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UBC Theses and Dissertations

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  We a c c e p t  this  OF PHYSICS  t h e s i s as conforming  to the r e q u i r e d  THE  standard  UNIVERSITY OF BRITISH March  ©Serge  COLUMBIA  1982  Theberge,  1982  In p r e s e n t i n g  t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of  requirements f o r an advanced degree at the  the  University  o f B r i t i s h Columbia, I agree t h a t  the L i b r a r y s h a l l make  it  and  f r e e l y a v a i l a b l e for reference  study.  I  further  agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may  be  department o r by h i s o r her  granted by  the head o f  representatives.  my  It is  understood t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l gain  s h a l l not  be  allowed without my  permission.  Department of  Pk^  The U n i v e r s i t y of B r i t i s h 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5  DE-6  (2/79)  5 c Cs Columbia  written  ii  ABSTRACT  In t h i s baryons,  the  a r e assumed static  which  theory  of  cribes  of  cavity  f o r the i n t e r n a l (CBM),  quarks called  permanently the  chiral  confined  in  surface  in  baryons,  a  a  resonance  in  moments o f t h e b a r y o n  of t h e n u c l e o n . A l l t h e s e r e s u l t s  experiment  when t h e o n l y  bag  i s chosen  true  anywhere  free  a  Hamiltonian  pion-nucleon octet  agree  parameter  i n t h e range  way  density  b a r y o n s c o u p l e d t o p i o n s i s o b t a i n e d . The model the  of  "Bag", and o f a c l o u d of  symmetry. When t h e CBM L a g r a n g i a n  on t h e s p a c e of c o l o u r l e s s  the magnetic  radius,  structure  i s p r e s e n t e d . The b a r y o n s  c o u p l e t o t h e q u a r k s a t t h e bag  successfully  tering, radii  t o be made  restores  projected  a new model  C l o u d y Bag Model  spherical  p i o n s which  is  thesis,  desscat-  and t h e c h a r g e  rather  well  with  of the theory, the  0.8 t o 1.1  fermi.  iii  TABLE OF  CONTENTS Page  Abstract  i i  List  of T a b l e s  vi  List  of F i g u r e s  ;  vii  Acknowledgements  ix  Chapters I) I n t r o d u c t i o n I I ) The a.  MIT  Bag  1 Model  Introduction  b. B a s i c  16  assumptions  c . The  MIT  d. The  static  Lagrangian  I I I ) The  MIT  Cloudy  a. C h i r a l  density  spherical  e. C o l o u r e d q u a r k s f . The  17  bag  18  solution  i n t h e MIT  bag  model  Hamiltonian Bag  and  density  QCD  b. The  linear  c . The  non-linear  sigma model  d. The  non-linear  CBM  e. The  CBM  32  sigma model  Lagrangian  23 25  Model L a g r a n g i a n  symmetry  19  Lagrangian density  35 38 density  40 43  iv  IV)  Hamiltonian a.  V)  f o r m u l a t i o n of  the  Cloudy  Bag  Model  Introduction  46  b. The  CBM  quark H a m i l t o n i a n  47  c . The  CBM  baryonic Hamiltonian  49  d. The  e i g e n s t a t e s of H  Renormalization a. E x p a n s i o n  of of  the  dressed  delta  c . The  nucleon  and  B a r e bag  Cloudy  H Bag  55 Model  the p h y s i c a l nucleon  b. The  d.  and  Q  58 60  delta  probability  self-energy Z^(E^)  61 63  AB  VI)  f u n c t i o n 1^  e. The  vertex  f . The  renormalized  coupling constants  66  g. The  renormalized  propagator  68  h. The  r e n o r m a l i z a t i o n procedure  i.  Expressions  j.  R e s u l t s of  The  P  3 3  f o r the  (E ,Eg)  70  r e n o r m a l i z a t i o n f u n c t i o n s ..  the c a l c u l a t i o n s  72 77  resonance  a. The  one-pion  eigenstate  b. The  T-matrix  for pion-nucleon  c . The  partial  d. The  64  A  wave T - m a t r i x  P „ resonance 33 e. R e s u l t s of t h e c a l c u l a t i o n s  87 scattering  89 94 100 105  V VII) E l e c t r o m a g n e t i c a.  Introduction  b.  The EM  form  p r o p e r t i e s of the baryon  109 factors G (q ) A  2  and G ( q A  c. F o r m a l e x p r e s s i o n  for jj^( )  i  r  n  )  110  t h e CBM  d.  The p i o n  contribution to jj^r)  e.  The p i o n  c o n t r i b u t i o n t o G ^ ( q ) and £  y  ) and  g. h.  The n u c l e o n c h a r g e r a d i u s The c e n t e r of mass c o r r e c t i o n s t o y  i.  Results  of the c a l c u l a t i o n s  114 1  2  f . The q u a r k c o n t r i b u t i o n t o G ( q  Bibliography  2  M  E ^  VIII) Conclusion  octet  A  17  A  123  y  124  and < r > 2  N  131 . 132 135 148 150  vi  L I S T OF TABLES  Table  I  The a , a and ag c o e f f i c i e n t s needed f o r the calculation of t h e c o l o u r magnetic i n t e r a c t i o n energy [32] D  O  o  s  S  )  29  /  Table  Table  II  Best f i t t o the hadron MIT bag model [ 3 2 ]  I I I The  TTAB  mass s p e c t r u m  coupling constants  31  f  A  B  /f  o  Table  IV  i n the  The 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  52 Q  with  j =1  .........  122  2 Table V  The q u a r k  magnetic  moment m a t r i x  T a b l e VI  The b a r y o n o c t e t magnetic S U ( 6 ) , MIT and CBM m o d e l s  elements  moments  in  130  the  T a b l e V I I C o n t r i b u t i o n o f t h e p i o n , q u a r k and c e n t e r of mass t o t h e b a r y o n magnetic moments i n t h e CBM  139  140  vii  L I S T OF FIGURES Fig.  Fig.  Fig.  Fig.  Fig.  1  2  3  4  5  E i g e n f r e q u e n c y fi(mR) o f t h e l o w e s t q u a r k i n t h e MIT bag model [ 3 2 ] M a g n e t i c g l u o n exchange of mR [ 3 2 ] The p o t e n t i a l and c =0 The fit,  6  22  function 28  e n e r g y d e n s i t y V(O,TT) w i t h  y  2  <0 37  CBM form factor l u ( k R ) | and a G a u s s i a n v(kR)=exp(-0. l 0 6 k R )  53  R a t i o of t h e b a r e t o t h e r e n o r m a l i z e d nNN c o u pling constant fp /f as 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  79  E n e r g y d e p e n d e n c e o f t h e JINN r e n o r m a l i z e d c o u pling 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 t h e r a n g e o f 0.5 to 1.0 fermi  81  Energy d e p e n d e n c e o f t h e r a t i o o f 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  B a r e b a r y o n bag p r o b a b i l i t y z f ( m ) v s t h e bag radius R f o r a l l members o f t h e b a r y o n o c t e t o b t a i n e d v i a eq.(5-93) but w i t h the renormalized coupling constants defined i n eq.(5117)  83  Ratio m /m  85  2  B  Fig.  energy M as a  mode  A  2  B  N  Fig.  Fig.  Fig.  7  8  9  A  oA  Fig.  Fig.  10  11  of the bare t o r e n o r m a l i z e d baryon mass f o r a l l members o f t h e b a r y o n o c t e t A  The s t r a n g e q u a r k mass m r e l a t i o n e q . (5-118)  which  obeys  t h e mass  Total cross section f o r pion nucleon scatt e r i n g i n the P 3 3 channel as a f u n c t i o n of the p i o n k i n e t i c e n e r g y . The t h i c k l i n e i s t h e CBM p r e d i c t i o n f o r bag r a d i i i n t h e r a n g e of 0.8 t o 1.1 f e r m i  86  107  viii  Fig.  Fig.  12  13  Best f i t (solid line) to the experimental (dashed l i n e ) t o t a l cross section f o r pion nucleon scattering i n the P33 channel as a f u n c t i o n of t h e p i o n k i n e t i c e n e r g y . In this f i t , f =0.24, R=0.72fm. and m =1232 Mev  108  Lowest order matrix cleon interaction  111  element  moment u  f o r e l e c t r o n nu-  Fig.  14  The q u a r k m a g n e t i c  (m R)  127  Fig.  15  The n u c l e o n t h e o r e t i c a l t o e x p e r i m e n t a l magnet i c moment r a t i o v / y^ p a s a f u n c t i o n o f t h e n u c l e o n bag r a d i u s R  141  The lambda m a g n e t i c moment r a t i o y / Vexp a s a f u n c t i o n o f t h e lambda bag r a d i u s Ry\ a n d t h e s t r a n g e q u a r k mass m  142  N  X  N  Fig.  16  A  s  Fig.  17  The dependence of t h e sigma m a g n e t i c moment r a t i o s y /y,§xp bag r a d i u s R £ u s i n g t h e r e c e n t v a l u e o f y | =-0 . 8 9 ± 0 . 1 4 n.m. [ 4 8 ] E  o  n  t  n  e  x p  Fig.  18  Same a s f i g . 17 b u t w i t h y| =-1 .41±0.25 n.m. [ 8 6 ]  the  old  value 144  xp  Fig.  19  The dependence o f t h e c a s c a d e m a g n e t i c r a t i o s y - / y | on t h e bag r a d i u s R_  moment  x p  Fig.  20  21  145  The n u c l e o n t h e o r e t i c a l t o e x p e r i m e n t a l c h a r g e radius r a t i o a s a f u n c t i o n o f t h e n u c l e o n bag r a d i u s R,  146  The n e u t r o n c h a r g e d i s t r i b u t i o n 4irr j°(r) v s the radial distance r (shaded area). Also shown a r e t h e q u a r k (Q) and t h e p i o n ( TT) charge distribution inside t h e n e u t r o n . The n e u t r o n c h a r g e r a d i u s i s s e t a t one f e r m i  147  T  Fig.  143  2  ix  ACKNOWLEDGEMENTS  I wish derable  t o thank my s u p e r v i s o r , A.W.  amount o f t i m e and e f f o r t  project  during  I am a l s o cussions  and  N.  search  indebted  like  h e l p i n g me w i t h  this  t o G.A. M i l l e r  Weiss  for" the  stimulating  of the Cloudy  dis-  Bag M o d e l .  a l s o t o thank D.S. B e d e r , J . Ng, J.M. M c M i l l a n  f o r t h e numerous d i s c u s s i o n s we had on t h i s r e -  project.  I am a l s o g r a t e f u l graduate four  he s p e n t  consi-  my s t a y a t U.B.C.  we h a d i n t h e d e v e l o p m e n t  I would  Thomas f o r t h e  students  years  of  of graduate  to a l l the f a c u l t y , the  Physics  staff  Department  s t u d i e s a t U.B.C.  an  members  f o r making  unforgettable  and these ex-  perience .  Finally, received  I greatly appreciate  from my  Financial Fellowship Council  t h e encouragement and h e l p I  wife.  assistance  from  o f Canada  the  in  Natural  i s gratefully  the  form  Sciences  of  a  Postgraduate  and E n g i n e e r i n g  acknowledged.  Research  1  I)  INTRODUCTION:  The Cloudy Bag Model [1 to 9 ] , denoted CBM, i s a model desc 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 b a r y o n s . portant  ingredients:  It  involves  the quark and p i o n f i e l d s .  moving f r e e l y i n s i d e a s t a t i c  two  im-  The quarks  are  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  "cloud"  of p i o n s , which c o u p l e t o the quarks at the bag  i n a way which p r e s e r v e s c h i r a l in  i t ' s present  symmetry. The Cloudy  surface  Bag  Model  form was developed m a i n l y 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 self.  Various  my-  a s p e c t s of the model have been s t u d i e d by us and  some o t h e r c o l l a b o r a t o r s but I s h a l l p r e s e n t  in this thesis  the a s p e c t s of the model i n which I brought  a  major  only  contribu-  tion. The 1940's  story  of  the  Cloudy  Bag Model t a k e s us back to the  and the d i s c o v e r y of the Yukawa p a r t i c l e , the p i o n  [10].  At t h a t t i m e , i t was understood t h a t the nucleon had an i n t e r n a l structure,  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  nucleon.  the d i s c o v e r y of the p i o n and o t h e r mesons, i t was b e l i e v e d the  With that  c o r r e c t n u c l e o n magnetic moment c o u l d be o b t a i n e d 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  virtual  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 t h e o r y of p o i n t l i k e nucleons  coupled  to  mesons  was soon found t o be i m p r a c t i c a l ,  s i n c e 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 p a n s i o n which d i v e r g e s  badly!  ex-  2  In tion  1954, G.F. Chew  to  tirely  the r e l a t i v i s t i c  n e g l e c t i n g nucleon  function cleon  [11 t o 15] p r e s e n t e d  p ( r ) t o spread  interaction.  would  meson t h e o r y recoil.  which  He a l s o  o u t i n space  t h e r e f o r e make t h e t h e o r y  i s obtained  introduced  i n t e g r a l s and  No s p e c i f i c  prescription  the  nucleon  approximate  i f three  theory  a  real  calculations duction  were  internal  r a t h e r than nucleon  i s the case,  physical such  fields  f o r the pion  t o be n o n l o c a l . I f t h i s have  nucleon  as p i o n  possible  nucleon  two a r e r e q u i r e d , an  interaction  With  such  scattering  and a g r e e d  factor u(k)  structure [11]:  the c u t - o f f  significance".  for  to cut-off at  mass. Chew was s u s p i c i o u s t h a t t h e form  example,  source  f o r the d i v e r g e n t  finite.  of the  a  namely u ( k ) ,  c o u l d be s u p p l i e d a n d i t was g e n e r a l l y c h o s e n  was a s i m p l i f i c a t i o n  by e n -  of p ( r ) ,  u(k)  "For  approxima-  t h e r e g i o n o f t h e p i o n nu-  The F o u r i e r t r a n s f o r m  provide a natural cut-off  a static  would  factor  reasonably  u ( k ) may  a finite  and p i o n  have  theory, photopro-  well with e x p e r i -  ments.  A few y e a r s sive the "the  later,  calculation context static  mental charge  of  G. Salzman the nucleon  of Chew's s t a t i c theory  electromagnetic  theory. His r e s u l t s  leads t o severe  neutron-electron  [16,17] p r e s e n t e d  disagreement  interaction.  i s assumed t o be s t a t i c a l l y  with  out  up  reasonable with  agreement c a n be a c h i e v e d . "  a physical  nucleon  consisting  in  that  the e x p e r i -  I f , however,  spread  exten-  properties in revealed  example w i t h a d e n s i t y p r o p o r t i o n a l t o t h e s o u r c e then  an  the  core  space,  for  function p ( r ) ,  So, t h e 1950's ended  o f an e x t e n d e d  core s u r -  3  rounded  by a c l o u d  The  way  changed  of  of p i o n s .  looking  dramatically  at  when,  the in  nucleon  internal  1964, G e l l - M a n n  structure  [ 1 8 ] and Zweig  [19] p r e s e n t e d t h e q u a r k model o f h a d r o n s . The i d e a being  made  unitary with  of  quarks  symmetry. F o r example,  respect  to strong  o t h e r words,  the strong  tion  doublet  of t h i s  this two  case  the  t h e p r o t o n and t h e n e u t r o n  interaction, interaction  in isospin  picture  metry  group  classified covery  t h e n u c l e o n . In  i s invariant  under any r o t a -  The  the  triplet  of the n u c l e o n d o u b l e t  model  [21] but r a t h e r  1/2 o b j e c t s  called  quarks. Their  with the r e s p e c t i v e  picture,  stable  which would  state.  However,  violate  This  t h e sym-  member  l e d to the  the  of the e i g h t f o l d  as  does  in  of f r a c t i o n a l l y  "flavours"  dis-  i n t h e d e c u p l e t of  i n the theory  are  not con-  the  Sakata  charged  up,  way,  down  spin and  c h a r g e s 2/3, -1/3 and -1/3. In t h i s  problems  arise  identical  quarks  Fermi  (e.g.  baryons with  b a r y o n s a r e made of t h r e e q u a r k s ,  i s made o f t h r e e then  hyperons  a l l low mass h a d r o n s c a n be  t h e lambda  of a t r i p l e t  strange  1=0  plus  of  in  matrices in  [20] t o e n l a r g e  the v a l i d i t y  of p a r t i c l e s  group  unitary  the s t a b l e  was t h e m i s s i n g  Assuming  symmetry  e n l a r g e d . G e l l - M a n n and  i n one o f t h e SU(3) m u l t i p l e t s .  fundamental  sist  be  way"  of SU(3). In f a c t ,  particles.  +  to  t o SU(3) and a s s o c i a t e d  o f t h e fi~ w h i c h  =3/2  needed  form,  a doublet,  space.  i n the " E i g h t f o l d  representation  hadrons  from c o n s i d e r a t i o n s of  i s SU(2), the group of t r a c e l e s s  Ne'eman p r o p o s e d  J  naturally  d i m e n s i o n s . However, w i t h t h e d i s c o v e r y  A,I,...),  8  follows  of  with the A  statistics!  each + +  i n t h e same The  i n the  f o r example, state,  simplest  and  and v e r y  4  fruitful  remedy  to t h i s  problem  i s to give  tional  quantum number, c o l o u r . Then,  the  delta  wavefunction  by  will  be  provided  w a v e f u n c t i o n . Q u a r k s come i n t h r e e and An  the  symmetry  extremely  known  In  are  colour  1973,  dynamics  i s SU(3)  important  hadrons  Therefore  group  could  be  Since  only  spectrum,  This  three the  extremely  large the  energies), results  tering, less  of  which are partons.  "infrared that  says that  the  finitely  the  feature  for very  energy  an  effective with  their  interaction separation  world.  that  similar of  the and  to  weak  hadron  the  eight  interaction  between  g l u o n s was  baptised  t h a t QCD  "asymptotic  distances freely.  free, spin  quark  i n s i d e hadrons.  realized  called  all  singlets.  model  SU(3)  the  soon  short  (or  freedom" for  very with  inelastic  1/2,  nearly  property  of  between q u a r k s  has  This agrees  quark confinement,  [25].  that  hadronic  to d e s c r i b e  "expected"  s l a v e r y " a l s o known as  is  colour  e l e c t r o n nucleon  c o n s i s t e n t with Finally,  [22].  colour  t o be  q u a r k s move a l m o s t  high  for colour  suggested  q u a r k s and  (QCD). I t was  interesting  [24,26], which  chosen  colour blue,  gauge p r i n c i p l e  seem n e c e s s a r y  the  g r e e n and  Weinberg-Salam  of c o l o u r e d  Quantum Chromodynamics an  a  g l u o n s would c a r r y  model  are  i n s i d e the  by  the  gauge g r o u p was  This  i . e . they  addi-  of  antisymmetric  observation  scheme would c o n f i n e  colours  gauge b o s o n s c a l l e d quarks.  and  an  [23,24,25]  governed  Quantum E l e c t r o d y n a m i c s interactions.  confined  authors  antisymmetry  where c s t a n d s  C  colourless,  various  q u a r k s an  c o l o u r s : red,  experimental  must be  the  scatmass-  QCD  which,  increases  is says inde-  5  Many quarks  phenomenological  permanently  models  confined  of h a d r o n s  have been d e v e l o p e d  t e e n y e a r s . Some have n o n - r e l a t i v i s t i c tial  model of N.  relativistic course  I s g u r e t a l . [27]  quarks  the Cloudy  The  story  such  Bag  of q u a r k  bag  quark  fined  inside  a spherical  move  freely.  boundary  model  Assuming  massive  i n the l a s t  s u c h as t h e  f o r example, and  in  t h e bag  coloured  models,  fif-  poten-  others  have  including  of  Model.  Bogolioubov  infinitely  as  quarks  made o f  models  begins  [28,29]. Quarks static the  outside,  cavity  quarks  in  1967  with  the  are  con-  in his picture  of r a d i u s  R  where  t o be m a s s l e s s  inside  B o g o l i o u b o v was  led  to  the  they R  and  linear  condition  (1-1 ) on  the  lying  bag  state  surface, can  be  and  written  the quark  wavefunction  analytically  f o r the l o w e s t  as  (1-2) w i t h -Q=  2.04  successful the  Roper  from  for  the l i n e a r  example  resonance  the  there  i s no d y n a m i c s  Finally, conserved plied face .  radius  we  mention at  R fixed  radial by  t h e bag  of  t o balance the D i r a c  1470  MeV  was for  nucleon),  t h e n u c l e o n mass. However,  to determine  since  model  the  t h e bag  i n t h e B o g o l i o u b o v bag,  surface  The  t h e mass of  excitation  fitting  i n t h i s model that  condition.  in predicting  (first  with  boundary  no  outside  radius.  momentum i s n o t p r e s s u r e i s sup-  p r e s s u r e of the q u a r k s  on  the  sur-  6  The  p r o b l e m s w i t h t h e B o g o l i u b o v model were s o l v e d  when A. Chodos and c o w o r k e r s Model" in  [29 t o 3 3 ] , t h e d e t a i l s  chapter  entirely  I I of t h i s contained  (eq.(2-24)). inside  a  closed  pressure  the D i r a c  the  (  for  the  MIT  invariant  gluons  V called  are  bag  "MIT at  an  Lagrangian equation L  boundary  the outward  -  the q u a d r a t i c  density  fVr>-) C^(nc )  to the  are  =  0  is  confined  hoc,  outside  e q u a t i o n s of  on t h e motion  ( i n t h e a b s e n c e of V  /X6V  >  (1-3)  which g u a r a n t e e s c o n f i n e m e n t ,  (n  M  surface)  <j ( o O  boundary  The  length  energy-momentum  ad  f o r the quarks i n s i d e  condition  normal  way.  