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The cloudy bag model Théberge, Serge 1982

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THE CLOUDY BAG MODEL by SERGE THEBERGE B. Sc. U n i v e r s i t e du Quebec a C h i c o u t i m i , 1978 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA March 1982 © S e r g e Theberge, 1982 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y a v a i l a b l e 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 p u b l i c a t i o n 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 Pk^ 5 c Cs  The University of B r i t i s h Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 DE-6 (2/79) i i ABSTRACT In t h i s t h e s i s , a new model f o r the i n t e r n a l s t r u c t u r e of baryons, the Cloudy Bag Model (CBM), i s presented. The baryons are assumed to be made of quarks permanently c o n f i n e d i n a s t a t i c s p h e r i c a l c a v i t y c a l l e d the "Bag", and of a c l o u d of pions which couple to the quarks at the bag su r f a c e i n a way which r e s t o r e s c h i r a l symmetry. When the CBM Lagrangian d e n s i t y i s p r o j e c t e d on the space of c o l o u r l e s s baryons, a Hamiltonian theory of baryons coupled to pions i s obt a i n e d . The model des-c r i b e s s u c c e s s f u l l y the resonance i n pion-nucleon s c a t -t e r i n g , the magnetic moments of the baryon o c t e t and the charge r a d i i of the nucleon. A l l these r e s u l t s agree r a t h e r w e l l with experiment when the only true f r e e parameter of the theory, the bag r a d i u s , i s chosen anywhere i n the range 0.8 to 1.1 f e r m i . i i i TABLE OF CONTENTS Page A b s t r a c t i i L i s t of Tables v i L i s t of F i g u r e s ; v i i Acknowledgements ix Chapters I) I n t r o d u c t i o n 1 II) The MIT Bag Model a. I n t r o d u c t i o n 16 b. B a s i c assumptions 17 c. The MIT Lagrangian d e n s i t y 18 d. The s t a t i c s p h e r i c a l bag s o l u t i o n 19 e. Coloured quarks i n the MIT bag model 23 f. The MIT Hamiltonian 25 I I I ) The Cloudy Bag Model Lagrangian d e n s i t y a. C h i r a l symmetry and QCD 32 b. The l i n e a r sigma model 35 c. The n o n - l i n e a r sigma model 38 d. The n o n - l i n e a r CBM Lagrangian d e n s i t y 40 e. The CBM Lagrangian d e n s i t y 43 iv IV) Hamiltonian f o r m u l a t i o n of the Cloudy Bag Model a. I n t r o d u c t i o n 46 b. The CBM quark Hamiltonian 47 c. The CBM baryonic Hamiltonian 49 d. The e i g e n s t a t e s of H Q and H 55 V) Renormalization of the Cloudy Bag Model a. Expansion of the p h y s i c a l nucleon 58 b. The dressed d e l t a 60 c. The nucleon and d e l t a s e l f - e n e r g y 61 d. Bare bag p r o b a b i l i t y Z^(E^) 63 AB e. The vertex f u n c t i o n 1^ (E A,Eg) 64 f. The renormalized c o u p l i n g c o n s t a n t s 66 g. The renormalized propagator 68 h. The r e n o r m a l i z a t i o n procedure 70 i . E x p r e s s i o n s f o r the r e n o r m a l i z a t i o n f u n c t i o n s .. 72 j . R e s u l t s of the c a l c u l a t i o n s 77 VI) The P 3 3 resonance a. The one-pion e i g e n s t a t e 87 b. The T-matrix f o r pion-nucleon s c a t t e r i n g 89 c. The p a r t i a l wave T-matrix 94 d. The P „ resonance 100 33 e. R e s u l t s of the c a l c u l a t i o n s 105 V VII) E l e c t r o m a g n e t i c p r o p e r t i e s of the baryon o c t e t a. I n t r o d u c t i o n 109 b. The EM form f a c t o r s G A ( q 2 ) and G A ( q 2 ) 110 E ^ M c. Formal e x p r e s s i o n f o r j j ^ ( r ) i n the CBM 114 d. The pion c o n t r i b u t i o n to j j^r) 1 17 e. The pion c o n t r i b u t i o n to G^(q 2) and y A 123 f. The quark c o n t r i b u t i o n to G £ ( q ) and y 124 g. The nucleon charge r a d i u s 131 h. The center of mass c o r r e c t i o n s to y A and <r 2> N . 132 i . R e s u l t s of the c a l c u l a t i o n s 135 VII I ) C o n c l u s i o n 148 B i b l i o g r a p h y 150 LIST OF TABLES vi Table I The a D O , a o s and ag S c o e f f i c i e n t s needed f o r the c a l c u l a t i o n of the c o l o u r magnetic i n t e r a c t i o n energy [32] 29 ) / Table II Best f i t to the hadron mass spectrum i n the MIT bag model [32] 31 Table III The TTAB c o u p l i n g c o n s t a n t s f A B / f 52 o Q Table IV The Clebsch-Gordan c o e f f i c i e n t s with j =1 ......... 122 2 Table V The quark magnetic moment matrix elements 130 Table VI The baryon o c t e t magnetic moments i n the SU(6), MIT and CBM models 139 Table VII C o n t r i b u t i o n of the p i o n , quark and cente r of mass to the baryon magnetic moments i n the CBM 140 v i i LIST OF FIGURES F i g . 1 Eigenfrequency fi(mR) of the lowest quark mode in the MIT bag model [32] 22 F i g . 2 Magnetic gluon exchange energy M as a f u n c t i o n of mR [32] 28 F i g . 3 The p o t e n t i a l energy d e n s i t y V(O,TT) with y2 <0 and c =0 37 F i g . 4 The CBM form f a c t o r lu(kR)| and a Gaussian f i t , v(kR)=exp(-0. l 0 6 k 2 R 2 ) 53 F i g . 5 R a t i o of the bare to the renormalized nNN cou-p l i n g constant f p B / f A B as a f u n c t i o n of the nucleon bag r a d i u s R 79 F i g . 6 Energy dependence of the JINN renormalized cou-p l i n g constant f ^ N as d e f i n e d i n eq.(5-114) f o r bag r a d i i i n the range of 0.5 to 1.0 fermi 81 F i g . 7 Energy dependence of the r a t i o of IINA and IINN c o u p l i n g c o n s t a n t s d e f i n e d i n eq.(5-116) 82 F i g . 8 Bare baryon bag p r o b a b i l i t y zf(m A) vs the bag ra d i u s R f o r a l l members of the baryon o c t e t obtained v i a eq.(5-93) but with the renorma-l i z e d c o u p l i n g c o n s t a n t s d e f i n e d i n eq.(5-117) 83 F i g . 9 R a t i o of the bare to renormalized baryon mass m /m f o r a l l members of the baryon o c t e t 85 oA A F i g . 10 The strange quark mass m which obeys the mass r e l a t i o n eq. (5-118) 86 F i g . 11 T o t a l c r o s s s e c t i o n f o r pion nucleon s c a t -t e r i n g i n the P 3 3 channel as a f u n c t i o n of the pion k i n e t i c energy. The t h i c k l i n e i s the CBM p r e d i c t i o n f o r bag r a d i i i n the range of 0.8 to 1.1 fermi 107 v i i i F i g . 12 Best f i t ( s o l i d l i n e ) to the experimental (dashed l i n e ) t o t a l c r o s s s e c t i o n f o r pion nucleon s c a t t e r i n g i n the P33 channel as a f u n c t i o n of the pion k i n e t i c energy. In t h i s f i t , f =0.24, R=0.72fm. and m =1232 Mev 108 F i g . 13 Lowest order matrix element f o r e l e c t r o n nu-cl e o n i n t e r a c t i o n 111 F i g . 14 The quark magnetic moment u (m R) 127 F i g . 15 The nucleon t h e o r e t i c a l to experimental magne-t i c moment r a t i o v N / y^ Xp as a f u n c t i o n of the nucleon bag r a d i u s R N 141 F i g . 16 The lambda magnetic moment r a t i o y A / Vexp as a f u n c t i o n of the lambda bag r a d i u s Ry\ and the strange quark mass ms 142 F i g . 17 The dependence of the sigma magnetic moment r a t i o s yE/y,§xp o n t n e bag r a d i u s R £ using the recent value of y | x p =-0 . 89±0. 1 4 n.m. [48] 143 F i g . 18 Same as f i g . 17 but with the o l d value y| x p=-1 .41±0.25 n.m. [86] 144 F i g . 19 The dependence of the cascade magnetic moment r a t i o s y - / y | x p on the bag r a d i u s R_ 145 F i g . 20 The nucleon t h e o r e t i c a l to experimental charge r a d i u s r a t i o as a f u n c t i o n of the nucleon bag r a d i u s R,T 146 F i g . 21 The neutron charge d i s t r i b u t i o n 4irr 2j°(r) vs the r a d i a l d i s t a n c e r (shaded a r e a ) . A l s o shown are the quark (Q) and the pion ( TT) charge d i s t r i b u t i o n i n s i d e the neutron. The neutron charge r a d i u s i s set at one fermi 147 ix ACKNOWLEDGEMENTS I wish to thank my s u p e r v i s o r , A.W. Thomas f o r the c o n s i -derable amount of time and e f f o r t he spent h e l p i n g me with t h i s p r o j e c t d u r i n g my stay at U.B.C. I am a l s o indebted to G.A. M i l l e r for" the s t i m u l a t i n g d i s -c u s s i o n s we had i n the development of the Cloudy Bag Model. I would l i k e a l s o to thank D.S. Beder, J . Ng, J.M. McMillan and N. Weiss f o r the numerous d i s c u s s i o n s we had on t h i s r e -search p r o j e c t . I am a l s o g r a t e f u l to a l l the f a c u l t y , s t a f f members and graduate students of the Ph y s i c s Department f o r making these four years of graduate s t u d i e s at U.B.C. an u n f o r g e t t a b l e ex-peri e n c e . F i n a l l y , I g r e a t l y a p p r e c i a t e the encouragement and he l p I r e c e i v e d from my wi f e . F i n a n c i a l a s s i s t a n c e i n the form of a Postgraduate F e l l o w s h i p from the N a t u r a l Sciences and En g i n e e r i n g Research C o u n c i l of Canada i s g r a t e f u l l y acknowledged. 1 I) INTRODUCTION: The Cloudy Bag Model [1 to 9], denoted CBM, i s a model des-c r i b i n g the i n t e r n a l s t r u c t u r e of baryons . I t i n v o l v e s two im-por tant i n g r e d i e n t s : the quark and p ion f i e l d s . The quarks are moving f r e e l y i n s i d e a s t a t i c s p h e r i c a l c a v i t y c a l l e d a "bag" in which they are permanently c o n f i n e d . The bag i s surrounded by a " c l o u d " of p i o n s , which couple to the quarks at the bag surface in a way which preserves c h i r a l symmetry. The Cloudy Bag Model in i t ' s present form was developed mainly by D r . A.W. Thomas of TRIUMF, D r . G .A . M i l l e r of the U n i v e r s i t y of Washington and my-s e l f . Var ious aspects of the model have been s t u d i e d by us and some other c o l l a b o r a t o r s but I s h a l l present i n t h i s t h e s i s only the aspects of the model in which I brought a major c o n t r i b u -t i o n . The s t o r y of the Cloudy Bag Model takes us back to the 1 9 4 0 ' s and the d i s covery of the Yukawa p a r t i c l e , the pion [ 1 0 ] . At that t ime , i t was understood that the nucleon had an i n t e r n a l s t r u c t u r e , and the most s t r i k i n g evidence of t h i s was the mea-surement of the anomalous magnetic moment of the n u c l e o n . With the d i s c o v e r y of the pion and other mesons, i t was b e l i e v e d that the c o r r e c t nucleon magnetic moment c o u l d be obta ined by adding the c o n t r i b u t i o n of the p o i n t l i k e nucleon core and the v i r t u a l meson c l o u d . However, a t o t a l l y r e l a t i v i s t i c theory of p o i n t l i k e nucleons coupled to mesons was soon found to be i m p r a c t i c a l , s ince q u a n t i t a t i v e p r e d i c t i o n s were based on a p e r t u r b a t i o n ex-pansion which d iverges bad ly ! 2 In 1954, G.F. Chew [11 to 15] presented a s t a t i c approxima-t i o n to the r e l a t i v i s t i c meson theory which i s obtained by en-t i r e l y n e g l e c t i n g nucleon r e c o i l . He a l s o i n t r o d u c e d a source f u n c t i o n p ( r ) to spread out i n space the r e g i o n of the pion nu-cl e o n i n t e r a c t i o n . The F o u r i e r transform of p ( r ) , namely u ( k ) , would provide a n a t u r a l c u t - o f f f o r the d i v e r g e n t i n t e g r a l s and t h e r e f o r e make the theory f i n i t e . No s p e c i f i c p r e s c r i p t i o n f o r u(k) c o u l d be s u p p l i e d and i t was g e n e r a l l y chosen to c u t - o f f at the nucleon mass. Chew was s u s p i c i o u s that the form f a c t o r u(k) was a s i m p l i f i c a t i o n of the nucleon i n t e r n a l s t r u c t u r e [11]: "For example, i f three f i e l d s r a t her than two are r e q u i r e d , an approximate theory f o r the pion nucleon i n t e r a c t i o n would have to be n o n l o c a l . I f t h i s i s the case, the c u t - o f f f a c t o r u(k) may have a r e a l p h y s i c a l s i g n i f i c a n c e " . With such a f i n i t e theory, c a l c u l a t i o n s such as pion nucleon s c a t t e r i n g and pion photopro-d u c t i o n were p o s s i b l e and agreed reasonably w e l l with e x p e r i -ments. A few years l a t e r , G. Salzman [16,17] presented an exten-s i v e c a l c u l a t i o n of the nucleon e l e c t r o m a g n e t i c p r o p e r t i e s i n the context of Chew's s t a t i c theory. His r e s u l t s r evealed that "the s t a t i c theory leads to severe disagreement with the e x p e r i -mental n e u t r o n - e l e c t r o n i n t e r a c t i o n . I f , however, the core charge i s assumed to be s t a t i c a l l y spread out i n space, f o r example with a d e n s i t y p r o p o r t i o n a l to the source f u n c t i o n p ( r ) , then reasonable agreement can be achieved." So, the 1950's ended up with a p h y s i c a l nucleon c o n s i s t i n g of an extended core sur-3 rounded by a c l o u d of p i o n s . The way of l o o k i n g at the nucleon i n t e r n a l s t r u c t u r e changed d r a m a t i c a l l y when, i n 1964, Gell-Mann [18] and Zweig [19] presented the quark model of hadrons. The idea of hadrons being made of quarks f o l l o w s n a t u r a l l y from c o n s i d e r a t i o n s of u n i t a r y symmetry. For example, the proton and the neutron form, with re s p e c t to strong i n t e r a c t i o n , a doublet, the nucleon. In other words, the strong i n t e r a c t i o n i s i n v a r i a n t under any r o t a -t i o n of t h i s doublet in i s o s p i n space. The symmetry group in t h i s case i s SU(2), the group of t r a c e l e s s u n i t a r y m a t rices in two dimensions. However, with the d i s c o v e r y of hyperons (e.g. A , I , . . . ) , the p i c t u r e needed to be enlarg e d . Gell-Mann and Ne'eman proposed i n the " E i g h t f o l d way" [20] to enlarge the sym-metry group to SU(3) and a s s o c i a t e d the s t a b l e baryons with the 8 r e p r e s e n t a t i o n of SU(3). In f a c t , a l l low mass hadrons can be c l a s s i f i e d i n one of the SU(3) m u l t i p l e t s . T h i s l e d to the d i s -covery of the fi~ which was the missing member i n the decuplet of J =3/2+ p a r t i c l e s . Assuming the v a l i d i t y of the e i g h t f o l d way, the fundamental t r i p l e t of p a r t i c l e s i n the theory does not con-s i s t of the nucleon doublet p l u s the lambda as i n the Sakata model [21] but ra t h e r of a t r i p l e t of f r a c t i o n a l l y charged s p i n 1/2 o b j e c t s c a l l e d quarks. T h e i r " f l a v o u r s " are up, down and strange with the r e s p e c t i v e charges 2/3, -1/3 and -1/3. In t h i s p i c t u r e , s t a b l e baryons are made of three quarks, each in the 1=0 s t a t e . However, problems a r i s e with the A + + f o r example, which i s made of three i d e n t i c a l quarks i n the same s t a t e , and would then v i o l a t e Fermi s t a t i s t i c s ! The simplest and very 4 f r u i t f u l remedy to t h i s problem i s to give the quarks an a d d i -t i o n a l quantum number, c o l o u r . Then, the antisymmetry of the d e l t a wavefunction w i l l be p r o v i d e d by an antisymmetric c o l o u r wavefunction. Quarks come in three c o l o u r s : red, green and blue, and the symmetry group i s SU(3) C where c stands f o r c o l o u r [22]. An extremely important experimental o b s e r v a t i o n i s that a l l known hadrons are c o l o u r l e s s , i . e . they are c o l o u r s i n g l e t s . T h e r e f o r e c o l o u r must be c o n f i n e d i n s i d e the hadronic world. In 1973, v a r i o u s authors [23,24,25] suggested that quark dynamics c o u l d be governed by a gauge p r i n c i p l e s i m i l a r to Quantum Electrodynamics and the Weinberg-Salam model of weak i n t e r a c t i o n s . T h i s scheme would c o n f i n e c o l o u r i n s i d e hadrons. Since only three c o l o u r s seem necessary to d e s c r i b e the hadron spectrum, the gauge group was chosen to be SU(3) and the e i g h t gauge bosons c a l l e d gluons would c a r r y the i n t e r a c t i o n between quarks. T h i s model of c o l o u r e d quarks and gluons was b a p t i s e d Quantum Chromodynamics (QCD). I t was soon r e a l i z e d that QCD has an extremely i n t e r e s t i n g f e a t u r e c a l l e d "asymptotic freedom" [24,26], which says that f o r very short d i s t a n c e s (or f o r very l a r g e e n e r g i e s ) , the quarks move almost f r e e l y . T h i s agrees with the r e s u l t s of high energy e l e c t r o n nucleon i n e l a s t i c s c a t -t e r i n g , which are c o n s i s t e n t with f r e e , s p i n 1/2, n e a r l y mass-l e s s partons. F i n a l l y , an "expected" pr o p e r t y of QCD i s " i n f r a r e d s l a v e r y " a l s o known as quark confinement, which, says that the e f f e c t i v e i n t e r a c t i o n between quarks i n c r e a s e s inde-f i n i t e l y with t h e i r s e p a r a t i o n [25]. 5 Many phenomenological models of hadrons made of c o l o u r e d quarks permanently c o n f i n e d have been developed i n the l a s t f i f -teen y e a r s . Some have n o n - r e l a t i v i s t i c quarks such as the poten-t i a l model of N. Isgur et a l . [27] f o r example, and others have r e l a t i v i s t i c quarks such as i n the bag models, i n c l u d i n g of course the Cloudy Bag Model. The s t o r y of quark bag models begins i n 1967 with the Bogolioubov quark model [28,29]. Quarks i n h i s p i c t u r e are con-f i n e d i n s i d e a s p h e r i c a l s t a t i c c a v i t y of r a d i u s R where they move f r e e l y . Assuming the quarks to be massless i n s i d e R and i n f i n i t e l y massive o u t s i d e , Bogolioubov was l e d to the l i n e a r boundary c o n d i t i o n on the bag s u r f a c e , and the quark wavefunction f o r the lowest l y i n g s t a t e can be w r i t t e n a n a l y t i c a l l y as with -Q= 2.04 from the l i n e a r boundary c o n d i t i o n . The model was s u c c e s s f u l f o r example in p r e d i c t i n g the mass of 1470 MeV f o r the Roper resonance ( f i r s t r a d i a l e x c i t a t i o n of the nucleon), with the r a d i u s R f i x e d by f i t t i n g the nucleon mass. However, there i s no dynamics i n t h i s model to determine the bag r a d i u s . F i n a l l y , we mention that i n the Bogolioubov bag, momentum i s not conserved at the bag s u r f a c e s i n c e no o u t s i d e pressure i s sup-p l i e d t o balance the D i r a c p r e s s u r e of the quarks on the sur-face . (1-1 ) (1-2) 6 The problems with the Bogoliubov model were s o l v e d i n 1974 when A. Chodos and coworkers at MIT presented the "MIT Bag Model" [29 to 33], the d e t a i l s of which are d e s c r i b e d at l e n g t h i n chapter II of t h i s t h e s i s . In b r i e f , the MIT bag model i s e n t i r e l y c o n t a i n e d i n a Lorentz i n v a r i a n t Lagrangian d e n s i t y (eq.