B). Next, we s p e c i a l i z e d our model to describe systems of N-p a r t i c l e s ('nucleons' without s p i n and i s o s p i n ) and r r - p a r t i c l e s ('pions' without' i s o s p i n ) . In correspondence to.the Cloudy Bag  Model we r e f e r r e d to B^ as c r e a t i n g a bare 'pion', F as c r e a t i n g a bare 'nucleon' and F/ as c r e a t i n g a bare 'A-p a r t i c l e ' . The operator Bf created a p h y s i c a l 'pion' while ff created a p h y s i c a l 'nucleon'. We noted that since F*\0> was an eigenket of the t o t a l Hamiltonian, F^ could not describe an unstable p a r t i c l e such as the a c t u a l ^ - p a r t i c l e . In order to 53 w o r k w i t h o n l y o n e s t a b l e f e r m i o n ( t h e N - p a r t i c l e ) we c o n s i d e r e d a l l p h y s i c a l s t a t e s t o c o n s i s t o n l y o f p r o d u c t s o f F / a n d B a c t i n g o n t h e v a c u u m . We saw t h a t t h e p h y s i c a l d r e s s e d H a m i l t o n i a n H ' ( F , , B ) c o n t a i n s t e r m s d e s c r i b i n g t h e s e t o f p o s s i b l e i n t e r a c t i o n s i n v o l v i n g o n l y p h y s i c a l N a n d ^ - p a r t i c l e s . T o t h i r d o r d e r i n A t h e i n t e r a c t i o n t e r m s a r e w h e r e V„ c o r r e s p o n d s t o N+N-*N+N, V / 5 t o H+^-^n+TT, V„" f l + v,"& t o N + N « N + N + 7 T a n d v j ^ c o r r e s p o n d s to-N+7T-* N+ff+TT. I n C h a p t e r 5 we l o o k e d a t p h y s i c a l N -7T s c a t t e r i n g i n t h e r e g i o n w h e r e t h e i n c i d e n t f r e e p a r t i c l e s h a v e a t o t a l e n e r g y n e a r t h e e n e r g y o f a b a r e / - - p a r t i c l e . I n t h i s r e g i o n t h e p h y s i c a l N - 7 T p o t e n t i a l yt(k,g,was a p p r o x i m a t e d a s s e p a r a b l e a n d a n e x p r e s s i o n f o r t h e T - o p e r a t o r w a s f o u n d . I n C h a p t e r 6 we i n t r o d u c e d t h e d - p a r t i c l e ( ' d e u t e r o n ' ) a n d u s e d a s e t o f m o d i f i e d F a d d e e v e q u a t i o n s t o f i n d e x p r e s s i o n s f o r t h e m u l t i c h a n n e l T - o p e r a t o r s f o r r e a c t i o n s o c c u r r i n g w i t h i n t h e NN, NN77" a n d 7Td c h a n n e l s . 54 Bibliography Goldberger, M.L. and Watson, J.M., 1964, C o l l i s i o n Theory, Wiley, New York. Greenberg, O.W. and Schweber, S.S., 1958, Nuovo Cim. 8, 378. Hearn, D., 1981 M.Sc. Thesis, University of B r i t i s h Columbia. Newton, R.G., 1966, Scattering Theory of Waves and P a r t i c l e s , McGraw-Hill, New York. Piskunov, V.N., 1974, Theor. Math. Phys. JJ5, 546. Rose, M.E., 1957, Elementary Theory of Angular Momentum, Wiley, New York. Schweber, S.S., 1961, An Introduction to R e l a t i v i s t i c Quantum  F i e l d Theory, Harper and Row, New York. Taylor, J.R., 1972, Scattering Theory, Wiley, New York. Theberge, S., Thomas, A.W., and M i l l e r , G., 1980, Phys. Rev. D 22, 2838. 55 Appendix A Calculation of the Dressed Hamiltonian In t h i s appendix the d e t a i l s of the dressing transformation outlined in Chapter 3 w i l l be shown. Recall that the dressed Hamiltonian was given to second order as Ho n{H, fa*, ] } +?(&b,vi[tHoA W 'A &7 To calculate [H0,D ] we l e t Hence fa., a 1 [n„, Q,]tf&,-AT] ' ' f a , A , V f a , o » y Now l e t where and (A. 1 ) (A.2) (A.3) (A.4) (A.5) 0^ ^}d\^(M)Ffy) Zf^fJdCf) (A.6) To f i r s t order we eliminate the t r i l i n e a r terms from the dressed Hamiltonian. Thus we choose (A. 7) 56 T h i s l e a d s t o T h e n e x t o r d e r t e r m i n t h e H a m i l t o n i a n i s Now a n d [H,Df/]= [ t f t H 1 , tftti] W h e r e (A.8) a n d </fa*> M ; . (A. 9 ) (A.10) (A.11 ) (A.12) +[44 hfttffy f) fifa-fidf?) Fife - f) Sty a n d 57 ft} 4 St'faf)faf)/Ctf) 3 %<) &$) ftp-* +sf) F u r t h e r m o r e -f4 4^^' *) F?W #W -J4 SfrffaiWfo -1+91 r)FA? -f+f) fa % > ( A . 