Bag  model  permanently  introducing  1974  Lagrangian density  t h e bag, and  invariant  c*Y-/n and  the  of which are d e s c r i b e d  Lorentz  and  in a Lorentz  gluons):  is  presented  w h i c h b a l a n c e s t h e q u a r k and g l u o n p r e s s u r e  from  linear  a  volume  derived  the  in  MIT  In b r i e f ,  i s g u a r a n t e e d by  B,  surface  thesis.  The q u a r k s  conservation  bag  at  in  =  "J  condition  f  /  X  )  0 < 6 S j  which g u a r a n t e e s  (1-4) stability  t h e bag  B = H - ~/>o.'i)[^(/x)<j(,*)] When g l u o n s a r e  present,  another  boundary  / x e S . (1-5) condition  can  be  der i v e d  /Y)  h  F*(tf) -  O  // 6 S  (1-6)  7  where F*^ tion  i s the gluon  leads  in  field  fact  tensor.  This  one must  use t h e s t a t i c  then  reduces  with  the extra  exactly  spherical  center  o f mass c o r r e c t i o n t e r m -Z /R,  fixed  at  bag  radius.  279  MeV,  pion  mass, p r e d i c t e d  o f 140 MeV.  1.09  f o r the a x i a l  center well  vector  success  constant  with  the  a major  experimental improvement  value  moments  vement  predicted  magnetic center  value  of  2.79 n.m.  any mechanism the  model  quark  exception  the  the  mass  prediction  rises  when  [ 3 3 ] . T h i s compares  very  with  , and  consti-  the n o n - r e l a t i v i s octet  i s a l s o an i m p o r t a n t  magimpro-  V I ) . However, t h e p r o t o n  1.9 n.m.  (2.24 below  n.m.  when  the e x p e r i concerns  inappropriate  in  t h e MIT bag model does not  for interactions  rather  problems [ 3 4 ] .  that  of  t o 1.27  Another major d i s c r e p a n c y  we m e n t i o n  is  experimental  r a d i u s w h i c h comes o u t t o be e x a c t l y z e r o  model. F i n a l l y ,  provide  dynamically  of the baryon  (see t a b l e  moment comes o u t t o be o n l y  neutron charge  makes  twice  o f mass c o r r e c t i o n s a r e i n c l u d e d ) , w e l l  mental  this  by t h e model  together  reasonable f i t  notable  o f 1.25+ 0.01  in comparison  on t h e SU(6) v a l u e s  a very  is  which  ingredients plus a  R  SU(6) p r e d i c t i o n of 5/3. The r a t i o  netic  which  g , which  of mass c o r r e c t i o n s a r e i n c l u d e d  tutes  sics  t o be 280 MeV,  A n o t h e r major  one,  and a s t r a n g e  t h e MIT g r o u p o b t a i n s  way,  approximation  a l l t h e above c  value  the  With  t h e mass s p e c t r u m o f h a d r o n s . The o n l y  the  in a covariant  boundary c o n d i t i o n , e q . ( 1 - 5 ) ,  the  tic  bag  condi-  f o r the bag.  t h e MIT bag model t o t h e B o g o l i o u b o v  fixes  for  boundary  t o the absence of net c o l o u r  However, t h e model c a n n o t be s o l v e d and  last  between  baryons,  f o r t r e a t i n g nuclear  which phy-  8  Among t h e many a t t e m p t s interesting T.D.  Lee  approach  [35],  G o l d f l a m and  is  which  phenomenogical  the s o l i t o n  has  L. W i l e t s  t o g e t bag  been  sigma  representation  field.  field  <J  l =  °,  AlKI  lowest  sigma and  state,  However,  sigma  field  mum:  the  choice  of  potential  it  when  potential  is a  is tilted  q u a r k s have dug  of p a r a m e t e r s ,  this  1N  then  quark  which  Friedberg  on  and  by  R.  introducing  a  i s assumed t o be a gluon  minima, a r e l a t i v e The  latter  identified field  very  self-interacting  two  = AT.  a  recently  i s based  the  has  an a b s o l u t e one, cr^  energy  state.  model  field,  phenomenological The  model of R.  investigated  [ 3 6 ] , The  scalar  m o d e l s f r o m QCD,  one  with  being the  the  i s p r e s e n t , the  ando~=0 becomes t h e  h o l e becomes i d e n t i c a l  vacuum  effective  true  a h o l e i n t h e vacuum. W i t h  one,  a  mini-  suitable  t o t h e MIT  bag  model!  An  i m p o r t a n t p r o p e r t y o f QCD  symmetry  [ 3 7 ] , In s i m p l e words,  eigenstates  and  the  of t h e q u a r k s  helicity  model, term to  confinement  flip  their  group  [38],  and  strong  and  L  partially when  R  after  In  introducing  surface,  thus  isotopic  spin  invariant,  the  an  MIT  bag  infinite  mass  forcing  the  quarks  by  less  symmetry  sym-  than of  7% the  symmetry". The c o r r e s p o n -  current  w i t h t h e weak a x i a l  helicity  However, t h e c h i r a l  " i t i s the b e s t  conserved a x i a l  identified  by  is chiral  are  is chiral  i s known t o be b r o k e n  as P a g e l s s t r e s s e s ,  interaction  Lagrangian  when r e f l e c t e d .  SU(2)  quarks  s h o u l d not c h a n g e .  is realized  helicity  SU(2) x  massless  t h e QCD  f o r t h e q u a r k s a t t h e bag  metry  ding  since  with massless quarks  (PCAC) i s d e n o t e d current,  its  A^(x)  divergence  9  is  given  by  TT(x)  where  is  decay c o n s t a n t the the  nucleon  the of  pion  field  93 Mev.  matrix  In  the  element  Goldberger-Treiman  of mass m^,  of  the  relation  i s the  and  i s the TTNK c o u p l i n g  the  w  8 i 2%  should  be  The model  the  level included  first  was  Since  nucleon  bag  i n bag  m o d e l s of  symmetry  surface,  at  SU(2) x  SU(2)  L  of  a  the  bag  symmetry  R  Lagrangian  one  sigma model o f G e l l - M a n n and quantize sically  the  f o r the  p e r t y TT(x)  In  meson  =  1979,  fields,  model  to  is right  feature  at  which  to  the  C.B.  MIT  Thorne  is violated  introduced  bag [40].  only  coupling  d i c t a t e d by  is  the  when e x p r e s s e d the  [ 4 1 ] . No the  analog attempt  model was  where the  fields  at  a multiplet  to  rather  "hedgehog" b a r y o n  constant  which couple  recovers  Levy  but  symmetry a  Thorne  The  relation  Chodos and  c o n d i t i o n , and  density,  rise  axial-vector  symmetry  bag  fields  surface.  gives  ^^=0,  hadrons.  A.  and  ( 0" ,TT ) of p h e n o m e n o l o g i c a l only  by  i n the MIT  Chodos  limit  (1-8)  This  chiral  chiral  i n 1975  pion  >  the  constant.  t o add  current  ^VA/TT  is  makes  proposed  chiral  mass, g  which  attempt  symmetric  axial  "rjn  where m  | f ^ | i s the  [39],  ^N  g  chiral  and  quarks the  in  terms  to the  linear  was  made  solved  to  clas-  have the  pro-  g(r)r.  G.E.  Brown and  M.  Rho  [42,43,44] p r o p o s e d  a  chiral  10  bag  model  of  the  vacuum. U n l i k e the  bag  field the  the  nucleon  quarks.  In  spontaneously  the  vacuum o u t s i d e  broken  to  quarks occurs  is  highly  bag,  the  i s the bag  use  bag.  sfully baryon  This  pion  calculated  the  and  the  s  boson,  F.  and  fermi  and in  pion  zero-point  t o be be  pion,  pion  field  effect  of  model Thorn  the  their  pro-  "shrinks" the  calculations. (~1.1fm),  a  stabilizes  [45,46] t o  succes-  In  their  f i t ,  each  the  quark energy  with  volume e n e r g y w i t h one-gluon  of  invalidates  found which  B'' = 137  MeV,  exchange e n e r g y  motion energy with  with  the  and  which  large  coworkers  m a g n e t i c moment, and  i n e x c e l l e n t agreement  is  the  quantitative  and  c o n t r i b u t i o n , the  symmetry  Chodos  thereby  B can  the  by  requirement  field  b a r y o n mass s p e c t r u m .  s  carried  b o u n d a r y . The  the  of pion  hedgehog b a r y o n . A n o t h e r the  combined  of  the  the  like  of  Myhrer  with  through  i s allowed  m =210 MeV,  the  coupling  surface  vacuum p r e s s u r e  is  cloud  i s a Goldstone  theory  radius  allowed  mass  <x =i.85 and  is  bag  is entirely  the  the  interior  chiral  f o r the  0.3  the  bag,  current  only  Thorne,  current  bag  in nature,  c a l c u l a t e the  <  the  axial  to about  the  m =m ( = lO MeV ti  the  phase p i c t u r e of  W i g n e r - W e y l s e n s e , no  c u r r e n t . The  enormous p r e s s u r e  i f the  for  axial  there  perturbation  However,  the  at  solved  radius  of  value  the  be  and  axial  nonlinear  i t can  blem  the  of  i n the  the  carries  continuity  symmetric  and  which the  a two  model of Chodos and  is chirally  i s present,  based' on  7_„ = 0.13.  result  experimental  of  value  They -0.62 of  with also n.m.  -0.614  n.m.  In revised  the the  same p e r i o d as bag  models  Brown and with  and  Rho's work, R.L. without  chiral  Jaffe  symmetry,  [47] and  11 developed models  a Lagrangian  depending  on  e q u a t i o n s of motion. radius ment  i s large, around  The models  which  Cloudy  Bag  discussed  First,  the  cussion, metry.  excluded  above,  and  realization  Thorn  model,  quarks  of c r e a t i n g distance  between q u a r k s ,  vacuum),  a  surface, the bag.  pion f i e l d ,  First,  i n n a t u r e . Due  which  will  and can  we  occur  this  that  restrains  dissym-  QCD  Chodos  There  behaves  the p i o n  field  phase  more  a  finite  will (the  symmetric  of t h e s u r f a c e  have true in-  will  have  of  the  o f the  bag,  motion  treatment  are  probability  for  (the c h i r a l l y  to  intro-  in  t h e bag.  to the dynamical  will  as  non-zero  in a covariant  pion f i e l d recall  if  a confining  treatment  bag  i t s own.  of not  (the g l u o n s ) , the  has  static  Finally,  field  The  is  the  i s then e q u i v a l e n t  the advantage  therefore  I f QCD  of  in Jaffe's  has  n o n c o n f i n i n g phase  time-averaged  nature, pion  and  of the bag), a c o r r e c t non  R  qq p i o n t y p e of o b j e c t s  treat-  incorporates chiral  L  for this.  bag  i s treated  ones of  i s d e s c r i b e d , as  field.  the  bag.  i s allowed to leak inside  t h e bag.  t o be  that  a perturbation  many i m p o r t a n t  which  bag  Lagrangian  the p i o n f i e l d  the  coupled with s t r i n g s  and  the  assume  o f SU(2) x SU(2)  sigma  to leak i n s i d e  terior  plus  model w h i c h  many c o m p e l l i n g r e a s o n s like  from  by a L a g r a n g i a n d e n s i t y  a fictitious  t o t h e above  Model p o s s e s s e s most p r o p e r t i e s  t h e n o n - l i n e a r sigma ducing  to  model. Again,  f o r m a l i s m of t h e CBM  The  is  t h e n makes p o s s i b l e  bag  field  reduces  t h e a p p r o x i m a t i o n s made on His approach  t h e MIT  as a c l a s s i c a l  f o r m a l i s m which  t o some e x t e n t l e a k i n s i d e  the  a l l t h e s e bag models a r e s t a t i c  in  us  t o low  t h e n be c o n s i s t e n t l y  energy  applications.  approximated  The  by a l o n g wave-  12  length way,  structureless object,  i . e . v i a a p l a n e wave  w h i c h can  be  expansion,  quantized  which  i n the  usual  necessarily  goes  through a l l space.  The add  next  a pion  king  step  mass t e r m t o the  chiral  symmetry  making a s m a l l ty  in c o n s t r u c t i n g  field  ( i . e . considering  density  which  interaction  i s the  t e r m of  Lagrangian  to r e s t o r e  pion  the  the  one  plus  Cloudy  density,  Bag  Model  i s to  explicitly  brea-  PCAC • r e s u l t  approximation  large bags), MIT  the  t o the  we  obtain  the  free  eq.(1-7).  Lagrangian  a simple pion  By  densi-  Lagrangian  one,  and  an  form  (1-9) The  explicit  with  the  III  of  d e s c r i p t i o n and  small  this  With  pion  approximation  Hamiltcnian  Lagrangian  formalism the  usual  density  f o r the  operator  s p a c e of  energy d e n s i t y  the  H  r e p l a c i n g TT (x) by  By  is  Lagrangian  treated  density  in  thus s i m p l i f i e d ,  Cloudy  canonical  Hamiltonian  sor.  the  chapter  thesis. the  following  field  a n a l y s i s of  Bag  Model. T h i s  quantization  is identified T ° ° ( x ) of  with  the  i t ' s quantized  rule  the  we  derive  is  realized  where  integral  energy  momentum  Fourier  transform  a  the  in a l l ten-  (1-10) and  the  quark  wavefunction  Hamiltonian  quark operator  Next,  most  and  by  given  importantly,  we  the  MIT  explicitly project  one, in  we  obtain  the  eq.(4-10,11,14).  t h i s Hamiltonian  quark  13  operator  onto  the space  us w i t h a H a m i l t o n i a n  of c o l o u r l e s s  baryon  bags.  This  leaves  of t h e form  H = H + H 0  ,  x  ( 1  _  n )  w i th  Ho = I where The  m  i s the bare  ofl  interaction  + I (A  ^ O F L R+0R0  Hamiltonian  v ^ (k) b e i n g  H  cleon  sector,  which  i s exactly  before,  with  Therefore, factor  i n the context  presence  of the Cloudy  of t h e q u a r k s  will  chapter  shall  bility  of H . 0  to  In  zero  the  _  1  3  )  nu-  mentioned  form  Bag M o d e l ,  significance:  factor.  the  cut-off  i t r e f l e c t s the  confined in a f i n i t e  s e c t o r of o u r b a r y o n i c  the nucleon  r e g i o n of  and  the  Hamiltonian,  delta  bags.  the p h y s i c a l  nucleon  from  Basically,  the p h y s i c a l  nucleon  has a  t h r e e quark  w i t h one p i o n  be a b a r e  1  the  build  t o be a b a r e nucleon  (  be t h e s u b j e c t of c h a p t e r I V .  t o both  eigenstates  + kc. ,  functions.  being  D  we  _12)  of t y p e A.  t h e same a s t h e one i n t h e Chew model  the strangeness  V,  (1  has t h e form  the p i o n c o u p l e s  P^r  ^ { \ h K  u(kR) = j ( k R ) + ^ ( k R )  A l l this  In  a bare  R!  u(kR) does have a p h y s i c a l  underlying space!  this  <yfe>,  axh  h a s t h e form  x  the a p p r o p r i a t e v e r t e x  0  k  o r MIT bag mass f o r t h e b a r y o n  H^IJAI with  ^  delta  bag, a p r o b a b i l i t y  "floating  around",  a  In  t h e bare proba-  P^, t o be  probability  bag w i t h a p i o n , a n d so o n . N o t i c e  that  14 Z^  i s , t o some e x t e n t ,  zation  approximation  the  pion  fm.,  Z*  field  In  expansion  fact,  find  brings  to the us  the  two  Low  judicious based  delta. the  nucleon  processes citly.  We  by  Model  the  P  perturbation  result  c o r r e c t i o n the to baryons.  resonance  33  has  the  which  we  pion  cloud  This  will  SU(6)  ratio  (also called  One  the  g r a p h of  crossed  have shown t o of  quark  the  Cloudy  and  the  pion  formation Bag  Model,  d e l t a bags  nucleon  reproduce the  parameters  t o the  data  ?J  substructure  the  that  our  formation  and  P^  of  hadrons,  and  decay  s i n c e the  calculation bag  resonance other  tells of  pion  in a well defined  for a nucleon that  s c a t t e r i n g which  i s made. The  i n t e r f e r e n c e t e r m s can  f e r m i , and the  P  s c a t t e r i n g . Up  the  r e s o n a n c e have p r e v a i l e d .  find  dominated  to  approxi-  Dodd e t a l . [9]  Bag  of p i o n  R>0.7  the  shall  1.1  0.8  weak p i o n  important  small  for  p i c t u r e s of  the  to  on  that  now,  [12]  in pion  plus  experimental  I t shows whether  to  i s due  In  L.R.  lineari-  elastic  the  resonance  this  Cloudy  Another  find  the  nucleon  choice  on  shall  of  q u a n t i t a t i v e c a l c u l a t i o n s t o use  resonance)  Chew and  the  constants  i s devoted  on  density.  justify  relatively  C h a p t e r VI  based  accuracy  work by  radii.  constants.  different  was  the  of  coupling  delta  to  some f o r m a l  coupling  i n our  Lagrangian  large  convergence  is  the  i s "weak". We  f o r l a r g e bag  shall  of  really  shown the  allow  t o our  is sufficiently  mation. also  a measure of  be  d e c a y of  way,  P  both  these  J5  to two  calculated expli-  i s compatible  resonance the  the  elementary  couples  r a d i u s anywhere  the  picture,  that  an  if a  delta  with  i n the is  bag.  in  the  range fact  15 In  chapter  properties study the  of t h e b a r y o n  the  nucleon,  nucleon  The  V I I , we  charge  overall  wide  bag  agreement  occurs  agreement  is also  is  one  mental  result  comes  out  of  result  sigma and  charge  is excellent.  range  1.15  range  0.85  to  fm.  standard d e v i a t i o n  from  of  [ 4 8 ] . The  -0.89*0.14 n.m. for  n.m.  within  10%  the  is consistent  within  12%  [ 4 9 ] . The  with  the e x p e r i m e n t a l  range with  of  0.85  those  proton  to  f o r the P  1.15  within  For  10%  the  I , +  prediction experi-  magnetic  moment  the  =.' t h e o r e t i c a l  the and  preliminary  neutron  results  charge  for a  fermi. This  resonance  15  nucleon  available  with  n.m.  The  The  =°  R =0.95*0.10 fm. , and  agrees  within  the best  moments,  fermi. Similar  0.8<R<1.2 fm.  f o r 0.9<R<1.1  shall  distribution.  values  of  we  magnetic  the e x p e r i m e n t a l  A  excellent  the e l e c t r o m a g n e t i c  cascade  the neutron  f o r ju i n t h e  -0.69*0.04  r a d i u s agree radius  with  radius  right  of -0.61  result  specifically,  More  agreement w i t h e x p e r i m e n t  in  value  octet.  r a d i u s and  moment a g r e e s  about  with  lambda,  magnetic the  are concerned  bag  latter  mentioned  above.  Finally, We  shall  use  lativistic and  use  a  word about  the B j o r k e n  quantities  such  the c o n v e n t i o n s  coefficients described  and  their  and  the c o n v e n t i o n Drell  as  conventions  in this  [50]  thesis.  for a l l  re-  four-vectors, Dirac matrices etc.,  of M.E.  Rose  identities.  in great d e t a i l  used  i n the  [51]  f o r the  Clebsch-Gordan  A l l other conventions  text  as we  need  them.  will  be  16 11)  THE MIT BAG MODEL  a.  Introduction.  In model  1974, t h e MIT g r o u p of  hadrons  the hadrons less  are composite  that  is  freely  accomplished  inside  a  of these  i n a Lorentz  B.  This  substructure phenomena manently  In carry  of h a d r o n s ,  [52], while  colour  keeping  quantum  When a p p l i e d  a l l physical  experimentally.  otherwise  t h e bag  way by a s s u m i n g per u n i t  vo-  context the f r e e - p a r t o n  in  the  Bjorken  scaling  these parton c o n s t i t u e n t s  Also,  degenerate assuming  t o t h e MIT  are colourless  i f the one-gluon a mass  hadrons  bag  are obtained  model,  splitting  is  per-  [32,33].  quarks  and  obtains  as observed  i s calculated  generated  such as the nucleon  mass s p e c t r u m  one  particles  t h e bag t o be a s p h e r i c a l ,  f o r the hadrons  (QCD), t h e  interaction  bag e q u a t i o n s c a n be s o l v e d e x a c t l y ,  sults  volume  number, and t h e g l u o n s a r e t h e v e c t o r  hadrons  t h e bag c a v i t y ,  By  observed  inside  energy  t h e c o n t e x t o f Quantum Chromodynamics  gauge b o s o n s .  inside  as  and mass-  restricted  invariant  i n a covariant  picture,  confined.  a  that  realizes  quarks  fields  t h e bag p o s s e s s e s a c o n s t a n t , p o s i t i v e  lume,  ties  o b j e c t s made o f l i g h t  t h e "Bag". The c o n f i n e m e n t  cavity  microscopic  t h e "MIT Bag M o d e l " . I n t h e i r  g l u o n s moving a l m o s t  called  MIT  called  [30,31] p r e s e n t e d a new  between  and t h e d e l t a .  static  c a v i t y , the  quantitative  and i t s o t h e r  static  re-  proper-  17  Basic  The  assumptions  MIT  bag  for  model  the  The  physical  the  2:  "Bag"  The  bag  positive 3:  has  action  to  be  pect  5:  6:  and  the  finite  action  S  with  is also  independent  surface  For  the  for  s i m p l i c i t y to  In  absence  region  following  gluon  of  space  respect fields  direction  unit  the  set  fields called  bag  independent,  volume  [30].  system  is  to v a r i a t i o n s  required  of  the  con-  with  res-  of  the  [30],  required  t o be  variations  of  stationary  the  position  [30].  lowest  l y i n g hadronic  of  be  the  static gluon  states,  and  field,  function  obeys the  free Dirac  volume;  and  spin-isospin  SU(6)  the  q u a r k and  t i m e and  energy B per  gluon  bag  the  by  quarks.  made of  S characterizing  stationary  to  in a  a constant,  potential  The  The  model.  [30].  f i n e d q u a r k and 4:  bag  hadrons are  permanently c o n f i n e d the  bag  i s b a s i c a l l y defined  assumptions concerning  1:  MIT  the  q u a r k model  [31].  