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UBC Theses and Dissertations

The effects of massive neutrinos and their mixings on muon decay Kalyniak, Patricia Ann 1982

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THE EFFECTS OF MASSIVE NEUTRINOS AND THEIR MIXINGS ON MUON DECAY  PATRICIA ANN M.Sc,  University  B.Sc,  KALYNIAK  of B r i t i s h  University  A THESIS SUBMITTED  Columbia, 1978  o f C a l g a r y , 1977  IN PARTIAL FULFILMENT OF  THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF  PHILOSOPHY in  THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS  We a c c e p t t h i s  t h e s i s as conforming  to the required  standard  THE UNIVERSITY OF B R I T I S H COLUMBIA July  Patricia  1 982  Ann K a l y n i a k ,  1982.  In p r e s e n t i n g  this  thesis i n partial  f u l f i l m e n t of the  r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e a t t h e of B r i t i s h Columbia, I agree that it  freely  t h e L i b r a r y s h a l l make  a v a i l a b l e f o r r e f e r e n c e and s t u d y .  agree that permission f o r extensive for  University  s c h o l a r l y p u r p o s e s may  for  financial  of  The U n i v e r s i t y o f B r i t i s h 1956 Main M a l l V a n c o u v e r , Canada V6T 1Y3  )E-6  n/R-n  Columbia  my  It is thesis  s h a l l n o t be a l l o w e d w i t h o u t my  permission.  Department  thesis  be g r a n t e d by t h e h e a d o f  copying or p u b l i c a t i o n of t h i s  gain  further  copying of t h i s  d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . understood that  I  written  i i  ABSTRACT  This Dirac  t h e s i s c o n t a i n s a study  and  Majorana  neutrinos  of and  s p e c t r u m f o r t h e muon d e c a y jx*~* * e  both  polarized  neutrino world range.  and  v  mixings  first-order  corrections  radiative  r a d i a t i v e d e c a y jm*-*e* i /  mass  of  are  Majorana  to  the  of  massive  in  the e  spectra  +  for  f o r the threein  the  MeV/c  2  calculated  and  neutrinos.  The  muon d e c a y a n d t h e  are included i n this analysis f o r  e  the case of a s i n g l e D i r a c  of  correlations  signature  n e u t r i n o o f mass i n t h e M e V / c  into the three-neutrino  correct ions.  their  v^Kv^. ) . The  e  with a s i n g l e neutrino  Electron-neutrino  regularization  effects  u n p o l a r i z e d muons a r e g i v e n  proposed as a p o s s i b l e  mixing  the  i s used f o r t h e  world.  The method o f  calculation  of  2  range  dimensional  the r a d i a t i v e  i ii  TABLE OF CONTENTS  page Abstract List  i i  of Figures  v  Acknowledgements  v i i  Chapters I Introduction II  1  Background and M o t i v a t i o n 1. E x p e r i m e n t a l 2. The G e n e r a t i o n  7  Situation of Neutrino  8 Masses  14  3. The E f f e c t i v e F o u r - f e r m i o n T h e o r y o f Weak Interactions  36  I I I Muon D e c a y W i t h M a s s i v e N e u t r i n o s 1 . The C a s e o f D i r a c  Neutrinos  45 46  2. The C a s e o f M a j o r a n a N e u t r i n o s  67  3. E l e c t r o n - n e u t r i n o  74  Correlations  I V The I n c l u s i o n o f R a d i a t i v e C o r r e c t i o n s V Summary a n d C o n c l u s i o n s  85 104  Bibliography  109  Appendices  113  A C o n v e n t i o n s and N o t a t i o n  113  B Majorana States  120  C Phase Space C o n s i d e r a t i o n s  126  D D e t a i l s o f M a j o r a n a Case C a l c u l a t i o n  141  E Dimensional Regularization  147  F Self-energy  154  and V e r t e x  Corrections  G Calculation  of  the  R a d i a t i v e Decay  166  V  L I S T OF FIGURES  page Fig.  1  The s c a l a r p o t e n t i a l V(<p)  Fig.  2  Fermion  Fig.  3  Neutrino  f o r ^_ <0 andyu?>0  18  2  mass c o n t r i b u t i o n due t o H i g g s d o u b l e t . Majorana  .  