15) a n d l ^ M l = 4 M$) *sb-i, f ) FX*) FX*-r r ) &r> -f4^fai)4ffif)F}+W£Cfi) Ff4*t 4 'hq)ctfo'*)F,%') Zfy-f) F^ffi'-f) W -foe^A 'KbWXri+f' r) Fifo-w'J&fy £40 ( A . i e) . B y c h o o s i n g t h e o p e r a t o r D-_ t o e l i m i n a t e t e r m s o f f o r m g i v e n i n ( 3 . 1 . 1 5 ) we c a n f i n d t h e d r e s s e d H a m i l t o n i a n t o 0(tf~). T h e p r o p e r e x p r e s s i o n f o r D-_ i s g i v e n b y D , _ = D _ _ - D _ £ w i t h tfrAA'MP* f ) F ? ( M ) Z(*-¥)B(q)B(f') ( A . 17) 58 where ( A . 1 8 ) 59 Appendix B The Function ) In t h i s appendix we give an expression for the function Y / ' % f o u n d in the interaction terms V;///fland V//b which were introduced in Chapter 3. Recall that (B.1) and (B.2) V'/fland Vf' are both t h i r d order terms in the dressed Hamiltonian and are found in the expression where Hp and H^  are defined in Chapter 2 and and are given in Appendix A. The operator i s chosen to eliminate a l l terms in the above expression of the form given in (3.1.15). Substituting the expressions for H^, H,,D/ and into (B.3) and summing up a l l terms of the form JVFfFf%¥FF we find that - (B.4) **j v r where 60 tfuw; As &M£&*-4*x f___L  f L _ 7 t / / (B.5) Fig. 18 VYA 61 Fig. 19 62 F i g . 20 'vjf, 63 r . i - /  (B.8) F i g . 21 64 (B.9) F i g . 22 65 F i g . 2 3 66 (B.11) F i g . 24 67 Appendix C B a s i c S c a t t e r i n g Theory In t h i s appendix we o u t l i n e some of the b a s i c t h e o r y f o r the p h y s i c a l NTT e l a s t i c s c a t t e r i n g d e s c r i b e d i n Chapter 5. The H a m i l t o n i a n H of Chapter 5 was g i v e n by - ^ , - (c.D where (C.2) and ^tfftMJKWtti^M^M^)ky<°/F,(*&«)B(if<-t') i C ' 3 ) The s c a t t e r i n g p r o c e s s i s such t h a t the N and / ^ - p a r t i c l e s a r e d e s c r i b e d by a s c a t t e r i n g s t a t e | ^ ( t ) > • lpfy~U(+)lf(°)y ( c ' 4 ) where • H-tM ( C ' 5 ) T h i s s t a t e has the p r o p e r t i e s t h a t f o r some |yf„(0)> and |^4/(0)> (C.6) -cx=> (C.7) where U J * ) - ^ " ^ * <C-E> 68 U o ( t ) | ^ . ( 0 ) > and U o ( t ) | ^ / ( 0 ) > are known r e s p e c t i v e l y as the i n and out asymptotes of the s c a t t e r i n g s t a t e . Equations (C .6 ) and (C .7 ) i n d i c a t e that the p a r t i c l e s move f r e e l y long before and a f t e r the s c a t t e r i n g . One can show that where (C .10) with Note that equation (C . 9 ) g i v e s the out-asymptote i n terms of the in-asymptote. The f u n c t i o n ^ ( p r = < _ _ f _ I J ^ v i (0)> i s the wave f u n c t i o n determined by the p r e p a r a t i o n apparatus and hence i t w i l l be sh a r p l y peaked about some p o i n t (£„ ,__.) where p e and g» are the average i n c i d e n t momenta of the p a r t i c l e s . In t h i s sense the s t a t e |y^ /,(0)> i s 'known' and i t turns out that knowledge of the S operator i s s u f f i c i e n t t o determine any measurable q u a n t i t y a s s o c i a t e d with the s c a t t e r i n g . I t can be shown ( s i m i l a r to the d e r i v a t i o n of T^ i n Appendix F) t h a t 69 where with (C.14) where (C.15) The t o t a l cross section <T i s defined by the rela t i o n s h i p (C.16) wheren fnC_ i s the number of incident p a r t i c l e s per unit area where the unit area i s perpendicular to the r e l a t i v e motion of the p a r t i c l e s and A/ S ( L i s the t o t a l number of p a r t i c l e s scattered. The number of p a r t i c l e s scattered into an element of s o l i d angle _-A-is given by (C.17) where it?