(2-24)). The quarks and gluons are permanently c o n f i n e d i n s i d e a c l o s e d volume V c a l l e d the bag, and energy-momentum c o n s e r v a t i o n i s guaranteed by i n t r o d u c i n g an ad hoc, o u t s i d e p ressure B, which balances the quark and gluon pressure on the bag s u r f a c e in a Lorentz i n v a r i a n t way. The equations of motion d e r i v e d from the Lagrangian d e n s i t y are ( i n the absence of g l u o n s ) : the Dirac equation f o r the quarks i n s i d e V ( L - fVr>-) C^(nc ) = 0 / X 6 V > (1-3) the l i n e a r boundary c o n d i t i o n which guarantees confinement, ( n M i s the outward normal to the s u r f a c e ) c * Y - / n <j (oO = " J f / X ) 0 < 6 S j (1-4) and the q u a d r a t i c boundary c o n d i t i o n which guarantees s t a b i l i t y f o r the bag B = H - ~/>o.'i)[^(/x)<j(,*)] / x e S . (1-5) When gluons are present, another boundary c o n d i t i o n can be der ived /Y)h F * ( t f ) - O // 6 S (1-6) 7 where F*^ i s the gluon f i e l d t e n s o r . T h i s l a s t boundary c o n d i -t i o n leads i n f a c t to the absence of net c o l o u r f o r the bag. However, the model can not be s o l v e d e x a c t l y in a c o v a r i a n t way, and one must use the s t a t i c s p h e r i c a l bag approximation which then reduces the MIT bag model to the Bogolioubov one, together with the e x t r a boundary c o n d i t i o n , eq.(1-5), which dynamically f i x e s the bag r a d i u s . With a l l the above i n g r e d i e n t s p l u s a center of mass c o r r e c t i o n term -Z c/R, and a strange quark mass f i x e d at 279 MeV, the MIT group o b t a i n s a very reasonable f i t for the mass spectrum of hadrons. The only notable exception i s the pion mass, p r e d i c t e d to be 280 MeV, twice the experimental value of 140 MeV. Another major success i s the p r e d i c t i o n of 1.09 f o r the a x i a l v e c t o r constant g R , which r i s e s to 1.27 when center of mass c o r r e c t i o n s are i n c l u d e d [33]. T h i s compares very w e l l with the experimental value of 1.25+ 0.01 , and c o n s t i -t u t e s a major improvement in comparison with the n o n - r e l a t i v i s -t i c SU(6) p r e d i c t i o n of 5/3. The r a t i o of the baryon o c t e t mag-n e t i c moments p r e d i c t e d by the model i s a l s o an important impro-vement on the SU(6) values (see t a b l e V I ) . However, the proton magnetic moment comes out to be only 1.9 n.m. (2.24 n.m. when center of mass c o r r e c t i o n s are i n c l u d e d ) , w e l l below the e x p e r i -mental value of 2.79 n.m. Another major dis c r e p a n c y concerns the neutron charge r a d i u s which comes out to be e x a c t l y zero i n t h i s model. F i n a l l y , we mention that the MIT bag model does not prov i d e any mechanism f o r i n t e r a c t i o n s between baryons, which makes the model r a t h e r i n a p p r o p r i a t e f o r t r e a t i n g nuclear phy-s i c s problems [34]. 8 Among the many attempts to get bag models from QCD, a very i n t e r e s t i n g approach i s the s o l i t o n model of R. F r i e d b e r g and T.D. Lee [35], which has been i n v e s t i g a t e d r e c e n t l y by R. Goldflam and L. W i l e t s [36], The model i s based on i n t r o d u c i n g a phenomenogical s c a l a r sigma f i e l d , which i s assumed to be a phenomenological r e p r e s e n t a t i o n of the s e l f - i n t e r a c t i n g gluon f i e l d . The sigma f i e l d p o t e n t i a l has two minima, a r e l a t i v e one, <JlAlKI= °, and an a b s o l u t e one, cr^ 1 N = AT. The l a t t e r one being the lowest energy s t a t e , i t i s then i d e n t i f i e d with the vacuum s t a t e . However, when a quark f i e l d i s p resent, the e f f e c t i v e sigma f i e l d p o t e n t i a l i s t i l t e d ando~=0 becomes the true m i n i -mum: the quarks have dug a hole i n the vacuum. With a s u i t a b l e c h o i c e of parameters, t h i s hole becomes i d e n t i c a l to the MIT bag model! An important p r o p e r t y of QCD with massless quarks i s c h i r a l symmetry [37], In simple words, massless quarks are h e l i c i t y e i g e n s t a t e s and s i n c e the QCD Lagrangian i s c h i r a l i n v a r i a n t , the h e l i c i t y of the quarks should not change. In the MIT bag model, confinement i s r e a l i z e d by i n t r o d u c i n g an i n f i n i t e mass term f o r the quarks at the bag s u r f a c e , thus f o r c i n g the quarks to f l i p t h e i r h e l i c i t y when r e f l e c t e d . However, the c h i r a l sym-metry group SU(2) Lx SU(2) R i s known to be broken by l e s s than 7% [38], and as Pagels s t r e s s e s , " i t i s the best symmetry of the strong i n t e r a c t i o n a f t e r i s o t o p i c s p i n symmetry". The correspon-ding p a r t i a l l y conserved a x i a l c u r r e n t (PCAC) i s denoted A^(x) and when i d e n t i f i e d with the weak a x i a l c u r r e n t , i t s divergence 9 i s given by where TT(x) i s the pion f i e l d of mass m^ , and | f ^ | i s the pion decay constant of 93 Mev. In the c h i r a l symmetric l i m i t ^^=0, the nucleon matrix element of the a x i a l c u r r e n t g i v e s r i s e to the Goldberger-Treiman r e l a t i o n [39], ^N "rjn ^VA / T T > (1-8) where m i s the nucleon mass, g i s the a x i a l - v e c t o r constant and g w i s the TTNK c o u p l i n g c o n s t a n t . T h i s r e l a t i o n i s r i g h t at the 8 i 2% l e v e l which makes c h i r a l symmetry a f e a t u r e which should be i n c l u d e d i n bag models of hadrons. The f i r s t attempt to add c h i r a l symmetry to the MIT bag model was proposed in 1975 by A. Chodos and C.B. Thorne [40]. Since c h i r a l symmetry in the MIT bag model i s v i o l a t e d only at the bag s u r f a c e , Chodos and Thorne introduced a m u l t i p l e t ( 0" ,TT ) of phenomenological f i e l d s which couple to the quarks only at the bag s u r f a c e . The c o u p l i n g i s d i c t a t e d by the SU(2) Lx SU(2) R symmetry c o n d i t i o n , and when expressed i n terms of a Lagrangian d e n s i t y , one r e c o v e r s the analog to the l i n e a r sigma model of Gell-Mann and Levy [41]. No attempt was made to q u a n t i z e the meson f i e l d s , but r a t h e r the model was s o l v e d c l a s -s i c a l l y f o r the "hedgehog" baryon where the f i e l d s have the pro-p e r t y TT(x) = g ( r ) r . In 1979, G.E. Brown and M. Rho [42,43,44] proposed a c h i r a l 10 bag model of the nucleon based' on a two phase p i c t u r e of the vacuum. U n l i k e the model of Chodos and Thorne, the i n t e r i o r of the bag i s c h i r a l l y symmetric in the Wigner-Weyl sense, no pion f i e l d i s present, and the a x i a l c u r r e n t i s e n t i r e l y c a r r i e d by the quarks. In the vacuum o u t s i d e the bag, c h i r a l symmetry i s spontaneously broken and there i s a Goldstone boson, the pion, which c a r r i e s the a x i a l c u r r e n t . The c o u p l i n g of the pion f i e l d to the quarks occurs at the bag s u r f a c e with the requirement of c o n t i n u i t y of the a x i a l c u r r e n t through the boundary. The model i s h i g h l y n o n l i n e a r in nature, and l i k e the Chodos and Thorn bag, i t can be solved only f o r the hedgehog baryon. Another pro-blem i s the enormous pressure of the pion f i e l d which " s h r i n k s " the bag r a d i u s to about 0.3 fermi and thereby i n v a l i d a t e s the use of p e r t u r b a t i o n theory i n q u a n t i t a t i v e c a l c u l a t i o n s . However, i f the bag r a d i u s i s allowed to be l a r g e (~1.1fm), a value f o r the vacuum pressure B can be found which s t a b i l i z e s the bag. T h i s allowed F. Myhrer and coworkers [45,46] to succes-s f u l l y c a l c u l a t e the baryon mass spectrum. In t h e i r f i t , each baryon mass i s the combined e f f e c t of the quark energy with mti=m<( = lO MeV and ms=210 MeV, the volume energy with B'' = 137 MeV, the pion c l o u d c o n t r i b u t i o n , the one-gluon exchange energy with <xs=i.85 and the z e r o - p o i n t motion energy with 7_„ = 0.13. They a l s o c a l c u l a t e d the magnetic moment, and t h e i r r e s u l t of -0.62 n.m. i s i n e x c e l l e n t agreement with the experimental value of -0.614 n.m. In the same p e r i o d as Brown and Rho's work, R.L. J a f f e [47] r e v i s e d the bag models with and without c h i r a l symmetry, and 11 developed a Lagrangian formalism which reduces to the above bag models depending on the approximations made on the Lagrangian equations of motion. His approach i s to assume that the bag r a d i u s i s l a r g e , which then makes p o s s i b l e a p e r t u r b a t i o n t r e a t -ment around the MIT bag model. Again, the pion f i e l d i s t r e a t e d as a c l a s s i c a l f i e l d excluded from the bag. The Cloudy Bag Model possesses most p r o p e r t i e s of the bag models d i s c u s s e d above, p l u s many important ones of i t s own. F i r s t , the formalism of the CBM i s d e s c r i b e d , as i n J a f f e ' s d i s -c u s s i o n , by a Lagrangian d e n s i t y which i n c o r p o r a t e s c h i r a l sym-metry. The r e a l i z a t i o n of SU(2)Lx SU(2)R i s then e q u i v a l e n t to the n o n - l i n e a r sigma model which has the advantage of not i n t r o -ducing a f i c t i t i o u s sigma f i e l d . The pion f i e l d , as i n Chodos and Thorn model, i s allowed to leak i n s i d e the bag. There are many co m p e l l i n g reasons f o r t h i s . F i r s t , i f QCD behaves more l i k e quarks coupled with s t r i n g s (the g l u o n s ) , the p r o b a b i l i t y of c r e a t i n g qq pion type of o b j e c t s i s non-zero f o r a f i n i t e d i s t a n c e between quarks, and t h e r e f o r e the pion f i e l d w i l l have to leak i n s i d e the bag. I f QCD has a c o n f i n i n g phase (the true vacuum), and a n o n c o n f i n i n g phase (the c h i r a l l y symmetric i n -t e r i o r of the bag), a c o r r e c t treatment of the s u r f a c e w i l l have to be non s t a t i c i n nature. Due to the dynamical motion of the s u r f a c e , which w i l l occur i n a c o v a r i a n t treatment of the bag, the time-averaged pion f i e l d w i l l to some extent leak i n s i d e the bag. F i n a l l y , we r e c a l l that a l l these bag models are s t a t i c i n nature, and t h i s r e s t r a i n s us to low energy a p p l i c a t i o n s . The pion f i e l d can then be c o n s i s t e n t l y approximated by a long wave-12 l e n g t h s t r u c t u r e l e s s o b j e c t , which can be q u a n t i z e d i n the usual way, i . e . v i a a plane wave expansion, which n e c e s s a r i l y goes through a l l space. The next step in c o n s t r u c t i n g the Cloudy Bag Model i s to add a pion mass term to the Lagrangian d e n s i t y , e x p l i c i t l y brea-king c h i r a l symmetry to r e s t o r e the PCAC • r e s u l t eq.(1-7). By making a small pion f i e l d approximation to the Lagrangian d e n s i -ty ( i . e . c o n s i d e r i n g l a r g e bags), we o b t a i n a simple Lagrangian d e n s i t y which i s the MIT one p l u s the f r e e pion one, and an i n t e r a c t i o n term of the form The e x p l i c i t d e s c r i p t i o n and a n a l y s i s of the Lagrangian d e n s i t y with the s m a l l p i o n f i e l d approximation i s t r e a t e d i n chapter III of t h i s t h e s i s . With the Lagrangian d e n s i t y thus s i m p l i f i e d , we d e r i v e a H a m i l t c n i a n formalism for the Cloudy Bag Model. T h i s i s r e a l i z e d f o l l o w i n g the usual c a n o n i c a l q u a n t i z a t i o n r u l e where the Hamiltonian operator H i s i d e n t i f i e d with the i n t e g r a l i n a l l space of the energy d e n s i t y T°°(x) of the energy momentum ten-s o r . By r e p l a c i n g TT (x) by i t ' s q u a n t i z e d F o u r i e r transform and the quark wavefunction by the MIT one, we o b t a i n the Hamiltonian quark operator given e x p l i c i t l y i n eq.(4-10,11,14). Next, and most importantly, we p r o j e c t t h i s Hamiltonian quark (1-9) (1-10) 13 operator onto the space of c o l o u r l e s s baryon bags. T h i s leaves us with a Hamiltonian of the form H = H0 + Hx , ( 1 _ n ) w i th Ho = I ^ O F L R + 0 R 0 + I ( A ^ k axh <yfe>, ( 1 _ 1 2 ) where mofl i s the bare or MIT bag mass f o r the baryon of type A. The i n t e r a c t i o n Hamiltonian H x has the form H ^ I J A I R! ^ { \ h K + kc. , ( 1 _ 1 3 ) with v 0^ (k) being the a p p r o p r i a t e vertex f u n c t i o n s . In the nu-cle o n s e c t o r , t h i s has the form which i s e x a c t l y the same as the one i n the Chew model mentioned before, with u(kR) = j D ( k R ) + ^ ( k R ) being the form f a c t o r . T h e r e f o r e , i n the context of the Cloudy Bag Model, the c u t - o f f f a c t o r u(kR) does have a p h y s i c a l s i g n i f i c a n c e : i t r e f l e c t s the u n d e r l y i n g presence of the quarks c o n f i n e d in a f i n i t e region of space! A l l t h i s w i l l be the sub j e c t of chapter IV. In the strangeness zero s e c t o r of our baryonic Hamiltonian, the pion couples to both the nucleon and the d e l t a bags. In chapter V, we s h a l l b u i l d the p h y s i c a l nucleon from the bare e i g e n s t a t e s of H 0. B a s i c a l l y , the p h y s i c a l nucleon has a proba-b i l i t y to be a bare three quark bag, a p r o b a b i l i t y P^, to be a bare nucleon with one pion " f l o a t i n g around", a p r o b a b i l i t y P^r to be a bare d e l t a bag with a p i o n , and so on. N o t i c e that 14 Z^ i s , to some extent, a measure of the accuracy of the l i n e a r i -z a t i o n approximation to our Lagrangian d e n s i t y . I t shows whether the pion f i e l d r e a l l y i s "weak". We s h a l l f i n d t h a t f o r R>0.7 fm., Z* i s s u f f i c i e n t l y l a r g e to j u s t i f y t h i s weak pion a p p r o x i -mation. In f a c t , some formal work by L.R. Dodd et a l . [9] has a l s o shown the convergence of the Cloudy Bag Model p e r t u r b a t i o n expansion f o r l a r g e bag r a d i i . Another important r e s u l t which we s h a l l f i n d i s the r e l a t i v e l y small c o r r e c t i o n the pion c l o u d b r i n g s to the c o u p l i n g c o n s t a n t s of pion to baryons. T h i s w i l l allow us i n our q u a n t i t a t i v e c a l c u l a t i o n s to use the SU(6) r a t i o of the c o u p l i n g c o n s t a n t s . Chapter VI i s devoted to the P 3 3 resonance ( a l s o c a l l e d the d e l t a resonance) in pion nucleon e l a s t i c s c a t t e r i n g . Up to now, two d i f f e r e n t p i c t u r e s of the P ? J resonance have p r e v a i l e d . One was based on the c r o s s e d graph of pion nucleon s c a t t e r i n g which Chew and Low [12] have shown to reproduce the P^ resonance i f a j u d i c i o u s c h oice of the parameters i s made. The other p i c t u r e , based on the quark s u b s t r u c t u r e of hadrons, t e l l s that the P J 5 resonance i s due to the formation and decay of an elementary d e l t a . In the Cloudy Bag Model, s i n c e the pion couples both to the nucleon and the d e l t a bags in a w e l l d e f i n e d way, these two processes p l u s the i n t e r f e r e n c e terms can be c a l c u l a t e d e x p l i -c i t l y . We s h a l l f i n d that our c a l c u l a t i o n i s compatible with the experimental data f o r a nucleon bag r a d i u s anywhere i n the range on 0.8 to 1.1 fermi, and that the resonance i s in f a c t dominated by the formation and decay of the d e l t a bag. 15 In chapter V I I , we are concerned with the e l e c t r o m a g n e t i c p r o p e r t i e s of the baryon o c t e t . More s p e c i f i c a l l y , we s h a l l study the nucleon, lambda, sigma and cascade magnetic moments, the nucleon charge r a d i u s and the neutron charge d i s t r i b u t i o n . The o v e r a l l agreement with experiment i s e x c e l l e n t . The nucleon magnetic moment agrees with the experimental values w i t h i n 10% in the wide bag r a d i u s range of 0.85 to 1.15 f e r m i . S i m i l a r agreement occurs f o r ju A in the range 0.8<R<1.2 fm. For the I + , agreement i s a l s o e x c e l l e n t f o r 0.9<R<1.1 fm. The p r e d i c t i o n i s about one standard d e v i a t i o n from the best a v a i l a b l e e x p e r i -mental r e s u l t of -0.89*0.14 n.m. [48]. The =° magnetic moment comes out r i g h t f o r R =0.95*0.10 fm. , and the =.' t h e o r e t i c a l value of -0.61 n.m. agrees w i t h i n 12% with the p r e l i m i n a r y r e s u l t of -0.69*0.04 n.m. [49]. The proton and neutron charge r a d i u s agree w i t h i n 10% with the experimental r e s u l t s f o r a bag r a d i u s w i t h i n the range of 0.85 to 1.15 f e r m i . T h i s l a t t e r r e s u l t i s c o n s i s t e n t with those f o r the P 1 5 resonance mentioned above. F i n a l l y , a word about the convention used i n t h i s t h e s i s . We s h a l l use the Bjorken and D r e l l conventions [50] f o r a l l re-l a t i v i s t i c q u a n t i t i e s such as f o u r - v e c t o r s , D i r a c matrices e t c . , and use the conventions of M.E. Rose [51] f o r the Clebsch-Gordan c o e f f i c i e n t s and t h e i r i d e n t i t i e s . A l l other conventions w i l l be d e s c r i b e d i n great d e t a i l i n the t e x t as we need them. 16 11) THE MIT BAG MODEL a. I n t r o d u c t i o n . In 1974, the MIT group [30,31] presented a new mic r o s c o p i c model of hadrons c a l l e d the "MIT Bag Model". In t h e i r p i c t u r e , the hadrons are composite o b j e c t s made of l i g h t quarks and mass-l e s s gluons moving almost f r e e l y i n s i d e a r e s t r i c t e d volume c a l l e d the "Bag". The confinement of these f i e l d s i n s i d e the bag c a v i t y i s accomplished i n a Lorentz i n v a r i a n t way by assuming that the bag possesses a consta n t , p o s i t i v e energy per u n i t vo-lume, B. T h i s r e a l i z e s i n a c o v a r i a n t context the f r e e - p a r t o n s u b s t r u c t u r e of hadrons, as observed in the Bjorken s c a l i n g phenomena [52], while keeping these parton c o n s t i t u e n t s per-manently c o n f i n e d . In the context of Quantum Chromodynamics (QCD), the quarks c a r r y a c o l o u r quantum number, and the gluons are the ve c t o r gauge bosons. When a p p l i e d to the MIT bag model, one ob t a i n s that a l l p h y s i c a l hadrons are c o l o u r l e s s p a r t i c l e s as observed e x p e r i m e n t a l l y . A l s o , i f the one-gluon i n t e r a c t i o n i s c a l c u l a t e d i n s i d e the bag c a v i t y , a mass s p l i t t i n g i s generated between otherwise degenerate hadrons such as the nucleon and the d e l t a . By assuming the bag to be a s p h e r i c a l , s t a t i c c a v i t y , the MIT bag equations can be so l v e d e x a c t l y , and q u a n t i t a t i v e r e -s u l t s f o r the hadrons mass spectrum and i t s other s t a t i c proper-t i e s are obtained [32,33]. 17 B a s i c assumptions f o r the MIT bag model. The MIT bag model i s b a s i c a l l y d e f i n e d by the f o l l o w i n g set assumptions concerning the bag and the quarks. 1: The p h y s i c a l hadrons are made of quark and gluon f i e l d s permanently c o n f i n e d i n a f i n i t e r e g i o n of space c a l l e d the "Bag" [30]. 2: The bag has a c o n s t a n t , time and d i r e c t i o n independent, p o s i t i v e p o t e n t i a l energy B per u n i t volume [30]. 