mass c o n t r i b u t i o n t o H i g g s  s i n g l e t -L*  30  Fig.  4  The  Fig.  5  Fig.  6  The h i g h - e n e r g y e n d o f t h e M i c h e l spectrum with h i e r a r c h i a l mixing Helicity considerations for the Michel s p e c t r u m o f JA* d e c a y w i t h m a s s l e s s n e u t r i n o s  Fig.  7  Michel  s p e c t r u m f o r u n p o l a r i z e d jx* d e c a y .  S e n s i t i v i t y of the Michel spectrum m i x i n g p a r a m e t e r Uy. f o r m = 5 M e V / c  ...  8  60 61 62  to the 63  2  3  Fig.  26  3  Sensitivity of t h e M i c h e l spectrum tothe mixing parameter U 3 f o r m =5 M e V / c and I U ^ l = .059  64  H i g h - e n e r g y end of M i c h e l spectrum f o r s p i n of jjl* a n t i p a r a l l e l t o p o s i t r o n momentum  66  2  e  3  2  Fig. Fig.  9 10  Energy-angle Majorana  correlations  f o r Dirac  (D) a n d  (M) n e u t r i n o s  80  Fig.  11  Energy-energy  c o r r e l a t i o n s f o r y=l/4  83  Fig.  12  Fig.  13  Feynman d i a g r a m s f o r t h e f r e e muon d e c a y a n d the v i r t u a l photon c o r r e c t i o n s Self-energy correction diagrams f o r the charged leptons  Fig.  14  Bremsstrahlung  c o r r e c t i o n s t o muon d e c a y  Fig.  15  The h i g h - e n e r g y e n d of the radiatively corrected Michel spectrum f o r a 5 MeV/c neutrino  86 86 86  2  Fig.  Fig.  16  17  The h i g h - e n e r g y e n d of corrected Michel spectrum neutrino  the radiatively f o r a 10 M e V / c  Limits of a l l o w e d mixing f u n c t i o n o f n e u t r i n o mass  of  99  2  IU/x | 3  2  as  a  100 101  vi  Fig.  18  High-energy end the spin of u* momentum /  of the Michel spectrum w i t h antiparallel to positron 1  02  vi i  ACKNOWLEDGEMENT  I abundant I  wish  to  t h a n k my r e s e a r c h s u p e r v i s o r , J o h n Ng, f o r h i s  h e l p a n d t h e t i m e he s p e n t w i t h me on t h i s also  thank  Doug  Beder  f o r acting  as  work. my  official  supervisor. My  thanks,  finally,  to  my  husband,  f o r h i s h e l p and  encouragement. Financial Engineering Columbia  assistance  from  the  Natural  Sciences  Research C o u n c i l and from t h e U n i v e r s i t y  i sgratefully  acknowledged.  and  of B r i t i s h  1  I.  The d e t e r m i n a t i o n  INTRODUCTION  of the nature  considerable  interest  grand u n i f i e d  t h e o r i e s [ 1 - 3 ] . Such  strong  and  traditionally definite  been d e s c r i b e d  as p a r t of t h e i r  theories  massless  chirality.  particle  attempt  Dirac  Many  a  phase.  We  to  of  the  t h i s phase  unify  are  have of  proposed neutrinos  A free massive Majorana  and i t s a n t i p a r t i c l e choose  of  particles  c o n t a i n massive Majorana  content.  been  Neutrinos  t o be an e i g e n s t a t e o f c h a r g e c o n j u g a t i o n  a Majorana p a r t i c l e  within  as  has  i n the context of  interactions.  schemes, however,  is defined  particularly  electroweak  (left-handed)  unification  is,  lately,  of n e u t r i n o s  field  [ 4 - 6 ] . That  the  same  to  such t h a t our Majorana  c o n d i t i o n c a n be s t a t e d a s f o l l o w s .  Majorana p a r t i c l e equation "c" a  and  (1.1) above,  antiparticle  whether  into  or not  described  by  i n d i s t i n g u i s h a b l e . In  IfJ i s a q u a n t u m f i e l d  indicates operation particle  are  and t h e s u p e r s c r i p t  of charge c o n j u g a t i o n ,  i t s antiparticle.  neutrinos  are  Dirac or Majorana  which  transforms  Thus t h e d e t e r m i n a t i o n  massive  and  whether  f i e l d s i s important  they  of are  i n deciding  w h i c h among c o m p e t i n g t h e o r i e s a r e v a l i d . E x p e r i m e n t s have n o t so f a r d e t e r m i n e d are  Dirac  or  Majorana  particles.  whether  Nor h a v e t h e y  neutrinos  r u l e d out the  2  possibility  of neutrinos  being  m a s s i v e . Thus, i t i s important  explore  t h e e f f e c t s of massive n e u t r i n o s and t o t r y  means  of  distinguishing  e x p e r i m e n t . Some o f neutrino  mass  or  C h a p t e r 2. A l s o theoretical  in  the to  Majorana  experiments  their  that  Chapter,  to facilitate  remainder  ordinary  interaction  of  [7-9]  this  muon  sensitive  to  examples  work  decay  of the  A brief  of electroweak  review  interactions generation.  