the  spherical the  quark  equation  bag  i s assumed  [31]. s p a t i a l wave-  inside  wavefunction  the  bag  i s given  by  18 c. The  MIT  Let gluon  Lagrangian  first  fields)  pects  a l l the  write  Density.  down t h e  f o r the  MIT  bag  assumptions of 3 3  (  Lagrangian density m o d e l , and  section  then  [47]  show t h a t  (without i t res-  2b:  '  (2-1)  t=i  In mass m , L  this 0  V  Lagrangian i s a step  e  density,  i s the  defined  by  n^  i s the  bag  O  outward  e  =  /*  /V)  /Y)^  2  _  2  )  V  a  A  (  surface  delta  function  >  S  bag  surface  - - 1  (  2  _  3  )  with  ,  (2-4)  action  S  =  \ o\"*  £  J  MIT  surface  v  is  s  4-normal t o t h e  demanding t h e  stationary  A  }  of  as  outside  volume and  3  be  field  >  /yj ' —  By  i t h quark  -  o and  i s the  function defined  v  where V  q^(x)  under  the  <*>  (2-5)  /HIT  v a r i a t i o n s of  the  fields  and  the  bag  (A-3,A-4),  9'  v  ^  9  V  +  6  A  s  (2-6)  19  we g e t t h e E u l e r - L a g r a n g e  equations  of motion.  ( i $ - An. ) c ^ U ) = O is  the free Dirac  1).  equation  First,  V G V  f o r the quarks  ,  inside  (2-7)  the c a v i t y  (A-  Next,  t is  Y-m  c^.Coc) -  known a s t h e bag l i n e a r  that  no q u a r k c u r r e n t  ^(o<>  S  nee  boundary c o n d i t i o n ,  crosses  t h e bag s u r f a c e ;  ,  ( 2  which  _  8 )  guarantees  and f i n a l l y ,  3 B  provides with  d.  stability  the Dirac  The s t a t i c If  C i  spherical  *  -i  B =  The eq.(2-l0)  (A-5)  of t h e q u a r k s P  bag  I  ^  V  -  that  /Kt. )  /*6 S ,  -X^.v[X  the  (A-2).  D  bag  cavity  a r e then  <j. ( A , t ) -  ^ . ( ^ o V  energy  in a spherical  (2-9)  t h e vacuum p r e s s u r e B  solution  0  O  ^ ^ ) ] of the free  volume o f r a d i u s  is  static  rewritten  =  Y • k  positive  D  solution.  the e q u a t i o n s of motion  1 ° \  =P  f o r t h e bag by e q u a t i n g  pressure  we assume  spherical,  I ^U)  = -±.».Zl  as  A  £  A  =• R  R  ^=R. Dirac  R i s given  and  ,  y  (2-10)  (2-11)  (2-12)  equation  by [ 5 3 ]  20  \-i * ; j - * ' / f O < 2  ?-v  with  X-  r  =  x-  -  + 1  by t h e c o m p o s i t i o n  monics of order  X. N-  s p h e r i c a l Bessel Since  UJ  with  J=l/2  I  ?  (2-15)  ± , / X  a n g u l a r momentum  1/2 w i t h  (JM)  the s p h e r i c a l  f a c t o r and  har-  (z) i s  o f o r d e r J? .  (e.g.  spin  to  the s p h e r i c a l s t a t i c  <  condition then  eq.(2-l2),  impose t h e q u a r k s  I f b- c a r r i e s t h e i n t e r n a l  and i s o s p i n ) , t h e n  V  quark  boundary  f o r the quarks w i l l  q  linear  of t o t a l  of a s p i n  the quadratic  of  The  (  i s the normalization  t o be i n t h e J! = 0 s t a t e . t  1  c  function  (2-14)  we assume t h e bag t o be s p h e r i c a l and B t o be d i r e c -  independent,  together  ,  (  t  formed  tion  /»7- R  1^(8 , f , s ) i s t h e f u n c t i o n  where  the  ss  wavefunction  the p o s i t i v e p a r i t y s o l u t i o n  bag i s  i,  {n h/R)d-.h L  t  f r e q u e n c y Sir i s d e r i v e d  boundary c o n d i t i o n  eq.(2-1l).  by u s i n g  This  eq.(2-16) i n the  condition  is satisfied  only i f  «l  ( ^ ) = * • -j.Cil;) .  ( - ) 2  l 7  21  If  the quark  i s massless,  then  (2-18) which il  i  is satisfied  as  f o r 11*2.0428.  We  give  i n f i g u r e 1 a g r a p h of  a f u n c t i o n of A-=m .R. N o t i c e  that  f o r m.  t  t o w a r d s T T , t h e non  From tion  relativistic  the quark w a v e f u n c t i o n  f a c t o r N-  by  imposing  l a r g e , _Q-  tends  limit.  (2-16),  we  get  the  normaliza-  the c o n d i t i o n  (2-19) where leads  the i n t e g r a t i o n i s performed  only.  This  to  (  1  N; = and  on t h e bag volume  a  f o r the m a s s l e s s  boundary  oc. (a- - i ) +  A using  (2-20)  .  case  N Finally,  - A*.)  a l l the p r e v i o u s  c o n d i t i o n eq.(2-12) a l l o w s  R t o t h e vacuum p r e s s u r e  (2-21 )  - 1 results  in  us t o r e l a t e  the  quadratic  t h e bag  radius  B: 2  (2-22) 4TT  R  i=i  iot-(a.-i)  +  fr  .  Fig. 1: Eigenfrequency n(mR) of the lowest quark mode in the MIT bag model [32],  23 e.  Coloured  quarks  i n the  MIT  Quantum Chromodynamics for  describing  volved  are  fields,  the  the  exactness  of  at  yet  In inside with  quark  being  [54]  model, the  First,  G£  are  where t h e  £ ^ a  the  i  and  covariant  ex  eight are  c  MIT  the  the  i s the  SU(3) bag  result  the  of  fields  much  almost large  i n s i d e hadrons  field  model w h i c h  of  quarks hadrons  tensor  &' .  coloured  structure  has  QCD.  existence  gluon  in-  boson  i n t e r a c t i o n between  Yang-Mills  the  f o r the  symmetry  quark  define  gauge  guaranteeing  c o n s e q u e n c e of  colour  fields  i n t e r a c t i n g at  +  Lagrangian density colour  of q u a r k s  candidate  the  of  - 3 . ^ t -U the  best  gluon  therefore  to prevent  let's  the  strongly  a direct  i s introduced  colour.  and  i s a theory  very  the  (a=1,...,8) and  with  and  t o be  net  the  symmetry. The  C  confinement  F,." =  SU(3)  SU(3)  been p r o v e d  hadrons  where  fields  i s t h a t QCD  However, t h e  bag  i s presently  m a s s l e s s and  distance  MIT  model.  i n t e r a c t i o n . In QCD,  underlying  work  short  distance. not  the  [54]  strong  coloured  latter  theoretical free  the  bag  "  (2  23)  gluon  fields  constants.  The  incorporates  this  following  field  q  i s a short  form  being  r e s p e c t i v e l y the  d e r i v a t i v e ( D ^ . is defined  for  f l a v o u r and as  colour  index;  24  C D, with  X* b e i n g  = 2 ^ 5 ,  ) the  Lagrange equations  eight  matrices.  i^a - ^ ) <J(#)  ^(^^—  A»  ii  3x3 G e l l - M a n n  this  =  O  The E u l e r -  Lagrangian  * 6 V ,  are  (2-28)  the boundary c o n d i t i o n s  t  The  (2-26)  C  w h i c h a r e c o n s e q u e n c e of  ( i P +^ with  - t <l )  conserved  F^Coc)  ^ CoO  =  //6S , (2-29)  O  / * 6 S . (2-31)  current i s  = 1- n ^-x  %e v M  Fw^Gr^ e  v  = ( D, F ^ )  e , < -32> v  2  with  7).V The  conserved  charge  Coc) =  O .  (2-33)  is  w h i c h , u s i n g e q . ( 2 - 3 2 ) becomes  d  and  when i n t e g r a t e d by p a r t s  ^  D  „ F« t o t ) G  v  ,  (2-35)  25 Q  =  a  w h i c h by e q . ( 2 - 3 1 )  H  -  *  vo  ^  F a (/X) A  (2-36)  ;  gives Q  =  O  •  u  (2-37)  Hence, a l l s o l u t i o n s t o t h e MIT bag singlets,  s  as observed  f . The MIT H a m i l t o n i a n  .equations  are  colour  experimentally.  H M I T  From next  the  step  identify  Lagrangian  density  £^  i s to obtain a Hamiltonian with  t h e e n e r g y component  i t  given  quark  i n eq.(2-24),  operator  H  M  the  w h i c h we  o f t h e stress-momentum  tensor  T°°  <A * l  where T°° i s g i v e n  in' a g e n e r a l  • form by  | When a p p l i e d t o i, ^ t h i s c  (2-38)  ^  •  (2-39)  gives  Mil  Having cribed  by  shown  i n the previous  s e c t i o n that a l l baryons  t h e MIT bag model a r e c o l o u r l e s s ,  des-  we now p r o j e c t t h e  A  operator  H  M i T  on t h e s p a c e  of  colourless  baryon  bags.  Also,  26  since  the  quark  baryons, tion  solution  we s h a l l  eq.(2-16)  then r e s t r i c t  i s valid  ourselves  only  f o r i =0  t o t h e J5j6 r e p r e s e n t a -  o f SU(6) a n d w r i t e  H  M , T  where B ( B o ) d e s t r o y s  \  =  BIB.  ,  (creates) a baryonic  Q  (2-4,)  bag  of  type  B  and  mass m . B  +  A+DAe  "*o*  The mass f o r m u l a i s d e r i v e d  = where E  y  l\l  +  f  oV  from eq.(2-40)  E v - E,  i s t h e volume  * ^ l\ V  0  +  EE  +  +  Q  (2-42)  t o be [ 3 2 ]  E „ • E. ,  <2-«)  energy  (2-44) is  t h e quark  energy E  which  can  be  low v a l u e s  vanishes  =  I  approximated  II-  for  Q  <J.  ^  o f m^R. E  identically  ,  (2-45)  by t h e s i m p l e  04+O.ltX-  i s the colour  f o r m u l a o f F. Myhrer  , X.  =/>K7  electric  ;  R,  energy  i f t h e q u a r k masses a r e a l l e q u a l .  (2  "  46)  which F o r an  27 admixture The  o f s t r a n g e and n o n - s t r a n g e q u a r k s , E  energy  E  M  i s the c o n t r i b u t i o n  i s about  E  of the c o l o u r  5  magnetic  Mev. inter-  action  =  where M  oo  strange  a  oc  + 0  magnetic  quarks,  is  M  o s  table  the  a  o, M  o  S  +  ^  I . However,  M  interaction  s  s  ,  (2-47)  between  two  non-  one between a s t r a n g e and a non-  i s t h e one between  ss  The M's c a n be r e a d from  mate  o  i s the colour  strange quark, and M  in  M  two  strange  f i g . 2 a n d t h e a's a r e g i v e n  quarks.  explicitly  i t i s more c o n v e n i e n t t o u s e t h e a p p r o x i -  formula [45]  E = — .  . * f a • o. ns + a Ao. ns - o.on  2> R  c L  °*  +a which  i s valid  f o r mR  s s  s  less  E  left  e  as  hadron  i s a positive a  totally  mass s p e c t r u m .  presented  to  than about written  1.5. E  i s the  0  ,  parameter  Since then,  of  (2-49)  c o n s t a n t . In t h e e a r l i e r free  center  i n the form  = -  0  calculate  (2-48)  s  ( o, ns - o.o^i An R ) J ,  mass e n e r g y a n d i s u s u a l l y  where Z  +  n  0 0  and v a r i e d  different  Z„ e x p l i c i t l y ,  versions, to give  models  notably  Z  D  was  the best  have  been  by L i u and Wong  [ 5 5 ] a n d K. J o h n s o n [ 5 6 ] .  For quadratic  t h e r a d i u s R o f t h e b a r y o n b a g , we c a n e i t h e r boundary  condition  solve the  e q . ( 2 - 3 0 ) , o r make u s e o f t h e  con-  dition  OR  0  R=R  &  (2-50)  F i g . 2: M a g n e t i c g l u o n exchange e n e r g y 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 t h e exchange e n e r g y between e q u a l - m a s s q u a r k s ( M and Mg ), t h e d a s h e d l i n e g i v e s t h e one between a mass m q u a r k and a massless one, M [32]. q o  S  s  o s  T a b l e I : The a , a and a 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 o f the c o l o u r magnetic interaction energy. 0 0  Hadron  a a a  oo  OS  N  -3  A  o s  E  =  -3  0 0 0  s s  1  0  -4  -4  0  0  A  3  £  1  0 1  =  0  2 0  fipKa)<j>7rK  0  2 0  2  0 1  0 3  0 2 0  2 0 0  0 0 0  -6 0 2  0 -6  0  0  30 which  i s equivalent to i t .  radius  which  equivalent baryon  In  balances  other  words,  finding  the v a r i o u s pressures  to minimizing  the  energy  the  bag  on t h e s u r f a c e i s  ( i . e . the  mass)  of  the  bag.  table  I I , we g i v e t h e b e s t  trum a s o b t a i n e d of  In  f i t t o t h e h a d r o n mass  by t h e MIT g r o u p when  using  the  spec-  following  set  parameters. i  Notice  in  this  p i o n mass w i t h is  a  to play  massless  145"  Z  1.8f  in  0  =  *c  ~  table  1*  Goldstone  the  i n t h e next  Mev.  Air  -  *. A  (2-51 )  the l a r g e discrepancy  the experimental  i n hadronic  discussion sented  B * =  of the c a l c u l a t e d  one. In m a s s l e s s  the  pion  boson and t h e r e f o r e has a s p e c i a l  role  physics. This w i l l context chapter.  be  of the Cloudy  QCD,  subject  of  further  Bag Model t o be p r e -  Table I I : Best f i t to the hadron MIT bag model w i t h t h e p a r a m e t e r s [32].  Hadron  M  R  exp  M, bag  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  o  E  mass s p e c t r u m i n t h e given in eq.(2-5l)  o  E,, V  E„ Q  R  E  E  32 I I I ) THE  CBM  a. C h i r a l  The  LAGRANGIAN DENSITY :  symmetry and  Lagrangian  massless  up and  density  down f l a v o u r  ^  =  where q  QCD,  -  i s the quark  for  of q u a r k s  \  -  isospin  Quantum  Chromodynamics  has  the  form  i cT  T  doublet  with  (3-D  wavefunction  -  %-W ) > and  J3,  F  are  the c o v a r i a n t d e r i v a t i v e  defined  p r e v i o u s l y by  density  i s invariant  e q . ( 2 - 2 3 ) and under t h e  (3 2)  and  eq.(2-26).  infinitesimal  gluon  tensor  field  This  Lagrangian  global  transforma-  tions  ~  0  % with £ being simal the  the  usual Pauli  constant  rise  of t h e  the  s  matrices  )<& and £  , being  and  infinite-  t o N o e t h e r ' s theorem  d e n s i t y under t h e  [57],  infinitesimal  fields t ,  ( 1-  to a conserved  t L- £ ) <p ,  current  _L I So,  £-1 Y /a  the Lagrangian  V* = give  *  parameter. According  i n v a r i a n c e of  transformations  < 1-  -  (3-3)  1 5  transformations given  (3-4)  J^(x):  _5)  ( L t )  in eq.(3-3) l e a d  (3  to the  conserved  33  vector  current  V (x)  and  p  axial-vector  current  A^(x)  [54],  (3-6)  R % > ="± with  the conserved  <j  ( / x )  we  ding  define  the l e f t -  these charges w i l l  and  Q  group  are SU(2)  There  try. that  and  *>  ,  (3-7)  R°(/x) . r i g h t - h a n d e d c h a r g e s 0_  L  L  and  accor-  By  + Q  ,  c  (3-8)  t h e n obey t h e commutation  relations  the g e n e r a t o r s of the c h i r a l  symmetric  R  a r e two  different The  first  1 o)  =  ways f o r t h e s e c h a r g e s possibility  o  Q  \oy  physical  t h e s e a r e not  -  o  Nambu-Goldstone  will  appear  i n n a t u r e ! The  realization  of c h i r a l  upon  (3-10)  of c h i r a l  [ 5 8 ] , s u c h a vacuum s t r u c t u r e  states  observed  to act  is  known as t h e W i g n e r - Weyl r e a l i z a t i o n  Coleman's theorem  the  Q  x SU(2) .  vacuum s t a t e .  is  =  L f  therefore  Q  This  V  to  Q  the  r  5  charges  CD = ( If  Y^Y  in parity  symmeimplies  d o u b l e t s ; but  other p o s s i b l e  way  is  the  symmetry,  (3-11  )  34  also  r e f e r r e d t o as  physical Goldstone triplet  will  theorem  remain  requires  boson, the the  current  pion,  broken c h i r a l  violated.  parity  [59,60] t h e r e  of p s e u d o - s c a l a r  Goldstone  ly  states  s p o n t a n e o u s symmetry  The  The  is slightly symmetry  i t ' s divergence  but  massless  only  candidate  massive L  a partially i s given  by  isospin  slight-  conserved  the  this  therefore  t o be  R  the  for  which  SU(2)  case,  by  a  SU(2) x  i s now  In t h i s  singlets,  exist  particles.  current  (PCAC) and  will  breaking.  PCAC  axial  relation  [61],  ^ where f  T  The strange  i s the  9 U)  pion  previous quarks  =  M  - ^  -rru)  decay c o n s t a n t  d i s c u s s i o n can as  (3-12)  )  (|f^ [ ~ 93  be  well. This w i l l  Mev).  extended generate  to the  include SU(3), x  the  SU(3)  chiral tion  group which,  of c h i r a l  symmetry  as  well  as  the  pion  and  why  we  and  therefore only  shall  when a s s u m i n g  pions. the  i n v o l v e s the  However, t h e  symmetry  restrict  the  kaon  pion  only field.  realiza-  of m a s s l e s s  kaons  i s much more m a s s i v e  i s t h e r e f o r e badly  ourselves  to the  Nambu-Goldstone existence  to the  violated. SU(2)  D  K  L  than  This  x SU(2)  is  group  35 b. The l i n e a r  In cal  embodies  of  the nucleon  t h e PCAC r e l a t i o n  Nambu-Goldstone  model four  involves boson  The ten  model.  1960, G e l l - M a n n and L e v y  model  the  sigma  the l i n e a r  mentioned  realization  an i s o s p i n  i n the previous  of c h i r a l  doublet  sigma model, section  symmetry.  of nucleon  which  The  field  and sigma  , and t h e  Lagrangian density  f o r the l i n e a r  sigma model  is  writ-  as [61]  For  c =0, t h i s T  tesimal  / A - T T S  I  1  with  i  Lagrangian  ,  *  a.  density  \  ,  a  i s invariant  a.a.  under  (3-13)  the i n f i n i -  transformation  cr'  =  cr  ( 3 - 1 4 )  the conserved current  ~ i s also  d  invariant v  and  called  theoreti-  f i e l d s CT and TT.  I  It  [41] proposed a f i e l d  >  ~  under  =  cr'  =  TT'  =  the conserved a x i a l  fl^,l(p/Yr^  ~  (3-15)  the transformation  ( i - ' r - ! v  s  / i ) ^  v" - e.TT -n- +  6  (3-16)  a-  current i s  o - y V -TT^cr .  0-17)  36  If  we  Lagrangian we  leave  the  density  symmetry  e q . ( 3 - 1 3 ) , V^  1  violating  and  now  we  eq.(3-l2)  identify  this  The  F o r ^fcO, mum  potential  remains c o n s e r v e d but  "  =  C  *  TT  the  for  A ,  ( / x )  r  TT  =  rr " V  relation  .  tL (<r*+ TT )  (o-\Tr )*+  i . e . a real  mass term, V ( V , T T )  O-=TT=0.  3 and  J  However,  the minimum a t  1  i f ^*<0, (OofH ) 0  DV/DTT)  -  ,  V(c,7r) w i l l now  =O  TT =  •.  e  mini-  look l i k e  (3-21) o .  must t h e n c o n s i d e r quantum  a r o u n d s" = 0 b u t a r o u n d v = \r  (3-20)  satisfies  '  o; i s n o n z e r o , we  i . e . around  fluctuations  t h e c l a s s i c a l minimum  Define  <T = replace  by  p o s s e s s e s a unique  0  the p o t e n t i a l .  (3-19)  e n e r g y d e n s i t y V(T,7T) i s g i v e n  l«-r«-  Since  (3-18)  »  l a s t e q u a t i o n w i t h t h e PCAC  V(vriT)=  when  figure  and  in  n  and C  of  c \r  have  *V 5^(o°  not  term  i t everywhere  + °".  ,  (3-22)  i n the Lagrangian d e n s i t y  ^  ~  eq.(3-13)  (3-23)  38  The  fermion  then  t h e masses  W , t h e cr f i e l d  field  ^  =  ""V where we  made use  finite  term  =  X  being  =  ^  the  nucleon  doesn't  c . The  that  except  w h i c h has  for  4TT  more  the  been  than  being  SU(2)  x SU(2)  variant  .  (3-25)  density: >  can  be  (3-26)  removed.  field  a new  and  sigma  relatively  field  stable  of  which  physical  model. approach  t o the  i s a function  field.  symmetry  relation  light  of t h e  picture  observed.  physical  sigma  (3-24)  implies  the presence  t o any  n o n - l i n e a r sigma  A  ' ^  ^ X in ,  +  in^n  =  f i e l d TT a c q u i r e  sigma model p r o v i d e s t h e n a w o r k a b l e  correspond  particle  TT^  independent,  linear  h  +  i n the L a g r a n g i a n  spacetime The  U  of e q . ( 3 - 2 l ) w h i c h  C  the pion  ^ U >  =  The  and  of  sigma model  the p i o n  However, s i n c e we  of t h e L a g r a n g i a n  available  ^  want  i s t o assume  field  ir  rather  to preserve  density,  the o n l y  the in-  is  +  J  ~  U  .  (3-27)  39 We c a n t h e n d e f i n e a new f i e l d  4>  s u c h t h a t ($=<*>/<$>)  (3-28)  and  the coupling  The  sigma  and w i t h  term  and p i o n  kinetic  e n e r g y t e r m become  d e f i n e d as  -yfaAt*  (^1)  eq.(3-30)  (D* *)* 4  i s easily The  then  4  (3-3D  ^(^t>\  =  v e r i f i e d using  (3-32)  4>'-(D4) 0« =  Lagrangian d e n s i t y of the n o n - l i n e a r  sigma  model  has  t h e form  ?t->^e We  ,  becomes  J-CD^Z+J-CD^T) which  reads  the covariant d e r i v a t i v e  •= then  i n the Lagrangian density  n o t i c e that^. (x) (r  transformation  T 7"  z~  +  i s invariant  4  under  +  f(T;).(3-33)  the i n f i n i t e s i m a l  vector  (eq.(3-28) i n eq.(3-14))  =  ( I - i r . « / a ) T (3"  which  leads  to the conserved vector  current  3 4 )  y  In  a  M  =  *  Y  similar  ^  way,  transformation  v  +  £^!"(x)  i  i s invariant  ( i f m =0 and  replacing  T  (3-35)  .  under  the  eq.(3-28)  infinitesimal  i n eq.(3-16))  (3-36)  which  leads  t o the c o n s e r v e d a x i a l  Y  f= If  m^  y  [  1  +  i s n o n - z e r o , the a x i a l  served  and a c c o r d i n g \  The  which  SU(2)  and  R  sigma  d. The  pion  =  sigma model  h* will 1  and ^  *  the  t^<M be no  (3-37)  longer  con-  eq.(3-l8), ( <r7^  ) .  (3-38)  therefore allows  chiral  symmetry  without r e q u i r i n g  us t o w r i t e group  a  SU(2) x L  the e x i s t e n c e  of  field.  we  resulting  current  ~ iv  t h e PCAC r e l a t i o n  add  field  CBM  i n a way  R  Allowing  Lagrangian density.  t o t h e MIT  theory  SU(2) xSU(2) L  H  incorporates  non-linear  If  %  A^  4>y> +  t o eq.(3-28)  non-linear  theory  the  ^ \  current  bag model  similar  described  to the n o n - l i n e a r  i s t h e C l o u d y Bag Model  non-linear  the pion  representation  field  in chapter  to e x i s t  which  sigma model, incorporates  of c h i r a l  inside  II,  a  the the  symmetry.  and o u t s i d e  t h e bag  41 for  reasons described  field  in  field to  eq.