3: The a c t i o n S c h a r a c t e r i z i n g the bag system i s r e q u i r e d to be s t a t i o n a r y with respect to v a r i a t i o n s of the con-f i n e d quark and gluon f i e l d s [30], 4: The a c t i o n S i s a l s o r e q u i r e d to be s t a t i o n a r y with r e s -pect to independent v a r i a t i o n s of the p o s i t i o n of the bag s u r f a c e [30]. 5: For the lowest l y i n g hadronic s t a t e s , the bag i s assumed fo r s i m p l i c i t y to be s t a t i c and s p h e r i c a l [31]. 6: In absence of the gluon f i e l d , the quark s p a t i a l wave-f u n c t i o n obeys the f r e e D i r a c equation i n s i d e the bag volume; and the s p i n - i s o s p i n wavefunction i s given by the SU(6) quark model [31]. 18 c. The MIT Lagrangian D e n s i t y . Let f i r s t w r i t e down the Lagrangian d e n s i t y [47] (without gluon f i e l d s ) f o r the MIT bag model, and then show that i t r e s -pects a l l the assumptions of s e c t i o n 2b: 3 (2-1) 3 ( ' t = i In t h i s Lagrangian d e n s i t y , q^(x) i s the i t h quark f i e l d of mass mL, 0 V i s a step f u n c t i o n d e f i n e d as e - > ( 2 _ 2 ) v O o u t s i d e V where V i s the bag volume } and A s i s a s u r f a c e d e l t a f u n c t i o n d e f i n e d by o e v = / * A S > ( 2 _ 3 ) and n^ i s the outward 4-normal to the bag s u r f a c e with /yj3' — /V) /Y)^ - - 1 , (2-4) By demanding the a c t i o n S = \ o\"* £ <*> (2-5) M I T J / H I T be s t a t i o n a r y under the v a r i a t i o n s of the f i e l d s and the bag s u r f a c e (A-3,A-4), 9 ' v ^ 9 V + 6 A s (2-6) 19 we get the Euler-Lagrange equations of motion. F i r s t , ( i $ - An. ) c ^ U ) = O V G V , ( 2 - 7 ) i s the f r e e D i r a c equation f o r the quarks i n s i d e the c a v i t y (A-1 ) . Next, t Y - m c^.Coc) - ^(o<> nee S , ( 2 _ 8 ) i s known as the bag l i n e a r boundary c o n d i t i o n , which guarantees th a t no quark c u r r e n t c r o s s e s the bag s u r f a c e ; and f i n a l l y , 3 B = -±.».Zl I ^ U ) = PD / * 6 S , ( 2 - 9 ) p r o v i d e s s t a b i l i t y f o r the bag by equating the vacuum pressure B with the D i r a c p r essure of the quarks P D ( A - 2 ) . d. The s t a t i c s p h e r i c a l bag s o l u t i o n . If we assume (A-5) that the bag c a v i t y i s s t a t i c and s p h e r i c a l , the equations of motion are then r e w r i t t e n as C i 1 ° \ * I ^ V - / K t . ) <j. ( A , t ) - O A £ R , ( 2 - 1 0 ) -i Y • k = A =• R y ( 2 - 1 1 ) B = -X ^ . v[X ^ . ( ^ o V 0 ^ ^ ) ] ^ = R . ( 2 - 1 2 ) The p o s i t i v e energy s o l u t i o n of the f r e e D i r a c equation e q . ( 2 - l 0 ) i n a s p h e r i c a l volume of r a d i u s R i s given by [ 5 3 ] 20 \-i * ; j - * ' / f O < 2 ?-v with X- ss /»7- R ( , (2-14) r = x - UJ 1 - + 1 c ( I ? ± , / X (2-15) where 1^(8 , f , s t ) i s the f u n c t i o n of t o t a l angular momentum (JM) formed by the composition of a spin 1/2 with the s p h e r i c a l har-monics of order X. N- i s the n o r m a l i z a t i o n f a c t o r and (z) i s the s p h e r i c a l B e s s e l f u n c t i o n of order J? . Since we assume the bag to be s p h e r i c a l and B to be d i r e c -t i o n independent, the q u a d r a t i c boundary c o n d i t i o n e q . ( 2 - l 2 ) , together with J=l/2 f o r the quarks w i l l then impose the quarks to be i n the J! = 0 s t a t e . If b- c a r r i e s the i n t e r n a l wavefunction of q t (e.g. s p i n and i s o s p i n ) , then the p o s i t i v e p a r i t y s o l u t i o n to the s p h e r i c a l s t a t i c bag i s V < i, {nLh/R)d-.ht The quark frequency Sir i s d e r i v e d by using eq.(2-16) i n the l i n e a r boundary c o n d i t i o n e q . ( 2 - 1 l ) . T h i s c o n d i t i o n i s s a t i s f i e d only i f «l ( ^ ) = * • -j.Cil;) . ( 2 - l 7 ) 21 If the quark i s massless, then (2-18) which i s s a t i s f i e d f o r 11*2.0428. We give i n f i g u r e 1 a graph of i l i as a f u n c t i o n of A-=mt.R. Not i c e that f o r m. l a r g e , _Q- tends towards TT, the non r e l a t i v i s t i c l i m i t . From the quark wavefunction (2-16), we get the normaliza-t i o n f a c t o r N- by imposing the c o n d i t i o n (2-19) where the i n t e g r a t i o n i s performed on the bag volume only. T h i s leads to N; = 1 and f o r the massless case N ( - A*.) a oc. (a- - i ) + . A - 1 (2-20) (2-21 ) F i n a l l y , using a l l the previous r e s u l t s in the q u a d r a t i c boundary c o n d i t i o n eq.(2-12) allows us to r e l a t e the bag radi u s R to the vacuum pressure B: 2 4 T T R i = i i o t - ( a . - i ) + fr . (2-22) Fig. 1: Eigenfrequency n(mR) of the lowest quark mode in the MIT bag model [32], 23 e. Coloured quarks in the MIT bag model. Quantum Chromodynamics [54] i s p r e s e n t l y the best candidate f o r d e s c r i b i n g the s t r o n g i n t e r a c t i o n . In QCD, the f i e l d s i n -v o l v e d are the c o l o u r e d quark f i e l d s and the gluon gauge boson f i e l d s , the l a t t e r being massless and t h e r e f o r e guaranteeing the exactness of the u n d e r l y i n g SU(3) C symmetry. The r e s u l t of much t h e o r e t i c a l work [54] i s that QCD i s a theory of f i e l d s almost f r e e at short d i s t a n c e and very s t r o n g l y i n t e r a c t i n g at l a r g e d i s t a n c e . However, the confinement of quarks i n s i d e hadrons has not yet been proved to be a d i r e c t consequence of QCD. In the MIT bag model, the c o l o u r i n t e r a c t i o n between quarks i n s i d e hadrons i s i n t r o d u c e d to prevent the e x i s t e n c e of hadrons with net c o l o u r . F i r s t , l e t ' s d e f i n e the gluon f i e l d tensor F,." = - 3 . ^ + t -U &' . (2"23) where G£ are the e i g h t Y a n g - M i l l s c o l o u r e d gluon f i e l d s (a=1,...,8) and £ a^ c are the SU(3) s t r u c t u r e c o n s t a n t s . The Lagrangian d e n s i t y f o r the MIT bag model which i n c o r p o r a t e s t h i s SU(3) c o l o u r symmetry i s the f o l l o w i n g where the quark f i e l d q i s a short form f o r with i and ex being r e s p e c t i v e l y the f l a v o u r and c o l o u r index; the c o v a r i a n t d e r i v a t i v e ( D ^ . i s d e f i n e d as 24 C D, ) = 2 ^ 5 , - t <l ) C (2 -26) with X* being the e i g h t 3x3 Gell-Mann m a t r i c e s . The E u l e r -Lagrange equations which are consequence of t h i s Lagrangian are ( i P + ^  i^a - ^ ) <J(#) = O * 6 V , (2-28) with the boundary c o n d i t i o n s t ^(^^ — ^ CoO //6S , (2-29) A » F^Coc) = O /*6S. (2-31) The conserved c u r r e n t i s ii = 1- n ^-x % e v M Fw^Gr^ e v = ( D, F ^ ) e v, <2-32> with V Coc) = O . (2-33) 7). The conserved charge i s which, u s i n g eq.(2-32) becomes d ^ D „ F« t o t ) G v , (2-35) and when i n t e g r a t e d by p a r t s 25 v o Q a = - H * ^ F a (/X) A s ; (2 -36) which by eq.(2-31) giv e s Q = O u • (2-37) Hence, a l l s o l u t i o n s to the MIT bag .equations are c o l o u r s i n g l e t s , as observed e x p e r i m e n t a l l y . f. The MIT Hamiltonian H M I T From the Lagrangian d e n s i t y £ ^ i t given i n eq.(2-24), the next step i s to o b t a i n a Hamiltonian quark operator H M which we i d e n t i f y with the energy component of the stress-momentum tensor T°° <Al* • (2-38) where T°° i s given in' a general form by | ^ • (2-39) When a p p l i e d to i,c ^  t h i s g i v e s M i l Having shown i n the previous s e c t i o n that a l l baryons des-c r i b e d by the MIT bag model are c o l o u r l e s s , we now p r o j e c t the A operator H M i T on the space of c o l o u r l e s s baryon bags. A l s o , 26 s i n c e the quark s o l u t i o n eq.(2-16) i s v a l i d only f o r i =0 baryons, we s h a l l then r e s t r i c t o u r s e l v e s to the J5j6 r e p r e s e n t a -t i o n of SU(6) and w r i t e H M , T = \ BIB. , (2-4,) where B Q(Bo) d e s t r o y s ( c r e a t e s ) a baryonic bag of type B and mass m B . + "*o* A + D A e + l \ l 0 * ^oVl\fVQ + (2-42) The mass formula i s d e r i v e d from eq.(2-40) to be [32] = Ev - E, + EE + E „ • E. , < 2 - « ) where E y i s the volume energy (2-44) i s the quark energy EQ = I , (2-45) which can be approximated by the simple formula of F. Myhrer I I - ^ <J. 0 4 + O . l t X - , X. =/>K7 ; R, ( 2" 4 6 ) f o r low values of m^R. E i s the c o l o u r e l e c t r i c energy which vanishes i d e n t i c a l l y i f the quark masses are a l l equal. For an 27 admixture of strange and non-strange quarks, E E i s about 5 Mev. The energy E M i s the c o n t r i b u t i o n of the c o l o u r magnetic i n t e r -a c t i o n = a o c M o 0 + a o , M o S + ^ M s s , (2-47) where M o o i s the c o l o u r magnetic i n t e r a c t i o n between two non-strange quarks, M o s i s the one between a strange and a non-strange quark, and M s s i s the one between two strange quarks. The M's can be read from f i g . 2 and the a's are given e x p l i c i t l y i n t a b l e I. However, i t i s more convenient to use the a p p r o x i -mate formula [45] E = — . * f a • o. ns + anAo. ns - o.on + . 2> R c L 0 0 °* s (2-48) + a s s ( o, ns - o.o^i AnsR ) J , which i s v a l i d f o r mR l e s s than about 1.5. E 0 i s the center of mass energy and i s u s u a l l y w r i t t e n i n the form E 0 = - , (2-49) where Z e i s a p o s i t i v e c o n s t a n t . In the e a r l i e r v e r s i o n s , Z D was l e f t as a t o t a l l y f r e e parameter and v a r i e d to gi v e the best hadron mass spectrum. Since then, d i f f e r e n t models have been presented to c a l c u l a t e Z„ e x p l i c i t l y , notably by L i u and Wong [55] and K. Johnson [56]. For the r a d i u s R of the baryon bag, we can e i t h e r s o l v e the q u a d r a t i c boundary c o n d i t i o n eq.(2-30), or make use of the con-d i t i o n OR R=R& 0 (2-50) F i g . 2: Magnetic gluon exchange energy M as a f u n c t i o n of mR. The s o l i d l i n e g i v e s the exchange energy be-tween equal-mass quarks ( M q o and MgS ), the dashed l i n e g i v e s the one between a mass m s quark and a massless one, M o s [32]. Table I: The a 0 0 , a o s and a s s c o e f f i c i e n t s needed f o r the c a l c u l a t i o n of the c o l o u r magnetic i n t e r a c t i o n energy. Hadron N A E = A £ = fipKa)<j>7rK a -3 -3 1 0 3 1 0 0 2 0 2 0 -6 0 oo a 0 0 -4 -4 0 2 2 0 0 2 0 0 0 -6 OS a 0 0 0 1 0 0 1 3 0 0 0 2 0 0 30 which i s e q u i v a l e n t to i t . In other words, f i n d i n g the bag ra d i u s which balances the v a r i o u s p r e s s u r e s on the s u r f a c e i s e q u i v a l e n t to minimizing the energy ( i . e . the mass) of the baryon bag. In t a b l e I I , we give the best f i t to the hadron mass spec-trum as obtained by the MIT group when using the f o l l o w i n g set of parameters. i B * = 145" Mev. Z 0 = 1.8f * c ~ 1* Air - * . A (2-51 ) Notice i n t h i s t a b l e the l a r g e d i s c r e p a n c y of the c a l c u l a t e d pion mass with the experimental one. In massless QCD, the pion i s a massless Goldstone boson and t h e r e f o r e has a s p e c i a l r o l e to play i n hadronic p h y s i c s . T h i s w i l l be sub j e c t of f u r t h e r d i s c u s s i o n i n the context of the Cloudy Bag Model to be pre-sented i n the next chapter. Table I I : Best f i t to the hadron mass spectrum i n the MIT bag model with the parameters given i n eq.(2-5l) [32]. Hadron M M, R E E,, E„ R E exp bag o o V Q E N 0, .938 0.938 0, .99 -0.367 0.234 1, .226 -0.155 0, .000 A 1. .116 1.105 0, .98 -0.371 0.227 1, .400 -0.156 0. .005 E 1. . 189 1.144 0. .98 -0.371 0.227 1, .400 -0.116 0, .005 = 1. .321 1.289 0. .97 -0.374 0.222 1. .572 -0.136 0. .005 A 1. .236 1.233 1. .08 -0.336 0.308 1. ,119 0.141 0. ,000 I 1. ,385 1.382 1. ,07 -0.338 0. 301 1. .292 0.122 0. .005 - 1. ,533 1.529 1. ,06 -0.341 0.293 1. .465 0.106 0. ,005 1. ,672 1.672 1. ,06 -0.343 0.287 1. ,636 0.092 0. ,000 P 0. ,776 0.783 0. ,93 -0.390 0.196 0. 868 0.110 0. ,000 K 0. 892 0.928 0. ,92 -0.395 0.189 1. ,039 0.091 0. 004 03 0. 783 0.783 0. ,93 -0.390 0.196 0. 868 0.110 0. 000 * 1. 019 1.068 0. ,91 -0.399 0.183 1. ,207 0.076 0. 000 K 0. 495 0.497 0. 64 -0.564 0.065 1. 407 -0.415 0. 003 TT 0. 139 0.280 0. ,66 -0.549 0.070 1. 222 -0.462 0. 000 32 I I I ) THE CBM LAGRANGIAN DENSITY : a. C h i r a l symmetry and QCD, The Lagrangian d e n s i t y f o r Quantum Chromodynamics with massless up and down f l a v o u r of quarks has the form = ^ - \ - T i c T (3-D where q i s the quark i s o s p i n doublet wavefunction % - W ) > (3-2) and J3, F are the c o v a r i a n t d e r i v a t i v e and gluon tensor f i e l d d e f i n e d p r e v i o u s l y by eq.(2-23) and eq.(2-26). T h i s Lagrangian d e n s i t y i s i n v a r i a n t under the i n f i n i t e s i m a l g l o b a l transforma-t i o n s 0 ~ 1 5 (3-3) % - < 1 - * £-1 Y s/a )<& , with £ being the usual P a u l i m a t rices and £ being and i n f i n i t e -s imal constant parameter. Acc o r d i n g to Noether's theorem [57], the i n v a r i a n c e of the Lagrangian d e n s i t y under the i n f i n i t e s i m a l t r a n s f o r m a t i o n s of the f i e l d s t , V* = ( 1- t L- £ ) <p , (3-4) give r i s e to a conserved c u r r e n t J ^ ( x ) : _L I ( L t ) (3_5) So, the t r a n s f o r m a t i o n s given in eq.(3-3) l e a d to the conserved 33 v e c t o r c u r r e n t V p ( x ) and a x i a l - v e c t o r c u r r e n t A^(x) [54], R % > ="± < j ( / x ) Y^Y 5 r V * > , with the conserved charges CD = ( R°(/x) . ( 3 - 6 ) ( 3 - 7 ) I f we d e f i n e the l e f t - and right-handed charges 0_L and a c c o r -ding to Q L f = Q + Q c , ( 3 - 8 ) these charges w i l l then obey the commutation r e l a t i o n s and Q are t h e r e f o r e the generators of the c h i r a l symmetric group SU(2) L x SU(2) R . There are two d i f f e r e n t ways f o r these charges to act upon the vacuum s t a t e . The f i r s t p o s s i b i l i t y i s Q 1 o ) = o Q \oy - o ( 3 - 1 0 ) T h i s i s known as the Wigner- Weyl r e a l i z a t i o n of c h i r a l symme-t r y . By Coleman's theorem [58], such a vacuum s t r u c t u r e i m p l i e s that the p h y s i c a l s t a t e s w i l l appear i n p a r i t y d o u b l e t s ; but these are not observed i n nature! The other p o s s i b l e way i s the Nambu-Goldstone r e a l i z a t i o n of c h i r a l symmetry, ( 3 - 1 1 ) 34 a l s o r e f e r r e d to as spontaneous symmetry breaking. In t h i s case, p h y s i c a l s t a t e s w i l l remain p a r i t y s i n g l e t s , but by the Goldstone theorem [59,60] there w i l l e x i s t a massless i s o s p i n t r i p l e t of pseudo-scalar p a r t i c l e s . The only candidate f o r t h i s Goldstone boson, the pion, i s s l i g h t l y massive which t h e r e f o r e r e q u i r e s the broken c h i r a l symmetry SU(2) Lx SU(2) R to be s l i g h t -l y v i o l a t e d . The c u r r e n t i s now a p a r t i a l l y conserved a x i a l c u r r e n t (PCAC) and i t ' s divergence i s given by the PCAC r e l a t i o n [61], ^ 9MU) = - ^ -rru) ) (3-12) where f T i s the pion decay constant (|f^ [ ~ 93 Mev). The previous d i s c u s s i o n can be extended to i n c l u d e the strange quarks as w e l l . T h i s w i l l generate the SU(3), x SU(3) D L K c h i r a l group which, when assuming the Nambu-Goldstone r e a l i z a -t i o n of c h i r a l symmetry i n v o l v e s the e x i s t e n c e of massless kaons as w e l l as p i o n s . However, the kaon i s much more massive than the pion and the symmetry i s t h e r e f o r e badly v i o l a t e d . T h i s i s why we s h a l l r e s t r i c t o u r s e l v e s only to the SU(2) x SU(2) group and t h e r e f o r e only to the pion f i e l d . 35 b. The l i n e a r sigma model. In 1960, Gell-Mann and Levy [41] proposed a f i e l d t h e o r e t i -c a l model of the nucleon c a l l e d the l i n e a r sigma model, which embodies the PCAC r e l a t i o n mentioned i n the p r e v i o u s s e c t i o n and the Nambu-Goldstone r e a l i z a t i o n of c h i r a l symmetry. The sigma model i n v o l v e s an i s o s p i n doublet of nucleon f i e l d , and the four boson f i e l d s CT and TT. The Lagrangian d e n s i t y f o r the l i n e a r sigma model i s w r i t -ten as [61] I / A - T T S 1 I i , * a . \ , a a .a . (3-13) For c T=0, t h i s Lagrangian d e n s i t y i s i n v a r i a n t under the i n f i n i -t e s i m a l t r a n s f o r m a t i o n c r ' = c r ( 3 - 1 4 ) with the conserved c u r r e n t ~ d ~ ~ (3-15) It i s a l s o i n v a r i a n t under the t r a n s f o r m a t i o n v > = ( i - ' r - ! v s / i ) ^ cr' = v" - e.TT (3-16) TT' = -n- + 6 a-and the conserved a x i a l c u r r e n t i s fl^,l(p/Yr^ o - y V -TT^cr . 0-17) 36 If we leave the symmetry v i o l a t i n g term cn\r i n the Lagrangian d e n s i t y eq.(3-13), V^ 1 remains conserved but f o r A r, we have *V 5^(o° = " C * T T ( / x ) » (3-18) and now we i d e n t i f y t h i s l a s t equation with the PCAC r e l a t i o n e q.(3-l2) and C T T = rr " V . ( 3 - 1 9 ) The p o t e n t i a l energy d e n s i t y V(T,7T) i s given by V ( v r i T ) = ( o - \ T r J ) * + tL (<r*+ TT1) - , (3-20) For ^fcO, i . e . a r e a l mass term, V(V , T T) possesses a unique m i n i -mum when O-=TT=0. However, i f ^*<0, V(c,7r) w i l l now look l i k e f i g u r e 3 and the minimum at (Oo fH 0 ) s a t i s f i e s l«-r«-0 ' (3-21) D V / D T T ) = O •. TT = o . Since o; i s nonzero, we must then c o n s i d e r quantum f l u c t u a t i o n s not around s" = 0 but around v = \re i . e . around the c l a s s i c a l minimum of the p o t e n t i a l . D e f i n e <T = + °". , (3-22) and r e p l a c e i t everywhere i n the Lagrangian d e n s i t y eq.(3-13) ^ ~ (3-23) 38 The fermion f i e l d W , the cr f i e l d and the pion f i e l d TT a c q u i r e then the masses ^ = ^ U > = X U + h ' ^ (3-24) ""V = ^ + ^ X in , where we made use of eq.(3-2l) which i m p l i e s = 4TT . (3-25) The f i n i t e term i n the Lagrangian d e n s i t y : C T T ^ = in^n > (3-26) being spacetime independent, can be removed. The l i n e a r sigma model p r o v i d e s then a workable p i c t u r e of the nucleon except f o r the presence of the sigma f i e l d which doesn't correspond to any l i g h t and r e l a t i v e l y s t a b l e p h y s i c a l p a r t i c l e which has been observed. c. The n o n - l i n e a r sigma model. A more p h y s i c a l approach to the sigma model i s to assume that the sigma f i e l d i s a f u n c t i o n of the pion f i e l d i r rather than being a new f i e l d . However, s i n c e we want to preserve the SU(2) x SU(2) symmetry of the Lagrangian d e n s i t y , the only i n -v a r i a n t r e l a t i o n a v a i l a b l e i s ^ + J ~ U . (3-27) 39 (3-28) We can then d e f i n e a new f i e l d 4> such that ($=<*>/<$>) and the c o u p l i n g term i n the Lagrangian d e n s i t y reads The sigma and pion k i n e t i c energy term become and with the c o v a r i a n t d e r i v a t i v e d e f i n e d as •= ( D * 4 * ) * -yfaAt* ( ^ 1 ) , (3-3D then eq.(3-30) becomes J-CD^Z+J-CD^T) 4 = ^(^t>\ (3-32) which i s e a s i l y v e r i f i e d using 4>'-(D4)=0« The Lagrangian d e n s i t y of the n o n - l i n e a r sigma model has then the form ? t - > ^ e T + 7" z4 ~ + f ( T ; ) . ( 3 - 3 3 ) We n o t i c e that^. ( r(x) i s i n v a r i a n t under the i n f i n i t e s i m a l v e c t o r t r a n s f o r m a t i o n (eq.(3-28) i n eq.