i s an i n v e s t i g a t i o n o f t h e  which  reaction  proceeds  -  that  v i a the  weak  e  +  v  (1.2)  _  T£  £  ( 1 . 2 ) , t h e muon a n t i n e u t r i n o i s r e p r e s e n t e d  conjugate  notation,  Majorana  particle.  , t o indicate the p o s s i b i l i t y For  for  consequently, interactions. experiments  a  clue  to  the  This which  current  as  the  V-A  theory  nature  of  description  decay  i s the  for deviations  minus a x i a l v e c t o r  four-fermion  effective  to  particular  the e f f e c t i v e  that  be u s e d . We h e r e s t u d y  theoretical  search  by t h e i ti sa  the case of Dirac n e u t r i n o s , the usual  a n t i n e u t r i n o n o t a t i o n , v, w i l l  neutrinos  by  [10].  In e q u a t i o n  "vector  give  a  a r e mentioned i n  i n one p a r t i c u l a r  -  (1.2)  nature  find  neutrinos  t h i s d i s c u s s i o n o f mass  e f f e c t s ,of m a s s i v e n e u t r i n o s the  we  are  schemes f o r n e u t r i n o mass g e n e r a t i o n .  is also given  of  Dirac  which  Majorana  o f t h e S U ( 2 ) X U ( 1 ) gauge t h e o r y  The  and  to  to  of  decay and  electroweak  focus the  of  most  standard  (V-A) s t r u c t u r e o f The  standard  i s assumed h e r e a n d t h e e f f e c t s o f  massive  on t h e d e c a y  weak i n t e r a c t i o n  neutrinos,  from  current"  the  (1.2)  theory.  a r e s t u d i e d . The g e n e r a l  effective  3  four-fermion  weak i n t e r a c t i o n  theory  is  briefly  reviewed  C h a p t e r 2 a s an i n t r o d u c t i o n t o t h e c a l c u l a t i o n s o f t h i s The  neutrinos  of  the  decay  for the general  case of N l e p t o n f a m i l i e s ,  by  a = e, jx, -c, . ..  v ,  where  a  ,  simply  number  L  that, denoted  a r e t h o s e p r o d u c e d a t a weak  s u c h t h a t L^=+1  a  "weak  means  the neutrinos  i n t e r a c t i o n v e r t e x . They a r e c o n v e n t i o n a l l y family  work.  (1.2) a r e o f t e n c a l l e d  interaction eigenstates". This designation  in  for  assigned a n d L^=-1  a  lepton  f o r i£ . I n  g e n e r a l , n e u t r i n o m a s s e s do n o t c o n s e r v e t h i s  l e p t o n number [ 1 1 ]  so t h a t a weak e i g e n s t a t e  well-defined  i s not a s t a t e  of  (mass e i g e n s t a t e ) . T h u s , m i x i n g among d i f f e r e n t occurs.  In previous  neutrinos [12].  mass  eigenstates  i=1,...,N. That  i s , neutrino  propagates  a  as  will v  L  is  be a  massive  denoted state  of  as  among  neutrinos v,  where  L  mass  m. L  It  s t a t e o f w e l l - d e f i n e d e n e r g y E- =-J~ I p l *+mf . I f L  the masses of t h e s e s t a t e s a r e not eigenstates  species  c a l c u l a t i o n s o f muon d e c a y , no m i x i n g  was i n c l u d e d e v e n when c o n s i d e r i n g  The  neutrino  mass  are  in  degenerate,  principle  then  distinguishable.  e i g e n s t a t e s c a n be e x p r e s s e d a s a l i n e a r  superposition  the  mass  The  weak  of  the  mass e i g e n s t a t e s a s f o l l o w s . (1.3) Equation  (1.3)  mixing with elements  of  is  IX • the a  an  expression  Pontecorvo  unitary  following orthonormality  of  mixing  transformation  the concept of n e u t r i n o parameters.  They  are  matrix  have  the  and  conditions. (1.4)  4  (1.5) For  the s p e c i a l  case  Kobayashi-Maskawa Pontecorvo  of  three  neutrino  parametrization  mixing  m a t r i x . That  families  [13]  is,  can  be  (N=3), used  the  f o rthe  XL -  (1.6)  040^/2  where mixing  a n d -Tf$6<1T. F o r t h e g e n e r a l  matrix  angles  can  and  neutrino  be  completely  (1/2)(N—1)(N-2)  mass  adopted  here  family  specified  phases is  N  6.  