(3-33)  to couple  the  i n the by  to the  the  I  quark  quarks only  following non-linear  1) —  i n t r o d u c t i o n , r e p l a c i n g the  CBM  field, at  the  and bag  Lagrangian  allowing surface,  ,  1  the  pion  leads  us  density:  e  /j ^ s / | l l  nucleon  1  V  (3-39)  1  % which,  i n the  limit  Demanding  of $=Q,  the  action  finitesimal  v a r i a t i o n s of  sent  Lagrangian  i n the  Euler-Lagrange  reduces t o be  the  of  MIT  bag  model.  i n v a r i a n t under a r b i t r a r y  bag  surface  density,  equations  t o the  we  and  obtain  of a l l f i e l d s to  the  inpre-  following  motion  (3- 40) 0(6 S ,  (3- 41 )  K6S,  (3- 42)  V,  *6  ^  (3- 43)  F; (or) = O  "1M 3  Equation with the  (3-40) i s the  the  gluons  pressure  lanced  by  of the  the  Dirac  equation  i n s i d e the  bag  q u a r k s and  vacuum p r e s s u r e  f o r the  volume; e q . ( 3 - 4 l )  g l u o n s on B.  quarks  the  bag  (3- 44)  interacting assures  surface  Eq.(3-42) i s the  is  linear  that baboun-  42 dary  condition  which  Indeed, the f l u x given  guarantees  of q u a r k  current  through  of  the  the quarks.  surface  being  by Ti^ ^ ( o O =  when r e p l a c i n g e q . ( 3 - 4 2 ) />0 and  confinement  using  J  (/x) =  U) -  h  in e q . ( 3 - 4 5 ) ,  -t  ^  e  + < <fi  0(6S  (3-45)  (3-46)  t  in eq.(3-45)  1f/x)  e  ,  we g e t  ^(*>  of e q . ( 3 - 4 2 )  the conjugate  sy> 3  ^(tf)Y-'* ^ ( o O  ^  6  gives  >  S  (3-47)  therefore  m  tion  (3-43)  <Jf*>  was a l r e a d y  non-linear  This finitesimal  density  implies  1 "  =  the conservation =  The L a g r a n g i a n d e n s i t y under  6  ,  (3-48)  S .  (3-49)  t o be t h e b o u n d a r y Finally,  equation  condi-  eq.(3-44)  is  a  f o r t h e <\> f i e l d .  i s i n v a r i a n t under  the g l o b a l i n -  transformation ^  which  O  fields.  Klein-Gordon  Lagrangian  =  mentioned  f o r c o n f i n i n g the gluon  horrible  /yeS  r  e  - c^(oc) Equation  J (oO - o  u  1  of t h e q u a r k b a r y o n i c  % (oc) Y^(*>  eq.(3-39)  the i n f i n i t e s i m a l  (3-  t  6  is  9  also  transformation  V  0  )  current  .  (3-51)  invariant  defined  5  (if  in eq.(3-34)  ^=0) and  43 eq.(3-36), tially  giving  conserved  the conserved axial  vector current  current  and t h e par-  :  and  e*4with  V£[+^M.<*H>+:>,*J><3-53>  r  the divergence  ^  All  2 ^  these h i g h l y  directly  a n d we  make t h e C l o u d y  order  -  +  shall  add a few more p l a u s i b l e  Bag M o d e l a w o r k a b l e  t osimplify  t h e CBM  small  that  order expansion  few  a first  wavefunction  the pion f i e l d .  around".  q (r*,t) <x  We w i l l  mations..  have t o v e r i f y  N e x t , we s h a l l  t h e r e f o r e make u s e o f  we r e c a l l  densi-  that  very  assume t h a t t h e  i s n o t p e r t u r b e d by t h e p r e s e n c e o f  a r e many c o m p e l l i n g r e a s o n s First,  consider the  of the Lagrangian  f u n c t i o n s a s g i v e n by t h e MIT bag s o l u t i o n  There  we  assume t h a t 4»/f„. i s r e l a t i v e l y  I n o t h e r words, we w i l l  pions are " f l o a t i n g  quark  first  formalism,  we  i s adequate.  hypotheses t o  density.  hypotheses:  ty  (3-54)  theory.  following such  •  non-linear equations are hopeless to solve  e. The CBM L a g r a n g i a n In  -  our b a s i c  t h e quark  wave-  (eq.(2-16)).  f o r making t h e s e a p p r o x i assumption  of t r e a t i n g the  44 pion  as  a pointlike  assumption pion, in  like  a long  here, to  any  this  of  faced  other  with  hadron,  internal  the our  multi-pion Lagrangian  structure  of  pion,  ties  to  of  the  the  treatment  of  bag  the  the  surface  pion  to  the be  pion,  as  the  the  as  a  the  first  pion,  we  simple  of  do  Next,  the But,  assume  i f we  also  such as  try  the  have  the  quark  w h i c h a l l add  suggest  and  we  will  this  long  consistent  one.  approximation,  now  crude  i n the n o n - l i n e a r i -  we  same o r d e r  i s why  rather  structure.  neglected.  eq.(3-39),  by  eq.(3-39)  density  of  internal  perturbation  model. T h i s  According Lagrangian  of  a  q u a r k model w h i c h r e q u i r e s  t o have an  density  effects  the  seem t o be  contribution implied  other  and  the  s t r u c t u r e can  consider  tion  T h i s may  wavelength•treatment  study  ties  when  particle.  4> = TT ,  to  internal wavefunccomplexi-  wavelength  and  the  reads s  (3-55) w h i c h we  rewrite  as  no IT  C6^  0*>  TT  x  (3-56)  '  ' ^ ( i i ^ — ^ V B - i F ^ f T j e . - i B ^ s , <3-57) 2  K  ^  „  j  -  _  ^  „  ^  (  equations  _  5  8  )  (3-59)  =  The  3  of m o t i o n  (i  eq.(3-37)  (]l>-»n)yo()  =  become  O  K£\J  }  (3-60)  45  t Y  /V)  ^(*> =  ^ (/y)  1 1 ^ ^  ^  FJ^CoO  % ( 2 + a  The  first  five  (eqs.(2-27 tion  to  AIOJ  F  *  )T(*; =  equations  'ij" ^  =  (Of) =  term  f«'  O  /x€ V  O  ,  (3-61 )  '  (3-62)  ,  (3-63)  S ,  -  ^  are  the  3 1 ) ) , and t h e l a s t  with a source  0<e S  A  MIT  bag  (3-64) (3-65)  .  S  model  equations  one i s t h e K l e i n - G o r d o n  on t h e bag s u r f a c e . The a x i a l  equa-  current i s  now  % Y VS Z  4  i s a g a i n t h e PCAC  relation  P ^ ( o O = -^1  H  ^  2> TT(*>  (3-66)  h  t  % and  i t ' s divergence  \ 9^' = - W 7 All  these equations  •  (3-55) t o (3-67) form  the core  (3  of  "  67)  the  Cloudy  Bag M o d e l and t h e y w i l l a l l o w us t o c a l c u l a t e e x p l i c i t l y  various  baryon  and  others.  properties  s u c h as m a g n e t i c moments, c h a r g e  radii  46  IV) HAMILTONIAN  a.  FORMULATION  OF THE CLOUDY BAG MODEL :  Introduction.  So  f a r , we have b u i l t  volving a pion for  quarks field  and g l u o n s c o n f i n e d  which  under  the  gave t h e p i o n  and  pions,  chiral  a mass t o embody  SU(2) x L  in-  introduced  a t t h e bag s u r f a c e and,  guarantees  group  interactions  i n a b a g . We a l s o  couples t o the quarks  massless quarks  theory  a theory of s t r o n g  invariance  SU(2) . R  of  the  Furthermore,  t h e PCAC c o n d i t i o n  on  the  we  axial  current [62].  We  have  the quarks  also  shown i n c h a p t e r I I t h a t  and g l u o n s  i n t h e bag l e a d s t o c o l o u r l e s s  mesons. T h e r e f o r e , i t w i l l Model  in  quark  space  the  space  make  sense  of c o l o u r l e s s  as d e s c r i b e d  baryon  stable  n u c l e o n d o u b l e t and t h e l o w e s t d e l t a  MeV).  We  energy  of  static  baryons  [63] i n v o l v i n g  perimentally. octet  bag and l o n g We  also  wavelength  neglect  the  we l i m i t  (m = A  the 1232  bags  and  high  w i t h our p i c t u r e  of a  ( i . e . low e n e r g y )  pion  influence  i s possible  ourselves  and d e c u p l e t of p h y s i c a l  to  of h i g h e r r a d i a l o r  to  particles.  of c o l o u r l e s s  i n the  b u t do n o t c o r r e s p o n d t o any s t a t e s  In s h o r t ,  for stran-  ourselves  distorted  are inconsistent  Bag  than the  quadruplet  the i n f l u e n c e  [ 4 7 ] , whose e x i s t e n c e  bag m o d e l s ,  baryon  neglect  p r o p a g a t o r s , which  approximations. exotic  f o r example, we r e s t r i c t  momentum s t a t e s  spherical,  bags r a t h e r  i n chapter I I I . Furthermore,  zero baryons  angular  b a r y o n s and  to express the Cloudy  geness  therefore  the confinement of  context  o b s e r v e d ex-  the  low  lying  47  The  procedure  down t h e q u a r k density  space  eq.(3-55),  ding  t o the  space  Hamiltonian  bag  interaction baryons  the  next  we  onto  will and  following:  will  operator  where H  is related  We  obtain  then  field  couple  this  space  will the  can  Lagrangian  derive  H according to *  Lagrangian  project bag  write  accorquark  using  the  generate  an  pion  to  the  H :  From t h e e x p r e s s i o n f o r t h e CBM we  baryon  will  way.  quark H a m i l t o n i a n  eq.(3-55) to eq.(5-39),  will  This procedure  which  defined  we  we  to the  q u a n t i z e the p i o n  the c o l o u r l e s s  Hamiltonian  first  corresponding  finally  wavefunctions.  in a well  CBM  be  Hamiltonian  usual rules,  model SU(6)  b. The  will  A  t o i. (oO v i a  /V  A  the  density given in  quark  Hamiltonian  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 e q . ( 2 - 1 6 )  \toc- ^(n.-ft/RK-A J and  expanding  Ttj ( r , t ) i n momentum s p a c e  t . o with  -  (  {a(W*  t h e u s u a l commutation  according to  +  + a  e I iW  (f)  (4-8)  rules  (4-9)  A  we g e t t h e MIT q u a r k H a m i l t o n i a n  the  free  pion  H  (4-10) M | T  Hamiltonian  H . [ ( J k % a\(h a^d) 3  and  the i n t e r a c t i o n  Hamiltonian  ,  quark o p e r a t o r  (4-u)  H  T  = - r f R* i « ) i ! ( r i , ) w ; (jn b*?.j! r.ir.ba  1)7!  J  1=1  ~  ~  (4-12)  I  Since  ^ (kR) + ^ ( l ? R ) ]  3> then  P^has t h e f i n a l  0  >  (4-13)  form  H = I j  | V.(?) a. (U) V*<1) a ' d b j ,  a  +  ( 4  -i ) 4  with 3  </.(?)=  [  c  t  b^.l?  r^b - , t  (4-15)  49 >  For  t h e sake  a.  _  (4-16)  of s i m p l i c i t y ,  up a n d down q u a r k s  J .=  we s h a l l  from now on t h a t t h e  a r e m a s s l e s s a n d we t h e r e f o r e  ^  Q  assume  -JL-  *  write  o.^  (4-18)  c . The CBM b a r y o n i c H a m i l t o n i a n . In  t h e space of c o l o u r l e s s  interaction  b a r y o n i c bags,  we c a n d e f i n e  an  Hamiltonian according to  j J fe { V 0 . ( ? ) ^(f)  = I  H  1  + V ^ ? ) a-(?)J  3  ,  (4-19)  A,  The  connection  between  H  and  H  x  i s g i v e n by e q . ( 4 - l 5 ) a n d  eq.(4-l9) as  v (l)= I 0)  R0+<R,| V / ? > | B , > B .  where A* c r e a t e s a b a r e c o l o u r l e s s destroys  a  quantities  great  quark  detail  Let's  bag  («-20)  bag o f t y p e A  o f t y p e B (We s h a l l  by a d d i n g a s u b s c r i p t , , ). hq  spin-isospin in  bare  baryon  ,  always  and  B^  and  indicate  B  e  bare  a r e t h e SU(6)  w a v e f u n c t i o n s o f type A and B which  aregiven  in ref.[64].  r e w r i t e V - (k) i n t h e f o l l o w i n g  form  Q  V„(1?>= I  H*. <  8  <f»B.  ,  (4-21)  50  Ml\ Since  ^  1  of SU(6),  S  •  ;  here  -7-  a r e members of t h e J56 r e p r e s e n -  tensors  I  =<RJ  n 6  6  and  w  are spherical  ii  =  "  '  T  v T  U  S  flB  and  o f rank 1  S^llfO ,  T"-<R.I i where s ^ and t  (4-22)  i t i s natural to rewrite the operators  as i r r e d u c i b l e  f i 6  R  the baryons concerned  tation T  (* > -  .  (4 23)  IB.>,  (  4- 4) 2  basis defined according to  ^  '  }  ^  £ l  '  ( 4  "  2 5 )  therefore  with  S  fl  jection  and s  fi  / u n .  being  respectively  the t o t a l  on t h e z- a x i s o f t h e b a r e  ly  for  can  apply  T  fi  the Wigner-Eckart  < = &  and t . S i n c e  fl  (4-26)  S^ and  s p i n and  baryonic  bag A , and s i m i l a r -  are irreducible  tensors,  we  t h e o r e m and g e t  ^r<^T,Jj\5TilS T ^>^S )( T oj" ' ,  B  6  /l  2  where t h e c T ' * " ^ a r e t h e u s u a l C l e b s c h - G o r d a n 1  eq.U-15)  i t ' s pro-  i n e q . ( 4 - 2 0 ) and e q . ( 4 - 2 7 ) g i v e s  fl+  1 ;  coefficients.  for f"  6  (4-28)  Using  51  Since  the s p i n - i s o s p i n  quark wavefunctions  tric,  we may as w e l l t a k e  third  q u a r k o n l y , and i f we c h o o s e )c=kz and e. =e  the matrix  i =3^<R,ib> r B  ;  The  coupling constants  plicitly In  f  iMB  are  symme-  e l e m e n t i n e q . ( 4 - 2 9 ) on t h e then  3  ,>.c; ;;: c^i. :  c a n be computed  0  totally  «-3.>  and we g i v e them ex-  in table I I I . summary, t h e C l o u d y  Bag M o d e l  baryonic  Hamiltonian  is  g i v e n by  H + H  H = H  -  D  I  />»  E  R!  R  R  ,  T  1  + D  (4-3D j J fe %  = I \j }k { V tf) a(h where t h e b a r e  masses  m  ofl  = =  \IV  (  t h e form  M normalized If  we  factor  (kR)  I  ^  l  i n e q . ( 2 - 4 3 ) and  F?I  B  s  ^(f) /w;.  b  .  '  ^  (?) B  k  >  (4-  Q  ,  (4-34)  6  ,  (4-35)  C  V  l  T  /  ^  (^."(F))*,  )  ,  (4-36) (4-37)  of the i n t e r a c t i o n , =  jjkR)  + -j (fef?) a  ,  t o u n i t y f o r k=0, i s shown i n f i g u r e now  (4-32)  y foxha.(h\  4  were g i v e n  R  C  =  w"(h and  I  a* <?)<*.("?) ,  J  consider  the nucleon  (4-38) 4.  s e c t o r of the theory,  we  Table I I I :  o  The  TTAB  bare c o u p l i n g  constants  r  AB, o Q* / I  *  *  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  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  Q  /  z  *  =  *  Fig. fit,  4 : The CBM f o r m f a c t o r v(kR)=exp(-0.l06k*R ). 2  |u(kR)|  and  a  Gaussian  54 have  ,  • < A/: I . This  i s the usual  nucleon can  except  be  then the f  D  form  form  with  form  quark  (4-40)  coupling  factor  to the  u(kR)  factor described  which  i n the i n -  s t r u c t u r e of the n u c l e o n  of the c o u p l i n g  f a c t o r . The s t r e n g t h  pion  o f t h e form  Chew's  The u n d e r l y i n g  o f 0.23 may  ,  o  (4-41 )  f o r the pseudoscalar  i n the strength  | |S/ >  3  (4-39)  O. 13  f o r the presence  identified  troduction.  ^  Z> ? r  {i.e.)  their  r a t i o and  o f t h e b a r e ITNN c o u p l i n g  constant  seem t o d i s a g r e e  with  constants,  shows  the experimental  one o f  (4-42) with if  g /4TT^14 J  center  12%  [ 6 5 ] . However, t h i s  o f mass c o r r e c t i o n s a r e i n c l u d e d ,  c a n be  which  that  20 Mev. w h i c h adds a n o t h e r o u r 7TNN c o u p l i n g  mental  one  when  constant  the  f  constant  f*  w  which  are  radius  R, by i t ' s p r e s e n c e  true  given  c  [ 6 7 ] . We  i s consistent  with  corrections just constants  are  then  in table  i n t h e form  left  mass  0  by of  conclude  the e x p e r i -  related  to  the  coefficients  III. Finally,  f a c t o r u(kR),  i n the Cloudy  f  mentioned are  v i a t h e SU(6) q u a r k model  i n our n o t a t i o n  f r e e parameter  e  necessary  made. N e x t , a l l t h e TTAB c o u p l i n g TTNN c o u p l i n g  2% t o f  explained  increase  [ 3 3 ] t o 20% [ 6 6 ] , and by g i v i n g t h e q u a r k s a s l i g h t  about  only  discrepancy  t h e bag  i s then the  Bag M o d e l .  55 d.  The e i g e n s t a t e s  The culate let's  study  of H  and H.  0  of the t o t a l  explicitly  Hamiltonian  are  Hamiltonian  H,  the  lowest  energy  and  we s h a l l  mention  discussion.  are present  where  |N,K.>4  nucleon. the  o f momentum With  of  only  of  first,  members the f u l l  t h e non s t r a n g e  f o r baryon  o °  _  43)  baryons  when n e c e s s a r y  which a t t = ;  to the  1,2  or  n  i s made o f a  i n d i c e s j and o f a p h y s i c a l  truncations described  relation  (4  s o l u t i o n s when  Ic and i n t e r n a l  i n s e c t i o n a, we have  number one  ^  "  J>  notation,  free Hamiltonian SU(6),  ... .  ones o n l y  the s c a t t e r i n g  }  i n a shorthand  plet and  the  completeness  The  consider  the strange  represent  "A or  |N>  There a r e a l s o s c a t t e r i n g  pions  pion  eigenstates  cal-  i.e.  convenience,  free  us t o  t h e p h y s i c a l p r o p e r t i e s o f b a r y o n s . But  H I N> =  (N,A)  allow  i n v e s t i g a t e t h e e i g e n s t a t e s o f H. The b a r y o n o c t e t  (N,A,I,-)  For  H will  H  i . e . both  the baryon decuplet  0  possesses the baryon  of s p i n  3/2  a l l members o c t e t of s p i n baryons.  For  of  the  56-  1/2 b a r y o n s strangeness  56 zero,  the  which  lowest  lying  s t a t e s a r e t h e bare  nucleon  and d e l t a  obey  (4-47)  H  c  I  A  e  >  ^  -  However, when t h e i n t e r a c t i o n delta  i s no l o n g e r  channel. bare  We w i l l  bag w i t h  I  A  > .  0  H  Hamiltonian  i s turned  z  s t a b l e and appear as a resonance  also  have e i g e n s t a t e s o f  H  on, t h e i n the P  consisting  D  of  a  n pions:  H. I w.,*,> = (  Mni  * I  u,. ) | . *> _ N  c-l  with  the completeness  relation  i = I { iN/o^X^^I OO  Notice  that the state  the p i o n c r e a t i o n  Model  first, bare  and t h i s  vertices  e  f o r example  a^(k)  to apply  (4-49)  6 ;  i sobtained  on t h e b a r e  state  p e r t u r b a t i o n theory  by a p p l y i n g  |N > e  t o t h e Cloudy  i s t h e o b j e c t of t h e f o l l o w i n g c h a p t e r s but  we d e s c r i b e a v e r y  i n Quantum  1-  e  operator  We a r e now r e a d y Bag  |N ,k>  + U ^><A /n|1  convenient  and p r o p a g a t o r s  very  graphic  similar  notation  t o t h e Feynman  Electrodynamics.  Incoming and o u t g o i n g  bare  baryon  A : |A > a n d <A J o  =  lfl„>  -  <n.l  f o rthe  0  graphs  I n c o m i n g and  outgoing  TTAB a b s o r p t i o n  and  Bare A p r o p a g a t o r  pion:  | k- > and  emission  : G (E 0  vertex  <k^ |  : v " ( i k ) and B  >  ) :  (e Pion  propagator  h  (already r  included  in H ) :  :  W',8(K>)  58 V) RENORMALIZATION OF THE CLOUDY  a. E x p a n s i o n  In bags,  of the p h y s i c a l  the Cloudy  i . e . that  Hamiltonian  which  nucleon  nucleon  bag,  etc.  basis  of the bare  of the  couples  :  the pion  nucleon  or d e l t a  e i g e n s t a t e s of H  0  be  a bare  energy  i s the p r o b a b i l i t y  w  (no p i o n p r e s e n t )  the  of  the  bag w i t h  interaction bags, t h e  time  one p i o n  the p h y s i c a l  a  on  A 1 Ni> ,  +  |N >  nucleon  and E ^ i s  0  ( 5  the  '  1 )  |N> t o nucleon  obeying  E =m N  projects  for a free  N  physical  (5-2)  nucleon.  A i s an o p e r a t o r  o u t a l l t h e components o f |N> w i t h a t l e a s t  A  =  A* = Using  the  according to  H |N>= Ej»V> , with  bare  " i n the  nucleon  f o r the physical  nucleon  are "dressed"  t o the bare  be p a r t  T h e r e f o r e we c a n expand  Z^(E )  of  field  | N > •= Z[(tj\ N O where  baryons  presence  f o r example w i l l  a bare  air",  nucleon  Bag M o d e l , t h e p h y s i c a l  in virtue  physical  BAG MODEL :  i  ~  I  |R >< R J ,  which  one p i o n :  (5-3)  e  A •  (5-4)  the i d e n t i t y  AlfN/>=r  ( EK  H )~' A ( E o  N  H ) 0  IN )  i  (5-5)  59 together  I and  with  e q . ( 5 - 1 , 2 ) and  G (E)  i s the  e  usual  gives  o  I/O  z^ej  > =  N  [A,H ]=0  bare  + £ ( A  1  >,  N  (5-6)  propagator  (5-7)  Expanding nucleon  the  integral  equation  eq.(5-6) f o r  the  physical  gives  8) w h i c h can  be  condensed  i n the  form  (5-9) Using  our  relates  graphic  |N>  to  notation defined  |N >  > = iteJ \.  1w  -  + • 4  —  4  —  * . — II  + •  then  eq.(5-9)  — »  J  S~  J>  •  >  ' i l l  \  fl  « 4- .  /  R  (5-10)  - + . . . {  + • •  with  > ,, +  ^  N  A  IV,  via  C  +  in chapter  +  A  (5-11 )  • • • J,  \  A  «  «  a  j  \  (5-13)  + where t h e  unspecified intermediate  baryon  lines  can  be  either  60 nucleon  b. The  or d e l t a  dressed delta  In  the  Hamiltonian the  lines.  .  previous H  physical  has  0  chapter,  two  delta  no-pion  mass  the  dressed  Hamiltonian neously  |A>  15)  can  delta  The the tion a  =  principal  A  f l  ?{  value  the  particle  case  i s not  model of an  comes from  be be  an  If  e i g e n s t a t e of t h e  allowed nucleon.  eq.(5-9),  have m >  7*(E  bare  (5-14)  we  to So,  would  decay  an  ni^+m^ and we  can  in a  which  in eq.(5-l6)  and  [69,70]  t h a t the d r e s s e d  t h e r e f o r e must  be  eq.(5dressed  satisfies  c  requirement  the  (5-15)  1 + P U ^ - H r A H ^ - ' r t j l A >.  unstable V-particle  state  >.  define  e i g e n s t a t e of H but  prescription  to  have  the p r o p a g a t o r  still  full  sponta-  similarly  a  really  the  ,  ^ A , " * "  a physical  v a n i s h . In t h i s  single  and  e  Z*(E f { 1 + ( E - H ^ A H / i ' H , ] I A  s t a t e which  Lee  e i g e n s t a t e s : |N >  would  expansion  -  we  |A>  that  H s i n c e i t would not  nucleon  However,  ^  delta  i n t o a p i o n and  physical  mentioned  satisfied  ^c,  then  we  (5-16)  originates and  it'.s motiva-  delta  time  from  should  reversal  be in-  variant .  