(3-14)) = ( I - i r . « / a ) T ( 3 " 3 4 ) which leads to the conserved v e c t o r c u r r e n t y M = Y * ^ v + i . ( 3 - 3 5 ) In a s i m i l a r way, £^!"(x) i s i n v a r i a n t under the i n f i n i t e s i m a l t r a n s f o r m a t i o n ( i f mT=0 and r e p l a c i n g eq.(3-28) i n eq.(3-16)) (3-36) which leads to the conserved a x i a l c u r r e n t A^ f = Y y ^ \ + 1 [ 4>y> + t^<M ( 3 - 3 7 ) If m^ i s non-zero, the a x i a l c u r r e n t h1* w i l l be no longer con-served and a c c o r d i n g to eq.(3-28) and e q . ( 3 - l 8 ) , \ %H = ~ iv * ^ ( <r7^ ) . (3-38) The n o n - l i n e a r sigma model t h e r e f o r e allows us to write a theory which i n c o r p o r a t e s the c h i r a l symmetry group SU(2) Lx SU(2) R and the PCAC r e l a t i o n without r e q u i r i n g the e x i s t e n c e of the sigma f i e l d . d. The n o n - l i n e a r CBM Lagrangian d e n s i t y . If we add to the MIT bag model d e s c r i b e d in chapter I I , a pion f i e l d i n a way s i m i l a r to the n o n - l i n e a r sigma model, the r e s u l t i n g theory i s the Cloudy Bag Model which i n c o r p o r a t e s the SU(2) LxSU(2) R n o n - l i n e a r r e p r e s e n t a t i o n of c h i r a l symmetry. A l l o w i n g the pion f i e l d to e x i s t i n s i d e and o u t s i d e the bag 41 e V (3-39) fo r reasons d e s c r i b e d i n the i n t r o d u c t i o n , r e p l a c i n g the nucleon f i e l d i n eq.(3-33) by the quark f i e l d , and a l l o w i n g the pion f i e l d to couple to the quarks only at the bag s u r f a c e , leads us to the f o l l o w i n g n o n - l i n e a r CBM Lagrangian d e n s i t y : 1) /j I — ^ s / | l l , 1 1 1 % which, in the l i m i t of $=Q, reduces to the MIT bag model. Demanding the a c t i o n to be i n v a r i a n t under a r b i t r a r y i n -f i n i t e s i m a l v a r i a t i o n s of the bag s u r f a c e and of a l l f i e l d s pre-sent i n the Lagrangian d e n s i t y , we o b t a i n to the f o l l o w i n g Euler-Lagrange equations of motion ^ F; (or) = O Equation (3-40) i s the D i r a c equation f o r the quarks i n t e r a c t i n g with the gluons i n s i d e the bag volume; eq.(3-4l) assures that the pressure of the quarks and gluons on the bag s u r f a c e i s ba-lanced by the vacuum pressure B. Eq.(3-42) i s the l i n e a r boun-(3- 40) 0(6 S , (3- 41 ) K 6 S , (3- 42) * 6 V , (3- 43) "31M (3- 44) 42 dary c o n d i t i o n which guarantees confinement of the quarks. Indeed, the f l u x of quark c u r r e n t through the s u r f a c e being given by T i ^ ^ ( o O = ^(tf)Y-'* ^ ( o O 0 ( 6 S , ( 3 - 4 5 ) when r e p l a c i n g e q . ( 3 - 4 2 ) in e q . ( 3 - 4 5 ) , we get />0 J (/x) = - t ^ e ^(*> t ( 3 - 4 6 ) and using the conjugate of e q . ( 3 - 4 2 ) i n e q . ( 3 - 4 5 ) g i v e s sy>h 3 U ) - + < <fi e 1f/x) ^ 6 S > ( 3 - 4 7 ) t h e r e f o r e mu Jr(oO - o /yeS , ( 3 - 4 8 ) - c (^oc) e <Jf*> = O 6 S . (3-49) Equation ( 3 - 4 3 ) was a l r e a d y mentioned to be the boundary c o n d i -t i o n f o r c o n f i n i n g the gluon f i e l d s . F i n a l l y , e q . ( 3 - 4 4 ) i s a h o r r i b l e n o n - l i n e a r Klein-Gordon equation f o r the <\> f i e l d . T h i s Lagrangian d e n s i t y i s i n v a r i a n t under the g l o b a l i n -f i n i t e s i m a l t r a n s f o r m a t i o n ^ = 1 " 1 6 t ( 3 - 5 0 ) which i m p l i e s the c o n s e r v a t i o n of the quark baryonic c u r r e n t = % ( o c ) Y^(*> 9 V . ( 3 - 5 1 ) The Lagrangian d e n s i t y e q . ( 3 - 3 9 ) i s a l s o i n v a r i a n t ( i f ^ = 0 ) under the i n f i n i t e s i m a l t r a n s f o r m a t i o n d e f i n e d i n e q . ( 3 - 3 4 ) and 43 eq.(3-36), g i v i n g the conserved v e c t o r c u r r e n t and the par-t i a l l y conserved a x i a l c u r r e n t : and e*4- r V£[+^M .<*H>+ : > ,*J> < 3- 5 3 > with the divergence ^ 2 ^ - - + • (3-54) A l l these h i g h l y n o n - l i n e a r equations are hopeless to so l v e d i r e c t l y and we s h a l l add a few more p l a u s i b l e hypotheses to make the Cloudy Bag Model a workable theory. e. The CBM Lagrangian d e n s i t y . In order to s i m p l i f y the CBM formalism, we co n s i d e r the f o l l o w i n g hypotheses: we f i r s t assume that 4»/f„. i s r e l a t i v e l y small such that a f i r s t order expansion of the Lagrangian d e n s i -ty i s adequate. In other words, we w i l l have to v e r i f y that very few pions are " f l o a t i n g around". Next, we s h a l l assume that the quark wavefunction q < x(r*,t) i s not perturbed by the presence of the pion f i e l d . We w i l l t h e r e f o r e make use of the quark wave-f u n c t i o n s as given by the MIT bag s o l u t i o n (eq.(2-16)). There are many compelling reasons f o r making these a p p r o x i -mations.. F i r s t , we r e c a l l our b a s i c assumption of t r e a t i n g the 44 pion as a p o i n t l i k e p a r t i c l e . T h i s may seem to be a rather crude assumption when faced with the quark model which r e q u i r e s the p i o n , l i k e any other hadron, to have an i n t e r n a l s t r u c t u r e . But, in a long wavelength•treatment of the pion, as we do assume here, t h i s i n t e r n a l s t r u c t u r e can be n e g l e c t e d . Next, i f we t r y to study the m u l t i - p i o n c o n t r i b u t i o n i m p l i e d i n the n o n - l i n e a r i -t i e s of our Lagrangian d e n s i t y eq.(3-39), we w i l l a l s o have to c o n s i d e r other e f f e c t s of the same order such as the i n t e r n a l s t r u c t u r e of the p i o n , the p e r t u r b a t i o n of the quark wavefunc-t i o n and of the bag s u r f a c e by the p i o n , which a l l add complexi-t i e s to the model. T h i s i s why we suggest t h i s long wavelength treatment of the pion as a simple and c o n s i s t e n t one. According to the f i r s t approximation, 4> = TT , and the Lagrangian d e n s i t y eq.(3-39) now reads s (3-55) which we r e w r i t e as C6^ no IT TT x ' (3-56) 0*> ' ^ ( i i ^ — ^ V B - i F ^ f T j e . - i B ^ s , <3-57) 2 K ^ „ j - _ ^ „ ^ ( 3 _ 5 8 ) = (3-59) The equations of motion eq.(3-37) become ( i (]l>-»n)yo() = O K£\J} (3 -60) 4 5 t Y /V) ^ (*> = ^ (/y) 0<e S , (3-61 ) 1 1 ^ ^ ' i j " ^ f«' ' (3-62) ^ FJ^CoO = O /x€ V , (3-63) % F * (Of) = O S , (3-64) ( 2 a+ AIOJ ) T ( * ; = - ^ A S . (3-65) The f i r s t f i v e equations are the MIT bag model equations (eqs.(2-27 to 31)), and the l a s t one i s the Klein-Gordon equa-t i o n with a source term on the bag s u r f a c e . The a x i a l c u r r e n t i s now P^(oO = -^1 % Y H V S Z 4 ^ 2>hTT(*> t (3-66) % and i t ' s divergence i s again the PCAC r e l a t i o n \ 9^' = - W 7 • ( 3"6 7 ) A l l these equations (3-55) to (3-67) form the core of the Cloudy Bag Model and they w i l l allow us to c a l c u l a t e e x p l i c i t l y v a r i o u s baryon p r o p e r t i e s such as magnetic moments, charge r a d i i and o t h e r s . 46 IV) HAMILTONIAN FORMULATION OF THE CLOUDY BAG MODEL : a. I n t r o d u c t i o n . So f a r , we have b u i l t a theory of strong i n t e r a c t i o n s i n -v o l v i n g quarks and gluons c o n f i n e d in a bag. We a l s o introduced a pion f i e l d which couples to the quarks at the bag sur f a c e and, for massless quarks and pi o n s , guarantees i n v a r i a n c e of the theory under the c h i r a l group SU(2) Lx S U ( 2 ) R . Furthermore, we gave the pion a mass to embody the PCAC c o n d i t i o n on the a x i a l c u r r e n t [62]. We have a l s o shown i n chapter II that the confinement of the quarks and gluons in the bag leads to c o l o u r l e s s baryons and mesons. T h e r e f o r e , i t w i l l make sense to express the Cloudy Bag Model in the space of c o l o u r l e s s baryon bags r a t h e r than the quark space as d e s c r i b e d in chapter I I I . Furthermore, f o r s t r a n -geness zero baryons f o r example, we r e s t r i c t o u r s e l v e s to the s t a b l e nucleon doublet and the lowest d e l t a quadruplet (mA= 1232 MeV). We t h e r e f o r e n e g l e c t the i n f l u e n c e of higher r a d i a l or angular momentum s t a t e s [63] i n v o l v i n g d i s t o r t e d bags and high energy propagators, which are i n c o n s i s t e n t with our p i c t u r e of a s p h e r i c a l , s t a t i c bag and long wavelength ( i . e . low energy) pion approximations. We a l s o n e g l e c t the i n f l u e n c e of c o l o u r l e s s e x o t i c baryons [47], whose e x i s t e n c e i s p o s s i b l e i n the context of bag models, but do not correspond to any s t a t e s observed ex-p e r i m e n t a l l y . In sh o r t , we l i m i t o u r s e l v e s to the low l y i n g baryon o c t e t and decuplet of p h y s i c a l p a r t i c l e s . 47 The procedure w i l l be the f o l l o w i n g : f i r s t we w i l l w r i t e down the quark space Hamiltonian corresponding to the Lagrangian d e n s i t y eq.(3-55), next we w i l l q u a n t i z e the pion f i e l d a c cor-ding to the usual r u l e s , and f i n a l l y we w i l l p r o j e c t t h i s quark space Hamiltonian onto the c o l o u r l e s s baryon bag space using the bag model SU(6) wavefunctions. T h i s procedure w i l l generate an i n t e r a c t i o n Hamiltonian which w i l l couple the pion to the baryons i n a w e l l d e f i n e d way. b. The CBM quark Hamiltonian H : From the e x p r e s s i o n f o r the CBM Lagrangian d e n s i t y given in eq.(3-55) to eq.(5-39), we can d e r i v e the quark Hamiltonian operator H a c c o r d i n g to * A /V A where H i s r e l a t e d to i. (oO v i a We then o b t a i n 48 When r e p l a c i n g q.(r,t) by i t ' s e x p r e s s i o n eq.(2-16) \toc- ^ (n.-ft/RK-A J and expanding Ttj ( r , t ) i n momentum space a c c o r d i n g to t . o - ( {a(W* + a + ( f ) eiW I (4 -8) = - r f R* i« ) a i ! ( r i , )w ; (jn b*?.j! r.ir.b-1)7! 1=1 J ~ ~ I Since 3> then P^has the f i n a l form with 3 (4-9) with the usual commutation r u l e s A we get the MIT quark Hamiltonian H M | T the f r e e pion Hamiltonian H . [ ( J3k % a\(h a^d) , ( 4 - u ) and the i n t e r a c t i o n Hamiltonian quark operator H T (4-10) ( 4 - 1 2 ) ^ 0(kR) + ^ ( l?R)] > ( 4 - 1 3 ) H a = I j | V.(?) a. (U) + V*<1) a ' db j , ( 4 - i 4 ) < / . ( ? ) = [ c t b ^ . l ? r^bt- , (4-15) > _ 49 a. (4-16) For the sake of s i m p l i c i t y , we s h a l l assume from now on that the up and down quarks are massless and we t h e r e f o r e w r i t e JQ.= ^ -JL- * o.^ (4-18) c. The CBM baryonic Hamiltonian. In the space of c o l o u r l e s s baryonic bags, we can d e f i n e an i n t e r a c t i o n Hamiltonian a c c o r d i n g to H1 = I j J3fe { V 0 . (?) (^f) + V ^ ? ) a - ( ? ) J , (4-19) A, The connection between H and Hx i s given by eq.(4-l5) and eq.(4-l9) as v0)(l)= I R 0 + < R , | V / ? > | B , > B . , ( « - 2 0 ) where A* c r e a t e s a bare c o l o u r l e s s baryon bag of type A and Be d e s t r o y s a bare bag of type B (We s h a l l always i n d i c a t e bare q u a n t i t i e s by adding a su b s c r i p t , , ). hq and B^ are the SU(6) s p i n - i s o s p i n quark wavefunctions of type A and B which are given i n great d e t a i l i n r e f . [ 6 4 ] . L e t ' s r e w r i t e VQ- (k) in the f o l l o w i n g form V„(1?>= I H*. < 8 < f » B . , (4-21) 50 Ml\ (* > - 1 ^ R ; • -7- (4-22) Since the baryons concerned here are members of the J56 represen-t a t i o n of SU(6), i t i s n a t u r a l to r e w r i t e the op e r a t o r s S f l B and T f i 6 as i r r e d u c i b l e tensors of rank 1 S n 6 = < R J I S ^l l fO , (4.23) T"-<R.I i IB.>, ( 4 - 2 4 ) where s ^ and t w are s p h e r i c a l b a s i s d e f i n e d a c c o r d i n g to 6 i i = T v T U ^ } ' ^ £ l ' ( 4 " 2 5 ) and t h e r e f o r e " ' / u n . (4-26) with Sfl and sfi being r e s p e c t i v e l y the t o t a l s p i n and i t ' s pro-j e c t i o n on the z- a x i s of the bare baryonic bag A , and s i m i l a r -l y f o r Tfl and t f i . Since S^ and are i r r e d u c i b l e t e n s o r s , we can apply the Wigner-Eckart theorem and get <& = ^r<^T,Jj\5Ti lS BT 6^>^S / l)( 2T f l +oj" ,' 1 ; (4-28) where the c T 1 ' * " ^ are the usual Clebsch-Gordan c o e f f i c i e n t s . Using e q . U - 1 5 ) i n eq.(4-20) and eq.(4-27) g i v e s f o r f " 6 51 Since the s p i n - i s o s p i n quark wavefunctions are t o t a l l y symme-t r i c , we may as w e l l take the matrix element i n eq.(4-29) on the t h i r d quark only, and i f we choose )c=kz and e. =e3 then i B=3^<R,ib> ;r i M B,>.c; :;;: c^i. «-3.> The c o u p l i n g c o n s t a n t s f 0 can be computed and we give them ex-p l i c i t l y i n t a b l e I I I . In summary, the Cloudy Bag Model baryonic Hamiltonian i s give n by H = H E + H T , ( 4 - 3 D H D - I />» R R! RD + 1 j JJfe % a* <?)<*.("?) , (4-32) = I \j }k { V tf) a(h 4 yfoxha.(h\ (4-33) where the bare masses mofl were given i n eq.(2-43) and = I F?I (^f) B Q , (4-34) \IV = I R b /w;. ( ? ) B 6 , (4-35) ( l ^ C s . ' ^ k > C V l T / ^ ) , (4-36) w"(h = ( ^ . " ( F ) ) * , (4-37) and the form f a c t o r of the i n t e r a c t i o n , M (kR) = jjkR) + -j a(fef?) , (4-38) normalized to u n i t y f o r k=0, i s shown i n f i g u r e 4. If we now co n s i d e r the nucleon s e c t o r of the theory, we Table I I I : The TTAB bare c o u p l i n g c o n s t a n t s AB, r o / IQ* o / Q N A A z * z * N 5 4/2 0 0 0 0 0 A 2/2 5 0 0 0 0 0 A 0 0 0 2/3 2/6 0 0 Z 0 0 -2 4/6/3 -4/3/3 0 0 * z 0 0 2 2/6/3 2/30/3 0 0 = 0 0 0 0 0 -1 -2/2 * 0 0 0 0 0 2 1/5/3 F i g . 4: The CBM form f a c t o r |u(kR)| and a Gau s s i a n f i t , v ( k R ) = e x p ( - 0 . l 0 6 k * R 2 ) . 54 have , {i.e.) (4-39) • < A/: I Z> ? r 3 | |S/o > , (4-40) . ^ O. 13 (4-41 ) T h i s i s the usual form f o r the pseudoscalar pion c o u p l i n g to the nucleon except f o r the presence of the form f a c t o r u(kR) which can be i d e n t i f i e d with Chew's form f a c t o r d e s c r i b e d i n the i n -t r o d u c t i o n . The u n d e r l y i n g quark s t r u c t u r e of the nucleon shows then i n the s t r e n g t h of the c o u p l i n g c o n s t a n t s , t h e i r r a t i o and the form f a c t o r . The s t r e n g t h of the bare ITNN c o u p l i n g constant f D of 0.23 may seem to d i s a g r e e with the experimental one of 12% [33] to 20% [66], and by g i v i n g the quarks a s l i g h t mass of about 20 Mev. which adds another 2% to f c [67]. We then conclude that our 7TNN c o u p l i n g constant f e i s c o n s i s t e n t with the e x p e r i -mental one when the necessary c o r r e c t i o n s j u s t mentioned are made. Next, a l l the TTAB c o u p l i n g c o n s t a n t s are r e l a t e d t o the TTNN c o u p l i n g constant f * w v i a the SU(6) quark model c o e f f i c i e n t s which are given in our n o t a t i o n i n t a b l e I I I . F i n a l l y , the bag r a d i u s R, by i t ' s presence i n the form f a c t o r u(kR), i s then the only true f r e e parameter l e f t in the Cloudy Bag Model. (4-42) with g J/4TT^14 [65]. However, t h i s d i s c r e p a n c y can be e x p l a i n e d i f c e n t e r of mass c o r r e c t i o n s are i n c l u d e d , which i n c r e a s e f 0 by 55 d. The e i g e n s t a t e s of H 0 and H. The study of the t o t a l Hamiltonian H w i l l allow us to c a l -c u l a t e e x p l i c i t l y the p h y s i c a l p r o p e r t i e s of baryons. But f i r s t , l e t ' s i n v e s t i g a t e the e i g e n s t a t e s of H. The baryon o c t e t members ( N , A , I , - ) are the lowest energy e i g e n s t a t e s of the f u l l H amiltonian H, i . e . H I N> = |N> ... . ( 4_ 4 3 ) For convenience, we s h a l l c o n s i d e r only the non strange baryons ( N , A ) and mention the strange ones only when necessary to the d i s c u s s i o n . There are a l s o s c a t t e r i n g s o l u t i o n s when 1,2 or n pions are present where |N , K . > 4 represent the s c a t t e r i n g which at t = ; o ° i s made of a f r e e pion of momentum Ic and i n t e r n a l i n d i c e s j and of a p h y s i c a l nucleon. With the t r u n c a t i o n s d e s c r i b e d i n s e c t i o n a, we have the completeness r e l a t i o n f o r baryon number one "A } ^ " J> or i n a shorthand n o t a t i o n , The f r e e Hamiltonian H 0 possesses a l l members of the 56-p l e t of SU(6), i . e . both the baryon o c t e t of s p i n 1/2 baryons and the baryon decuplet of s p i n 3/2 baryons. For strangeness 56 zero, the lowest l y i n g s t a t e s are the bare nucleon and d e l t a which obey ( 4 - 4 7 ) H c I A e > - ^ I A 0 > . However, when the i n t e r a c t i o n Hamiltonian Hz i s turned on, the d e l t a i s no longer s t a b l e and appear as a resonance i n the P channel. We w i l l a l s o have e i g e n s t a t e s of H D c o n s i s t i n g of a bare bag with n p i o n s : H. I w.,*,> = (Mni* I u,. ) | N . *> _ c - l with the completeness r e l a t i o n OO i = I { i N / o ^ X ^ ^ I + U e ^><A 6 ; /n|1 ( 4 - 4 9 ) N o t i c e that the s t a t e |Ne,k> for example i s obtained by a p p l y i n g the pion c r e a t i o n operator a^(k) on the bare s t a t e |Ne> We are now ready to apply p e r t u r b a t i o n theory to the Cloudy Bag Model and t h i s i s the o b j e c t of the f o l l o w i n g chapters but f i r s t , we d e s c r i b e a very convenient graphic n o t a t i o n f o r the bare v e r t i c e s and propagators very s i m i l a r to the Feynman graphs i n Quantum El e c t r o d y n a m i c s . 1- Incoming and outgoing bare baryon A : |Ao> and <A 0J = l f l „ > - <n.l Incoming and outgoing p i o n : | k- > and <k^  | TTAB a b s o r p t i o n and emission v e r t e x : v" B(ik >) and W ' , 8 ( K > ) Bare A propagator : G 0 ( E ) : (e Pion propagator (already i n c l u d e d i n H :) : h r 58 V) RENORMALIZATION OF THE CLOUDY BAG MODEL : a. Expansion of the p h y s i c a l nucleon : In the Cloudy Bag Model, the p h y s i c a l baryons are "dressed" bags, i . e . that in v i r t u e of the presence of the i n t e r a c t i o n Hamiltonian which couples the pion f i e l d to the bare bags, the p h y s i c a l nucleon f o r example w i l l be pa r t of the time a bare nucleon bag, a bare nucleon or d e l t a bag with one pion " i n the a i r " , e t c . Therefore we can expand the p h y s i c a l nucleon on the b a s i s of the bare e i g e n s t a t e s of H 0 a c c o r d i n g to |N> •= Z[(tj\ N O + A 1 Ni> , ( 5 ' 1 ) where Z^(E w) i s the p r o b a b i l i t y f o r the p h y s i c a l nucleon |N> to be a bare (no pion present) nucleon |N0> and E^ i s the nucleon energy obeying H | N > = E j » V > , (5 -2) with EN=mN f o r a f r e e p h y s i c a l nucleon. A i s an operator which p r o j e c t s out a l l the components of |N> with at l e a s t one p i o n : A = i ~ I | R e > < R J , (5 -3) A* = A • (5 -4) Using the i d e n t i t y A l f N / > = r ( EK- H o )~ ' A ( EN- H 0 ) I N ) i (5 -5) 59 together with eq.(5-1,2) and [A,H o]=0 g i v e s I N > = z^ej I/O + £ ( A 1 N >, and G e(E) i s the usual bare propagator (5-6) (5-7) Expanding the i n t e g r a l equation eq.(5-6) f o r the p h y s i c a l nucleon g i v e s 8) which can be condensed i n the form (5-9) Using our graphic n o t a t i o n d e f i n e d in chapter IV, then eq.