case,  with  The  the  (1/2)N(N-1)  assumption  of  i n d e p e n d e n t o f any p a r t i c u l a r  gauge m o d e l . As m e n t i o n e d a b o v e , t h e p r o c e s s d e s c r i b e d by t h e V-A theory  [14,15],  effective  The  choice  interaction  i s made i n o r d e r  neutrinos;  that  i s , the  independently  of  interactions.  The  this calculation of e l e c t r o w e a k  any  a  neutrino  of massless  (1.2)  (known a s t h e M i c h e l  with  the  mass  left-handed  structure  four-fermion  neutrinos, the positron  theory  been  and w i t h t h e f u l l  t h e weak i n t e r a c t i o n s  v e c t o r b o s o n s [ 1 6 ] . The r e s u l t s  are  of these  massive  are studied  in  the  weak  i s chosen f o r gauge  reason.  spectrum  spectrum) has  of  effects  f o r the following  theory  interaction  predominantly  the renormalizable SU(2)XU(1)  interactions  effective  weak  t o i s o l a t e the e f f e c t s  non-standard  case  theory, wherein  four-fermion of  effective  over  ( 1 . 2 ) i s h e r e assumed t o be  theory  For  the  f o r t h e decay  calculated  both  SU(2)xU(1)  gauge  mediated  by  two c a l c u l a t i o n s  massive differ  5  by M  terms of the o r d e r  w  is  the  mass  experimental  of  of  (m^/M ^ ), where  the  precision  exchanged  is  d i f f e r e n c e b e t w e e n t h e two have  been  included  i s t h e muon mass  2  in  not  v e c t o r b o s o n . The  good  enough  to  and  current  detect  the  theories. Also, radiative corrections these  the  c a s e of  m a s s l e s s n e u t r i n o s s i n c e i t i s w e l l known t h a t t h e y a r e  large in  t h e measurement of t h e M i c h e l electromagnetic radiative  to  effective  e  7k  fine  SU(2)XU(l) the  family  So,  effective  and  the  of  are  massless  results  (1.7)  of  [16,17] of  to f i r s t  the  two  the  order  theory  is  radiative and  the  indistinguishable,  This  case i n d i c a t e s t h a t the e f f e c t i v e  i n the  of full  w  theories  the  and 2  experimentally neutrinos.  results  ( a m £ / M ) , where a  four-fermion  Chapter  3,  the  small difference for  the  massless  four-fermion  decay  (1.2)  theory  even  of m a s s i v e D i r a c and case.  The  spectra  M i c h e l spectrum i s c a l c u l a t e d f o r Majorana neutrinos  and  i n the g e n e r a l  f o r the decay of b o t h  u n p o l a r i z e d muons a r e g i v e n . We world  (1.2)  virtual  for  neutrinos.  In cases  process  theory  p r o v i d e a good d e s c r i p t i o n of  massive  order  ]  i s of the o r d e r  gauge t h e o r y  between t h e  will  [18]  the  case  neutrinos  first  the d i f f e r e n c e between the  structure constant.  corrections,  +  four-fermion  SU(2)xu(1) theory  for  the  for  decay  h a v e been c a l c u l a t e d and  the  s p e c t r u m . The  corrections  -  the  calculations  s p e c i a l i z e to the  the N-  p o l a r i z e d and three-neutrino  d i s p l a y t h e e f f e c t s o f m a s s i v e n e u t r i n o s on  the  Michel  6  s p e c t r u m f o r v a r i o u s n e u t r i n o m a s s e s and Majorana  cases  outgoing  e  are v  and  +  &  compared.  Such  J a r l s k o g as  a  interaction  for  were  right-handed  along  w i t h the  method  is  a l l cancel  gauge  F.eynman g a u g e . B e s i d e s  of  proposed  by  in  the  weak  c o r r e c t i o n s to the  decay  r a d i a t i v e decay  the  method  leave  and  we  of  i n which  the  a  the  which  Mev/c  occur  result. 't  Hooft-  bremsstrahlung  necessary photon  for  dimensional  finite  the  information  for  detected.  We  t h e i r mixings  for  is  e f f e c t s o f m a s s i v e n e u t r i n o s and of n e u t r i n o mass i n  (1.7)  work i n t h e  spectrum,  i s a l s o g i v e n . T h i s w o u l d be  various values  range  2  and  for  mixings.  