What  is  the  mass  m  A  of  the  dressed  d e l t a ? At  first,  one  61 would  t h i n k t h a t m^  channel. delta  However,  bag d o e s n ' t  channel.  should the  formation  account  In f a c t ,  be t h e r e s o n a n c e  in  and t h e d e c a y  f o r the f u l l  c r o s s e d graphs  energy  the  of the d r e s s e d  c r o s s - s e c t i o n i n the  such  P  P  J3  as  (5-17) also  c o n t r i b u t e t o the P  bution and  will  be shown  therefore  resonance  we  i n the next  shall  assume  | a* a  ^  u  will  never  appear  i n our e q u a t i o n s  since  at  cance  the d r e s s e d  t= + °° ,  delta  propagators.  dressed  delta  bag e x p a n s i o n  mediate  s t a t e s . We  in  chapter,  c . The n u c l e o n  In  we  mass  t o be t h e  For  a s an delta  incoming  this  eq.(5-l6)  delta  or  out-  must have d e c a y e d .  have i t ' s p h y s i c a l reason  we  signifi-  shall  use t h e  only  for describing  inter-  d i s c u s s the d e l t a  i n more d e t a i l  later  one on p i o n n u c l e o n  scattering.  self-energy.  of the i n t e r a c t i o n  and d e l t a  i s that the dressed  the  bag w i l l  and i n t h e next  and d e l t a  absence  nucleon  shall  (5-18)  we c o n t i n u e  inside  this  delta  small  t  note  state,  before  AA eV.  final  However,  contri-  t o be r e l a t i v e l y  the dressed  One  going  the  chapter  However, t h i s  mass: />v)  bag  cross section.  would be s i m p l y  t u r n on t h e c o u p l i n g , t h e p h y s i c a l  Hamiltonian m  otJ  H , T  and m .  nucleon  oA  mass  t h e mass o f  However, when becomes  m  N  62 and  the dressed delta bag mass moves to m . The difference betA  ween the physical and bare masses is the  self-energy. For  the  physical nucleon, we have the self-energy 2. (m) given by N  £  N  ("»  w  )  =  ^ - ^ )  o  W  ,  (5-19)  IH - H j l N / > ,  =  (5-20)  and from the nucleon expansion eq.(5-9) we have I*  = < K > I H, ( E - H - A H / ) " ' H - J f ^ M  (5-2D  e  and similarly for the dressed delta bag, = < ol A  P(E-H-AH.A)"'^ |A > l 0  Graphically, the first few terms of *L  (5-22)  for example are given  by  £"('»O = S  -  LCJI  +  +  +  ..  (5-23)  and we use the notation  I. <0 fl  =  HH  ,  (5-24)  where the square bubble contains all the one-particle irreducible graphs, i.e. the graphs which do not separate in two disjoint  graphs when cutting an intermediate baryon line anywhere  inside the graph. Similarly, the dressed delta is given graphically by  bag self-energy  63  We delta of  shall  give  explicit  self-energies correct  this  then  eigenstates w i l l probability  in H  4  in  section  i  B  i < & =  nucleon  wavefunction  e q . ( 5 - 9 ) of t h e p h y s i c a l  nucleon  to unity,  on t h e b a r e  g i v e the formal e x p r e s s i o n f o r the bare  nucleon  z"(E ): w  [  1  +  l/v/>] '  <K\H (£ H -r)H f\f H  Z "(e„)= [ I " £ 4  and  order  Z (E ).  we n o r m a l i z e t h e p h y s i c a l  the expansion  z  t o second  f o r t h e n u c l e o n and  chapter.  d. Bare bag p r o b a b i l i t y , If  expressions  i  J  Ar  a  I  J  0  <^.|H (E-H -/\^"H l^>] I  0  1  we r e c o g n i z e i n t h e b r a c k e t t h e e x p r e s s i o n f o r  ( -26) 5  <" > 5  >  the  27  nucleon  s e l f - e n e r g y . So,  (5-28) and  more  generally,  (5-29) Again, sions  we  shall  for Z  (E ). f l  wait  to section  i t o g i v e second  order  expres-  64  e. The v e r t e x f u n c t i o n  z" (E ,E ). 6  n  B  In c h a p t e r I V , we d e f i n e d the bare v e r t e x  function  v"* (k)  according to eq.(4-21)  = < R J V .(l?> |B >  irJfih  However,  we s h a l l encounter the  between  s t a t e s . We  the  shall  therefore  shown [15]  and  BQ,B  t h a t v " 8 {t) ^  (5-30)  such as p i o n nucleon  matrix  elements  define  a  so-called  of  V . (k)  have  renormalized  to  = < R I V.^(f) |B>  «r**tf)  A Q ., A  .  d r e s s e d s t a t e s r a t h e r than between the bare  vertex function v„^(k) according  Since  e  i n p h y s i c a l p r o c e s s e s of i n t e r e s t  scattering, taken  0  .  (5-3D  the same quatum numbers,  i t can be  i s p r o p o r t i o n a l t o v"^(k*) =  ^  ( E „ , E ) AT ^ ( k)  ,  a  8  (5-32)  RB  where the p r o p o r t i o n a l i t y c o n s t a n t the  magnetic quantum numbers s z  s i d e r the ding  the  NNTT  °) ( E , E ) i s independent fl  and t 2 .  6  For example,  renormalized vertex function v ^ ( i t ) .  physical  nucleon  states  let's  When  of con-  expan-  | N > and | N ' > a c c o r d i n g t o  e q . ( 5 - 9 ) , we get *  1  (5-33)  7  • V..(S>(E,-W.-AM)"'^ IW >j . 0  But a c c o r d i n g t o e q . ( 5 - 3 2 ) , the l a s t b r a c k e t  is proportional  to  65  «  I H ( E r H - A H / ) V . ( 1 ) ( £ - H - A H ^ ) " H,! W*> = AT^(1> X " ( \ , , e , ; > ( 5-34) 2  and  w  w  i f we d e f i n e t h e v e r t e x  ^(•WU  f u n c t i o n Z, ( E ^ E ^ ) a c c o r d i n g t o  - [1* X '( ,,e„)]",' w  £  WW  then  ^ (E .,E ) i s r e l a t e d W  t o Z * ( E ) and Z.*V*w* ( E ^ , E )  W  W  W  via  I ,  _  / ^.'.tj In in  general, analogy  ^ (S)  i f we c o n s i d e r  Rj  again  phic using  ft  x  (5-36)  A a n d B t o be d r e s s e d  £- H-  fl  . < *A) = E  the p r i n c i p a l  [ 1  value  t  b a g s , we have  /\H/\)\\ (5-37) ;  * x'Vvl"!  of the i n t e g r a l s  whenever t h e p r o p a g a t o r s  vanish. F i n a l l y ,  r e p r e s e n t a t i o n of the r e n o r m a l i z e d  vertex  (5-38)  involved we g i v e function  is  imp-  the grav (ic) R 6  e q . ( 5 - 3 2 ) and e q . ( 5 - 3 9 )  /ir and  -  t o eq.(5-34) t o eq.(5~36),  z  lied  r  x"< e e ) = < H. I H ( £ -H.- A H.A)' V  B  and  w  I  from-  bubbles  (k) eq.(5-37)  i s g i v e n by  7 j C £ f l )  and  Z  ' »> ( g  .  eq.(5-38),  a  /  the vertex  ,  (5-40) with  all  its  66  f . The renormalized c o u p l i n g c o n s t a n t s . The  bare NNTT vertex f u n c t i o n was given i n e q . ( 4 ~ 4 l ) (I?) = i  /V  xx(t?R)  • J » _  . < y j  to be  f. | / ^ _ K  If we now w r i t e the renormalized vertex f u n c t i o n  o  (5-42)  >  v^ (k) a c c o r -  ding t o i  A T ^ ( k ) ^ c ^  then  f^  .<N'.l?-f  ^IW.>,  i s the renormalized NNTT c o u p l i n g constant and i t i s  r e l a t e d t o the bare c o u p l i n g constant  i,  (  „ , , o  £  f„ a c c o r d i n g t o  *:•».;>* W . 4. .  =  -  (5  F o l l o w i n g Schweber [15], we now show that f^ i s r e l a t e d experimental  proof  44)  t o the  one v i a  -£«f=  The  (5-43)  ^  ^  5  c  ^  ^  -  ^  i  r  r  ^  (n,,^) .  (5-45)  f o l l o w s from w r i t i n g the pion f i e l d p o t e n t i a l  i n the  p h y s i c a l nucleon. T h i s i s d e f i n e d by <TT.(5h>= and  \  I  J  ^  ~  U ( h J  ;  '  i  ^ a \ c h e  i  U  \ \ h l >  )  (5-46)  since  I [ H,a- lo] I N >  < M  t  < N / | w „ a*(k) + Voi(l?)  =  IW>=  O,  ( -47) 5  then < T T ^ >  and  =  using e q . ( 5 - 4 3 )  <  N  |  V  >  e  W  +  V..<f>«r-'W|N>  (5-48)  f o l l o w e d with a few simple m a n i p u l a t i o n s , 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 t h e bag r a d i u s , o n l y  k will  c o n t r i b u t e and u ( k R )  finally,  we  So,  for  large  distances,  whose s t r e n g t h  constant  t (m ,m ). We y  constant  We  with  conclude  coupling  this  constant  we  ,<is/J  by  the  one.  therefore identify  Z  analogy  >  (  .?3  Z  So  ^  ^  1/sQ  .  i s a Yukawa  renormalized  coupling  the renormalized  cou-  one.  s e c t i o n by d e f i n i n g t h e  renormalized  TTAB  to eq.(5-44):  (  C  »  }  of the r e n o r m a l i z e d  • {. ,  vertex  (5-51)  function in eq.(5-  reads  ^ 0 0 = 'ivJiaJ^) and  i s given  in direct  the expression  32) now  by  the pion p o t e n t i a l  the experimental  i» and  approximated  U">».~>^  potential  pling  be  have  <TT<*)>=  JU  can  t h e s m a l l v a l u e s of  represent  AT  the  r  pling  constant.  s  this graphically  . ("£)  =  below t h e v e r t e x Finally,  bubbles as d e s c r i b e d  T  J  (5-52)  (5-53)  %  G  refering  the the dressed  i n eq.(5-4l)  i  by •  ^  FI ^  "1  with  C \A^t)C*V \?:-l )  *(HR>  t o the renormalized vertex  with  a l l  couit's  c a n be r e w r i t t e n as e q . ( 5 - 5 4 )  68  1  12 Z>*,e )  (5-54)  a  g. The r e n o r m a l i z e d  When e x p a n d i n g states, tor  we s h a l l  G ( E ) between  apparent malized  propagator.  t h e t - m a t r i x on t h e b a s i s  encounter  of the f u l l  baryons.  For reasons  i n t h e next  section,  we d e f i n e l i k e  propagator  G*(e.) A/  elements  bare  nucleon  w i t h E =m  matrix  for on-shell  function  =  which  will  Chew  eigen-  propagabe  more  [11] a renor-  G^(E) a c c o r d i n g t o  CrU > I N  z " ( e )"' < A / J  nucleons,  of the bare  and G(E)  is  0  >}  the  (5-55),  usual  full  propagator  = (e-H)". Now, e x p a n d i n g  ( S  0  o  realizing  contribute  S 6 )  G(E) i n terms of G ( E ) a c c o r d i n g t o  6 (e) + 6-(f)H6„(e) + 6 (£) H 6;(e)H 6 (e) + ... , and  .  that  t o G*(E),  I  only  0  I  J  t e r m s w i t h an even  and u s i n g t h e f o r m a l  (5-57)  (l  number  o f H.J w i l l  expression  eq.(5-21)  for 1 ( E ) gives  z / W ' < ^ J 6.U-sce>) |/v/> , D  =  ^ E r ^ ^ K e ^ - K ^ ' l N / . ) .  (5  _  58)  (5-59)  69  Now,  f o r Earni^,  i . e . the free  = l^ey' and  we c a n expand  <N and  a  then &  0  eq.(5-58) reduces t o  (E-^-^A/.II(e)k,>)'  <N„|I(E)|N„>  I lie) |iv>  nucleon,  i n Taylor  serie  * (e-^).^ r  s  (5-60)  v  (e)  +•••  ,  (5-61)  eq.(5-60) reads >  >  z : w [i-^tW<e-*i>''+A.<».,  =  (  5  _  6  2  )  A/ A/  and  since  the bracket  6 So  the  physical to  W V  U)  renormalized  i s simply  Z ( m ) , we a r e l e f t 3  ( £-^i'  =  propagator  nucleon propagator  w  +  (5-63)  ' Ji.o. .  function  <N|G(E)|N>  with  G^(E) r e d u c e s t o t h e  i n the l i m i t  of  E  close  m. w  To expand  obtain  a  graphical  representation  w h i c h c a n be w r i t t e n  (^Uje)^ 0 (e)l(e)^e)+- I A/.>  (6)  3=  A/  A/  A/  A/  (5-64)  ..... I(5-65)  .  Finally, e  ,  g r a p h i c a l l y as  A/  |N >  first  eq.(5-58) t o g e t  G"(e)= Z > ) ' '  G.  o f G,,(E), we  (5-66)  we c a n r e p l a c e  everywhere the bare  e v e r y w h e r e by t h e b a r e s t a t e  |A > where A 0  nucleon  state  i s any b a r e bag  70  state G  (including  the d e l t a ) t o get the renormalized  A  propagator  (E) d e f i n e d as  &?(e) = z 4 V ) ' ' < n U ( a | R > o  and  similarly  to  6  (5-67)  ;  eq.(5-59)  G?(e>= Z*(e)~' < f l . | ( e - H l ( f ) ) ' | f l . > , A  which  i n the l i m i t  of E c l o s e t o m  reduces  fl  ( 5 - 6 8 )  r  to  (5-69) and  the graphic  r e p r e s e n t a t i o n o f G*(E) i s  G\E)=  .  h.  The r e n o r m a l i z a t i o n  The as  next  pion  step  nucleon  perturbation the  L  . = « ~'-7  «(£>  Y  renormalized  >  K  '  fl  •  •  (  5  _  7  0  )  ft  procedure.  i s t o show t h a t scattering  expansion  ^  (0  for physical  such  f o r example, we c a n a l w a y s make a  of the t-matrix  vertex  processes  function  which w i l l  v  involve  only  (k) and t h e r e n o r m a l i z e d \  v  propagator  G .(E). )  A typical w h i c h we w i l l  t(N'  t\  Nfa  example o f i n t e r e s t show i n c h a p t e r  =  <N  I V\k)  i s the pion  nucleon  t-matrix  VI t o be g i v e n by  6(E) V  (IO  |W>+  ••• .  (5-71)  71  If  we expand  sion  the p h y s i c a l  e q . ( 5 - 9 ) , and  relation  insert  between e a c h  OO  ,  f*\-o  \  |M m></v/ ^| e  0  completeness  lA /h><A.,*il  +  ,  D >  (5-72)  J  t h e t - m a t r i x element  graphical  operator the  eq.(4-49) f o r t h e bare e i g e n s t a t e s  1=1 then  n u c l e o n s a c c o r d i n g t o t h e u s u a l expan-  i n eq.(5-71) w i l l  take  the  form  following  ^  (5-73) A"  A>  L e t ' s w r i t e a g e n e r a l term  of the t-matrix expansion as  ' where  the v e r t i c a l  pion l i n e s  '  c a n be e i t h e r  p i o n s . However, a c c o r d i n g t o t h e d e f i n i t i o n (eq.(5-54)) and t h e bubble  propagator  = and  therefore  a  <  1  and all  the r e s u l t i n g  graph  the bare v e r t i c e s  renormalized nucleon  ones.  j  t  of t h e bubble  -  .  vertex  ( 5  i  //  ! . »  the f i r s t  "  7 5 )  with  <  i  (5-76)  A/ bare graph  and bare p r o p a g a t o r s a r e  replaced  two terms  of  where by t h e  the pion  read  \*'  \T  >  — - — *  A7  M  emitted or absorbed  i s t h e n an i r r e d u c i b l e  t-matrix i n eq.(5-7l) w i l l  i  a  a n d we a r e l e f t  - i >i  F o r example  •2 (e ) (5-74)  ( e q . ( 5 - 7 0 ) ) we have  • - i  a l l the Z 's cancel  '  A  A/  »  I?  .,  -+  ,  (5-77)  72  where t h e nucleon  on  In us  intermediate delta  the  the  irreducible  pagators  and  We order bag  bare  vertices  Second o r d e r  elements  in  this  f o r the  be  those  for other  equations  First energy  of  z"(E ), the  given  i n the  be  consider  in eq.(5-2l)  and  n  function Z  R B (  pro-  (E ,E ) n  B  second  the  bare  and  the  ( E , E ) . A l l these q u a n t i t i e s R  6  f o r the  nucleon from  formal  since a l l  the  expression  )  z  value  I (E)  -  o  the odd  1  A  L e t ' s a l s o d e f i n e the  an  in  general  must be  taken  o  propagator  number of  H  T  1  I  whenever  operator  H A6 (E)AH I  f o r the  self-  eq.(5-22)  vanishes.  terms w i t h  n  only  the  principal  since  to  self-energy X ( E ) ,  derived easily  where t h e  expanding  involve renormalized  H ( E - H.- f\H l\) U  fl  by  processes  text.  X"(E } - <R.|  Then,  f^  explicitly  b a r y o n s can  let's  physical  allow  renormalization functions.  the  vertex  n  calculated  given  either  described  section correct expressions  coupling constants  will  be  only.  in perturbation theory  renormalized  of  graphs which  expressions  present  probability  can  r e n o r m a l i z a t i o n procedure  t-matrix  t e r m s of  i.  propagators  ones.  summary,  to express  renormalized  1  1 (E) 0  R.> the  ,  (5-78) propagator  according  .  to (5-79)  f u n c t i o n i n eq.(5-78) will  not  contribute  and to  73  I (E  fi  ) , we g e t  lV )  = < f l j H (E,- H„- I e (E^V |R >  f l  3  To o b t a i n an e x p l i c i t be an i n t e g r a l  equation,  H where  H  D  C  expression  we s h a l l  + £ (E )  i s the bare  .  (5-80)  f o r I (E ) which w i l l  not  fl  f u r t h e r assume t h a t  ^  Hamiltonian  Ho , with  (5-81 ) the bare  masses  replaced  by t h e p h y s i c a l ones fj All  this  =• 1  /»>^ f{\ R  g i v e s us a s e c o n d o r d e r  T*(E ) A  =  I Replacing  = H I  a (^)a.(1) t  k  >  expression  .  (5-82)  f o r I ( E ) as A  f l  0  explicitly  Jk l  (5-83)  &  u s i n g eq.(4-33) f o r H : 2  /v^Cl?)  v . (k) by i t ' s e x p l i c i t o  identity  w  <R I H, (^-HJ-'H, |R >,  w h i c h c a n be e v a l u a t e d  the  + %  e  (E^-^-u,^"  expression  e q . ( 4 - 3 7 ) f o r w ^ (k) we a r e t h e n 0<  /w.  (*>.  (5-84)  e q . ( 4 - 3 6 ) and u s i n g left  with X  (5-85)  But  •  l^Ac^JuriHiJ^  y  -  t  rnr/i  ,  .  (s  86)  74  therefore,  and  t h e b a r e mass o f t h e b a r y o n  A i s then g i v e n  approximatively  by  (5-89)  For  the physical  nucleon  ru,,, = - — 1 (—) (  f o r example,  we have  (**  Jk  V ( k R )  (5-90,  where we d e f i n e d  Having  now  we c a n w r i t e down bare baryon  a second  order expression f o r the self-energy,  immediately t h e s e c o n d - o r d e r  A probability  expression f o r the  Z (E„) as A  (5-92) and  w i t h eq.(5-88)  i ti s given e x p l i c i t l y  by  -1  2_ 8  For bare  example,  \ ">* /  the p r o b a b i l i t y  n u c l e o n bag Z^(E ) N  1 +  U7T  U7J  (5-93)  A  f o r the physical  nucleon'  to  be  a  w i l l be  (5-94)  75  Next, cording  we c o n s i d e r  the vertex  function  t o eq.(5-37) and eq.(5~38)  Aft  -I  fit  A B _»  ,  Z  (  ( E , E ) which a c f t  6  i s g i v e n by  /  ,  -i  _  (5-95)  • V .(l?)(VH.-A« A) H IB > ,  e  or  6  ;I  I  g r a p h i c a l l y by e q . ( 5 - 4 l ) . AB  .1  2, And  AB _»  ^ ( similarly  f  > =  e  '  +  to the self-energy  in  ^  + ...  eq.(5-78),  we  (5-96)  approximate  e q . ( 5 - 9 5 ) by  = ' S < * ' <'tlH (e fi )" V(f)(E -fi )-VlB.> +  ,  ir  1  w h i c h c a n be e v a l u a t e d /  - 7 AB  \~'  (2, (e ,E )) A T A  1  (5-97)  l  explicitly  = AT (k) +  s  +  t  A & -9  _  (  >  A-  ,  „  ,  (  5  "  9  8  in  Again,  replace  the  v  0  's  by  their  formal  expression of  e q . ( 4 - 3 6 ) and y o u g e t  ^ 7_A*  (>»> (E  E  >  )  =  AT .($) + ^  1 L 1° ^  4* /«  ft^R^  .£ . V (5-99)  76  and t h e i n t e g r a l i n t h e s p i n t e r m e q . ( 5 - 1 0 0 )  can  be  done  and  gives  £  C \  . iOLf^-k),  ( 5  _  1 0 2 )  w h i c h h a s t h e same s t r u c t u r e a s t h e i s o s p i n t e r m e q , ( 5 - 1 0 1 ) . Now use t h e 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 [ 5 1 ] (5-103)  where t h e {...}  are the usual  I^^Jf,  6 - j symbols and d e f i n e  K=  / "=  1.1  (5-104)  u  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 (5-106)  3 = and  r .  c  v  u a j . ^ j j,  <B-IO7>  a l t o g e t h e r , t h e s e e q u a t i o n s g i v e us t h e f o l l o w i n g f i n a l e x -  pression  for Z  Aft (  (E„,E ) 6  c o r r e c t t o second o r d e r :  Z>„^=  ^  1  •  "V  [1*1  / V 4. ;  ^ ( E ^ ) ] "  1  ,  _  U(T < -,T.-,T f ,T > )-!- l U  (5-108)  (5-109)  fVifH  77  For  example t h e v e r t e x  f u n c t i o n Z,  (E .,E„)  (iT). .!_._!_ U  t  4  take  t h e form  NN  I  ^..,0= | 1 +  will  f^V>  +  I  J  3t  ^T7  ^V(fR)  ( dp  • /  (5-110)  Having lized can  now f o r m a l e x p r e s s i o n s  coupling  constant  be c a l c u l a t e d  7  A  a  ^  Z  f  lowing pling in  (  U  £  A  &  I  E  s i n c e by d e f i n i t i o n  {.  )  6  dependence  .  (5-111)  >  R e s u l t s of the c a l c u l a t i o n s . The  and  j  explicitly  renorma-  (  &  order  7,  j.  Z , the  f ^ ( E , E ) and i t ' s e n e r g y  t o second  U A A ) =  i  f o r 1^ and  second  order  expressions  (E ,E ) obtained A  f c  parameters: constant  t h e form  f  c  factor  for I(E„),  i n the previous  the  dressed  section  Z (E ),Z i  f l  (E ,E ) f l  &  involve the f o l -  bag mass m , t h e b a r e TTAB C O U fl  and t h e bag r a d i u s R, w h i c h a r i s e s u(kR).  l  explicitly  78  The if  dressed  bag mass m  A i s a s t a b l e baryon  mass  i f A i s unstable  constants III  (such  (such  t o be t h e  the  bare  f"  t o the quark  w  as t h e d e l t a ) .  coupling  frequency  physical  mass  a s t h e n u c l e o n ) and t h e r e s o n a n c e The b a r e  a r e 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  to  lated  i s chosen  fl  constant.  and t h e  pion  f£  given  Notice  that  decay  coupling  B  in  table  f^ " i s r e 1  constant  via  eq.(4-18)  \ However, tions, masses  as  pion  =  0  field  of  malized traint is  how  constant  (5-112)  IV, c e n t e r  n o n - z e r o up shall  o f mass c o r r e c and  therefore  such t h a t  down  leave  v  w  eq.(5-1l3). close  second-order  Z"" a s a  ^  - l T =  S-O  ,  (5-113)  t h e r a d i u s R of t h e bag i s l e f t  so t h a t  f  The i m p o r t a n t  f^"" i s t o f " . We of the v e r t e x  as a  the dependence of the p r e d i c t i o n s  constant  w  the  satisfied  studied.  5 shows t h e r a d i u s d e p e n d e n c e o f t h e b a r e  ITNN c o u p l i n g  quark  eq.(5-45) l i n k i n g  f " ( m „ , m ) t o t"" be  Bag Model on i t c a n be  renormalization our  choose  Finally,  f r e e parameter  Figure  . We  =  f ^ 0.081.  the Cloudy  .  