(5-9) r e l a t e s |N> to |NC> v i a 1 w> = iteJ \. + • * + • ^ > ,, + + - 4 — — 4 . — I I — » - + . . . { N J > + • • • + • • • J , (5-10) (5-11 ) with A \ S~ \ / A \ J > « 4- . ' i l l + fl R A « « a j (5-13) where the u n s p e c i f i e d i n t e r m e d i a t e baryon l i n e s can be e i t h e r 60 nucleon or d e l t a l i n e s . b. The dressed d e l t a . In the p r e v i o u s chapter, we mentioned that the bare Hamiltonian H 0 has two no-pion e i g e n s t a t e s : |Ne> and If the p h y s i c a l d e l t a mass s a t i s f i e d ^ c , ^ ^ A , " * " , (5-14) then the dressed d e l t a would be an e i g e n s t a t e of the f u l l H amiltonian H s i n c e i t would not be allowed to decay sponta-neously i n t o a pion and a p h y s i c a l nucleon. So, s i m i l a r l y to the p h y s i c a l nucleon expansion e q . ( 5 - 9 ) , we would have | A > - Z * (E a f { 1 + ( E - H ^ A H / i ' H , ] I A >. (5-15) However, we r e a l l y have mA> ni^+m^ and the propagator i n e q . ( 5 -15) can v a n i s h . In t h i s case we can s t i l l d e f i n e a dressed d e l t a s t a t e which i s not an e i g e n s t a t e of H but which s a t i s f i e s |A> = 7 * ( E f l ? { 1 + P U ^ - H r A H ^ - ' r t j l A c>. (5-16) The p r i n c i p a l value p r e s c r i p t i o n i n e q . ( 5 - l 6 ) o r i g i n a t e s from the Lee model of an u n s t a b l e V - p a r t i c l e [69,70] and it'.s motiva-t i o n comes from the requirement that the dressed d e l t a should be a s i n g l e p a r t i c l e s t a t e and t h e r e f o r e must be time r e v e r s a l i n -v a r i a n t . What i s the mass mA of the dressed d e l t a ? At f i r s t , one 61 would think that m^  should be the resonance energy i n the P channel. However, the formation and the decay of the dressed d e l t a bag doesn't account f o r the f u l l c r o s s - s e c t i o n i n the P J 3 channel. In f a c t , c r o s s e d graphs such as (5-17) a l s o c o n t r i b u t e t o the P c r o s s s e c t i o n . However, t h i s c o n t r i -b u t i o n w i l l be shown i n the next chapter to be r e l a t i v e l y small and t h e r e f o r e we s h a l l assume the dressed d e l t a mass to be the resonance mass: />v)u ^ | a* a AA eV. t (5-18) One f i n a l note before we continue i s that the dressed d e l t a bag w i l l never appear i n our equations as an incoming or out-going s t a t e , s i n c e at t= + °° , the d e l t a must have decayed. However, the dressed d e l t a bag w i l l have i t ' s p h y s i c a l s i g n i f i -cance i n s i d e propagators. For t h i s reason we s h a l l use the dressed d e l t a bag expansion eq.(5-l6) only f o r d e s c r i b i n g i n t e r -mediate s t a t e s . We s h a l l d i s c u s s the d e l t a i n more d e t a i l l a t e r i n t h i s chapter, and i n the next one on pion nucleon s c a t t e r i n g . c. The nucleon and d e l t a s e l f - e n e r g y . In absence of the i n t e r a c t i o n Hamiltonian H T, the mass of the nucleon and d e l t a would be simply motJ and m o A. However, when we t u r n on the c o u p l i n g , the p h y s i c a l nucleon mass becomes mN 62 and the dressed delta bag mass moves to mA . The difference bet-ween the physical and bare masses is the self-energy. For the physical nucleon, we have the self-energy 2. (mN) given by £ N ( " » w ) = ^ - ^ ) o W , (5-19) = I H - H j l N / > , (5-20) and from the nucleon expansion eq.(5-9) we have I * = <K> I H, ( E -H e -A H / ) " ' H - J f ^ M (5-2D and similarly for the dressed delta bag, = < A o l P (E -H-AH.A) " '^ | A 0 > l (5-22) Graphically, the first few terms of *L for example are given by £"('»OS= - LCJI + + +.. (5-23) and we use the notation I.fl<0 = HH , (5-24) where the square bubble contains all the one-particle irredu-cible graphs, i.e. the graphs which do not separate in two dis-joint graphs when cutting an intermediate baryon line anywhere inside the graph. Similarly, the dressed delta bag self-energy is given graphically by 63 We s h a l l give e x p l i c i t e x p r e s s i o n s f o r the nucleon and d e l t a s e l f - e n e r g i e s c o r r e c t to second order i n H i n s e c t i o n i of t h i s chapter. d. Bare bag p r o b a b i l i t y , Z 4 ( E B ) . If we normalize the p h y s i c a l nucleon wavefunction to u n i t y , then the expansion eq.(5-9) of the p h y s i c a l nucleon on the bare e i g e n s t a t e s w i l l g i v e the formal e x p r e s s i o n f o r the bare nucleon p r o b a b i l i t y z " ( E w ) : z i < & = [ 1 + <K\HJ(£ArHa-r)HIf\fiHJ l/v/0>] ' ( 5 - 2 6 ) Z4"(e„)= [ I " £ <^.|H I(E-H 0-/\^"H 1l^>] > <5"27> and we recognize i n the bracket the expr e s s i o n f o r the nucleon s e l f - e n e r g y . So, and more g e n e r a l l y , Again, we s h a l l wait to s e c t i o n i to give second order expres-s i o n s f o r Z ( E f l ) . (5-28) (5-29) 64 e. The ver tex f u n c t i o n z " 6 ( E n , E B ) . In chapter I V , we d e f i n e d the bare ver tex f u n c t i o n v"* (k) accord ing to eq.(4-21) irJfih = < R J V0.(l?> |B e> . (5-30) However, i n p h y s i c a l processes of i n t e r e s t such as p ion nucleon s c a t t e r i n g , we s h a l l encounter the matr ix elements of V . (k) taken between the dressed s t a te s r a ther than between the bare s t a t e s . We s h a l l there fo re de f ine a s o - c a l l e d renormal ized ver tex f u n c t i o n v „ ^ ( k ) accord ing to «r**tf) = < R I V.^(f) |B> . ( 5 - 3 D Since A Q . , A and B Q , B have the same quatum numbers, i t can be shown [15] that v " 8 {t) i s p r o p o r t i o n a l to v"^(k*) ^ = ^ ( E „ , E 8 ) ATa^ ( k) , (5-32) R B where the p r o p o r t i o n a l i t y constant °) (E f l,E 6) i s independent of the magnetic quantum numbers s z and t 2 . For example, l e t ' s con-s i d e r the NNTT r enormal ized ver tex f u n c t i o n v ^ ( i t ) . When expan-ding the p h y s i c a l nucleon s t a te s | N > and | N ' > a ccord ing to e q . ( 5 - 9 ) , we get * 1 7 (5-33) • V . .(S>(E,-W.-AM)"'^ IW 0>j . But accord ing to e q . ( 5 - 3 2 ) , the l a s t bracket i s p r o p o r t i o n a l to 65 « I H2( Ewr H - AH/ ) V . ( 1 ) ( £ w - H- AH^)" H,! W*> = AT^(1> X" ( \ , , e , ; >( 5-34) and i f we d e f i n e the vertex f u n c t i o n Z, ( E ^ E ^ ) a c c o r d i n g t o ^ ( • W U - [1* Xw'(£,,e„)]",' then ^ (E W.,E W) i s r e l a t e d to Z*(E W) and Z*** (E^,E W) v i a (5-36) In g e n e r a l , i f we con s i d e r A and B to be dressed bags, we have i n analogy to eq.(5-34) to eq.(5~36), ^ B(S) x"< eRje f t) = < H. I Hx( £fl-H.- A H.A)' V £t- H - / \ H / \ ) \ \ ; ( 5 - 3 7 ) WW . V w I , _ I / ^ . ' . t j wr -z . < E *A) = [ 1 * x'Vvl"! (5-38) and again the p r i n c i p a l value of the i n t e g r a l s i n v o l v e d i s imp-l i e d whenever the propagators v a n i s h . F i n a l l y , we g i v e the gra-phic r e p r e s e n t a t i o n of the renormalized vertex f u n c t i o n v R 6 ( i c ) using eq.(5-32) and eq.(5-39) / i r (k) - 7 j C £ f l ) Z ' ( g » > . a / , (5-40) and from- eq.(5-37) and eq.(5-38), the vertex with a l l i t s bubbles i s given by 66 f. The renormalized coupling constants. The bare NNTT vertex function was given in e q . ( 4 ~ 4 l ) to be /V (I?) = i • J » _ xx(t?R) . < y j f. | K/o^>_ ( 5 - 4 2 ) If we now write the renormalized vertex function v^ (k) accor-ding to i A T ^ ( k ) ^ c ^ .<N'.l?-f ^ IW.>, ( 5 - 4 3 ) then f^ i s the renormalized NNTT coupling constant and i t i s related to the bare coupling constant f„ according to i, ( £ „ , , o = *:•».;>* W . 4. . (5-44) Following Schweber [15], we now show that f^ is related to the experimental one via - £ « f = ^ 5 ^ c ^ ^ - ^ i r r ^ (n,,^) . ( 5 - 4 5 ) The proof follows from writing the pion f i e l d potential in the physical nucleon. This i s defined by <TT.(5h>= \ I J ^ ~ U ( h J ; ' i ^ a \ c h e i U \ \ h l > ) (5-46) and since < M I [ H , a - t l o ] I N > = < N / | w „ a*(k) + Voi(l?) I W > = O, ( 5 - 4 7 ) then < T T ^ > = < N | V > e W + V . . < f > « r - ' W | N > (5-48) and using e q . ( 5 - 4 3 ) followed with a few simple manipulations, we 67 get <TT(^)>= \HTT7 i^-^ <Nj?.y r^N/tt>( A_ *(fcg)c' * ( 5 _ 4 9 ) TT For |/i|=r>>R where R i s the bag r a d i u s , only the small values of k w i l l c o n t r i b u t e and u(kR) can be approximated by one. So f i n a l l y , we have JU < T T < * ) > = U">».~>^ ,<is/J Z.?3 1/sQ . (5-50) So, f o r l a r g e d i s t a n c e s , the pion p o t e n t i a l i s a Yukawa p o t e n t i a l whose s t r e n g t h i s given by the renormalized c o u p l i n g constant ty (m ,m ). We t h e r e f o r e i d e n t i f y the renormalized cou-p l i n g constant with the experimental one. We conclude t h i s s e c t i o n by d e f i n i n g the renormalized TTAB c o u p l i n g constant in d i r e c t analogy to eq.(5-44): i » Z > ( ^ ^ ( C » } • {. , (5-51) and the ex p r e s s i o n of the renormalized vertex f u n c t i o n i n eq . ( 5 -32) now reads ^ 0 0 = ' ivJiaJ^) *(HR> C s \ A ^ t ) C * V T \ ? : - l i ) J (5-52) and we represent t h i s g r a p h i c a l l y by • AT . ("£) = ^ % (5-53) "1 FI ^ G with the r below the vert e x r e f e r i n g to the renormalized cou-p l i n g c o n s t a n t . F i n a l l y , the the dressed vertex with a l l i t ' s bubbles as d e s c r i b e d i n eq.(5-4l) can be r e w r i t t e n as eq.(5-54) 68 12 1 ( 5 - 5 4 ) Z > * , e a ) g. The renormalized propagator. When expanding the t-matrix on the b a s i s of the bare eigen-s t a t e s , we s h a l l encounter matrix elements of the f u l l propaga-t o r G(E) between bare baryons. For reasons which w i l l be more apparent i n the next s e c t i o n , we d e f i n e l i k e Chew [11] a renor-malized nucleon propagator f u n c t i o n G^(E) a c c o r d i n g to G*(e.) = z " ( e )"' < A / J CrU > I N 0 > } ( 5 - 5 5 ) , with E =mA/ f o r o n - s h e l l nucleons, and G(E) i s the usual f u l l propagator = (e-H)". ( S . S 6 ) Now, expanding G(E) i n terms of G 0(E) ac c o r d i n g to 6 (e) + 6-o(f)HI6„(e) + 60(£) HI6;(e)HJ6(l(e) + ... , ( 5 - 5 7 ) and r e a l i z i n g that only terms with an even number of H.J w i l l c o n t r i b u t e to G*(E), and using the formal e x p r e s s i o n e q . ( 5 - 2 1 ) f o r 1 ( E ) g i v e s z / W ' < ^ J 6.U-sce>) |/v/D> , ( 5 _ 5 8 ) = ^ E r ^ ^ K e ^ - K ^ ' l N / . ) . ( 5 - 5 9 ) 69 Now, f o r Earni^, i . e . the f r e e nucleon, eq.(5-58) reduces to = l^ey' (E-^-^A / . I I ( e )k ,> ) ' (5-60) and we can expand <N„|I(E)|N„> in T a y l o r s e r i e <Na I lie) |iv0> s * (e-^).^ r v ( e ) + • • • , (5-61) and then eq.(5-60) reads & > > = z : w [ i - ^ t W < e - * i > ' ' + A . < » . , ( 5 _ 6 2 ) A/ A/ and s i n c e the bracket i s simply Z3 (m w), we are l e f t with 6 V W U ) = ( £-^i' + ' J i .o . . (5-63) So the renormalized propagator f u n c t i o n G^(E) reduces to the p h y s i c a l nucleon propagator <N|G(E)|N> i n the l i m i t of E c l o s e to mw. To o b t a i n a g r a p h i c a l r e p r e s e n t a t i o n of G,,(E), we f i r s t expand eq.(5-58) to get G"(e)= Z > ) ' ' (^Uje)^ 0(e)l(e)^e)+- I A/.> , (5-64) which can be w r i t t e n g r a p h i c a l l y as G . (6) 3= . A/ A / A/ A/ A/ ..... I (5-65) (5-66) F i n a l l y , we can re p l a c e everywhere the bare nucleon s t a t e |Ne> everywhere by the bare s t a t e |A0> where A i s any bare bag 70 s t a t e ( i n c l u d i n g the d e l t a ) to get the renormalized A propagator G (E) d e f i n e d as &?(e) = z 4 V) ' ' <n o U(a|R 6 > ; ( 5 - 6 7 ) and s i m i l a r l y to e q . ( 5 - 5 9 ) G?(e>= Z*(e)~' < f l . | ( e A - H r l ( f ) ) ' | f l .> , ( 5 - 6 8 ) which i n the l i m i t of E c l o s e to mfl reduces to (5-69) and the graphic r e p r e s e n t a t i o n of G*(E) i s G\E)= . L . =7«(0~'-- ^ • • ( 5 _ 7 0 ) Y «(£> >K ' fl ft h. The r e n o r m a l i z a t i o n procedure. The next step i s to show that f o r p h y s i c a l processes such as pion nucleon s c a t t e r i n g f o r example, we can always make a p e r t u r b a t i o n expansion of the t-matrix which w i l l i n v o l v e only the renormalized vertex f u n c t i o n v (k) and the renormalized v\ propagator G ).(E). A t y p i c a l example of i n t e r e s t i s the pion nucleon t-matrix which we w i l l show i n chapter VI to be given by t(N' t\ Nfa = < N I V\k) 6 ( E ) V (IO |W>+ ••• . (5-71) 71 If we expand the p h y s i c a l nucleons a c c o r d i n g to the usual expan-si o n eq.(5-9), and i n s e r t between each operator the completeness r e l a t i o n eq.(4-49) f o r the bare e i g e n s t a t e s OO , 1 = 1 |Mem></v/0^| + l A D > / h > < A . , * i l , (5-72) f*\-o \ J then the t-matrix element i n eq.(5-71) w i l l take the f o l l o w i n g g r a p h i c a l form ^ (5-73) A" A> L e t ' s w r i t e a general term of the t-matrix expansion as ' ' A ' • 2 a ( e M ) i ( 5 - 7 4 ) where the v e r t i c a l pion l i n e s can be e i t h e r emitted or absorbed p i o n s . However, a c c o r d i n g to the d e f i n i t i o n of the bubble vertex (eq.(5-54)) and the bubble propagator (eq.(5-70)) we have = • - i - . ( 5 " 7 5 ) and t h e r e f o r e a l l the Z a's c a n c e l and we are l e f t with < j t i < 1 - i > i / / ! . » i (5-76) A/ and the r e s u l t i n g graph i s then an i r r e d u c i b l e bare graph where a l l the bare v e r t i c e s and bare propagators are r e p l a c e d by the renormalized ones. For example the f i r s t two terms of the pion nucleon t-matrix i n eq.(5-7l) w i l l read \T \*' — - — * A7 > . , I? A/ » - + , (5-77) 72 where the intermediate renormalized propagators can be e i t h e r nucleon on d e l t a ones. In summary, the r e n o r m a l i z a t i o n procedure d e s c r i b e d allow us to express the t-matrix elements of p h y s i c a l processes i n terms of i r r e d u c i b l e bare graphs which i n v o l v e renormalized pro-pagators and v e r t i c e s o n l y . i . Second order e x p r e s s i o n s f o r the r e n o r m a l i z a t i o n f u n c t i o n s . We present i n t h i s s e c t i o n c o r r e c t e x p r e s s i o n s to second order i n p e r t u r b a t i o n theory of the s e l f - e n e r g y X n ( E n ) , the bare bag p r o b a b i l i t y z"(E n), the vertex f u n c t i o n Z ( R B ( E n , E B ) and the renormalized c o u p l i n g c o n s t a n t s f^ ( E R , E 6 ) . A l l these q u a n t i t i e s w i l l be c a l c u l a t e d e x p l i c i t l y only f o r the nucleon s i n c e a l l those f o r other baryons can be d e r i v e d e a s i l y from the general equations given i n the t e x t . F i r s t l e t ' s c o n s i d e r the formal e x p r e s s i o n f o r the s e l f -energy given i n eq.(5-2l) and eq.(5-22) X"(Efl} - <R.| Hz ( EA- H.- f\H1l\))U1 I R.> , (5-78) where the p r i n c i p a l value must be taken whenever the propagator v a n i s h e s . L e t ' s a l s o d e f i n e the operator 1 0(E) a c c o r d i n g to I o(E) - H IA6 o(E)AH 1 . (5-79) Then, by expanding the propagator f u n c t i o n i n eq.(5-78) and s i n c e terms with an odd number of H T w i l l not c o n t r i b u t e to 73 I (E f i) , we get l V f l ) = <flj H3 (E,- H„- Ie(E^V |RC> . (5-80) To o b t a i n an e x p l i c i t e x p r e s s i o n f o r I (E f l) which w i l l not be an i n t e g r a l equation, we s h a l l f u r t h e r assume that H + £ (E ) ^ Ho , (5-81 ) where H D i s the bare Hamiltonian with the bare masses r e p l a c e d by the p h y s i c a l ones fj =• 1 /»>^ f{\ Re + % w k a t >(^)a . ( 1 ) . (5-82) A l l t h i s g i v e s us a second order e x p r e s s i o n f o r I A ( E f l ) as T * ( E A ) = <R0I H, (^-HJ-'H, |R&>, (5-83) which can be evaluated e x p l i c i t l y using eq.(4-33) f o r H 2: I = H I J lk /v^Cl?) ( E ^ - ^ - u , ^ " /w. ( * > . ( 5 - 8 4 ) R e p l a c i n g v o. (k) by i t ' s e x p l i c i t e x p r e s s i o n e q . ( 4 - 3 6 ) and using the i d e n t i t y eq.(4-37) f o r w0<^  (k) we are then l e f t with X (5-85) But • y - t l^Ac^JuriHiJ^ rnr/i , (s.86) 7 4 t h e r e f o r e , and the bare mass of the baryon A i s then given a p p r o x i m a t i v e l y by (5-89) For the p h y s i c a l nucleon f o r example, we have ru,,, = - — 1 (—) (Jk ( * * V ( k R ) (5-90, where we d e f i n e d Having now a second order e x p r e s s i o n f o r the s e l f - e n e r g y , we can w r i t e down immediately the second-order e x p r e s s i o n f o r the bare baryon A p r o b a b i l i t y Z A(E„) as (5-92) and with eq.(5-88) i t i s given e x p l i c i t l y by 8 \ ">* / U7J A 2_ -1 (5-93) For example, the p r o b a b i l i t y f o r the p h y s i c a l nucleon' to be a bare nucleon bag Z^(EN) w i l l be (5-94) 1 + U7T 75 Next, we c o n s i d e r the vertex f u n c t i o n Z( (E f t,E 6) which ac-co r d i n g to eq.(5-37) and eq.(5~38) i s given by Aft -I AB _» fit , / , - i _ (5-95) • Ve.(l?)(VH.-A«IA),H;I IB6> or g r a p h i c a l l y by e q . ( 5 - 4 l ) . AB .1 AB _» 2, ^ ( f e > = ' + ^ + ... (5-96) And s i m i l a r l y to the s e l f - e n e r g y i n eq.(5-78), we approximate eq.(5-95) by = ' S < * ' + <'tlH 1(e i r f i >)" ,V(f)(E t-fi 1)-VlB.> l (5 -97) which can be eval u a t e d e x p l i c i t l y / - 7 AB \ ~ ' _ A & - 9 (2, (eA,Es)) A T = AT (k) + ( A - , „ , ( 5 " 9 8 ) + in Again, r e p l a c e the v 0 's by t h e i r formal e x p r e s s i o n of eq.(4-36) and you get ^ (>»> = AT .($) + 1 L 1° 4* /« . £ . V 7_A*(E E > ^ ^ ft^R^ (5-99) 76 and the i n t e g r a l i n the s p i n te rm e q . ( 5 - 1 0 0 ) can be done and g i v e s £ C \ . i O L f ^ - k ) , ( 5 _ 1 0 2 ) which has the same s t r u c t u r e as the i s o s p i n te rm e q , ( 5 - 1 0 1 ) . Now use the i d e n t i t y on C l e b s c h - G o r d a n c o e f f i c i e n t s [51] (5-103) where the {...} a re the u s u a l 6 - j symbols and d e f i n e I ^ ^ J f , K= 1.1 /u"= (5-104) and d e f i n e a U - f u n c t i o n a c c o r d i n g t o U ( fl ot • A 6 ) — (-1 ) v(2B + i ) U * f i ) 1 ( (5-105) then 3 = c r . v u a j . ^ j j , <B-IO7> and a l t o g e t h e r , t he se e q u a t i o n s g i v e us the f o l l o w i n g f i n a l e x -Aft p r e s s i o n f o r Z ( ( E „ , E 6 ) c o r r e c t t o second o r d e r : Z > „ ^ = [ 1 * 1 ^ ( E ^ ) ] " 1 , (5-108) 1 ^ " V / V 4. ; _ (5-109) • U(T<-,T.-,T f,T>)-!- lU f V i f H (5-106) 77 For example the vertex f u n c t i o n Z, (E .,E„) w i l l take the form I N N ^..,0= | 1 + (iT). .!_._!_ U f^V> + t 4 I J 3t • ^T7 / ( dp ^ V ( f R ) (5-110) Having now formal e x p r e s s i o n s f o r 1^ and Z( , the renorma-l i z e d c o u p l i n g constant f^ ( E A , E & ) and i t ' s energy dependence can be c a l c u l a t e d to second order e x p l i c i t l y s i n c e by d e f i n i t i o n i U A A ) = 7 a ^ Z j ( £ & ) {. . (5-111) 7, U A I E 6 > j . R e s u l t s of the c a l c u l a t i o n s . The second order e x p r e s s i o n s f o r I ( E „ ) , Z i ( E f l ) , Z l (E f l,E &) and f (E A,E f c) obtained i n the p r e v i o u s s e c t i o n i n v o l v e the f o l -lowing parameters: the dressed bag mass mfl , the bare TTAB C O U -p l i n g constant f c and the bag r a d i u s R, which a r i s e s e x p l i c i t l y i n the form f a c t o r u(kR). 