-* and are the average momenta of the incident p a r t i c l e s . In the l i m i t as A-T- approaches zero we define the d i f f e r e n t i a l cross-section e/c/Jj^ by rfctJL*~dJL (C.18) 70 j F o r c e n t e r o f m a s s r e a c t i o n s w h e r e k' i s t h e s a m e d i r e c t i o n as</yi/. 71 Appendix D The d - p a r t i c l e In t h i s appendix we look for an eigenket of the dressed Hamiltonian that describes a bound state containing two physical N-particles (we w i l l assume that the parameters of our model have been chosen such that the bound state a c t u a l l y e x i s t s ) . We w i l l c a l l the bound state the d - p a r t i c l e . We write (D.1 ) where D*(k) creates a physical d - p a r t i c l e of momentum k while the operator D(k) annihilates such a p a r t i c l e . The eigenket |D(k)> s h a l l be chosen to s a t i s f y the equation. (D.2) where H'is defined in Chapter 4 and , (D.3) rife*)* with m^  being the rest mass of the d - p a r t i c l e . While our dressed Hamiltonian H has the property that (D.4) due to ^ - p a r t i c l e production terms such as v'j6 , i t also has the property that (D.5) for any function E(2,g). Hence the d - p a r t i c l e cannot contain just two N-particles, but instead the operator w i l l also contain terms of the form 72 (D .6) However to second order H i s given by (D.7) and i t i s possible to treat the d- p a r t i c l e as so l e l y consisting of two N-particles. To t h i s order we write (D.8) and %#)*&<*)foCi) (D'9) where with ^ ( J O being the wave function for r e l a t i v e momentum k. For normalization we require (D.11) This can be achieved by l e t t i n g (D.12) and I ' jftjfmfct ( D . i 3 ) Using equation (2.1.1) we may rewrite equation (D.10) as 73 where Using equation (D.13) we have Jcl'r/p/1* / < D - , 6 > Thus fe) i s the d-pa r t i c l e wave function for the r e l a t i v e coordinate r. By taking the N-particles to be n o n - r e l a t i v i s t i c , that i s _•/(_>) =mc 2 +p 2 /2m, and approximating the N-N potential (given on page 25) as (D.17) equation (D.2) may be written to f i r s t order as (fi't vt;)/z>»>-*;(o)/6°(o)> (D'18) where and l// = £fKAtJK {/(/*-«'/) (fat') (D. 20) One can show that equation (D.18) y i e l d s the non-r e l a t i v i s t i c Schrodinger equation for fat) "T^vyrc)* * fe) ( D - 2 1 } 74 where (D.22) 4- "2- 2-and Considering the problem to the next order in ^ , there i s also a small p r o b a b i l i t y of the d-p a r t i c l e containing two N-p a r t i c l e s and one -//"-particle. To next order we write and [ 5 ° C k ) 7 f * f 6 ' m ~ ? f (D.25) W~ fJtOcWVteH Oft) Equation (D.2) now becomes * ( t / » ) /A SOm] (fi W > F * (D. 26) where the terms & ,% >%,B,%'/B r V ; / | B > < B | and ^ | F > < F | are defined on page 38. Matching powers of /\ gives " ' (D.27) ^ ^ (D.28) 75 (D.31 ) and Equation (D.28) gives the f i r s t order energy correction as zero and equation (D.29) gives the three p a r t i c l e state in terms of the dominant two p a r t i c l e state. Using these equations we can write (D.30) where Expanding fiti) as where we may write equation (D.30) as from which the d-p a r t i c l e may be represented graphically as below (D.32) (D.33) 76 F i g . 2 5 T h e d - p a r t i c l e T h e s o l i d l i n e r e p r e s e n t s a n N - p a r t i c l e a n d t h e b r o k e n l i n e r e p r e s e n t s a " 7 7 - p a r t i c l e 77 A p p e n d i x E The M o d i f i e d F a d d e e v E q u a t i o n s In t h i s a p p e n d i x we o u t l i n e t h e p r o o f o f e q u a t i o n s ( 6 . 