i n chapter  modify  coupling  f ( ^ ™ » )  totally  *  D  renormalization,  w h i c h we s h a l l  renormalized  where  {  we m e n t i o n e d  etc. will  parameter  3,  w  w  /f^  w  result  to  which s a t i s f i e s revealed  by  function  i s small  c a l c u l a t i o n s very p l a u s i b l e .  and  the cons-  this  can t h e r e f o r e c o n c l u d e  renor-  graph  t h a t the  that  makes  _|  O.S  !  0.6  ,  ,  0.7  !  O.B  0.9  ,  1.0  !  1.1  ,  1.2  R  F i g . 5: R a t i o o f t h e b a r e t o t h e r e n o r m a l i z e d irNN c o u pling c o n s t a n t f ^ / f ^ a s a f u n c t i o n o f t h e n u c l e o n bag radius . N  N  r  fm.  80  In  is  shown.  this  in  f i g u r e 6,  Notice  allows  our  late  the  us  e n e r g y d e p e n d e n c e of  that  i n good a p p r o x i m a t i o n  c a l c u l a t i o n of  the  ratio  t h i s dependence  the  CBM  £ (e) d e f i n e d v  to  f^E)  defined  is relatively  as  weak  and  write  p r e d i c t i o n s . In  f i g u r e 7 we  calcu-  as (5-116)  to v e r i f y close that  i f the  to the f o r a bag  close  t o one  renormalized  TTNN  b a r e ones w h i c h a r e radius and  we  of  0.8  shall  fermi  and  TTNA  given  coupling  in table  o r more, t h i s  therefore  use  ratio  I I I . We ratio  i n our  is  remark  is  very  further calcula-  - ( £ ) I ' W o - (C) • -" i03  Up i.e.  t o h e r e , we  that  baryon  not  a l w a y s assumed t h e  t o o many p i o n s  probability  R  is  reasonable.  b a r y o n s . From t h i s is  expected  In  the  for smaller  ters  that  CBM  CBM  g r a p h , we  conclude  around".  is consistent  in  that  the  n  see  0.8  i n the  experiment  Bag  fermi  bare i f the eq.(3-  Z*(m ) f o r t h e  shall  with  The  give  Cloudy  7)  "small",  i n d i c a t o r to v e r i f y  f o r l a r g e bags of  o n e s . L u c k i l y , we  t o be  Lagrangian density  f i g u r e 8 we  t o work b e t t e r  than  the  of  field  "floating  Z^(m ) i s a good  one-pion approximation 55)  are  pion  (5  or  stable Model more,  next  chap-  f o r R>0.8  fermi.  F i g . 6: E n e r g y d e p e n d e n c e o f t h e TTNN 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 f£ as d e f i n e d i n eq.(5-114) for bag r a d i i i n t h e r a n g e o f 0.5 t o 1.0 f e r m i .  82  Fig.  7:  coupling  Energy  dependence  constants  defined  of  in  the  ratio  of  eq.(5-1l6).  trNA  and  irNN  m o'  ol 0.5  1  1  1  1  1  0.6  0.7  0.6  0.9  1.0  1 1.1  1 1.2  R  F i g . 8: B a r e b a r y o n bag p r o b a b i l i t y Z^dn^) v s t h e bag radius R f o r a l l members o f t h e b a r y o n o c t e t o b t a i n e d via eq.(5-93) but w i t h the r e n o r m a l i z e d c o u p l i n g constants d e f i n e d i n eq.(5-117).  fm.  84  In  figure  physical large  9, we g i v e  mass r a t i o  f o r the s t a b l e baryons.  pionic effects  order  perturbation The  ference radii  t h e r a d i u s dependence o f t h e b a r e t o  f o r small  theory  Again,  we  bags w h i c h t h e r e f o r e make  q u a r k mass m  between  t h e E and t h e I b a r e m a s s e s . A s s u m i n g  eq.(2-43)  with  second  i n a d e q u a t e when a p p l i e d t o s m a l l  strange  t o be e q u a l ,  observe  c a n be e x t r a c t e d  $  <* »1 .5  [ 4 6 ] , then  s  from  the  bags.  the  dif-  t h e i r bag  mass  formula  gives ^ - " " . i  =  - O.O^/v^  (  (5-118)  R which  can  function equal  be  solved  for m .  of R i s given  to  one  fermi  in figure will  p e r o n m a g n e t i c moments In be  summary,  derived  radius  larger  o f 144 MeV  i n the c a l c u l a t i o n  s  at  in R  o f t h e hy-  i n chapter V I I .  second  o f t h e Cloudy. Bag M o d e l c a n order  seems a d e q u a t e when  than a t l e a s t  model works b e t t e r w i t h i s much weaker. We  chapter  be u s e d  c o n s i s t e n t l y and  t o t h e model  malization  10. The v a l u e  the r e n o r m a l i z a t i o n  applied  field  A g r a p h of t h e s o l u t i o n f o r m  s  of the Cloudy  0.7  strange  perturbation  theory  t r e a t i n g baryon  bags of  fermi. Notice  baryons  since  finally  that the  f o r those  the pion  s t o p here our d i s c u s s i o n of the Bag M o d e l and c o n s i d e r  t h e CBM p r e d i c t i o n f o r p i o n  nucleon  renor-  i n the f o l l o w i n g  scattering.  tn  F i g . 9: R a t i o of t h e b a r e t o r e n o r m a l i z e d b a r y o n m ./m . f o r a l l members o f t h e b a r y o n o c t e t .  mass  F i g . 10: The strange quark mass r e l a t i o n e q . ( 5 - 1 1 8 ) .  mass m  s  w h i c h obeys t h e  87  VI ) THE P  RESONANCE :  3 3  a. The o n e - p i o n  In state (for  eigenstate  the previous  of the f u l l X =0).  eigenstate  We  chapter,  Hamiltonian shall  now  d e n o t e d a s |N,  free  pion  cleon  |N>. We t h e r e f o r e  o f momentum  we s t u d i e d  the lowest  H: t h e f r e e p h y s i c a l consider  t h e one p i o n  lying  eigen-  nucleon + one  |N>  nucleon  > w h i c h , a t t i m e t = + «> c o n s i s t s  of  t  a  k* and i s o s p i n j , and a f r e e p h y s i c a l nuwrite  I N,^>  H with  :  =  ±  E  ( 6  '  1 )  the on-shell condition  E = ^  Following posed  Wick  +  .  (6-2)  [ 1 4 ] , t h e boundary c o n d i t i o n a t t=+°°is  im-  by w r i t i n g  (6-3) |"X>  where  +  (t = -<»).  ( |"X>.  )  has  As i n r e g u l a r  only  outgoing  (ingoing)  s c a t t e r i n g theory,  this  waves a t t= °°  amounts  to  chan-  +  ging  E by E ± i e = E ~ To  obtain  i n the propagators  an e x p l i c i t (E--  and t a k i n g  expression H  ,  l K / J  f o r |"X>+  7  >  =  the l i m i t  , we f i r s t 0  ,  fe^O. write (6-4)  88  .(E*-H  1N>  ( E - H ^ 1-X>  =  1  +  ±  o .  (6-5)  But:  H and  aUh  according  I N> - [ H  |N>  H lh/>  + aVh)  to eq.(5-47),  L H , ^ ( k ) ] = % a t ( k > + V (?) . Using  eq.(6-6,7)  i n eq.(6-5)  - V {h IIM> 0)  and  the formal  the usual  4  eq.(6~9)  ( E - H ) I7>  0 ,  (6-8)  f o r |N,k:>, e q . ( 6 - 3 ) , h a s now t h e form  -  of the f u l l  (Eti6-H)  _  ,  =  (  6  "  9  )  propagator  i ^ ' - H ) ' ' ,  (6-10)  c a n be r e w r i t t e n a s  Equation  (6-11)  |N,t- > t o t h e n o - p i o n  cal  =  4  lA/,^X =  this  f  * 0 o |j\J> * ( £ - H ) " ' V . . t f ) | N > .  a  definition  G (E)  (6-7)  gives J  +  expression  |Nk-X= With  (6-6)  |N> +  which  relates  eigenstate  chapter.  In t h e n e x t  information  from i t .  |N>,  V..(£)lN>.  the  one-pion  i s the  s e c t i o n s , we s h a l l  key  ( 6  "  1 1 )  eigenstate  equation  for  e x t r a c t the p h y s i -  89  b. The T-matrix f o r pion-nucleon s c a t t e r i n g . The amplitude f o r the t r a n s i t i o n the  state  |N' ,k*|. >  S(N',k*J. |N,k^) which  is  from the s t a t e  given  by  |N,kj> to  the S-matrix  element  i s d e f i n e d a c c o r d i n g to [70]  sU'.Kj.K^) =  (6-12)  <N\i-,\N,$.y^.  From eq.(6-9), _<N',k^. | i s r e l a t e d to <N',k^»| v i a +  <N'X\  =  £N'X,\  </S/ |V (k')[&V)-6"(E')l ,  +  t  (6-13)  #  But  G (E') -<T(E') +  l(E'-H)  -27TC  ,  (6-14)  so  S(A/;S;.IW ^)= <A/;i ^^> -^c5( -e)<N'i v;..rf';iw^> -i5) j  +  r  +  c  +>(6  and a c c o r d i n g t o the o r t h o g o n a l i t y of the s c a t t e r i n g  Si^k-ANJ.) with  = ^^.S^U'-H) — i T T I  states,  Ue-e;T(N/' ^.|iv> lE) /  i  TC^'P-.^.V1e > = <^KV* AX> )lw  where T i s the t r a n s i t i o n matrix with the  usual  )  ( -ie) 6  (6-17)  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...(f')at(E> |N/> . t  (6-18)  90  Since  a ^ ( k ) commutes w i t h V ^.(k') a n d 0  a*(K)H  </v' I then  =  <V I t ^ U \ H ]  u s i n g e q . ( 6 - 7 ) f o r t h e commutator  + Hat(?)  and a  ,  few  (6-19)  manipulations  result in  <M' which  =  < N  leads t o the exact  ,  W  form  o  i  (E-^-^.-H  ( ^  )''  f o r the pion-nucleon  (6-20)  s c a t t e r i n g T-  matrix  T ( ^ l W , ^ | E ) = <N'l V |. (H )G (E)V.^S)lN/> ,  <  +  l  (6-21 )  Next, the  we  expand  explicitly  this  e i g e n s t a t e s o f H . I f we i n s e r t  the  0  eq.(4-49) ding the  between  to their  each o p e r a t o r ,  formal  expansion  T-matrix  completeness  a n d expand  eq.(5-9),  on t h e b a s i s o f relation  |N'> a n d |N> a c c o r -  the T-matrix  takes  then  form  T ( N X \ X . • V +  e  +  v  e  C^-rl.-A^A)"'^!  1  7  U ) l f l ^ > < f t ^ | (c -H -H )" |B ^'><B y) J  MS) ]-  z ^ > < ^ l{ i - HJ +  0  o  J  lN/ >Z^ra e  I.  z ' u , / <  ,  o  o  v ao j 1 + o>  (6-22)  +  I{ 1 + H ,  H - AH A )"']•• e  X  91  All  the terms  generated  in this  expansion are  represented  gra-  p h i c a l l y as  Notice  that  crossing that  a l l the graphs  version  i n the second b r a c k e t a r e simply the  [61] of the f i r s t  a l l i n t e r m e d i a t e baryon  bracket. F i n a l l y ,  lines  we  recall  a r e summed o v e r n u c l e o n s a n d  deltas.  If  fl,  we l i m i t e d  we would c a l c u l a t e  However, and  ourselves to c a l c u l t i n g  G.F. Chew  a  propagator  H i s argument  present, result  Chew c a l c u l a t e d  a n d was a b l e t h a t  which  (e)  that  in  i s based  large.  With  a l l graphs  to order  eq.(6-23)  neglecting  i n a graph can v a n i s h ,  tude of t h e graph a n o m a l o u s l y parameters,  ( a ) and  [11] argues  (c) i s i n c o r r e c t .  when  graphs  the T-matrix  on  terms the  only.  l i k e (b)  fact  that  t h i s makes t h e m a g n i a  suitable  s e t of  ( a - b - c - e ) , w i t h no d e l t a  way t o g e n e r a t e t h e P  was i m p o s s i b l e t o g e t from g r a p h  3 i  resonance,  (e) a l o n e .  a  92  Following of  the  T-matrix  contains For  Chew's p r e s c r i p t i o n , following:  2n TTAB v e r t i c e s  example,  of o r d e r  as  graphs  X . We  and  therefore  phically  a  T  i  i  if it  p  have  a  pole.  (e) i n e q . ( 6 - 2 3 ) a r e a l l  expand t h e T - m a t r i x t  lambda-expansion  i s of o r d e r X"'  a graph  ( a ) , ( b ) , ( c ) and  i s of o r d e r X' , and  U )  define  p p r o p a g a t o r s which  T ( A / ; K ' K S . |E) = where T  we  \ N  ,  as  x \ N X \ ^ )  T*"(N',k*^ | N , K ^ | E ) i s  (6-24)  i  given  gra-  by  (6-25)  <K)\ where we bar  specified  the  intermediate delta  by a  double  propagator. F o r a l l terms  renormalization of  explicitly  i n eq.(6-25) except  procedure  the  (b) one,  therefore contribute  T - m a t r i x . We  shall  explicitly  extract  apply  the  d e s c r i b e d i n c h a p t e r 5. However, some  t h e b u b b l e s c o n t a i n e d i n (b) have p o l e s i n  and  we  them from  t o the  their  propagator  imaginary part  the bubbles  and  of  the  define  • 2 % v / , (6-26) with  93  0  A*  A  v  A  •  \  /'^  A  A  A " A  -x... (6-27)  A * A  (6-28)  &'  A  ' - A I T *  '  *  and  A '  A  A/  a ' A/'  A  Af  A ' A *  (6-29)  AJ  A M  1  (6-30)  N  where  G.(Ej  and  Z°  N  (E^ , E^ ) a r e  w h i c h have no p o l e s . F o r renormalized  truncated  convenience,  coupling constant  renormalized  functions  we d e f i n e a t r u n c a t e d  ft"(E  ,m )  TT NA  as  'fe (6-31  - a« Z,  The  T-matrix  to order  X  T(N;*-.I^IE)  (  ^  A  ,  ^  )  >  now r e a d s  _A  =  N  N"  1  Z_ tcf, +  N  _  (6  »  32)  w i t h u> =E-m.. and A/  A/"  A/  A/"  A/"'  X-  N'  N"  A/*  W"  A,'  A/""  A/  A  "  H  Al'  A»'  A  A/°  A "  Ai  ^~  A/'"  A/  W"  A/"  (6-33) -<£ A  Aj"  hi  N  A/'  A  A/*  A1  N  94 We  recognize  equation  t(t  eq.(6-33)  in  a  Lippmann-Schwinger  t y p e of  obeying  =  AriVt-yUj)  ( ^ - ^ o ' t c f ; ( 6 - 3 4 )  +  with  W  {VJ •<*>) =  ,ir(W  •_,<*>) =  being  the d r i v i n g  discern also  /ir^(  X'S k'fe-u>) +  potential,  the c r o s s e d  =4._\_i_Zl_  +  Arjk'j;^)  and we  +  /v;C^>'^)  used the  g r a p h s from t h e u n c r o s s e d  (6-35)  , (6-36)  subscript  "c"  to  o n e s . Note  that  we  defined  (6-37) To  make  define  the n o t a t i o n  the f i r s t  more s y m m e t r i c  t e r m of e q . ( 6 - 3 2 )  i n n u c l e o n s and d e l t a s ,  we  as  s N  /if ( I ? u O and  \  \  , /  (6-38)  therefore  T( W*,E-.lN/,fe- I E)  c.  =  The p a r t i a l - w a v e  According tential  =  /ir^Ci?;  T-matrix T  2 I  k • u.)  2 J  +  U S ^ u . ) * (?(\')  by  ( 6  -39)  (k',k;w).  t o e q . ( 6 - 3 5 ) and e q . ( 6 - 3 8 ) , t h e  terms a r e g i v e n  |  non-crossed  po-  95  "r.(i\$-><") and  more  \  =  g /  (6-40)  explicitly  =  /iral^f-u.) ir-Ck.k^uO  [ ^ ] " ' ^ ' ( ^ >  /v_ , < £ ' ) [ l-M r I (E)]  -  (6-41)  J  AT. (I?)  (6-42)  The c r o s s e d t e r m s a r e g i v e n by  (T^-u,) =  AT  (6-43)  (1?) [<*-«v*v]  ^,.(l?,k>> =•  ()?>)  »  irA.U',lr,<")  Since lar  momentum  total v  =  the f u l l  /v;  a  (fe)  [V"A-«"ir VJ  I,  ^.,0?*)  u  .  (6-45)  H a m i l t o n i a n H commutes w i t h t h e t o t a l  J and i t ' s p r o j e c t i o n  isospin  (6-44)  we a r e l e a d  on e a c h o f t h e a l l o w e d  J , and  similarly  z  to project  angu-  with  the p o t e n t i a l s  the  v, and  channels:  cm  I  r ^ V A a ^  = IP  (  J  U > ^ >  /,MP^C!JV*;U,)  ,  (6-46)  ,  (6-47)  1 J '"J'  where  v^' ( k ' , k ; u j )  v^ ( k ' , k ; U J ) operator  in  i s the the  amplitude  channel  i n the channel  21, 2J  of and  the p a  i4  J  *  projection s t  n  e  projection  21,2J obeying  i L ^ P ^ V ; ^ ) ^ ; ! / ^ - . ^ ) =^.S^.P^V.i-M) Furthermore, can  since  factorize  p J 3a J  spin  of  and i s o s p i n  as a product  space  .  (6-48)  are disconnected,  of a i s o s p i n p r o j e c t i o n  we  opera-  96  tor  (p ( j ' ; j ) and a s p i n  The q u i c k e s t elements  of  ,3=5^ that  graph  way t o d e r i v e  P  since  v^ ( k ' , k ; t o )  Therefore  be  a  M£ k, O= u  ,  )  where  i s a pure conserve  pure  the  matrix  t h e v (k',k;ui) a r e pure  i n eq.(6-40)  state  state both  with  we  notice  w i t h no p i o n  I and J, t h e n t h e 1=^  and  eq.(6-40,41,42)  P (^M^Cfc',.  with the r e s t r i c t i o n  J=S . M  ^  (6-5,)  (  ^  V  f o r the non-crossed  i  M  ^  *  ^  ^  ^  c a n be d e r i v e d  explicitly  Allied) Min'n^kMk'k J ^zr k  graphs:  >  5  (FJ;<") = ^ ( V / ^ ^ ) ^ ^ ' , * , ^  t h e v (k',k;oo) fc  ^  we g e t  K  We t h e n have t h e immediate r e s u l t  / ^ c ^ ' f • , « « > =  eq.(6-48),  (6-50)  Qc*'--«> ,  =  and  for  we w r i t e /  From  lies  the v e r t i c e s  must  $?(k';k)  the expression  the intermediate state  and  operator  i s to recognize that  s t a t e s . The p r o o f  since  present,  projection  (  ,  1  J  >  _  5  4  )  (6-55)  t o be  r  6  r  1  "  (6  56)  97  AA(k'R)A(kR)k>k r _ ^ _  kit,<»>„))  Next,  we  explicitly  consider  E  the  i n eq.(6-43,44,45).  crossed In t h e i r  graphs which general  *  7  - j " ' (6-57)  were  form,  defined  they  are  w r i t t e n as  ^ which,  (*',!?  =  when expanded  ^  L ^ - % - V  ^ v  V  ( ^ ) ,  (6-58)  become  I H , , E - ^ ( l ? ' r ? ) ^ ( l ? f ? ) fr'fe  with  V  which given  are  obviously  i n eq.(6-52,53).  M O ' S'i>i " }  V  V  V  different However,  v  from  ^  V  -  < 6  the p r o j e c t i o n  '  6 0 )  operators  i f we make u s e o f t h e i d e n t i t y  L^*')(^v.)J-  h)  and  ^  . 5Hj,r i,n  '  c  v  (  define -1 '  • )  1  >  1,  1-  1  sA  (6-63)  1  then  '<*  1  V  I  'A,  ).  V I 'A  'd/ '  'A  '  98  where t h e U - f u n c t i o n  was d e f i n e d  eq.(6-64) i n eq.(6-60), projection  operator  i n eq.(5-105).  and use t h e d e f i n i t i o n  in isospin  space,  If  we  replace  eq.(6-52) f o r the  then  = I u ( l , l , T P , X ) P^v-.-j) , 1(1  for  the spin s e c t o r .  (6-65)  Therefore  wi t h  Finally, crossed each  u  As  ,Pn  ,P  3|  I and t o t a l  T-marix T  a T  isospin  ^ ( k' , k ;u>) a s  nucleon  to  present  channels  (|  eq.(6-67),  and P ^  contributes  delta  channel  t e r m s v (k',k;w) and v a  on e a c h c h a n n e l  the to  i n c o n t r a s t t o t h e un-  and i n t e r m e d i a t e  t o the P  the p o t e n t i a l  p r o j e c t the T-matrix  according  and Pi}>  nucleon  respectively  for  that  intermediate  intermediate  contribute  spin  mention  graph with  of the P  crossed  can  we  ones  which  only.  C K  of s p e c i f i c  (k*',k;<W, we total  iso-  J . We t h e r e f o r e d e f i n e t h e p a r t i a l - w a v e  99  with  as  order  X , a n d remember t h a t v^ ( k ' ,lt ;LU) c o n t r i b u t e s o n l y  P  usual  channel,  n  E^m^+u, i t we u s e e q . ( 6 - 3 9 )  = t  (k',k;u>)  S  ^  ^  l  obeying  K  W  + t^;*)  the  partial-wave  to  the f u l l  driving  1  P  the partial-wave  T°"(U,) = J  (6-72)  v,  (6-73)  (k',k;ui) i s r e l a t e d i n eq.(6-36) v i a  /IT (fe'.fe-cu)  we w r i t e e x p l i c i t l y  (6-74)  the  T - m a t r i x and t h e p h a s e  k  =  t ' ^ i i l  relationshifts:  S (u,)  (6-75)  t o the o n - s h e l l c o n d i t i o n cu , fe  (6-76)  }  directly t « h 5 Cu.) - 1 ^ ( J  finally,  section  4  v(k*',k;w) g i v e n  d ? > - C o = ^ ) = -~4—  U; = tu  and  ,  ^  a j  potential  where t h e s u p e r s c r i p t "on" r e f e r s  or more  '  ( ^ V i i i )  To c o m p l e t e t h e p i c t u r e , s h i p between  3 J  driving  potential  T/-(J',?-,uO=  * d(\>)  (see eq.(6-34))  t l K t ^ - r K t ^ k ^ + U k ' t f  and  to the  then  T with  f o r t h e T-matrix of  via  the [72]  phase  O A /  shifts  (UJ))/^  (TT  are related  0 A /  CCOO  to the t o t a l  (6-77)  cross-  100  U L k*  +1  d. The P For  (6-78)  resonance.  3 3  the P  channel,  the T-matrix  T(k;k;^)= t <*>',uO with  t  eq.(6-72)  + C3(  X) 1  reads  ,  (6-79)  (k',k;<*j) o b e y i n g  t ^ k - . l . ^ ) - ^ ! , ' ! , ; ^ • jjk' k'^JK^ and v  making  use o f eq.(6-55) and eq.(6-67) t o eq.(6-70) g i v e s f o r  (k' , k ; t u ) :  AT^(fe'  k • t u )  =  ( k^k  AT  ; uo  a i i  From t h e d e f i n i t i o n AT: ( k > • uv) =  According about 82)  j  +  u  C i,i  to their  t h e v..  greater  k > +  (6-81)  we g e t  /v- (fe;'I? - uu) 4 _ L /IT U ' f e j u , )  expression  than  ,(  s i  e q . (6-69, 70 ) , v  e q u a l . However, t h e c o n t r i b u t i o n o f v  neglect  c a  eq.(5-105) of t h e U - f u n c t i o n ,  Ar-Ck'fe- ««) +  i s 16 t i m e s  ,I)AT  t h e one o f v  C N  and v  to v  .We  (6-82)  in  shall  C A  are  eq.(6-  therefore  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  Let's is  now r e c o n s i d e r  v (k',k;ui). A  According  t o eq.(6-57), i t  g i v e n by . AW  (6-84)  /TV)  K AN  We a l s o a p p r o x i m a t e  f^. ( m  + u w  \m )  OA * - n a  by i t ' s v a l u e  M  '  atuj=i*^  A M  (6-85) and  d e f i n i n g a f u n c t i o n h-(k) as  =  h-U) then  AW L U7T  \ A ^  A  v^(k*,k;u;) i s a s e p a r a b l e  p o t e n t i a l o f t h e form  *•  Next, we c o n s i d e r v  (6-86) T  °*  L  the crossed  nucleon  ATT  J  *  term c o n t r i b u t i o n  to  g i v e n by  (6-88)  AY)  Again,  we  constant  shall  t h e e n e r g y dependence o f t h e c o u p l i n g  and w r i t e  f Furthermore,  (<n,^) =  K.n^-^N-^o  t * /  k  , ) " '  =  .  (6-89)  i n eq.(6-88) c a n be r e w r i t t e n a s  the propagator  k  that  A/A/  , A/V  ( U J - U) -  Notice  neglect  -  ^  •+  U  t h e s e c o n d term on t h e r i g h t - h a n d  h  )(g>-u. Q h  ^  (6-90)  s i d e , which has a  102  two-meson c u t , v a n i s h e s whenever on-shell. has  G.A. M i l l e r ,  found  smaller find  than  that  decay  that  of  a  comes from fore  Miller, side  "the  contribution  o f t h e P%1 dressed  sense  resonance  delta  to neglect  from  reflects  based  the only  term  i s much  S i n c e we  a small  on  the  and  portion  it  and t h e r e s u l t s  term  shall  formation  on e q . ( 6 - 8 8 ) ,  argument  the second  second  term".  bag, and t h a t  this  entirely  of t h i s  of the f i r s t  t h e Chew-Low e x p a n s i o n  makes  or outgoing pion i s  i n h i s Chew-Low t y p e o f c a l c u l a t i o n [ 7 3 ] ,  the c o n t r i b u t i o n  most  the ingoing  thereof G.A.  right-hand  of e q . ( 6 - 8 8 ) and w r i t e  4_ AT (k'k'Uj)  Now,  like  -  11  CkJ  ~±L / 1"" V M(k'R)^(kK; q ' — I 1  f o r v , we have a s e p a r a b l e p o t e n t i a l A  k'kuj (6-91)  o f t h e form  = - ^ V ^>U>> k>  J^cJ-K*-^  .  (6-92)  with  3 As  (6-93)  Hi  a result  of our approximations, v  ( k ' , k ; ^ ) c a n be w r i t -  ten as  The the  solution rank-2  cally  of  the Lippmann-Schwinger  separable potential  equation eq.(6-80) f o r  e q . ( 6 - 9 4 ) c a n be w r i t t e n  analyti-  [74] as  t^u; k ^  cu> = N / U ^ - U O / D C U O ,  (6-95)  103 with  (6-96) WW  D(uO  A A  = D  D_  /  . (UJ) + UJ  N  W A  ( D - (uo)  1  (6-97)  t h e D's a r e g i v e n b y  (6-98) ^^  /  Uj - u j , OO  A £  _  r-  f  A  D-.(uO = E - / ^ - I J E ) - P U  ^ ** I  lAih! -* ^) ^  /  T  (6-99)  r  OO  r By r e p l a c i n g (U;) F  a  0  i  UJ - CO.  h (q) ?  by i t ' s f o r m a l  dusing the d e f i n i t i o n  n  D..(«*>) = E -/»*»« yy  l  A t l  A  If  A  yy  A  (  U  0  "i 2  we a p p r o x i m a t e  i n  A  INK C6)+iTr«,^l,-(o ti v- i  (E) -  on  eq.(6-86)  eq.(5-88) f o r l . _ ( E ) , we o b t a i n  and u s i n g eq.(5-68,69), we a r e l e f t  D  expression  (.6-100)  t  , (6-101)  with  -i  a  i  (""O ( E " ^ ) ^<]>MV +  .  A  t h e TTAN r e n o r m a l i z e d  a  ,. f c (6-102)  coupling constant  accor-  ding t o AFJ  *  t h e n we c a n r e w r i t e t  V  >  (6-103)  (k',k;u») a s (6-104)  104  D„CuO rv  W  W  =  D ^ < U i ) D**(u,)  A  1 /  _y A  -i  r .  tAJ -  +  tu( D  - .  n  W A  u ,  (6-105)  Ctu>y  V  (6-106)  v  W  i  (  6  -  1  0  7 )  CO  s  (6-108) 00  .  , i (6-109)  ; ( c o W > _  For  0 0 = 1 ^ = 1 ^ . ,  t^(k',k;cu)  is  (u>) and  77),  the p h a s e - s h i f t s are given v i a  according  to  eq.(6-  (6-110) and  finally  the t o t a l  cross-section  i n t h e 33 c h a n n e l 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  solving  eq.(6-104) t o eq.(6-109),  v o l v e d a r e t h e bag r a d i u s R, t h e d r e s s e d renormalized  coupling constants  Following take  f*  the parameters i n -  delta  mass m  A  and  the  and f " .  w  v  our d i s c u s s i o n a t t h e end of c h a p t e r  V, we  shall  f o r f*^ the value  (6-1 12) i.e.  we assume t h e r a t i o  constants  to  of the r e n o r m a l i z e d  be e q u a l . F o r  , we  simply  and  bare  use t h e  coupling  experimental  value of / The  dressed  tioned the free  the  (6-i 13)  as unknown. However,  chapter,  i t i s expected  as  men-  t o be c l o s e t o  Mev. The bag r a d i u s R i s  e a c h bag r a d i u s R between show t h a t t h e r e e x i s t  experimental  /YV)  fig.  previous  &* 1 .  left  as  a  parameter.  lations  The  mass m„ i s l e f t  r e s o n a n c e mass o f 1232  For  the  in  delta  •= L \Tir • o.as  {{{)  theoretical 11.  results  a delta  =: l l l o H/\<»  +  model,  mass m ( R ) A  which  cross-sections A  f i t best  empirically  by  (6-114)  5  f o r R>0.8 fm., m (R)  most  fermi, the c a l c u -  41]««.f^/R .  s o n a n c e mass, we t h e r e f o r e c o n c l u d e Chew-Low  a n d 1.2  [ 7 5 ] , and i t i s g i v e n  and e x p e r i m e n t a l  Since,  0.7  are  given  in  i s very c l o s e to the r e -  that  in  contrast  of t h e c o n t r i b u t i o n t o t h e P„  to  the  resonance  106  comes  from  Finally, is  we m e n t i o n  not s i g n i f i c a n t  which  fit and are  we l e a v e  M =1232 MeV A  decay  of  a  that the discrepancy  s i n c e we n e g l e c t  both  i  y  to the experimental  factor  and m data,  (see f i g . 1 2 ) .  a  dressed  the r e c o i l i n that  free,  of  t o such  kinetic  This context another  a f i t which  nucleon  region.  f =0.24,  the best  R=0.72  fm.  since r e c o i l corrections  is particularly  t o o much  sensitive  signi-  to  high  energies.  concludes  our s t u d y  of the Cloudy aspect  baryon  bag.  energies  the  and t r y t o f i n d  we o b t a i n  However,  delta  f o r high pion  n o t i n c l u d e d i n t h e m o d e l , one must n o t g i v e  ficance pion  and  i s s u r e l y an i m p o r t a n t If  the  the formation  Bag M o d e l . We  of the t h e o r y :  o c t e t which  of pion-nucleon shall  scattering  now  concentrate  the e l e c t r o m a g n e t i c  i s t h e o b j e c t o f t h e next  i n the on  p r o p e r t i e s of  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 scattering in the P c h a n n e l as a f u n c t i o n , of the p i o n k i n e t i c e n e r g y . The t h i c k l i n e i s the CBM prediction for bag radii i n the r a n g e of 0.8 t o 1.1 f e r m i . The dashed l i n e r e p r e s e n t s the e x p e r i m e n t a l d a t a [ 7 5 ] . 3 3  F i g . 12: B e s t f i t ( s o l i d line) t o the experimental (dashed line) total cross 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 c h a n n e l as a function of the pion kinetic e n e r g y . In t h i s f i t , f =0.24, R=0.72fm. and m =1232 Mev. 3 3  A  109  V I I ) ELECTROMAGNETIC PROPERTIES OF  a.  BARYON OCTET:  Introduction.  The of  THE  photon-hadron  the Cloudy  magnetic  Bag  Model,  properties  tention  to obtaining  radius  and  the  interaction,  when a n a l y z e d i n t h e c o n t e x t  leads to p r e d i c t i o n s  o f h a d r o n s . We  shall  on  moment  electro-  devote  our a t -  nucleon  charge  in fact  f o r m a l e x p r e s s i o n s f o r the  magnetic  the  o f a l l members o f t h e  baryon  octet.  We when  mentioned  studying  that  Bag  The  will  and  Salzman,  to obtain  Model,  to q u a n t i t a t i v e  this  to the pion extended  core  results  on  derive  the  in a well the  elec-  no  was  model simply  source function p ( r ) . i s the t h r e e quark  c o u p l e t o both the quarks  field,  1955,  conclusion  o f t h e n u c l e o n . However, s i n c e  therefore  to the pion  in  the c o r r e c t  f o r the n u c l e o n c o r e , the c o r e d e n s i t y  the Cloudy  bag,  extended  t o be p r o p o r t i o n a l  photon  that  model o f Chew, came t o t h e  t o be  properties  available  assumed  introduction  the s t a t i c  t h e c o r e needed  tromagnetic was  i n the  defined  way,  electromagnetic  In  bag.  inside  the  leading  then  properties  of  hadrons.  We form  first  factor  making use  G (q £  ) and  the magnetic  of t h e Cloudy  Bag M o d e l  quark  and  simple  center-of-mass c o r r e c t i o n  of  Donoghue  pion  formal e x p r e s s i o n s f o r the  and  contributions  Johnson  to  form  factor  formalism, Gp (q ). i  M  prescription  [33]. F i n a l l y ,  we  G (q M  we We  electric  ) . Then, extract  also  based  on  by the  suggest a the  work  evaluate e x p l i c i t l y  110  the  proton  all  members o f t h e b a r y o n  b.  and n e u t r o n c h a r g e  The e l e c t r o m a g n e t i c  In  order  we c o n s i d e r spin  1/2  photon read  first  octet  form  t o study  and t h e m a g n e t i c moment  as a f u n c t i o n  of  o f t h e bag r a d i u s .  f a c t o r s : G ^ q ^ ) and G ^ q * ) .  the electromagnetic  the e l a s t i c  b a r y o n s . To l o w e s t  i s exchanged  radius  s c a t t e r i n g of order  form  factors  electrons  G^qM, on  free  i n the i n t e r a c t i o n , a s i n g l e  ( f i g . 13 ) . The t r a n s i t i o n a m p l i t u d e  can  be  o f f t o be  where  p,m  (p',m') a r e t h e 4-momentum a n d s p i n  projection  of the  incoming  ( o u t g o i n g ) b a r y o n A, a n d k,s ( k ' , s ' ) a r e t h e 4-momentum  and  projection  spin The  electron  where u ^ ( k ) field. tor  of the incoming  current  (outgoing)  i n known f r o m QED t o have t h e f o r m  i s the positive-energy  Dirac  From t r a n s l a t i o n a l i n v a r i a n c e ,  i ( x ) c a n be r e w r i t t e n  A  M  '  =  spinor  f o r the electron  the baryon c u r r e n t  opera-  as  M  A  electron.  e  •  A  fi.  M  P  e  ^  >  (7-3)  A  where  is  the  e q . ( 7 - 1 ) becomes  4-momentum  operator.  The n u c l e o n  current i n  Ill  F i g . 13: Lowest o r d e r cleon interaction.  matrix  element  for electron  nu-  112  where T £ i s a 4x4 m a t r i x  i n Lorentz  From t h e r e q u i r e m e n t condition  o f gauge  of r e l a t i v i s t i c  the  H  a definite  and  the  = o  knowledge t h a t t h e s t a t e s  presentation  covariance,  i n v a r i a n c e which r e q u i r e s t h a t  q J and  space.  (7-5) |p,t»> form  a basis for a re-  o f t h e inhomogeneous L o r e n t z g r o u p c o r r e s p o n d i n g t o  spin,  t h e most g e n e r a l  form  of  the  operator  T£  is  [ 76 , 77 ]  where tic  p ^(q ) a r e r e a l (  moment  of the baryon  Following tion of  value  the baryon  f u n c t i o n s and K  fl  A.  E r n s t e t a l . [ 7 8 ] , we n e x t  o f t h e m a g n e t i c moment  few  consider the  expecta-  operator M i n the rest  frame  A  M , « <FKp=»)l ^Prl'li*) Substituting  i s t h e anomalous magne-  in eq.(7-7)  ? ( ? ) from  I RCf-o)> .  eq.(7-3,4,6)  gives  (7-7) after  a  manipulations  TZ L ^ * ^ ^ ] U; 5> ? c £ ( 3 , , ( 7  (  8 )  or  ^ft =  e  ^ (  o  )  ^ •  (  7  _  9  )  113 This  leads  t o the d e f i n i t i o n  of the magnetic  form  factor  G (q ) R  A  as  (7-10)  We a l s o requires radius for  intend  that  we compute t h e e x p e c t a t i o n  operator  a spin  <R„ i n t h e r e s t  charge  value  radius.  of  the  f rame o f t h e n u c l e o n .  This  charge  In g e n e r a l ,  1/2 b a r y o n A, we have  K Again,  t o c a l c u l a t e the nucleon  - <R(p=5)|jj3i  (7-11 )  lR(p>>  s u b s t i t u t i n g i n eq.(7-11) t h e e x p r e s s i o n  f o r 5°(r) q i v e s  .fl  And  from  this  geometrical  tric  form  f a c t o r G^q" ) as  (7-12) result,  Ernst  et a l . define  the e l e c -  1  (7-13) Sachs  [79]  teraction magnetic frame of  has shown t h a t of  the  fields.  the G  c  baryon A with  Finally,  (q ) a r e a measure o f a weak and s t a t i c  S a c h s a l s o showed t h a t  e l e c t r i c and  in  (where q =0), G ^ ( q ) a r e r e l a t e d t o t h e F o u r i e r e  the s p a t i a l  the, Breit transform  current v i a  i ° ^ - < i r«< >i«> = - £ } \ \ r  f l  the i n -  j  J  6  V> " " e  6  v  I ?<*>!"> = < ^ j ^ < t l ' ) < % l ' l l % > ^ J  <'-'«> (7-15)  114 We  shall  need t h e i n v e r s e o f t h e s e  &  MH'V'VV -  » ^ 77 (  relations, i . e .  =  1  where J * ^ ) i s t h e F o u r i e r t r a n s f o r m  <7  of j£(r) i n the B r e i t  "'  7)  frame (7-18)  c.  Formal e x p r e s s i o n If  we  Lagrangian  f o r jp (?)  introduce density  invariance,  the photon  using  the usual  rf>  A  ^ T i  field  Bag M o d e l .  i n the Cloudy  requirement  of  Bag Model  local  gauge  we g e t  W M«*)y*;J a  i n the Cloudy  s  +  i  i  (7-19)  "''  with  ^ This mal  local  "^("V*^'^^) .  Lagrangian  density  i s invariant  (7-20)  under  the i n f i n i t e s i -  transformation  </(oc)  ^  <t («"> - i6M  e <((#) ,  (7-21)  115 The  conserved  current  associated  with  this transformation  is  (7-22) satisfying  (7-23) with  j£(x)  being  the  quark  c o n t r i b u t i o n t o the  current  i  (7-24) -A 1 .  and  j  (x)  i s the  pion c o n t r i b u t i o n  TT  (7-25)  N e x t , we as  given  in  quantize  the  This  f o r m of  pion  contribution  The  pion  field  according  t o the  usual  way  eq.(4-8)  TT-t,o) then  the  f/  current  D  to the  key  by  +  equation  electromagnetic  current  ]£(x)  a,(k)e  a.(k)e  (7-26)  i n e q . ( 7 - 2 5 ) becomes  ( r ) i s the  quark  eq.(7-24) f o r  =  for determining  form  c o n t r i b u t i o n to  using  the  MIT  the  pion  factors. j!*(r)  quark  i s extracted  wavefunctions  from  eq.(2-  116 16)  (7-29) to get  -  J  r  1  (7-30) (7-31 )  Finally, expectation Dj(r)  since  the  v a l u e of the  quantity baryon  of p h y s i c a l  current  interest  operator,  we  i s the define  as  (7-32) where  |A>  is  the  physical  define  the q u a n t i t i e s  on-shell  ( r ) and  b a r y o n A s t a t e . We  also  ( r ) as  (7-33) (7-34) with  (7-35)  We the  a r e now  ready  electromagnetic  to c a l c u l a t e e x p l i c i t l y  properties  of the baryon  j j ( r ) to octet.  obtain  117 d. The p i o n  contribution  to j ( r ) K  R  The  formal expression  f o r the pion  current operator  eq.(7-  27,28) c a n be r e w r i t t e n a s  with  t * Si(^,^)ju)= (a-.tJ^+a^^Ca.^-^atc-i?)) and  we r e m i n d  step  t h a t g°°=1  i s to rewrite  eq.(6~6).  o r more  =-1  i n our c o n v e n t i o n s .  The n e x t  0  We use t h e H e r m i t i a n  conjugate  + V^.(k)  H] =  <^  a. it)  = - \ / (ft + CL(1o H  k  a^(h  of e q . ( 6 - 7 )  ,  which  reads  (7-38)  explicitly,  h  tx^a^k)  ( H+ These  t l  (7-37)  i n t e r m s o f t h e o p e r a t o r s V . a s we d i d i n  ( H + uj ) Adding  and g  ,  on b o t h  UJ,, +  identities  a (k) }  (7-39)  sides gives  uj ,) c<.(t) = - Ve -(k)  lead  .  (H +  +  k  _  (7-40)  t o the r e s u l t s [17]  lfl>=  (^-wW)"' r  (7-41)  V ^ c f t |fl> ,  and  ,  +  4  \  {  7  _  4  2  )  118 The physical  expectation baryon  ("k^,, k- JJH)  v a l u e of the o p e r a t o r  between  s t a t e s A h a s t h e n t h e form  (7-43) (7-44) with  i  <  fl  t  +  (7-45)  l^v^')K^.-w)" K-^-H)" v'(fe)lF)> ,  ,  (7-46)  " f <A I V.(-I)K—fe-H)'^-^,^)" V*,(-f) |R> , 0  C(^V»^" ~^{<lf If  we expand  eigenstates ness  (7 47)  the p h y s i c a l  b a r y o n s A on  of the bare H a m i l t o n i a n H ,  relation  D  f o r the bare e i g e n s t a t e s  eq.(7-45,46,47),  we w i l l  the  and i n s e r t  basis  of  the  the complete-  between e a c h o p e r a t o r s i n  generate the following  s e t of diagrams:  + i,.o. (7-48) >  + J,.o. (7-49) ;  + />.o.(7-50)  119  Applying V-h,  the renormalization procedure  a l l the bubbles  tities  will  a n d we g e t t o o r d e r  I  described  i n chapter  be r e p l a c e d by t h e r e n o r m a l i z e d  quan-  f*:  i.  - R > -*  +  6  A  i  r  £  v ft  &  (7-51 )  Afc  -Z &  A, ^ AB  = I  ATvjW)/*^  *  _i  1  8> > " A  ft *  8  1 " A  A& _  (fe)  &A  M « /^(-fo/W^'C-E')  (7-52)  &  fl *• & " A AS  J  r-R  f\ *• & f A  &A  (7-53)  ftb  With t h e h e l p of the i d e n t i t i e s A&  ^  ^ ^ t S (Jt) = - ^ ( ^ ,  "= ^ /IT, (-1?) = -AT A  Y)  we c a n c o m b i n e a l l t h e p r e v i o u s  s  (5;,!.-o)= X ( (  (7-54)  (fe), results  to get  3 (7-55)  (^h H'X^ +^k)(^+H') +  M  ) ^ C f e ) - ^ ( T ) ^  * K/fc  v  ( k  >j j ^  (7-56)  (H ^ .)(HA^O(^+H0 +  b  L^.(fe w ^ c f e ) * ^ . C^-Xfe')]  J  120 The n e x t  step c o n s i s t s  55,56). F o r t h i s  in evaluating  p u r p o s e , we d e f i n e  the brackets  i n eq.(7-  the quantity  AB \ A k X ) and  -  I  ^A5w"0h  ,  (7-57)  since * , .  •  6  AT  J  A  . . / u « x  B  * * * * * *  .  t * / » t .  C ^\z .hC;y\V.e^  *(hR>  s  :  (7-58)  then  ^(^•) = (r  / V)  f - j ^ w )  C  3*,  • . ( 7  s s )  with  cS,*)« I t d ( r . t )  We c a n f u r t h e r S =1/2 r  simplify  f o r A since  4  (  l  (k,k' )  ^ , ,  by  using  , - , 7  the  i t belongs t o the baryon o c t e t .  60  property  F o r S =1/2, B  we u s e t h e i d e n t i t y t  to g e t  i For  H  ( V  k  ' ) -  <^l(-^)(?fe')  I^A> .  (7-63)  S =3/2, we u s e e q . ( 7 - 6 2 ) a n d t h e i d e n t i t y &  Z_ L .  to  f  get after  .  C .  .  a few m a n i p u l a t i o n  - d >  5  '  (7-64)  121  i  1  r"( Evaluating spin  case  the isospin  i f the isospin  more s i m p l e  t o sum r i g h t  eq.(7-36)  forf*(r).  ty  via  t (B)  VV-± (?.$)(?.*') U,>  =y  term  !J^., would  be  ^  (7-65)  similar  to the  o f A was 1/2 (N a n d ^ ) . However, away o v e r  Therefore,  j  and  j ' as  i t is  required  we a r e l e d t o d e f i n e  in  the quanti-  H  (7-66) Doing  and  t h e sum e x p l i c i t l y  from t a b l e  IV  l e a v e s us w i t h  (extracted  f r o m r e f . [ 8 0 ] ) we have  t CB) = ^ t«/T ( i) .T T = T n  fl  V  fe  - *«/(T 0 A+  Combining  a l l those  ^  results  T = v fe  ( 7 A  "  6 8 )  1.  i n eq.(7-34)  f o rj ^ ( r ) gives  for TTrt  '(14>)J>  t  (7-69) i +H')(%^^)(u> e  and  with  f o r ?„„(?)  s (B) d e f i n e d fl  according to  6fl+  u> ,) k  Table IV: The Clebsch-Gordan j =1 and m =m-m [80]. 2  1  coefficients C l 2 12 m  m  J  2  J  with  m  3  m =-l  m =l  2  2  (j -m+l) (jj+m+1) 1  V  1  L(2j I)(2j 2)J 1+  1+  L (2j D(j + 1 +  1  1) J  .C2J +1)C2J +2D 1  1  (jj-m) (j +m+l)'  m  1  2J CJ +1) 1  h'  1  (j j-m)(J 1  L •2j (2j +l) 1  1  J  j ( 2 j + 1) 1  1  2j (2j +l) 1  1  123  <f and  5  (7-71 )  _3_  t h e momenta a r e a l l r e l a t e d v i a  (7-72)  e. The p i o n  c o n t r i b u t i o n t o G ^ q " ) and 1  The p i o n  c o n t r i b u t i o n to the nucleon  Gg(q*) i s g i v e n  electric  form  factor  i n e q . ( 7 - 1 6 ) and e q . ( 7 - ! 8 ) t o be  G  V) = en  Replacing  ^".  j ° ( r ) by w  (7-73)  *  i t ' s formal  expression  eq.(7-69) g i v e s  after  integration  (7-74)  or more  explicitly  (7-75)  i<.T7  3  1> (7-76)  72TT  J  (7-77) Notice all  t h a t 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  members o f t h e b a r y o n o c t e t . However, s i n c e  known  for  the strange  h a d r o n s , we  shall  limit  form f a c t o r G*(q ) a  ourselves  is  of un-  to the  124 nucleon  doublet  For  only.  the pion  from eq.(7-17,18)  contribution  t o t h e m a g n e t i c moment, we  that  « <> = ir IF <+»\"V**> • \** U*>'*'*>  & (  and  If  r e p l a c i n g ^.(r)  we d e f i n e  netic  ^  recall  by e q . ( 7 - 7 0 )  -»  (7 78  gives  t o be t h e p i o r i c o n t r i b u t i o n  t o t h e b a r y o n A mag-  moment, t h e n 3.  « __ « 6  C o )  s  0 0  _ L I ( j £ ] t (w^cw f  feV(^)(^ c +3  A  As a s p e c i f i c  example,  we c o n s i d e r  the nucleon  case  fe)  ( 7  _  8 Q )  f o r which  (7-81) (7-82)  ^  f.  ( A )  =  ^ ( - ^ /  i^  The q u a r k c o n t r i b u t i o n  ^ W ^ ^ ^ . < M l ^ ^ >  t o G * ( q ) a n d ^u".  The q u a r k c o n t r i b u t i o n  a  to the nucleon  G ^ ( q ) was m e n t i o n e d i n e q . ( 7 - 1 6 , 1 7 ) A  of  the quark d e n s i t y  j° (?) w  (7-83)  electric  form  t o be t h e F o u r i e r  factor  transform  125  and  from  eq.(7-30), ( ^ ( ^ / « ) + ^ C ^ / / ? > ) e ( R - A ) <M|1  where  we  assumed  t h e up a n d down q u a r k s  our L a g r a n g i a n d e n s i t y and  since  e b^b a  (7-85)  ;  t o be m a s s l e s s . S i n c e  eq.(7-19) c o n s e r v e s t h e  s i m p l y c o u n t s t h e quark  a  e ^ l j N >  electric  c h a r g e , we  charge  therefore  have  "• > ,  (7-86)  ?i  where C  i s determined  w  G" i0) 9  and  Q  w  by t h e c h a r g e c o n s e r v a t i o n  +  <^(o) =  -  w  ( J p  (7-87)  j  i s t h e n u c l e o n charge d o u b l e t i n u n i t s of e. E q u a t i o n ( 7 -  84) f o r G^iq*)  h a s t h e n t h e form  &"<*(p ' As m e n t i o n e d tion  Q  condition  i n section  t o the baryon  pectation  w  j  c of t h i s  spatial  J *(^(iln/J?)+-J, (HA/R))©(i?- ) i  chapter, the  quark  i s the p h y s i c a l  contribu-  baryon  A ex-  operator ^ ( r ) : I  <R  1  A  current  v a l u e of t h e quark  3  C  |R>  (7-89)  >  with  |^)- I The  quark  e  a  NI i i k ^CA */R)^,(^W' >W-A){b>«k]  magnetic  ?  k  moment o p e r a t o r yu^ i s r e l a t e d  . (7-90)  to f ^ t r )  via  .  126  N The  = 4-  \<f*  * * f«<*>  •  i n t e g r a t i o n can be performed e x a c t l y  (7-91)  [32] and g i v e s  3  with '3«A-l) + U A graph of  vs the quark mass m R  .  a  is  A  (7-93) shown  in  f i g . 14.  We  s h a l l denote fA = JA f o r the massless up and down quarks and jA = ^ a  for  0  a  the massive strange quark. The  contribution  of  the  quarks to the baryon A magnetic  moment can be c a l c u l a t e d as being the e x p e c t a t i o n value z-component  of  the  quark  magnetic  baryon s t a t e A of spin p r o j e c t i o n  K =<H,vil With  ^ R  2  £  We  need  operator ^ i n the  +1/2  / V  ^  ^  However, the matrix elements of ^ bags.  I R , ^ T > ,  k  7  "  9  4  )  z  t h e r e f o r e to expand  (7-95)  are known only f o r the  bare  the p h y s i c a l baryon A waveo  a c c o r d i n g to  formal expansion e q . ( 5 - 9 )  |R> then  (  .  