78 The dressed bag mass mfl i s chosen to be the p h y s i c a l mass i f A i s a s t a b l e baryon (such as the nucleon) and the resonance mass i f A i s unstable (such as the d e l t a ) . The bare f£ B c o u p l i n g c o n s t a n t s are a l l r e l a t e d v i a SU(6) c o e f f i c i e n t s given i n t a b l e III t o the bare f " w c o u p l i n g c o n s t a n t . N o t i c e t h a t f^1" i s r e -l a t e d to the quark frequency and the pion decay constant v i a eq.(4-18) \ 0 = 3, {D * . (5-112) However, as we mentioned i n chapter IV, center of mass c o r r e c -t i o n s , pion f i e l d r e n o r m a l i z a t i o n , non-zero up and down quark masses e t c . w i l l modify . We s h a l l t h e r e f o r e leave Z"" as a parameter which we s h a l l choose such that eq.(5-45) l i n k i n g the renormalized c o u p l i n g constant f" v(m„,m w) to t"" be s a t i s f i e d f ( ^ ™ » ) = - l T = ^ S-O , (5-113) where f ^ 0.081. F i n a l l y , the r a d i u s R of the bag i s l e f t as a t o t a l l y f r e e parameter so that the dependence of the p r e d i c t i o n s of the Cloudy Bag Model on i t can be s t u d i e d . F i g u r e 5 shows the r a d i u s dependence of the bare to renor-m a l i z e d ITNN c o u p l i n g constant f w w / f ^ w which s a t i s f i e s the cons-t r a i n t e q . ( 5 - 1 l 3 ) . The important r e s u l t r e v e a l e d by t h i s graph i s how c l o s e f^"" i s to f" w. We can t h e r e f o r e conclude that the r e n o r m a l i z a t i o n of the vertex f u n c t i o n i s small and that makes our second-order c a l c u l a t i o n s very p l a u s i b l e . _| ! , , ! , ! , O.S 0.6 0.7 O.B 0 .9 1.0 1.1 1.2 fm. R F i g . 5: R a t i o of t he ba re t o the r e n o r m a l i z e d irNN c o u -p l i n g c o n s t a n t f ^ N / f ^ N a s a f u n c t i o n of the n u c l e o n bag r a d i u s . r 80 In f i g u r e 6, the energy dependence of f ^ E ) d e f i n e d as i s shown. No t i c e that t h i s dependence i s r e l a t i v e l y weak and t h i s allows us i n good approximation to w r i t e i n our c a l c u l a t i o n of the CBM p r e d i c t i o n s . In f i g u r e 7 we c a l c u -l a t e the r a t i o £ (e) d e f i n e d as v (5-116) to v e r i f y i f the renormalized TTNN and TTNA c o u p l i n g r a t i o i s c l o s e to the bare ones which are given i n t a b l e I I I . We remark that f o r a bag r a d i u s of 0.8 fermi or more, t h i s r a t i o i s very c l o s e to one and we s h a l l t h e r e f o r e use i n our f u r t h e r c a l c u l a -- ( £ ) I ' W o - i03(C) • (5-"7) Up to here, we always assumed the pion f i e l d to be " s m a l l " , i . e . that not too many pions are " f l o a t i n g around". The bare baryon p r o b a b i l i t y Z^(m R) i s a good i n d i c a t o r to v e r i f y i f the one-pion approximation of the CBM Lagrangian d e n s i t y i n eq.(3-55) i s reasonable. In f i g u r e 8 we give Z*(mn) f o r the s t a b l e baryons. From t h i s graph, we conclude that the Cloudy Bag Model i s expected to work b e t t e r f o r l a r g e bags of 0.8 fermi or more, than for s m a l l e r ones. L u c k i l y , we s h a l l see i n the next chap-t e r s that the CBM i s c o n s i s t e n t with experiment f o r R>0.8 f e r m i . F i g . 6: Energy dependence of the TTNN renormalized cou-p l i n g constant f£ as d e f i n e d i n eq.(5-114) f o r bag r a d i i i n the range of 0.5 to 1.0 f e r m i . 82 F i g . 7: Energy dependence of the r a t i o of trNA and i r N N c o u p l i n g constants d e f i n e d i n e q . ( 5 - 1 l 6 ) . m o' o l 1 1 1 1 1 1 1 0.5 0.6 0.7 0.6 0.9 1.0 1.1 1.2 fm. R F i g . 8: Bare baryon bag p r o b a b i l i t y Z^dn^) vs the bag r a d i u s R f o r a l l members of the baryon o c t e t obtained v i a eq.(5-93) but with the renormalized c o u p l i n g con-s t a n t s d e f i n e d i n eq.(5-117). 84 In f i g u r e 9, we gi v e the r a d i u s dependence of the bare to p h y s i c a l mass r a t i o f o r the s t a b l e baryons. Again, we observe l a r g e p i o n i c e f f e c t s f o r small bags which t h e r e f o r e make second order p e r t u r b a t i o n theory inadequate when a p p l i e d to small bags. The strange quark mass m$ can be e x t r a c t e d from the d i f -ference between the E and the I bare masses. Assuming t h e i r bag r a d i i to be equal, with <*s»1 .5 [46], then the mass formula eq.(2-43) g i v e s ^ - " " . i = ( - O.O^/v^ (5-118) R which can be s o l v e d f o r ms. A graph of the s o l u t i o n f o r ms i n f u n c t i o n of R i s given i n f i g u r e 10. The value of 144 MeV at R equal to one fermi w i l l be used i n the c a l c u l a t i o n of the hy-peron magnetic moments i n chapter V I I . In summary, the r e n o r m a l i z a t i o n of the Cloudy. Bag Model can be d e r i v e d c o n s i s t e n t l y and second order p e r t u r b a t i o n theory a p p l i e d to the model seems adequate when t r e a t i n g baryon bags of ra d i u s l a r g e r than at l e a s t 0.7 f e r m i . N o t i c e f i n a l l y that the model works b e t t e r with strange baryons s i n c e f o r those the pion f i e l d i s much weaker. We stop here our d i s c u s s i o n of the renor-m a l i z a t i o n of the Cloudy Bag Model and c o n s i d e r i n the f o l l o w i n g chapter the CBM p r e d i c t i o n f o r pion nucleon s c a t t e r i n g . tn F i g . 9: R a t i o of t he ba re t o r e n o r m a l i z e d ba ryon mass m ./m . f o r a l l members of t he ba ryon o c t e t . F i g . 10: The strange quark mass ms which obeys the mass r e l a t i o n eq.(5-118). 87 VI ) THE P 3 3 RESONANCE : a. The one-pion e i g e n s t a t e : In the pr e v i o u s chapter, we s t u d i e d the lowest l y i n g e i g e n -s t a t e of the f u l l Hamiltonian H: the f r e e p h y s i c a l nucleon |N> (f o r X =0). We s h a l l now c o n s i d e r the one pion + one nucleon e i g e n s t a t e denoted as |N, >t which, at time t = + «> c o n s i s t s of a f r e e pion of momentum k* and i s o s p i n j , and a f r e e p h y s i c a l nu-cle o n |N>. We t h e r e f o r e w r i t e H I N , ^ > ± = E ( 6' 1 ) with the o n - s h e l l c o n d i t i o n E = ^ + . (6-2) F o l l o w i n g Wick [14], the boundary c o n d i t i o n at t=+°°is im-posed by w r i t i n g (6-3) where |"X>+ ( |"X>. ) has only outgoing (ingoing) waves at t= °° (t = -<»). As i n r e g u l a r s c a t t e r i n g theory, t h i s amounts to chan-+ ging E by E±ie=E~ i n the propagators and t a k i n g the l i m i t fe^O. To o b t a i n an e x p l i c i t e x p r e s s i o n f o r |"X>+ , we f i r s t w r i t e ( E - - H , l K / J 7 > = 0 , (6-4) 88 But: . ( E * - H 1N> + ( E 1 - H ^  1-X>± = o . (6-5) H aUh I N> - [ H |N> + a V h ) H lh/> (6-6) and a c c o r d i n g to e q . ( 5 - 4 7 ) , L H , ^ ( k ) ] = % a t ( k > + V (?) . (6-7) Using eq.(6-6,7) i n eq.(6-5) g i v e s - V0){h IIM> + ( E J - H ) I7> f = 0 , (6-8) and the formal e x p r e s s i o n f o r |N,k:>, e q . ( 6 - 3 ) , has now the form | N k - X = a * 0 o |j\J> * ( £ 4 - H ) " ' V . . t f ) | N > . ( 6 " 9 ) With the usual d e f i n i t i o n of the f u l l propagator G 4 ( E ) - ( E t i 6 - H ) _ , = i ^ ' - H ) ' ' , (6-10) eq.(6~9) can be r e w r i t t e n as l A / , ^X = |N> + V . . ( £ ) l N > . ( 6 " 1 1 ) Equation (6-11) which r e l a t e s the one-pion e i g e n s t a t e |N,t- > to the no-pion e i g e n s t a t e |N>, i s the key equation f o r t h i s chapter. In the next s e c t i o n s , we s h a l l e x t r a c t the p h y s i -c a l i n f o r m a t i o n from i t . 89 b. The T-matrix for pion-nucleon scattering. The amplitude for the t r a n s i t i o n from the state |N,kj> to the state |N' ,k*|. > is given by the S-matrix element S(N',k*J. |N,k^) which i s defined according to [70] sU'.Kj.K^) = <N\i-,\N,$.y^. (6-12) From eq.(6-9), _<N',k^ . | is related to +<N',k^»| via <N'X\ = £N'X,\ + </S/ , |V # t (k ' )[&V)-6"(E ' ) l (6-13) But G+(E') -<T(E') - - 2 7 T C l(E'-H) , (6-14) so S(A/;S;.IWj^ )= +<A/;i r^^> +-^c5( c-e)<N'i v;..rf';iw^>+>(6-i5) and according to the orthogonality of the scattering states, Si^k-ANJ.) = ^ ^.S^U'-H) — i T T I U e - e ; T ( N / ' / ^ . | i v > i l E ) ) ( 6 - ie) with TC^'P-.^.V1 e > = <^KV*)lwAX> (6-17) where T i s the t r a n s i t i o n matrix with the usual convention of Goldberger and Watson [71]. If we now replace |N,k^ > by i t ' s formal expansion eq.(6-11), we get + <A/'I V.t..(f')at(E> |N/> . (6-18) 90 Since a^(k) commutes with V 0^.(k') and </v' I a*(K)H = <V I t ^ U \ H ] + H a t ( ? ) , (6-19) then using eq.(6-7) f o r the commutator and a few manipulations r e s u l t i n <M' = < N , W o i ( ^ (E-^-^.-H )'' (6-20) which leads to the exact form f o r the pion-nucleon s c a t t e r i n g T-matrix T ( ^ l W , ^ | E ) = <N'l V<|.l(H,)G+(E)V.^S)lN/> (6-21 ) Next, we expand e x p l i c i t l y t h i s T-matrix on the b a s i s of the e i g e n s t a t e s of H 0. I f we i n s e r t the completeness r e l a t i o n eq.(4-49) between each operator, and expand |N'> and |N> accor-ding to t h e i r formal expansion eq.(5-9), the T-matrix takes then the form T ( N X \ X . z ^ > 7 < ^ l{ i - HJ MS)1]-• V e + v U J ) l f l e ^ > < f t 0 ^ | (c +-H o-H J )" ,|B o^'><B oy) v o > ao j 1 + + C^ - r l.-A^A)"'^! lN/ e >Z^r a + ( 6 - 2 2 ) I. z ' u , / < I { 1 + H , H e - AHXA )"']•• 91 A l l the terms generated in t h i s expansion are represented gra-p h i c a l l y as No t i c e that a l l the graphs i n the second bracket are simply the c r o s s i n g v e r s i o n [61] of the f i r s t b r a c k e t . F i n a l l y , we r e c a l l t h a t a l l i ntermediate baryon l i n e s are summed over nucleons and d e l t a s . If we l i m i t e d o u r s e l v e s to c a l c u l t i n g the T-matrix to order fl, we would c a l c u l a t e graphs (a) and (e) i n eq.(6-23) only. However, G.F. Chew [11] argues that n e g l e c t i n g terms l i k e (b) and (c) i s i n c o r r e c t . His argument i s based on the f a c t that when a propagator i n a graph can va n i s h , t h i s makes the magni-tude of the graph anomalously l a r g e . With a s u i t a b l e set of parameters, Chew c a l c u l a t e d a l l graphs (a-b-c-e), with no d e l t a present, and was a b l e that way t o generate the P 3 i resonance, a r e s u l t which was impossible to get from graph (e) alone. 92 F o l l o w i n g Chew's p r e s c r i p t i o n , we d e f i n e a lambda-expansion of the T-matrix as f o l l o w i n g : a graph i s of order X"'p i f i t c o n t a i n s 2n TTAB v e r t i c e s and p propagators which have a p o l e . For example, graphs ( a ) , (b), (c) and (e) i n eq.(6-23) are a l l of order X . We t h e r e f o r e expand the T-matrix as T(A/;K 'KS. |E) = t T i i \ N , x \ N X \ ^ ) i (6-24) where T U ) i s of order X' , and T * " ( N ',k*^ | N , K ^ | E ) i s given gra-p h i c a l l y by (6-25) < K ) \ where we s p e c i f i e d e x p l i c i t l y the i n t e r m e d i a t e d e l t a by a double bar propagator. For a l l terms i n eq.(6-25) except the (b) one, we apply the r e n o r m a l i z a t i o n procedure d e s c r i b e d i n chapter 5. However, some of the bubbles contained i n (b) have p o l e s i n t h e i r propagator and t h e r e f o r e c o n t r i b u t e e x p l i c i t l y to the imaginary p a r t of the T-matrix. We s h a l l e x t r a c t them from the bubbles and d e f i n e with • 2 % v / , ( 6 - 2 6 ) 93 0 \ A A* A • /'^ -x... (6-27) A A " A A * A A v &' A ' - A I T * ' * (6-28) and A ' A M 1 A / A a ' A / ' A f A A ' A * AJ (6-29) (6-30) N where G.(Ej and Z° N (E^ , E^ ) are t r u n c a t e d renormalized f u n c t i o n s which have no p o l e s . For convenience, we d e f i n e a tru n c a t e d TT NA renormalized c o u p l i n g constant f t "(E ,m ) as 'fe - a« Z, ( ^ A , ^ > (6-31 ) The T-matrix to order X now reads T ( N ; * - . I ^ I E ) = _ A Z_ + tcf, (6_32) N1 N" N » with u> =E-m.. and A/ A / " A/ A, ' A " H A » ' A " A i A/" A/"' A/"" A/ ~^ A l ' A A/° A/'" A/ X -N' N" W" A/* W" A/" A Aj" -<£ hi N (6-33) A/ ' A A/* A 1 N 94 We recognize i n e q . ( 6 - 3 3 ) a Lippmann-Schwinger type of equation obeying t ( t = AriVt-yUj) + ( ^ - ^ o ' t c f ; ( 6 - 3 4 ) with W {VJ •<*>) = X ' S + = 4 . _ \ _ i _ Z l _ (6-35) , i r ( W •_,<*>) = /ir^( k'fe-u>) + Arjk'j;^) + /v;C^>'^) , (6-36) being the d r i v i n g p o t e n t i a l , and we used the s u b s c r i p t "c" to d i s c e r n the c r o s s e d graphs from the uncrossed ones. Note that we a l s o d e f i n e d (6-37) To make the n o t a t i o n more symmetric i n nucleons and d e l t a s , we d e f i n e the f i r s t term of eq.(6-32) as s N \ /if ( I ? u O = \ , / (6-38) and t h e r e f o r e T( W*,E-.lN/,fe- I E) = /ir^Ci?; k • u.) + U S ^ u . ) * ( ? ( \ ' ) | ( 6-39) c. The p a r t i a l - w a v e T-matrix T 2 I 2 J(k',k;w). According to eq.(6-35) and eq.(6-38), the non-crossed po-t e n t i a l terms are given by 95 "r.(i\$-><") = \ g / (6 -40) and more e x p l i c i t l y /iral^ f-u.) = [ ^ ] " ' ^ ' ( ^ > J (6 -41) i r - C k . k ^ u O - /v_ , < £ ' ) [ l-M r I (E)] AT. (I?) (6 -42) The c r o s s e d terms are given by AT (T^-u,) = (6 -43) ^,.(l?,k>> =• (1?) [<*-«v*v] ()?>) (6 -44) » a i r A . U ' , l r , < " ) = /v; (fe) [ V " A - « " i r u V J ^ . , 0 ? * ) . (6 -45) Since the f u l l Hamiltonian H commutes with the t o t a l angu-l a r momentum J and i t ' s p r o j e c t i o n J z , and s i m i l a r l y with the t o t a l i s o s p i n I, we are lea d to p r o j e c t the p o t e n t i a l s v, and v on each of the allowed channels: cm I r ^ V A a ^ J U > ^ > , (6 -46) = I P ( / , M P ^ C ! J V * ; U , ) , (6 -47) 1 J '"J' where v^' ( k ' , k ; u j ) i s the amplitude of the p r o j e c t i o n of v^ ( k ' , k ;U J) i n the channel 21, 2J and p a i 4 J * s t n e p r o j e c t i o n operator i n the channel 21,2J obeying i L ^ P ^ V ; ^ ) ^ ; ! / ^ - . ^ ) = ^ . S ^ . P ^ V . i - M ) . (6-48) Furthermore, s i n c e s p i n and i s o s p i n space are disconnected, we can f a c t o r i z e p J 3 a J as a product of a i s o s p i n p r o j e c t i o n opera-96 t o r (p ( j ' ; j ) and a spin p r o j e c t i o n operator $?(k';k) The q u i c k e s t way to d e r i v e the expression f o r the matrix elements of P i s to reco g n i z e that the v (k',k;ui) are pure ,3=5^ s t a t e s . The proof l i e s i n eq.(6-40) where we n o t i c e that s i n c e the intermediate s t a t e i s a pure s t a t e with no pion present, and s i n c e the v e r t i c e s conserve both I and J, then the graph v^ ( k ' , k ; t o ) must be a pure s t a t e with 1=^ and J=S M . Theref o r e we w r i t e /M£ , )k , u O= P (^M^Cfc',. (6-50) From eq.(6-40,41,42) with the r e s t r i c t i o n eq.(6-48), we get = QKc*'--«> , (6-5,) We then have the immediate r e s u l t f o r the non-crossed graphs: / ^ c ^ ' f • , « « > = ^ ( ^ V i M ^ * ^ ^ ^ 5 > ( 6 _ 5 4 ) ^ (FJ;<") = ^ ( V / ^ ^ ) ^ ^ ' , * , ^ , (6-55) and the vfc(k',k;oo) can be d e r i v e d e x p l i c i t l y to be Allied) Min'n^kMk'k r r 1 J ^zrk 1 J > (6"56) 97 kit,<»>„)) AA(k'R)A(kR)k>k r E _ ^ _ 7 * - j " ' (6-57) Next, we co n s i d e r the c r o s s e d graphs which were d e f i n e d e x p l i c i t l y i n eq.(6-43,44,45). In t h e i r g eneral form, they are w r i t t e n as ^ (*',!? = ^ L ^ - % - V ^ v V ( ^ ) , (6-58) which, when expanded become I H , , E - ^ ( l ? ' r ? ) ^ ( l ? f ? ) fr'fe with V ^ V V V v ^ V - < 6 ' 6 0 ) which are o b v i o u s l y d i f f e r e n t from the p r o j e c t i o n o p e r a t o r s given i n eq.(6-52,53). However, i f we make use of the i d e n t i t y M O ' S'i>i " }h) L^*')(^v.)J- . v 5 H j , r c i , n ' ( and d e f i n e -1 ' • ) 1 1, 1 > 1-s A 1 (6-63) then '<* 1 V I ' A , ). V I 'A 'd/ ' ' A ' 98 where the U - f u n c t i o n was d e f i n e d i n eq.(5-105). I f we r e p l a c e eq.(6-64) i n eq.(6-60), and use the d e f i n i t i o n eq.(6-52) f o r the p r o j e c t i o n operator i n i s o s p i n space, then = I u ( l , l , T P , X ) P^v-.-j) , (6-65) 1(1 f o r the spin s e c t o r . Therefore wi th F i n a l l y , we mention that a c c o r d i n g to eq.(6-67), the cro s s e d graph with intermediate nucleon present c o n t r i b u t e s to each of the P u ,Pn ,P3| and Pi}> channels i n c o n t r a s t to the un-cro s s e d intermediate nucleon and int e r m e d i a t e d e l t a ones which c o n t r i b u t e r e s p e c t i v e l y to the P(| and P^ channel o n l y . As f o r the p o t e n t i a l terms v a (k',k;w) and v C K (k*',k;<W, we can p r o j e c t the T-matrix on each channel of s p e c i f i c t o t a l i s o -s p i n I and t o t a l i s o s p i n J . We t h e r e f o r e d e f i n e the p a r t i a l - w a v e T-marix T a T ^ ( k' , k ;u>) as 99 with as usual E^m^+u, i t we use eq.(6-39) f o r the T-matrix of order X , and remember t h a t v^ ( k ' ,lt ;LU) c o n t r i b u t e s only to the P n channel, then T = S ^ ^ l K W + t^;*) * d(\>) , ( 6 - 7 2 ) with t (k',k;u>) obeying (see eq.(6-34)) t l K t ^ - r K t ^ k ^ + U k ' t f 3 J' a j 4 ^ (6-73) and the p a r t i a l - w a v e d r i v i n g p o t e n t i a l v , (k',k;ui) i s r e l a t e d to the f u l l d r i v i n g p o t e n t i a l v(k*',k;w) given i n eq.(6-36) v i a T/-(J',?-,uO= 1 P ( ^ V i i i ) /IT (fe'.fe-cu) (6-74) To complete the p i c t u r e , we w r i t e e x p l i c i t l y the r e l a t i o n -s h i p between the p a r t i a l - w a v e T-matrix and the phase s h i f t s : T°"(U,) = J d?>-Co=^) = -~4— t ' ^ i i l S (u,) (6-75) where the s u p e r s c r i p t "on" r e f e r s to the o n - s h e l l c o n d i t i o n U; = tuk = cu f e, } (6-76) or more d i r e c t l y t«h 5 Cu.) - 1 ^ ( J O A / ( U J ) ) / ^ ( T T 0 A / CCOO (6-77) and f i n a l l y , the phase s h i f t s are r e l a t e d to the t o t a l c r o s s -s e c t i o n v i a [72] 100 +1 U L k* (6-78) d. The P 3 3 resonance. For the P channel, the T-matrix eq.(6-72) reads T(k;k;^)= t <*>',uO + C3( X1) , (6-79) with t (k',k;<*j) obeying t^ k-.l.^)-^!,'!,;^ • jjk' k'^JK^ j (6.80) and making use of eq.(6-55) and eq.(6-67) to eq.(6-70) g i v e s f o r v (k' , k ; t u ) : A T ^ ( f e ' k • t u ) = AT ( k^ k ; uo + u C i , i , I ) A T c a , ( k > + a i i s i (6-81) From the d e f i n i t i o n eq.(5-105) of the U - f u n c t i o n , we get AT: ( k> • uv) = Ar-Ck' fe - ««) + /v- (fe;'I? - uu) 4 _L /IT U'feju,) (6-82) According to t h e i r e x p r e s s i o n eq. (6-69, 70 ) , v and v C A are about e q u a l . However, the c o n t r i b u t i o n of v C N to v i n eq.(6-82) i s 16 times g r e a t e r than the one of v .We s h a l l t h e r e f o r e n e g l e c t the v.. c o n t r i b u t i o n and w r i t e ^ ( k ' , k • u»> = A^(k',fc • u,) + i L / i r ^ C ^ f e - u,) t ( 6- 8 3) 101 L e t ' s now re c o n s i d e r v A(k',k ; u i ) . According to eq.(6-57), i t i s given by . AW AN /TV) K OA *- an ' (6-84) We a l s o approximate f^. (m w + u\m M) by i t ' s value a t u j = i * ^ A M and d e f i n i n g a f u n c t i o n h-(k) as h - U ) = AW L U 7 T A \ A ^ T then v^(k*,k;u;) i s a separable p o t e n t i a l of the form (6-85) (6-86) *• L °* ATT J * Next, we con s i d e r the cro s s e d nucleon term c o n t r i b u t i o n to v given by AY) (6-88) Again, we s h a l l n e g l e c t the energy dependence of the c o u p l i n g constant and wr i t e , A / V A/A/ f K . n ^ - ^ N - ^ o (<n,^) = . (6-89) Furthermore, the propagator i n eq.