4 . 2 ) , ( 6 . 4 . 3 ) , (6.4.5) and ( 6 . 4 . 6 ) . P r o o f o f e q u a t i o n s (6.4.1) and (6.4.2) c a n be d e r i v e d a n a l o g o u s l y . A l l e x p r e s s i o n s u s e d i n t h i s a p p e n d i x a r e d e f i n e d i n C h a p t e r 6. The o p e r a t o r s T ^ Z ) and T Q e <(Z) a r e g i v e n by Now We d e f i n e ( E . 1 ) (E.2) where _ (E.3) (E.4) (E.5) (E.6) h e n c e and .rVL + \hCfr)Tdi(*) (E-7) (E.8) 78 H o w e v e r a l l m a t r i x e l e m e n t s o f V . G - ( Z ) V . a r e e q u a l t o z e r o /•A >-f o r i = 5 , 6 . H e n c e £(*>%t s-'wt- ( E , 9 ) We a l s o d e f i n e a n d H e n c e ( E . 1 0 ) ( E . 1 1 ) ( E . 1 2 ) ( E . 1 3 ) ( E . 1 4 ) a n d U s i n g ( E . 1 3 ) b u t s i n c e = %*)4&%iHC*) ( E . 1 6 ) ( E . 1 5 ) 79 which proves equation (6.4.6). Using the equation (E.17) Ccp) - G&) y - a) equation (E.10) becomes (E.18) Using equations (E.13) and (E.10) the above equation can be written (E.19) Now (E.20) Hence (E.21) which i s equation (6.4.5). 80 Appendix F Some Multichannel Scattering Theory In t h i s appendix we outline some of the basic multichannel scattering theory (see Taylor, 1972) used in Chapter 6. The Hamiltonian H used in Chapter 6 i s given by (F.1) where (F.2) (F.3) and As in the single channel case, the p a r t i c l e s involved are described by a scattering state |^(t)> (F.5) where •n*/* ( F - 6 ) This state has the properties that for some |jfA ( 0 ) ^ and 1^/(0)^ in the channel <=* (°t*q4<f) subspace 31 One refers to the terms | ^ (-)>_, and gttL<i/fc |^(0)>, respectively as the channel "^components of the in and out asymptotes. These equations r e f l e c t the fact that in general the asymptotes may contain components in a l l channels. One can show that (F.9) where (F.10) with - , (F.11) Equation (F.9) gives the out-asymptote state in channel -6 in terms of the in-asymptote in channel OL . Knowledge of the operator 5^ i s s u f f i c i e n t to determine any measurable quantity associated with a reaction o r i g i n a t i n g in channel— and ending in channel _>. We now show how may be expressed to include the operator "7_/?) used in Chapter 6. The operator may be written SM*4~>' e>fc#efo# eJ*. (F'12) (F.13) 4-*- — where 32 By l e t t i n g i0 =- 4 both l i m i t s may be taken simultaneously and hence where fe)9 e M 4 M €-asfa/* Using the fundamental theorem of calculus Now £(*) - / (F.15) (F.16) (F.17) (F.18) where / i s the unit operator and (F 19) Hence (F.20) Taking the matrix elements of Sj^ between an eigenket \f'<* > of the channel a Hamiltonian and an eigenket |^^> of the 33 channel b Hamiltonian we f i n d that where <JXf-g') represents a product of three dimensional delta functions. Now *o a ( F . 2 2 ) o with ( F . 2 3 ) and ( F . 2 4 ) Thus ( F . 2 1 ) may be written r - ( F . 2 5 ) One defines ( F . 2 6 ) and ( F . 2 7 ) &J*)*fr-Ho)' where T/A i s the T-operator for a reaction going from channel a to channelb . Now 84 ( F . 2 8 ) and VkG&* tifa&fG*)W*))-fati&)K)&i)- (F'29) ( F . 3 0 ) Hence t£«i/<v*)fc-W«> <F'3° Now i KtHfiumsj?) w)w>)fr°> v&£' v^)<? > and which i s zero for on-shell = ) reactions, Note that since 85 ( F . 3 4 ) & b I ( V « - O for y*fj, t h e o p e r a t o r T ^ Z ) d e f i n e d a s %:(*>Tti*)+frK * ^ V . G r C ^ ( F * 3 5 ) w i l l h a v e t h e s a m e o n - s h e l l m a t r i x e l e m e n t s a s t h e o p e r a t o r V z )-T h u s e q u a t i o n ( F . 2 1 ) b e c o m e s ( F . 3 6 ) 

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