f u n c t i o n on the b a s i s of the bare e i g e n s t a t e s of H the  the  ^  3  ' h=  moment  of  2 W{ a  1 + (/^-H.-AH^Xj IFO^  (7-96)  127  mR  Fig.  14: The  quark magnetic  moment  y a  (  m  R a  )«  128  + < A. 1 H (*v w nw^)"'.  K - *W\<Vf*J 2  and  t h e two o t h e r  since  £  9  1  no bag.  terms n o t p r e s e n t  in  does not c r e a t e o r d e s t r o y  Next, mation,  x  we make a "no more t h a n  i . e . we c o n s i d e r  more t h a n  one p i o n  only  r  this  (7  expansion  _  97}  vanish  any p i o n s . onepion  i n the a i r " approxi-  t h e terms i n eq.(7-97) which  i n t h e a i r when t h e p h o t o n c o u p l e s  have to the  L e t ' s d e f i n e as i n eq.(5-79)  I„( > = E  H AG (E)AH 1  o  I  )  (  7  _  9  8  and / V * ^ then  <&.»  S  / \ j O  (7-99)  f  e q . ( 7 - 9 7 ) becomes  1  C  3  J  (7-100)  or g r a p h i c a l l y  M Q = The n e x t did  ZaW{  I  approximation  1  +  consists into  replacing H +I 0  i n e q . ( 5 - 8 l ) . A f t e r a few m a n i p u l a t i o n s ,  hi  =  A V ^  +  Z  (7-101)  0  by H  eq.(7-100)  /C<B,C>  0  a s we  reads  (7-102)  t  with =  ^(B^C^Z'C^VI  Z \/M ) yMcpzCA,^ a  fl  j.^k  ,  A T ^ ^ C ^ O A ^ M )  (7-103) (7-104)  )  129 where we a l s o made u s e o f t h e c h a p t e r Replacing  v  and  w  by  their  V  main  result:  f" ^f" 8  B  formal expressions gives f o r  eq.(7-104)  S  Def i n i n g  ,  „  ^  A . •^-0  A vV,, A  7a Vl  -* ^  - * RK  R/V(B,C) s ^ then a f t e r  2  V'Aj  .,  „  ( 6 U ^ ) , C ( v i ) ) ,  ( 7  "  1 0 6 )  a few m a n i p u l a t i o n s ,  (7-107)  with  (7-108)  a  n  d  AV  ( B  ' ) i s given e x p l i c i t l y C  i n table  V with  the  symmetry  relation /V(B,C) =  As  an  yu (c,B) . 92  e x a m p l e , we c o n s i d e r t h e n u c l e o n  quark c o n t r i b u t i o n t o the eq.(7-102) t o be  magnetic  moment  ( 7  _  1 0 9  )  case f o r which the i s extracted  from  130  Table V: The quark magnetic moment matrix elements Pq(A,B) as defined in eq.(7-106). Z  y  (A ,B ) +  0 Z  +  V °' (A  B0)  V ~> ~) A  B  -2y /3 Q  2^y /3  2/2y /3  Q  2y /3 o  V  o  3  -y /3 o  -P /3 s  J/3  -y  /2y //3  o  (8y  o +  y )/9 s  v^"(4y +2y )/9 o  s  (4y -w )/9 0  s  (2y +y )/9 o  /2(y  o +  s  2y )/9 s  (P -P )/9 O  s  (-2y -4y )/9 Q  /2(4y  s  o +  2y )/9 s  2(y -v )/9 Q  s  C-4y y )/9 o +  s  2v^"(y -y )/9 o  s  C-2y -y )/9 Q  s  (y -4y )/9 Q  s  2/2(y -y )/9 o  s  (-y -2y )/9 0  s  131  (7-110)  or '-111)  with  P,  (7-112)  and t h e p r o b a b i l i t y c o n s e r v a t i o n  ™  g. The n u c l e o n c h a r g e  The  first  (7-113)  Aa-rr  radius.  nucleon e l e c t r i c  F o r q^ s m a l l a n d  The  Vw77  condition  form f a c t o r  spherically  i s defined as  s y m m e t r i c , we h a v e  term i s s i m p l y t h e n u c l e o n charge Q . F o r t h e second w  t e r m , t h e a n g u l a r i n t e g r a t i o n c a n be done a n d we a r e l e f t  j&  * * ^ ( » )  ]  with  + <S(f>,  (7-116)  (7-117)  and t h e n u c l e o n c h a r g e r a d i u s c a n be w r i t t e n a s  132  (7-118) If  we d e f i n e < r > " and < r > £ a  A  t o be r e s p e c t i v e l y t h e p i o n  quark c o n t r i b u t i o n t o the nucleon  O  and  from  =  the pion  and  we  cleon  charge  C  N  j J* 3  (7-119)  j  A* (^{ah/R) ^(£l7ilR))d(R-h)  contribution  and <r > i  Bag M o d e l c o n s i s t s o f a s t a t i c  of p i o n s .  However, t h i s when t r y i n g  static  give  the e f f e c t s  the  bag  calculations.  that  even  though the s t a t i c can s t i l l  Their  3-quark  the center  [33] p r e s e n t e d  starting  bag s t a t e  MIT  bag  brings o f mass  properties.  of the c e n t e r  be d e s c r i b e d  .  N  approximation  to estimate  t o the baryons electromagnetic  to estimate  it  eq.(7-118)  i n g r e d i e n t s t o c a l c u l a t e t h e nu-  1979, Donoghue a n d J o h n s o n  cription  . (7-120)  +  o f mass c o r r e c t i o n s t o ^  some c o m p l i c a t i o n s  state,  /-ry  then  radius.  Cloudy  a cloud  In  9  have a l l t h e n e c e s s a r y  The  in  =  c o n t r i b u t i o n , eq.(7-74,76,77) w i t h  h. The c e n t e r  with  > "  radius,  eq.(7-86)  <^>* For  a  charge  and t h e  point  i s not a  a simple  pres-  o f mass m o t i o n on i s the o b s e r v a t i o n momentum  by a w a v e p a c k e t  eigen-  o f bag s t a t e s  133 w h i c h a r e momentum  eigenstates:  \ *tVe'  I C(X)> = where  |A*"(]£)> i s a s t a t i c  at  s  the  H  i s the spin  VV>  (7-122)  3-quark b a g ( i . e . a b a r e bag) c e n t e r e d  o f A, a n d m  i s t h e 3-quark bag mass,  ofl  b a r e mass. F o r s i m p l i c i t y ,  eq.(7-122) i s then  F  only  X=s  n  i s considered  r  the spin  then a s c a l a r The  index  function  removed a l t o g e t h e r  (7-,23) a n d "X(p)  is  o f t h e momentum p .  d  H  fl  ofl  show  , Donoghue a n d J o h n s o n  that  the order  baryons electromagnetic  h i for  h a s been  X  mean momentum s p r e a d <p > i n t h e w a v e p a c k e t i s  A s s u m i n g <p > « m appendix  and  reduced t o  I R.<X)>- j J f ( j ^ ) y < p > e " ' | R „ ( f > ) > where  i.e.  <p '> /m* a  H  properties  = -i-  ^  >  in their fl  publication  corrections  t o the  a r e g i v e n by  • r"  Q  (7-125)  t h e m a g n e t i c moment, a n d  of?  for  the charge  The of  radius.  simplest  estimate  f o r <p*>  i s t o add t h e  fl  contribution  e a c h q u a r k momentum  <P  r  I  ill/R  1 ( 7  .  1 2 7 )  134  w h i c h , when combined cription  For  f o r l* u  w i t h e q . ( 7 - 1 2 5 , 1 2 6 ) , g i v e an e x p l i c i t  <r* >*. and <r > .  f t cin  CM  the nucleon,  e q . ( 7 - 1 2 7 ) becomes  N  and  3& /R  "  fYY\  b a r e mass i n f i g . 9 by  / R ,  (7-129)  w i t h -Q =2.04, we g e t o  =  ° -  <A >£ =  O.U  a  and  (7-128)  the nucleon  =  OK)  ,  0  s i n c e we c a n a p p r o x i m a t e  then  pres-  this As  form  correction a result  with 130).  A  JU^given  <^>"= w i t h <r >^, <r >£ a  tively.  A  a  i s of a reasonable  A magnetic  h i  =  the nucleon  < 0 : and  +  ML  ^  (7-131)  size.  we have t h e f o l l o w i n g  >  final  ( ?  f\  yu^ i n e q . ( 7 - 1 0 2 ) and charge  < V > ~  <ra>cM  (7-130)  moment  +  +  i n eq.(7-80),  Similarly,  <4 >£  of our d i s c u s s i o n ,  f o r the baryon  h  >  U  given  ^  f\ C h n  in  I 3 2 )  eq.(7-  radius i s  + <!»»)£,  ,  i n e q . (7-1 21 ,1 20 ,1 31)  (7-,33) respec-  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  members  of the baryon o c t e t , the parameters  renormalized stants  f o rthe electromagnetic  mass o f t h e b a r y o n s ,  f " , the strange  As  we d i d i n t h e p r e v i o u s  s  chosen  the  r e s o n a n c e masses f o r t h e u n s t a b l e  SU(6)  a r e : the  coupling  con-  and t h e bag r a d i u s R.  chapters,  are  coupling  involved  the renormalized  q u a r k mass m  B  p r o p e r t i e s of the  t h e r e n o r m a l i z e d masses  t o be t h e p h y s i c a l masses f o r t h e s t a b l e b a r y o n s , and  constants  factor,  f " , we s i m p l y  ones. F o r t h e r e n o r m a l i z e d  use f "  B  w  times  the appropriate  i . e . we u s e t h e r e l a t i o n (7-134)  with  For  t */i  0  given  the strange  m =144  Mev,  s  i n table  which  o f F. M y h r e r  MIT v a l u e left of  three  that  specific  comes o u t o f o u r CBM c a l c u l a t i o n the best  baryon  e t a l . [45,46];  mentioned  as a f r e e  i s such  y  q u a r k mass, we c o n s i d e r  m^=2l0 Mev., w h i c h g i v e work  I I I , and t  i n chapter  parameter  mass  i n order  (chap. V ) ;  spectrum  a n d m =279 Mev., w h i c h s  II.Finally, t o study  cases:  i n the i s the  t h e bag r a d i u s R i s t h e r a d i u s dependence  the c a l c u l a t i o n s . Let's  moments  first  consider  of the baryon  theoretical  the r e s u l t s  o c t e t . We  to experimental  ratio  show  obtained in  f o r t h e magnetic  f i g . 15  of the magnetic  to  moment  19 t h e fl R ^ /^*t*p  136  in  function  various in  SU(6),  explicitly  Figure the  nucleon  tal  values  the quark, center  to  bag r a d i u s  within  fermi  R=1.0fm.)  is  result  pion  field For  was c o n s i s t e n t  results are quite  the  whole r a d i u s  A  fore  well with  the experimeno f 0.85 t o  our a n a l y s i s o f t h e P within  data.  c o n t r i b u t i o n o f 0.6  i tcontributes  with  Notice  n.m. ( f o r  roughly  on  contrasts  quarter  with  the  [32,33] w h i c h does n o t c o n t a i n any  a lambda bag r a d i u s  radius  f i g .  Again,  since  for  r a n g e o f 0.8 t o 1.0 fm., t h e t h e o r e t i c a l p r e d i -  agrees  within  the strange  i n the theory  in  o f one f e r m i .  i n s e n s i t i v e t o t h e bag  t o 210 Mev. c h a n g e s ju that  re-  3 3  t h e r a n g e o f 0.8  the experimental  o f 2.65 n.m. T h i s  10%  =-0.614*0.005 n.m. N o t i c e  Mev.  ter  A  R  contribution.  the  e xp  with  the pion  o f ^ = 2 . 2 4 n.m.  are consistent  jU  c o n t r i b u t i o n t o JU  t h e lambda m a g n e t i c moment, o u r r e s u l t s shown  |A  give  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  with  e s s e n t i a l since  16  ction  that  VI where a bag r a d i u s  f  one  o f one f e r m i .  i s consistent  the t h e o r e t i c a l value  MIT  experimental  10% f o r t h e wide r a n g e o f bag r a d i i  f r o m t a b l e VI t h a t  of the  Bag m o d e l s . In t a b l e V I , we  m a g n e t i c moment a g r e e s v e r y  i n chapter  1.1  also  V shows t h e r a t i o  o f mass a n d p i o n  15 shows c l e a r l y  fermi. This  sonance  of  MIT a n d C l o u d y  the s p e c i f i c  1.15  R. T a b l e  b a r y o n m a g n e t i c moments t o t h e p r o t o n  the  for  o f t h e bag r a d i u s  ft  that  with  the experimental  a 25% d e c r e a s e o f m  by o n l y  s  v a l u e of from  10% (0.1 fm.) showing  q u a r k mass i s n o t s u c h a c r i t i c a l  presented  279  thereparame-  here.  Next comes t h e sigma m a g n e t i c moment. F o r L , +  the t h e o r e t i -  137  cal ^e*  p r e d i c t i o n s agree = 2 P  •  3 3 ±  of m .  0•  1  n.m.  expects quark  R  and  A  content.  via  exotic  has  changed  n.m.  [48].  17 and than  18 one  bag  t o be  t h e 1",  atom  ^  theoretical  standard  0.7  of  the  ="  the  cellent n.m.,  [ 8 4 ] . Our agreement  provided  with  t h a t the  H" m a g n e t i c moment, our with with  R=1.0*0.1 the  new  N e x t , we  experimental  charge charge figure result  radius.  The  r a d i u s f o r both 20. agrees  Accepting  is  slightly  the  1.2  new  p  experimental  =",  more  value  f e r m i . More a c c u r a t e  t a b l e VI We  shown  for  measure-  shows c l e a r l y  welcome  the  on  m =144 Mev, s  of  are  of  agrees  of  the  proton  the  results  discrepancy  the  neutron  of  for a nucleon  in  ex-  -1.250*0.014 fm.  For  the n.m.,  reasonably  well  [49].  for  experimental  and  decay  -0.61*0.02  of -0.69*0.04 n.m.  theoretical  deter-  the  r a d i u s i s R=0.95±0.10 value  the  o  i n f i g . 19 f o r ju~ value  that  recent  moments b a s e d  experimental  value  with experiment  in f i g .  welcomed.  ratio  a  value  and  results,  theoretical  c o n s i d e r the  determined  shown  =° bag  experimental  same  n.m.  of  magnetic  and  have t h e  -1.05  the  fermi,  one  result  be  results  since  is  —  asymmetry  independtly  t o /u^, =-0.89*0. 1 4  is negligible. 5° and  s i n c e they  of  = - 1 . 4 1 ± 0 . 2 5 n.m.  and  =° and  value  lambda c a s e  [81,82,83],  two  would c e r t a i n l y  contribution  equal  d e v i a t i o n from  f o r the  the  fm.,  whose m a g n e t i c moment  from  r a d i u s between  mination  about  techniques  recently  Finally, pion  R^  l i e between t h e s e  ments of JJ}  r a d i u s R=1.0-0.1  i s c o n s i s t e n t with  For  Our  w e l l . w i t h the e x p e r i m e n t a l  f o r a bag  This result  s  any  3  very  the to  nucleon  theoretical  is  shown  10%,  our  bag  radius  in  theoretical in  the  138  range of tant  0.85  1.15  to  difference  fm.  with  In t h e  t h e MIT  neutron bag  case,  this  model w h i c h ,  i s an  impor-  in i t s simplest  form, p r e d i c t s <r> =0.0 fm. a  Finally, 4fTr ^°(r) is  obtained  Fourier  sum  tail  adding  in  fig.  of  sign  an  One get  last  the b e s t  there  are  which  moments and  graph  shows c l e a r l y  the p i o n . N o t i c e  to s t r e s s  overall gests  agreement  neutron  on  [27],  of our  of the  internal  and  charge  electric  the  nucleon  fact  bag  t h a t we  electromagnetic  sea q u a r k s  form  which a negadensity  of b a r y o n s . results  with  Bag  factor  d i d not  been  [85],  the  the e x p e r i m e n t a l  baryons.  to  example, such  structure,  t o the  Model g i v e s a v e r y  s t r u c t u r e of  would  try  included  pion  Nevertheless,  that  radius.  d a t a . For  a l l c o n t r i b u t e t o some e x t e n t  s t r o n g l y t h a t the Cloudy  picture  t h a t the  e f f e c t s w h i c h have not  radii  core  r a d i u s , which t h e r e f o r e suggests  measurement of t h e  many o t h e r  a positive  the p i o n c o n t r i b u t i o n ,  f i t to a l l those  charge  fermi. This  form  a t t h e bag  will  r a d i u s of one  of t h e p i o n c o n t r i b u t i o n t o t h e e l e c t r i c  as c o n f i g u r a t i o n m i x i n g etc.  distribution  the quark c o n t r i b u t i o n e q . ( 7 - 8 6 )  t h e q u a r k and  note  charge  the  The  excellent  the  w i t h a bag  p r e c i s e measurement of t h e  supply  21  and  s u p p l i e d by  c h a n g e s of a  by  transform  the  tive  show  the neutron  eq.(7-74).  factor is  we  for  i  > ,  magnetic excellent data  sug-  reasonable  139  Table VI: S U ( 6 ) , MIT  A,  n  P  A SU(6)  1.00  A MIT  1.00  magnetic  E"  moments  A  i n the  =0  _-  -0.67  1.00  -0.33  -0.33  -0.67  -0.33  +  -0.67  0.97  -0.36  -0.26  -0.56  -0.23  0.95  -0.73  0.84  -0.39  -0.22  -0.46  -0.19  A CBM(210)  0.95  -0.73  0.84  -0.38  -0.23  -0.47  -0.20  A CBM(144)  0.95  -0.73  0.84  -0.38  -0.24  -0.49  -0.22  1.00  -0.68  0.83  -0.321.05  -0.22  -0.45  -0.25±.01  y  y  P  baryon octet CBM m o d e l s .  +  y  P  p exp  The and  CBM  A P exp  ( 2 7 9 )  140  T a b l e V I I : C o n t r i b u t i o n o f t h e p i o n , q u a r k and center of mass t o t h e b a r y o n m a g n e t i c moments i n t h e CBM w i t h R =1 fm. and m =144 Mev.  y (n.m.)  A  A y  y  Q A CM  y  A exp  =0  _-  P  n  0.60  -0.60  0.34  -0.34  1.74  -1.22  1.73  -0.62  -0.57  -1.16  -0.54  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  2.79  -1.91  -0.61  -1.25  A  2.33±.13 -.89±.14  0.00  -0.02  0.02  -.69±.04  141  F i g . 15: The n u c l e o n t h e o r e t i c a l n e t i c moment ratio y / yN n u c l e o n bag r a d i u s R . a  s  a  t o e x p e r i m e n t a l magfunction of the  142  F i g . 16: The lambda m a g n e t i c moment r a t i o y / y as a function of t h e lambda bag r a d i u s and t h e s t r a n g e q u a r k mass m . s  143  F i g . 17: The d e p e n d e n c e o f t h e sigma magnetic moment ratios y / y|xp t h e bag r a d i u s R^ u s i n g t h e r e c e n t v a l u e of y § x p ' « « « [ 4 8 ] . The dash-dot and dash-dot-dot lines are the experimental l i m i t s for y and y ~ respectively. exp exp E  o  =  z +  E  _  0  n  8  9  ±  0  1  4  n  m  144  18: Same as f i g . 17 =-1.41*0.25 n.m. [ 8 6 ] .  but  with  the  old  value  R_ F i g . 19: The d e p e n d e n c e o f t h e c a s c a d e m a g n e t i c moment ratios y= / y | on t h e bag r a d i u s R=. The d a s h - d o t and dash-dot-dqt l i n e s are the experimental limits for u and v" respectively. exp exp x p  E  =  146  F i g . 20: The nucleon t h e o r e t i c a l to c h a r g e r a d i u s r a t i o as a f u n c t i o n of t h e radius R . XT  experimental nucleon bag  147 E  m  CM  0.0  0.25  0.5  0.75  1 .0  1 .25  1 .5  ~l 1 .75 f m.  F i g . 21: The n e u t r o n c h a r g e d i s t r i b u t i o n 4 ^ r j ° ( r ) vs the r a d i a l d i s t a n c e r ( s h a d e d a r e a ) . Also shown are the quark (Q) and t h e p i o n (TT) c h a r g e d i s t r i b u t i o n i n s i d e t h e n e u t r o n . The n e u t r o n c h a r g e r a d i u s i s set a t one f e r m i . 2  148  VIII)  CONCLUSION:  In  the  previous  radius  in  the  obtain  very  low-lying  chapters  range  of  we  0.8  found to  1.1  good p r e d i c t i o n s f o r t h e baryons.  The  here  restricted  us  than  0.7  which  fermi,  weak  pion  to c o n s i d e r i n g  that  fermi,  as  two  pion  itself,  c o n t r i b u t i o n from h e a v i e r  the  baryon  face,  sea  length  Lagrangian  recoil  quarks,  pion  density,  c o r r e c t i o n s , the e t c . However,  field  with  our  radii  results.  have been  p h y s i c a l baryon  of  the  our  approximation  bags  with  linearities  to  i n our  (such  low  as  such non-  the  the  the  energy  these  course,  the  s t r u c t u r e of  mesons  larger  Of  expansion,  the  the  included,  f l u c t u a t i o n s of  approximation,  possible  assumed  many c o r r e c t i o n s w h i c h c o u l d i n the  bag  field  are  term  i t was  the  of  there the  keeping  physical properties  only  is consistent  by  pion  kaon),  bag  long  corrections  surwave-  are  ne-  glected.  Let's Model.  s t r e s s again  First,  ingredients: volume to  of  hadronic  the  confinement  s p a c e , and  the  world,  baryons coupled underlying the form  strength factor  main  i t incorporates  i n t e r a c t v i a the  "translate"  the  Cloudy  presence of  the  u(kR).  of  field. Bag  This Model  is  Cloudy two a  way  to  constants  perturbation  restricted the  baryons when  quark world formalism  the  pion  reveals and  theory  to  we the  involving  field.  itself  the  Bag  important  transparent  from t h e  quarks then  IIAB c o u p l i n g Next,  way  inside  a Hamiltonian  defined the  quarks  the  symmetry, w h i c h a l l o w s  thus o b t a i n i n g in a well  of  in a dynamical of  chiral  pion  features  The  through  interaction  applied  to  the  149  Cloudy  Bag  allowing there  Model  quantitative  i s only  radius  works  remarkably calculations  one t r u l y  free  found  mation  to  the P  that  tromagnetic field  resonance  most of t h e P ^  and d e c a y the  pion  will  matter  pion  In  properties is  sics.  the  fermi, Finally,  model,  of the Cloudy  the  bag  to  summary,  Physics  a few  the Cloudy  from  respect  obtain  correct  the f o r extent  t o the e l e c -  shown  that  baryon  o f t h e CBM  the  magnetic  f o r the neutron charge  nucleon-nucleon i n t e r a c t i o n  to c i t e  radius.  f o r nuclear  and p i o n  photo-  examples.  Bag M o d e l ,  by  providing  a  bridge  P h y s i c s w o r l d o f q u a r k s and g l u o n s , and t h e  world  for attacking  arises  we have  o f c o u r s e be many i m p l i c a t i o n s  just  i m p o r t a n t one.  b a g , and t o a l e s s e r  graph. With  essential  Bag M o d e l , t h e  i s a very  section  of t h e b a r y o n s ,  necessary  between t h e P a r t i c l e Nuclear  cross  nucleon crossed  calculations,  production,  proach  R > 0.7  be p e r f o r m e d .  in  dilemna  of a d r e s s e d d e l t a  moments, and a b s o l u t e l y There  for  R.  solution  from  to  parameter  Among t h e v a r i o u s a c h i e v e m e n t s  We  well  o f n u c l e o n s and p i o n s , o f f e r s  the u n s o l v e d problems  a new ap-  o f medium e n e r g y  phy-  150  BIBLIOGRAPHY  1.  G.A. M i l l e r , A.W. 192 ( 1 9 8 0 ) .  2.  S. T h e b e r g e , A.W. Thomas and G.A. 2839 ( 1 9 8 0 ) ; 23, 2106(E) ( 1 9 8 1 ) .  3.  A.W. 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