(6-88) can be r e w r i t t e n as ( U J - U)k- t * / k , ) " ' = - ^ • + U h )(g > - u. hQ ^ (6-90) N o t i c e that the second term on the right-hand s i d e , which has a 102 two-meson c u t , vanishes whenever the ingoing or outgoing pion i s o n - s h e l l . G.A. M i l l e r , i n h i s Chew-Low type of c a l c u l a t i o n [73], has found that "the c o n t r i b u t i o n of t h i s second term i s much smaller than the c o n t r i b u t i o n of the f i r s t term". Since we s h a l l f i n d that most of the P%1 resonance r e f l e c t s the formation and decay of a dressed d e l t a bag, and that only a smal l p o r t i o n comes from the Chew-Low expansion based on eq.(6-88), i t th e r e -f o r e makes sense from t h i s argument and the r e s u l t s of G.A. M i l l e r , to n e g l e c t e n t i r e l y the second term on the right-hand s i d e of eq.(6-88) and w r i t e 4_ AT (k'k'Uj) - ~±L 1 / 1"" V M(k'R)^(kK; k'kuj CkJ 1 1 q ' — I . (6-91) Now, l i k e f o r v A , we have a separable p o t e n t i a l of the form J^cJ-K*-^ = - ^ V k > ^>U>> (6-92) with 3 (6-93) Hi As a r e s u l t of our approximations, v (k',k;^) can be w r i t -ten as The s o l u t i o n of the Lippmann-Schwinger equation eq.(6-80) f o r the rank-2 separable p o t e n t i a l eq.(6-94) can be w r i t t e n a n a l y t i -c a l l y [74] as t^u; k ^ cu> = N/U^-UO/DCUO , (6-95) 103 w i t h WW A A / W A N 1 D(uO = D D_ . (UJ) + UJ ( D - (uo) (6-96) (6-97) the D's ar e g i v e n by ^ ^  / U j - u j , O O A £ r- _ A f ^ ** I ^ / T D-.(uO = E-/^- I J E ) - P U lAih! -*r^) OO r 0 UJ - CO. (6-98) (6-99) i t (.6-100) By r e p l a c i n g h ? ( q ) by i t ' s f o r m a l e x p r e s s i o n eq.(6-86) i n ( U ; ) F a n d u s i n g the d e f i n i t i o n eq.(5-88) f o r l . A _ ( E ) , we o b t a i n D..(«*>) = E -/»*» - l A t l ( E ) - I C6)+iTr«,^l,-(o yy « o n NK ti v- i and u s i n g eq.(5-68,69), we are l e f t w i t h A A A -i a i D y y ( U 0 " 2i (""O (E"^ A) + ^ <]>MV . a , .fc (6-102) , (6-101) I f we approximate the TTAN r e n o r m a l i z e d c o u p l i n g c o n s t a n t a c c o r -d i n g t o * V > (6-103) A F J then we can r e w r i t e t (k',k;u») as (6-104) 104 r . -i _y A (6-105) D„CuO = D^<Ui) D**(u,) + tu ( D W A Ctu>y (6-106) rv W W A / tAJ - COs 1 - . n u , V v W i ( 6 - 1 0 7 ) 00 (6-108) . , i (6-109) ; ( c o W > _ For 0 0 = 1 ^ = 1 ^ . , t^(k',k;cu) i s (u>) and a c c o r d i n g to eq.(6-77), the p h a s e - s h i f t s are given v i a (6-110) and f i n a l l y the t o t a l c r o s s - s e c t i o n in the 33 channel i s (6-111) 105 e. R e s u l t s of the c a l c u l a t i o n s . When s o l v i n g eq.(6-104) to eq.(6-109), the parameters i n -volved are the bag r a d i u s R, the dressed d e l t a mass mA and the renormalized c o u p l i n g constants f * w and f " v . F o l l o w i n g our d i s c u s s i o n at the end of chapter V, we s h a l l take f o r f*^ the value (6-1 12) i . e . we assume the r a t i o of the renormalized and bare c o u p l i n g c o n s t a n t s to be equal. For , we simply use the experimental value of / •= L \Tir • o.as &* 1 . (6-i 13) The dressed d e l t a mass m„ i s l e f t as unknown. However, as men-t i o n e d i n the previous chapter, i t i s expected to be c l o s e to the resonance mass of 1232 Mev. The bag r a d i u s R i s l e f t as a f r e e parameter. For each bag r a d i u s R between 0.7 and 1.2 f e r m i , the c a l c u -l a t i o n s show that there e x i s t a d e l t a mass m A(R) which f i t best the experimental r e s u l t s [75], and i t i s given e m p i r i c a l l y by /YV) {{{) =: l l l o H/\<» + 4 1 ] « « . f^ / R 5 . (6-114) The t h e o r e t i c a l and experimental c r o s s - s e c t i o n s are given in f i g . 11. S i n c e , f o r R>0.8 fm., m A(R) i s very c l o s e to the r e -sonance mass, we t h e r e f o r e conclude that i n c o n t r a s t to the Chew-Low model, most of the c o n t r i b u t i o n to the P„ resonance 106 comes from the formation and decay of a dressed d e l t a bag. F i n a l l y , we mention that the di s c r e p a n c y f o r high pion e n e r g i e s i s not s i g n i f i c a n t s i n c e we neg l e c t the r e c o i l of the nucleon which i s s u r e l y an important f a c t o r i n that r e g i o n . If we leave both iy and ma f r e e , and t r y to f i n d the best f i t to the experimental data, we o b t a i n f =0.24, R=0.72 fm. and MA=1232 MeV (see f i g . 1 2 ) . However, s i n c e r e c o i l c o r r e c t i o n s are not i n c l u d e d i n the model, one must not give too much s i g n i -f i c a n c e to such a f i t which i s p a r t i c u l a r l y s e n s i t i v e to high pion k i n e t i c e n e r g i e s . T h i s concludes our study of pion-nucleon s c a t t e r i n g i n the context of the Cloudy Bag Model. We s h a l l now concentrate on another aspect of the theory: the el e c t r o m a g n e t i c p r o p e r t i e s of the baryon o c t e t which i s the o b j e c t of the next chapter. F i g . 11: T o t a l c r o s s s e c t i o n f o r pion nucleon s c a t -t e r i n g i n the P 3 3 channel as a function, of the pion k i n e t i c energy. The t h i c k l i n e i s the CBM p r e d i c t i o n f o r bag r a d i i i n the range of 0.8 to 1.1 f e r m i . The dashed l i n e r e p r e s e n t s the experimental data [75]. F i g . 12: Best f i t ( s o l i d l i n e ) to the experimental (dashed l i n e ) t o t a l c r o s s s e c t i o n f o r pion nucleon s c a t t e r i n g in the P 3 3 channel as a f u n c t i o n of the pion k i n e t i c energy. In t h i s f i t , f =0.24, R=0.72fm. and mA=1232 Mev. 109 VII) ELECTROMAGNETIC PROPERTIES OF THE BARYON OCTET: a. I n t r o d u c t i o n . The photon-hadron i n t e r a c t i o n , when analyzed i n the context of the Cloudy Bag Model, leads to p r e d i c t i o n s on the e l e c t r o -magnetic p r o p e r t i e s of hadrons. We s h a l l i n f a c t devote our a t -t e n t i o n to o b t a i n i n g formal e x p r e s s i o n s f o r the nucleon charge r a d i u s and the magnetic moment of a l l members of the baryon o c t e t . We mentioned i n the i n t r o d u c t i o n that Salzman, i n 1955, when studying the s t a t i c model of Chew, came to the c o n c l u s i o n that the c o r e needed to be extended to o b t a i n the c o r r e c t e l e c -tromagnetic p r o p e r t i e s of the nucleon. However, s i n c e no model was a v a i l a b l e f o r the nucleon core, the core d e n s i t y was simply assumed to be p r o p o r t i o n a l to the pion source f u n c t i o n p ( r ) . In the Cloudy Bag Model, t h i s extended core i s the three quark bag. The photon w i l l t h e r e f o r e couple to both the quarks i n s i d e the bag, and to the pion f i e l d , i n a w e l l d e f i n e d way, l e a d i n g then to q u a n t i t a t i v e r e s u l t s on the e l e c t r o m a g n e t i c p r o p e r t i e s of hadrons. We f i r s t d e r i v e the formal e x p r e s s i o n s f o r the e l e c t r i c form f a c t o r G £ ( q ) and the magnetic form f a c t o r G M ( q ). Then, by making use of the Cloudy Bag Model formalism, we e x t r a c t the quark and pion c o n t r i b u t i o n s to G p M ( q i ) . We a l s o suggest a simple center-of-mass c o r r e c t i o n p r e s c r i p t i o n based on the work of Donoghue and Johnson [33]. F i n a l l y , we e v a l u a t e e x p l i c i t l y 110 the proton and neutron charge r a d i u s and the magnetic moment of a l l members of the baryon o c t e t as a f u n c t i o n of the bag r a d i u s . b. The el e c t r o m a g n e t i c form f a c t o r s : G^q^) and G ^ q * ) . In order to study the el e c t r o m a g n e t i c form f a c t o r s G ^ q M , we co n s i d e r f i r s t the e l a s t i c s c a t t e r i n g of e l e c t r o n s on f r e e s p i n 1/2 baryons. To lowest order i n the i n t e r a c t i o n , a s i n g l e photon i s exchanged ( f i g . 13 ). The t r a n s i t i o n amplitude can be read o f f to be where p,m (p',m') are the 4-momentum and spin p r o j e c t i o n of the incoming (outgoing) baryon A, and k,s (k',s') are the 4-momentum and s p i n p r o j e c t i o n of the incoming (outgoing) e l e c t r o n . The e l e c t r o n c u r r e n t i n known from QED to have the form where u^(k) i s the p o s i t i v e - e n e r g y D i r a c s p i n o r f o r the e l e c t r o n f i e l d . From t r a n s l a t i o n a l i n v a r i a n c e , the baryon c u r r e n t opera-t o r i M ( x ) can be r e w r i t t e n as A • fi. A M ' A M P ^ = e e > (7-3) A where i s the 4-momentum ope r a t o r . The nucleon c u r r e n t i n eq.(7-1) becomes I l l F i g . 13: Lowest order matrix element f o r e l e c t r o n nu-c l e o n i n t e r a c t i o n . 112 where T£ i s a 4x4 matrix i n Lorentz space. From the requirement of r e l a t i v i s t i c c o v a r i a n c e , and the c o n d i t i o n of gauge i n v a r i a n c e which r e q u i r e s that q JH = o (7-5) and the knowledge that the s t a t e s |p,t»> form a b a s i s f o r a r e -p r e s e n t a t i o n of the inhomogeneous Lorentz group corresponding to a d e f i n i t e s p i n , the most general form of the operator T£ i s [ 76 , 77 ] where p( ^ (q ) are r e a l f u n c t i o n s and Kfl i s the anomalous magne-t i c moment of the baryon A. F o l l o w i n g E r n s t et a l . [78], we next c o n s i d e r the expecta-t i o n value of the magnetic moment operator M i n the r e s t frame of the baryon A M , « <FKp=»)l ^Prl'li*) I RCf-o)> . (7 -7) S u b s t i t u t i n g i n eq . ( 7 - 7 ) ?(?) from eq. ( 7 - 3 , 4 , 6 ) g i v e s a f t e r a few manipulations TZ L ^ * ^ ^ ] U;(5> ? c £ ( 3 , , ( 7 - 8 ) or f^t = e ^ ( o ) ^ • ( 7 _ 9 ) 113 T h i s leads to the d e f i n i t i o n of the magnetic form f a c t o r G R ( q A ) as (7-10) We a l s o i n t e n d to c a l c u l a t e the nucleon charge r a d i u s . T h i s r e q u i r e s that we compute the e x p e c t a t i o n value of the charge r a d i u s operator <R„ in the r e s t f rame of the nucleon. In g e n e r a l , for a s p i n 1/2 baryon A, we have K - <R(p=5)|jj3i l R ( p > > (7-11 ) Again, s u b s t i t u t i n g i n eq.(7-11) the exp r e s s i o n f o r 5°(r) qi v e s .fl (7-12) And from t h i s g e o m e t r i c a l r e s u l t , E r n s t et a l . d e f i n e the e l e c -t r i c form f a c t o r G^q" 1) as (7-13) Sachs [79] has shown that the G c (q ) are a measure of the i n -t e r a c t i o n of the baryon A with a weak and s t a t i c e l e c t r i c and magnetic f i e l d s . F i n a l l y , Sachs a l s o showed that i n the, B r e i t frame (where q =0), G e ^ ( q ) are r e l a t e d to the F o u r i e r transform of the s p a t i a l c u r r e n t v i a i ° f l ^ - <r i r«<j>i«> = - £ } \ J \ 6 6 V > e " v " <'-'«> I ?<*>!"> = < ^ j ^ < t l ' ) < % l J ' l l % > ^ (7-15) 114 We s h a l l need the i n v e r s e of these r e l a t i o n s , i . e . & » ( ^ = 77 M H ' V ' V V 1 - <7"'7) where J * ^ ) i s the F o u r i e r transform of j£(r) i n the B r e i t frame (7-18) c. Formal e x p r e s s i o n f o r jp (?) i n the Cloudy Bag Model. If we introduce the photon f i e l d i n the Cloudy Bag Model Lagrangian d e n s i t y using the usual requirement of l o c a l gauge i n v a r i a n c e , we get Wrf> M«*)y*;J A s + i i (7-19) ^ T i a " ' ' with ^ " ^ ("V*^'^^) . (7-20) T h i s Lagrangian d e n s i t y i s i n v a r i a n t under the i n f i n i t e s i -mal l o c a l t r a n s f o r m a t i o n </(oc) ^ <t («"> - i6M e <((#) , (7-21) 115 t r a n s f o r m a t i o n i s (7-22) s a t i s f y i n g with j£(x) being the quark c o n t r i b u t i o n to the c u r r e n t i -A 1 . and j (x) i s the pion c o n t r i b u t i o n TT Next, we quantize the pion f i e l d a c c o r d i n g to the usual way as given i n eq.(4-8) T T - t , o ) = f/ a,(k)e + a.(k)e (7-26) then the pion c u r r e n t i n eq.(7-25) becomes T h i s form of D (r) i s the key equation f o r determining the pion c o n t r i b u t i o n to the e l e c t r o m a g n e t i c form f a c t o r s . The quark c u r r e n t c o n t r i b u t i o n to j!*(r) i s e x t r a c t e d from eq.(7-24) f o r ]£(x) by using the MIT quark wavefunctions eq.(2-The conserved c u r r e n t a s s o c i a t e d with t h i s (7-23) (7-24) (7-25) 116 16) (7-29) to get - J r 1 (7-30) (7-31 ) F i n a l l y , s i n c e the q u a n t i t y of p h y s i c a l i n t e r e s t i s the e x p e c t a t i o n value of the baryon c u r r e n t o p e r a t o r , we d e f i n e D j ( r ) as (7-32) where |A> i s the p h y s i c a l o n - s h e l l baryon A s t a t e . We a l s o d e f i n e the q u a n t i t i e s (r) and (r) as with (7-33) (7-34) (7-35) We are now ready to c a l c u l a t e e x p l i c i t l y j j ( r ) to o b t a i n the e l e c t r o m a g n e t i c p r o p e r t i e s of the baryon o c t e t . 117 d. The pion c o n t r i b u t i o n to j K ( r ) R The formal e x p r e s s i o n f o r the pion c u r r e n t operator eq.(7-27,28) can be r e w r i t t e n as with t * S i ( ^ , ^ ) j u ) = ( a - . t J ^ + a ^ ^ C a . ^ - ^ a t c - i ? ) ) , ( 7 - 3 7 ) and we remind that g°°=1 and g t l =-1 i n our conventions. The next step i s to r e w r i t e i n terms of the operato r s V0. as we d i d i n e q . ( 6 ~ 6 ) . We use the Hermitian conjugate of e q . ( 6 - 7 ) which reads H ] = < k^ a^(h + V ^ . ( k ) , (7-38) or more e x p l i c i t l y , ( H + ujh) a. it) = - \ / (ft + CL(1o H . (7-39) Adding t x ^ a ^ k ) on both s i d e s g i v e s ( H + UJ,, + ujk,) c<.(t) = - Ve+-(k) (H + _ (7-40) These i d e n t i t i e s l e a d to the r e s u l t s [17] a}(k) l f l > = (^ -wrW)"' V ^ c f t |fl> , (7-41) and , + 4 \ { 7 _ 4 2 ) 118 The e x p e c t a t i o n value of the operator ("k^ ,, k- JJH) between p h y s i c a l baryon s t a t e s A has then the form (7-43) (7-44) with i t + (7-45) <fl l^v^')K^.-w)",K-^-H)",v'(fe)lF)> (7-46) " f <A I V0.(-I)K—fe-H)'^ -^ ,^ )" V*,(-f) |R> , C(^V»^ " ~^ {<fl (7 47) I f we expand the p h y s i c a l baryons A on the b a s i s of the e i g e n s t a t e s of the bare Hamiltonian H D, and i n s e r t the complete-ness r e l a t i o n f o r the bare e i g e n s t a t e s between each operators i n eq.(7-45,46,47), we w i l l generate the f o l l o w i n g set of diagrams: + i,.o.>(7-48) + J,.o.;(7-49) + />.o.(7-50) 119 A p p l y i n g the r e n o r m a l i z a t i o n p r o c e d u r e d e s c r i b e d i n c h a p t e r V - h , a l l t he bubb l e s w i l l be r e p l a c e d by the r e n o r m a l i z e d quan -t i t i e s and we get t o o r d e r f * : I 6 -Z & i . - R > -* + i £ A r & v ft Afc = I & A, ^ 8> >" A AB 1 _ i * 1 ft * 8 " A A& _ &A ATvjW)/*^ (fe) M « / ^ ( - f o / W ^ ' C - E ' ) fl *• & " A AS &A - r - R J f\ *• & f A ftb (7-51 ) (7-52) (7-53) W i t h t he h e l p of the i d e n t i t i e s A& ^  ^ ^ t S "= ^ (Jt) = - ^ ( ^ , /IT,A (-1?) = -ATY) ( fe ) , we can combine a l l t he p r e v i o u s r e s u l t s t o ge t s (5;,!.-o)= X ( 3  ( (^h +H'X^M+^k)(^+H') * K/fc ) ^ C f e ) - ^ ( T ) ^ v ( k > j j ^ ( H + ^ b . ) ( H A ^ O ( ^ + H 0 L^ .(fe w^cfe) *^ . C^-Xfe')] J (7 -54) (7-55) (7-56) 120 The next step c o n s i s t s i n e v a l u a t i n g the brackets i n eq.(7-55,56). For t h i s purpose, we d e f i n e the q u a n t i t y AB \ A k X ) - I ^A5w "0h , (7-57) and s i n c e * 6 , • J A B . . / u « x * * * * * * . t * / » t . A T . *(hR> C s ^ \ z : . h C ; y \ V . e ^ (7-58) then ^(^•) = (r f - j ^ w ) /CV) 3 * , • ( 7 . s s ) with cS,*)« I t d ( r . t ) ( l ^ , , , 7 - 6 0 , We can f u r t h e r s i m p l i f y 4 (k,k' ) by usi n g the property S r=1/2 f o r A s i n c e i t belongs to the baryon o c t e t . For S B=1/2, we use the i d e n t i t y t to get i H ( V k ' ) - f < ^ l ( - ^ ) ( ? f e ' ) I^A> . (7-63) For S &=3/2, we use eq.(7-62) and the i d e n t i t y Z_ L . . C . . - d > 5 ' (7-64) to get a f t e r a few manipulation 121 i 1 r"( = y VV-± (?.$)(?.*') U,> ^ (7-65) E v a l u a t i n g the i s o s p i n term !J^., would be s i m i l a r to the s p i n case i f the i s o s p i n of A was 1/2 (N and ^  ) . However, i t i s more simple to sum r i g h t away over j and j ' as r e q u i r e d i n eq.(7-36) f o r f * ( r ) . T h e r e f o r e , we are l e d to d e f i n e the q u a n t i -ty t (B) v i a H Doing the sum e x p l i c i t l y leaves us with (7-66) and from t a b l e IV ( e x t r a c t e d from r e f . [80]) we have tnCB) = ^  t«/Tfl(Vi) .T Tfe= T A ( 7" 6 8 ) - *«/(T A +0 ^ Tfe= v 1 . Combining a l l those r e s u l t s i n eq.(7-34) f o r j ^ ( r ) g i v e s f o r TTrt t'(14>)J> (7-69) i e+H')(%^^)(u> 6 f l +u> k,) and f o r ?„„(?) with s f l(B) d e f i n e d a c c o r d i n g to Table IV: The Clebsch-Gordan c o e f f i c i e n t s C m l m 2 m with j2=1 and m1=m-m2 [80]. J1J23 m2=l m2=-l V 1 h' 1 L(2j 1 +I)(2j 1 +2)J 2J 1CJ 1+1) (j j-m)(J L •2j 1(2j 1+l) J (j1-m+l) (jj+m+1) L ( 2 j 1 + D ( j 1 + 1) J m j 1 ( 2 j 1 + 1) .C2J1+1)C2J1+2D (jj-m) (j1+m+l)' 2j 1(2j 1+l) <f 5 _3_ 123 (7-71 ) and the momenta are a l l r e l a t e d v i a (7-72) e. The pion c o n t r i b u t i o n to G^q" 1) and ^". The pion c o n t r i b u t i o n to the nucleon e l e c t r i c form f a c t o r Gg(q*) i s given in eq.(7-16) and eq.(7-!8) to be G V ) = en * (7-73) Repl a c i n g j° w(r) by i t ' s formal e x p r e s s i o n eq.(7-69) giv e s a f t e r i n t e g r a t i o n (7-74) or more e x p l i c i t l y i<.T73 72TTJ (7-75) 1 > (7-76) (7-77) N o t i c e that we c o u l d a l s o c a l c u l a t e the e l e c t r i c form f a c t o r of a l l members of the baryon o c t e t . However, s i n c e G*(q a) i s un-known f o r the strange hadrons, we s h a l l l i m i t o u r s e l v e s to the 124 nucleon doublet only. For the pion c o n t r i b u t i o n to the magnetic moment, we r e c a l l from eq.(7-17,18) that &« (<>  = ir IF <+»\"V**> • \** U*>'*'*>  (7- 78» and r e p l a c i n g ^.(r) by eq.(7-70) g i v e s I f we d e f i n e ^ to be the piori c o n t r i b u t i o n t o the baryon A mag-n e t i c moment, then 3. 0 0 « _  6 « C o ) s _ L I ( j £ ] tA(w^cw f feV(^)(^+3cfe) ( 7 _ 8 Q ) As a s p e c i f i c example, we c o n s i d e r the nucleon case f o r which (7-81) (7-82) ^ ( A ) = ^ ( - ^ / i ^ ^ W ^ ^ ^ . < M l ^ ^ > (7-83) f . The quark c o n t r i b u t i o n to G*(q a) and ^u". The quark c o n t r i b u t i o n to the nucleon e l e c t r i c form f a c t o r G^(q A) was mentioned i n eq.(7-16,17) to be the F o u r i e r transform of the quark d e n s i t y j° w(?) 125 and from eq.(7-30), ( ^ ( ^ / « ) + ^ C ^ / / ? > ) e ( R - A ) <M|1 e ^ l j N > ; (7-85) where we assumed the up and down quarks to be massless. Since our Lagrangian d e n s i t y eq.(7-19) conserves the e l e c t r i c charge and s i n c e e ab^b a simply counts the quark charge, we t h e r e f o r e have "•?i> , (7-86) where C w i s determined by the charge c o n s e r v a t i o n c o n d i t i o n G"9i0) + < ^ ( o ) = Q w - ( p J j (7-87) and Q w i s the nucleon charge doublet i n u n i t s of e. Equation (7-84) f o r G^iq*) has then the form &"<*(p ' C w j J i*(^(iln/J?)+-J, 1(HA/R))©(i?- A) . (7-88) As mentioned i n s e c t i o n c of t h i s chapter, the quark c o n t r i b u -t i o n to the baryon s p a t i a l c u r r e n t i s the p h y s i c a l baryon A ex-p e c t a t i o n value of the quark operator ^ ( r ) : 3 <R I |R> > (7-89) with |^)- I ea NI i ik ^CAk*/R)^,(^W' ?>W-A){b>«k] . (7-90) The quark magnetic moment operator yu^ i s r e l a t e d to f ^ t r ) v i a 126 N = 4 - \<f* * * f«<*> • (7-91) The integration can be performed exactly [32] and gives 3 with ' 3 « A - l ) + U a . (7 -93) A graph of vs the quark mass mAR i s shown in f i g . 14. We s h a l l denote fAa= JA0 for the massless up and down quarks and jAa= ^ for the massive strange quark. The contribution of the quarks to the baryon A magnetic moment can be calculated as being the expectation value of the z-component of the quark magnetic moment operator ^ i n the baryon state A of spin projection +1/2 K = < H , v i l / V I R , ^ T > , ( 7 " 9 4 ) W i t h ^ 3 ^ ' h2= R £ ^ ^ k . (7 -95) However, the matrix elements of ^ z are known only for the bare bags. We need therefore to expand the physical baryon A wave-function on the basis of the bare eigenstates of H o according to the formal expansion eq . ( 5 - 9 ) |R> - 2aW{ 1 + (/^-H.-AH^Xj IFO^ (7 -96) then 127 m R F i g . 14: The quark magnetic moment y a ( m a R ) « 128 K - 2*W\<Vf*J + < A. 1 H x (*v wr nw^ )"'. ( 7_ 9 7 } and the two other terms not present i n t h i s expansion v a n i s h s i n c e £ 9 1 does not c r e a t e or de s t r o y any pio n s . Next, we make a "no more than onepion i n the a i r " a p p r o x i -mation, i . e . we co n s i d e r only the terms in eq.(7-97) which have no more than one pion i n the a i r when the photon couples to the bag. L e t ' s d e f i n e as i n eq.(5-79) I„(E> = H 1 A G o ( E ) A H I ) ( 7 _ 9 8 ) and / V * ^ S <&.» / \ j O f (7-99) then eq.(7-97) becomes C 1  J 3 (7-100) or g r a p h i c a l l y M Q = ZaW{ I + 1 (7-101) The next approximation c o n s i s t s i n t o r e p l a c i n g H 0 + I 0 by H 0 as we d i d i n e q . ( 5 - 8 l ) . A f t e r a few mani p u l a t i o n s , eq.(7-100) reads hi = A V ^ + Z /C<B,C> t (7-102) with = Z a\/M f l) yMcpzCA,^ , (7-103) ^ ( B ^ C ^ Z ' C ^ V I j.^ k A T ^ ^ C ^ O A ^ M ) (7-104) 129 where we a l s o made use of the c h a p t e r V main r e s u l t : f " 8 ^ f " B R e p l a c i n g v and w by t h e i r f o r m a l e x p r e s s i o n s g i v e s f o r eq.(7-104) S , „ ^ A . AV, 7a -*K ^ 'A „ ., •^-0 A v , Vl - * R VjDef i n i n g R/V(B,C) s ^ 2 ( 6 U ^ ) , C ( v i ) ) , ( 7 " 1 0 6 ) then a f t e r a few m a n i p u l a t i o n s , (7-107) w i t h (7-108) a n d A V ( B ' C ) i s g i v e n e x p l i c i t l y i n t a b l e V w i t h the symmetry r e l a t i o n / V ( B , C ) = yu 9 2 (c,B) . ( 7 _ 1 0 9 ) As an example, we c o n s i d e r the n u c l e o n case f o r which the quark c o n t r i b u t i o n t o the magnetic moment i s e x t r a c t e d from eq.(7-102) t o be 130 Table V: The quark magnetic moment matrix elements PqZ(A,B) as defined in eq.(7-106). y 0 Z ( A + , B + ) V(A°'B0) VA~>B~) 2 y o / 3 2 ^ y Q / 3 V 3 ( 8 y o + y s ) / 9 v^"(4y o+2y s)/9 (4y 0-w s)/9 -2y Q/3 2/2y /3 o -P s/3 -y J/3 /2y //3 o ( 2 y o + y s ) / 9 / 2 ( y o + 2 y s ) / 9 ( P O - P s ) / 9 ( - 2 y Q - 4 y s ) / 9 / 2 ( 4 y o + 2 y s ) / 9 2 ( y Q - v s ) / 9 -y o/3 C-4y o +y s)/9 2 v ^ " ( y o - y s ) / 9 C-2y Q-y s)/9 ( y Q - 4 y s ) / 9 2 / 2 ( y o - y s ) / 9 ( - y 0 - 2 y s ) / 9 131 ( 7 - 1 1 0 ) or ' - 1 1 1 ) w i t h P, and the p r o b a b i l i t y c o n s e r v a t i o n c o n d i t i o n ™ Vw77 Aa-rr ( 7 - 1 1 2 ) ( 7 - 1 1 3 ) g. The nu c l e o n charge r a d i u s . The n u c l e o n e l e c t r i c form f a c t o r i s d e f i n e d as For q^ s m a l l and s p h e r i c a l l y symmetric, we have The f i r s t term i s s i m p l y the n u c l e o n charge Q w. For the second term, the a n g u l a r i n t e g r a t i o n can be done and we a r e l e f t w i t h j & * * ^ ( » ) ] + <S(f>, (7-116) ( 7 - 1 1 7 ) and the n u c l e o n charge r a d i u s can be w r i t t e n as 132 (7-118) If we d e f i n e <r a>" and <r A>£ to be r e s p e c t i v e l y the pion and the quark c o n t r i b u t i o n to the nucleon charge r a d i u s , then O a > " = 9 /-ry j (7-119) and from eq.(7-86) <^>* = CN j J3* A* (^{ah/R)+^(£l7ilR))d(R-h) . (7-120) For the pion c o n t r i b u t i o n , eq.(7-74,76,77) with eq.(7-118) give and we have a l l the necessary i n g r e d i e n t s to c a l c u l a t e the nu-cl e o n charge r a d i u s . h. The cente r of mass c o r r e c t i o n s to ^ and <ri>N . The Cloudy Bag Model c o n s i s t s of a s t a t i c 3-quark MIT bag with a c l o u d of p i o n s . However, t h i s s t a t i c approximation b r i n g s in some c o m p l i c a t i o n s when t r y i n g to estimate the center of mass c o n t r i b u t i o n to the baryons e l e c t r o m a g n e t i c p r o p e r t i e s . In 1979, Donoghue and Johnson [33] presented a simple pres-c r i p t i o n to estimate the e f f e c t s of the center of mass motion on the bag c a l c u l a t i o n s . T h e i r s t a r t i n g p o i n t i s the obs e r v a t i o n that even though the s t a t i c bag s t a t e i s not a momentum eigen-s t a t e , i t can s t i l l be d e s c r i b e d by a wavepacket of bag s t a t e s 133 which are momentum e i g e n s t a t e s : I C(X)> = \ * t V e' FVV> (7-122) where |A*"(]£)> i s a s t a t i c 3-quark bag ( i . e . a bare bag) centered at s H i s the spin of A, and mofl i s the 3-quark bag mass, i . e . the bare mass. For s i m p l i c i t y , o n l y X=s n i s co n s i d e r e d and eq.(7-122) i s then reduced to I R.<X)>- j J f ( j^)y<p>e " r ' X |R„(f>)> (7-,23) where the sp i n index has been removed a l t o g e t h e r and "X(p) i s then a s c a l a r f u n c t i o n of the momentum p. The mean momentum spread <pd >H i n the wavepacket i s Assuming <p >fl « mofl , Donoghue and Johnson i n t h e i r p u b l i c a t i o n appendix show that the order <pa'>H/m*fl c o r r e c t i o n s to the baryons ele c t r o m a g n e t i c p r o p e r t i e s are given by h i = -i- ^ > • r"Q (7-125) fo r the magnetic moment, and of? f o r the charge r a d i u s . The simplest estimate f o r <p*>fl i s to add the c o n t r i b u t i o n of each quark momentum <Pr I i l l / R 1 ( 7 . 1 2 7 ) 134 which, when combined with eq.(7-125,126), give an e x p l i c i t pres-l*cin and <r >CM. c r i p t i o n f o r u f t * *For the nucleon, eq.(7-127) becomes N " 3&0/R , (7-128) and s i n c e we can approximate the nucleon bare mass i n f i g . 9 by fYY\ OK) = / R , (7-129) then with -Qo=2.04, we get = ° - U > (7-130) < A a > £ = O . U < 4 a > £ ^ (7-131) and t h i s c o r r e c t i o n i s of a reasonable s i z e . As a r e s u l t of our d i s c u s s i o n , we have the f o l l o w i n g f i n a l form f o r the baryon A magnetic moment h = h i + + ML > ( ? - I 3 2 ) A f\ f\ with J U^ given i n eq.(7-80), yu^ i n eq.(7-102) and ^ C h n in eq.(7-130). S i m i l a r l y , the nucleon charge r a d i u s i s <^>"= < 0 : + < V > ~ + < !»» )£ , , ( 7 - , 33 ) with <r a>^, <r A>£ and < r a > c M given i n eq. (7-1 21 ,1 20 ,1 31) respec-t i v e l y . 135 i . R e s u l t s of the c a l c u l a t i o n s . In the equations f o r the ele c t r o m a g n e t i c p r o p e r t i e s of the members of the baryon o c t e t , the parameters i n v o l v e d a r e : the renormalized mass of the baryons, the reno r m a l i z e d c o u p l i n g con-s t a n t s f " B , the strange quark mass ms and the bag r a d i u s R. As we d i d i n the pr e v i o u s chapters, the reno r m a l i z e d masses are chosen t o be the p h y s i c a l masses f o r the s t a b l e baryons, and the resonance masses f o r the unstable ones. For the renormalized c o u p l i n g c o n s t a n t s f " B , we simply use f " w times the a p p r o p r i a t e SU(6) f a c t o r , i . e . we use the r e l a t i o n (7-134) with t */i0 given i n t a b l e III, and ty i s such t h a t For the strange quark mass, we conside r three s p e c i f i c cases: ms=144 Mev, which comes out of our CBM c a l c u l a t i o n (chap. V); m^=2l0 Mev., which give the best baryon mass spectrum i n the work of F. Myhrer et a l . [45,46]; and ms=279 Mev., which i s the MIT value mentioned in chapter I I . F i n a l l y , the bag r a d i u s R i s l e f t as a f r e e parameter i n order to study the r a d i u s dependence of the c a l c u l a t i o n s . L e t ' s f i r s t c o n s i d e r the r e s u l t s obtained f o r the magnetic moments of the baryon o c t e t . We show i n f i g . 15 to 19 the fl R t h e o r e t i c a l to experimental r a t i o of the magnetic moment ^ /^*t*p 136 i n f u n c t i o n of the bag r a d i u s R. Table V shows the r a t i o of the v a r i o u s baryon magnetic moments to the proton experimental one in the SU(6), MIT and Cloudy Bag models. In t a b l e VI, we give e x p l i c i t l y the quark, center of mass and pion c o n t r i b u t i o n to JUR f o r the s p e c i f i c bag r a d i u s of one f e r m i . F i g u r e 15 shows c l e a r l y that the t h e o r e t i c a l p r e d i c t i o n f o r the nucleon magnetic moment agrees very w e l l with the experimen-t a l v a l u e s w i t h i n 10% f o r the wide range of bag r a d i i of 0.85 to 1.15 f e r m i . T h i s i s c o n s i s t e n t with our a n a l y s i s of the P 3 3 r e -sonance i n chapter VI where a bag r a d i u s w i t h i n the range of 0.8 to 1.1 fermi was c o n s i s t e n t with the experimental data. N o t i c e a l s o from t a b l e VI th a t the pion c o n t r i b u t i o n of 0.6 n.m. ( f o r R=1.0fm.) i s e s s e n t i a l s i n c e i t c o n t r i b u t e s roughly on qua r t e r of the t h e o r e t i c a l value of 2.65 n.m. T h i s c o n t r a s t s with the MIT r e s u l t of ^ f=2.24 n.m. [32,33] which does not c o n t a i n any pion f i e l d c o n t r i b u t i o n . For the lambda magnetic moment, our r e s u l t s shown i n f i g . 16 are c o n s i s t e n t with a lambda bag r a d i u s of one f e r m i . Again, the r e s u l t s are q u i t e i n s e n s i t i v e to the bag r a d i u s s i n c e f o r the whole r a d i u s range of 0.8 to 1.0 fm., the t h e o r e t i c a l p r e d i -c t i o n |AA agrees w i t h i n 10% with the experimental value of jUeAxp=-0.614*0.005 n.m. Notice that a 25% decrease of ms from 279 Mev. to 210 Mev. changes juft by only 10% (0.1 fm.) showing t h e r e -fore that the strange quark mass i s not such a c r i t i c a l parame-te r i n the theory presented here. Next comes the sigma magnetic moment. For L+, the t h e o r e t i -137 c a l p r e d i c t i o n s agree very w e l l . w i t h the experimental value of ^e* P = 2• 3 3 ± 0 • 1 3 n.m. f o r a bag r a d i u s R=1.0-0.1 fm., independtly of ms. T h i s r e s u l t i s c o n s i s t e n t with the lambda case s i n c e one expects R A and R^ to be about equal s i n c e they have the same quark content. For the 1", whose magnetic moment i s determined v i a e x o t i c atom techniques [81,82,83], the experimental value has changed r e c e n t l y from ^ =-1.41±0.25 n.m. to /u^,p =-0.89*0. 1 4 n.m. [48]. Our t h e o r e t i c a l r e s u l t of -1.05 n.m. shown in f i g . 17 and 18 l i e between these two r e s u l t s , and i s s l i g h t l y more than one standard d e v i a t i o n from the new experimental value f o r any bag r a d i u s between 0.7 and 1.2 f e r m i . More ac c u r a t e measure-ments of JJ} would c e r t a i n l y be welcomed. F i n a l l y , f o r the =° and =", t a b l e VI shows c l e a r l y that the pion c o n t r i b u t i o n i s n e g l i g i b l e . We welcome the recent d e t e r -mination of the 5° and =" magnetic moments based on the decay — o asymmetry [84]. Our r e s u l t s shown in f i g . 19 f o r ju~ are i n ex-c e l l e n t agreement with the experimental value of -1.250*0.014 n.m., p r o v i d e d that the =° bag r a d i u s i s R=0.95±0.10 fm. For the H" magnetic moment, our t h e o r e t i c a l value of -0.61*0.02 n.m., with R=1.0*0.1 fer m i , and ms=144 Mev, agrees reasonably w e l l with the new experimental value of -0.69*0.04 n.m. [49]. Next, we c o n s i d e r the t h e o r e t i c a l r e s u l t s f o r the nucleon charge r a d i u s . The r a t i o of the experimental to t h e o r e t i c a l charge r a d i u s f o r both the proton and the neutron i s shown in f i g u r e 20. A c c e p t i n g a d i s c r e p a n c y of 10%, our t h e o r e t i c a l r e s u l t agrees with experiment f o r a nucleon bag r a d i u s i n the 138 range of 0.85 to 1.15 fm. In the neutron case, t h i s i s an impor-tant d i f f e r e n c e with the MIT bag model which, in i t s simplest form, p r e d i c t s <ra>>,=0.0 fm. F i n a l l y , we show in f i g . 21 the charge d i s t r i b u t i o n 4fTr i^°(r) f o r the neutron with a bag r a d i u s of one f e r m i . T h i s i s obtained by adding the quark c o n t r i b u t i o n eq . ( 7 - 8 6 ) and the F o u r i e r transform of the pion c o n t r i b u t i o n to the e l e c t r i c form f a c t o r e q . ( 7 - 7 4 ) . The graph shows c l e a r l y a p o s i t i v e core which i s the sum of the quark and the pion c o n t r i b u t i o n , and a nega-t i v e t a i l s u p p l i e d by the p i o n . N o t i c e that the charge d e n s i t y changes of s i g n at the bag r a d i u s , which t h e r e f o r e suggests that a p r e c i s e measurement of the neutron e l e c t r i c form f a c t o r would supply an e x c e l l e n t measurement of the nucleon bag r a d i u s . One l a s t note to s t r e s s on the f a c t that we d i d not t r y to get the best f i t to a l l those e l e c t r o m a g n e t i c data. For example, there are many other e f f e c t s which have not been i n c l u d e d such as c o n f i g u r a t i o n mixing [ 2 7 ] , sea quarks [ 8 5 ] , pion s t r u c t u r e , e t c . which w i l l a l l c o n t r i b u t e to some extent to the magnetic moments and charge r a d i i of baryons. N e v e r t h e l e s s , the e x c e l l e n t o v e r a l l agreement of our r e s u l t s with the experimental data sug-gests s t r o n g l y that the Cloudy Bag Model giv e s a very reasonable p i c t u r e of the i n t e r n a l s t r u c t u r e of baryons. 139 Table VI: The baryon o c t e t magnetic moments i n the S U ( 6 ) , MIT and CBM models. A, p exp P n E" A = 0 _-A ySU(6) 1.00+ -0.67 1.00 -0.33 -0.33 -0.67 -0.33 A yMIT 1.00+ -0.67 0.97 -0.36 -0.26 -0.56 -0.23 y C B M ( 2 7 9 ) 0.95 -0.73 0.84 -0.39 -0.22 -0.46 -0.19 A PCBM(210) 0.95 -0.73 0.84 -0.38 -0.23 -0.47 -0.20 A PCBM(144) 0.95 -0.73 0.84 -0.38 -0.24 -0.49 -0.22 A P exp 1.00 -0.68 0.83 -0.321.05 -0.22 -0.45 -0.25±.01 140 Table V I I : C o n t r i b u t i o n of the pi o n , quark and center of mass to the baryon magnetic moments i n the CBM with R =1 fm. and m =144 Mev. y (n.m.) P n A = 0 _-A 0.60 -0.60 0.34 -0.34 0.00 -0.02 0.02 A yQ 1.74 -1.22 1.73 -0.62 -0.57 -1.16 -0.54 A yCM 0.31 -0.22 0.27 -0.09 -0.10 -0.18 -0.09 2.65 -2.04 2.34 -1.05 -0.67 -1.36 -0.61 A y exp 2.79 -1.91 2.33±.13 -.89±.14 -0.61 -1.25 -.69±.04 141 F i g . 15: The nucleon t h e o r e t i c a l to experimental mag-n e t i c moment r a t i o y / yN a s a f u n c t i o n of the nucleon bag r a d i u s R . 142 F i g . 16: The lambda magnetic moment r a t i o y / y as a f u n c t i o n of the lambda bag r a d i u s and the strange quark mass ms. 143 F i g . 17: The dependence of the sigma magnetic moment r a t i o s y E / y|xp o n the bag r a d i u s R^ using the recent value of y § x p = _ 0 ' 8 9 ± 0 « 1 4 n « m « [48]. The dash-dot and dash-dot-dot l i n e s are the experimental l i m i t s f o r y z + and y E ~ r e s p e c t i v e l y . exp exp 144 18: Same as f i g . 17 but with the o l d value =-1.41*0.25 n.m. [86]. R_ F i g . 19: The dependence of the cascade magnetic moment r a t i o s y= / y | x p on the bag r a d i u s R=. The dash-dot and dash-dot-dqt l i n e s are the experimental l i m i t s f o r u E and v=" r e s p e c t i v e l y . exp exp 146 F i g . 20: The nucleon t h e o r e t i c a l to experimental charge r a d i u s r a t i o as a f u n c t i o n of the nucleon bag r a d i u s RXT. 147 E m CM ~ l 1 .75 f m. 0 . 0 0.25 0.5 0.75 1 .0 1 .25 1 .5 F i g . 21: The neutron charge d i s t r i b u t i o n 4^r 2j°(r) vs the r a d i a l d i s t a n c e r (shaded a r e a ) . A l s o shown are the quark (Q) and the pion (TT) charge d i s t r i b u t i o n i n s i d e the neutron. The neutron charge r a d i u s i s set at one f e r m i . 148 V I I I ) CONCLUSION: In the p r e v i o u s chapters we found that by keeping the bag r a d i u s i n the range of 0.8 to 1.1 f e r m i , i t was p o s s i b l e to o b t a i n very good p r e d i c t i o n s f o r the p h y s i c a l p r o p e r t i e s of the l o w - l y i n g baryons. The weak pion f i e l d approximation assumed here r e s t r i c t e d us to c o n s i d e r i n g only bags with r a d i i l a r g e r than 0.7 f e r m i , which i s c o n s i s t e n t with our r e s u l t s . Of course, there are many c o r r e c t i o n s which c o u l d have been i n c l u d e d , such as the two pion term in the p h y s i c a l baryon expansion, the non-l i n e a r i t i e s of our Lagrangian d e n s i t y , the s t r u c t u r e of the pion i t s e l f , the c o n t r i b u t i o n from h e a v i e r mesons (such as the kaon), the baryon r e c o i l c o r r e c t i o n s , the f l u c t u a t i o n s of the bag sur-f a c e , sea quarks, e t c . However, in our low energy long wave-l e n g t h pion f i e l d approximation, these c o r r e c t i o n s are ne-g l e c t e d . L e t ' s s t r e s s again the main f e a t u r e s of the Cloudy Bag Model. F i r s t , i t i n c o r p o r a t e s i n a dynamical way two important i n g r e d i e n t s : the confinement of quarks i n s i d e a r e s t r i c t e d volume of space, and c h i r a l symmetry, which allows the baryons to i n t e r a c t v i a the pion f i e l d . T h i s i s t r a n s p a r e n t when we " t r a n s l a t e " the Cloudy Bag Model from the quark world to the hadronic world, thus o b t a i n i n g a Hamiltonian formalism i n v o l v i n g baryons coupled i n a w e l l d e f i n e d way to the pion f i e l d . The u n d e r l y i n g presence of the quarks then r e v e a l s i t s e l f through the s t r e n g t h of the IIAB c o u p l i n g c o n s t a n t s and the i n t e r a c t i o n form f a c t o r u(kR). Next, p e r t u r b a t i o n theory a p p l i e d to the 149 Cloudy Bag Model works remarkably w e l l f o r R > 0.7 fermi, a l l o w i n g q u a n t i t a t i v e c a l c u l a t i o n s to be performed. F i n a l l y , there i s only one t r u l y f r e e parameter in the model, the bag ra d i u s R. Among the v a r i o u s achievements of the Cloudy Bag Model, the s o l u t i o n to the P resonance dilemna i s a very important one. We found that most of the P ^ c r o s s s e c t i o n a r i s e s from the f o r -mation and decay of a dressed d e l t a bag, and to a l e s s e r extent from the pion nucleon c r o s s e d graph. With res p e c t to the e l e c -tromagnetic p r o p e r t i e s of the baryons, we have shown that the pion f i e l d i s necessary to o b t a i n c o r r e c t baryon magnetic moments, and a b s o l u t e l y e s s e n t i a l f o r the neutron charge r a d i u s . There w i l l of course be many i m p l i c a t i o n s of the CBM f o r nuclear matter c a l c u l a t i o n s , nucleon-nucleon i n t e r a c t i o n and pion photo-p r o d u c t i o n , j u s t to c i t e a few examples. 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