Most  of  the  details  of  these  r e l e g a t e d to the appendices. A l s o a  to  invariant the M i c h e l  a muon d e c a y e x p e r i m e n t  of  a  nature  i s used to d e a l w i t h the d i v e r g e n c e s  [ 2 0 ] . These d i v e r g e n c e s  various  Majorana  currents  the v i r t u a l p h o t o n i c  calculated  regularization  display  the  [19].  are  spectrum  and  p r o p o s e d as  originally  t h e c a s e of m a s s i v e D i r a c n e u t r i n o s . The  This  Dirac  c o r r e l a t i o n s between  the D i r a c or  correlations  test  I n C h a p t e r 4, (1.2)  The  a r e c a l c u l a t e d i n C h a p t e r 3 and  p o s s i b l e means o f d e t e r m i n i n g neutrinos.  Also,  mixings.  more  conventions Majorana  general and  included are  First,  An  i s given  Chapter 5 contains  three  Appendix  n o t a t i o n . Appendix B c o n t a i n s  fields.  regularization  nature.  calculations  A a  our  E.  conclusions.  been  appendices contains  our  discussion  of  i n t r o d u c t i o n t o t h e method of i n Appendix  have  dimensional  7  II.  BACKGROUND AND  I n t h i s w o r k , we s t u d y muon  MOTIVATION  the e f f e c t s  of massive n e u t r i n o s  d e c a y . As m o t i v a t i o n f o r t h i s i n t e r e s t  on  i n nonzero n e u t r i n o  mass, we g i v e t h e e x p e r i m e n t a l  limits  on t h e m a s s e s o f t h e t h r e e  neutrinos  muon,  and  of  experiments Section  the  w h i c h have s e t t h e s e  2.1  sensitive  electron,  along  to  with  neutrino  a  families.  l i m i t s are b r i e f l y  few  mass  tau  other  or  to  described i n  experiments  the  The  which a r e  Majorana  nature  of  neutrinos. This  discussion  by a d e s c r i p t i o n nonzero  of the experimental  of t h e t h e o r e t i c a l  neutrino  mass.  We  review,  situation  means  of  accommodating  i n Section_2.2,  s p o n t a n e o u s symmetry b r e a k i n g a n d t h e m i n i m a l theory  of electroweak  the Higgs neutrino  sector mass  interactions.  and  are  the  given.  The  sector  wide  a  the idea of  SU(2)XU(1)  Extensions  fermion  i s followed  gauge  of t h i s model i n  which  generate  a  range i n t h e t h e o r e t i c a l  e s t i m a t e s o f t h e m a g n i t u d e o f n e u t r i n o mass i s p o i n t e d o u t . The  r e d u c t i o n of the SU(2)XU(1) theory  four-fermion  theory  of  weak  S e c t i o n 2.3, t h e V-A t h e o r y calculations  of the next  interactions  i s reviewed  Chapter.  t o t h e V-A is  effective  described.  t o s e t the stage  In  f o rthe  8  2.1 E x p e r i m e n t a l  If  one  Situation  has  any  doubt  c a l c u l a t i n g and t e s t i n g doubt  should  experimental electron  99%  when  there  mass [ 2 1 , 2 2 ] ,  i s motivation  of massive  "confronted  is  neutrinos, with  For  the  preliminary  The l i m i t s  the case  mass  Only  upper  tau type  TVL^ <  limits  Hb  of  the  for a are, at  e.V/c>  n e u t r i n o s . These a r e ,  [23-25]  yn  <  0.S0  KeV/c>  (2.1.2)  7YL  <  3USO  I^AeV/c*-  (2.1.3)  experiments which a r e s e n s i t i v e  Majorana nature  explores  sensitive  to  i n t o two t y p e s . The f i r s t the  limits  as  well  nature  neutrino  mass  type, p r i m a r i l y  can  be  weak  of t h e n e u t r i n o r a t h e r i n d i r e c t l y .  the neutrinos are detected. This category  oscillation  experiments  and  some  below.  second type,  correlation  d e c a y s . The i d e a o f n e u t r i n o o s c i l l a t i o n s section.  as  t o n e u t r i n o mass o r t o t h e  of n e u t r i n o s a r e d i s c u s s e d b r i e f l y  Experiments divided  (2.1.1)  have been s e t on t h e m a s s e s o f t h e muon a n d  e x p e r i m e n t s w h i c h have s e t t h e s e  other  this  this  present  evidence  s e t on t h e  for  C.L., IH- <  The  there  on n e u t r i n o m a s s e s .  neutrino,  nonvanishing  the e f f e c t s  vanish  limits  that  roughly decays, In the includes  m e a s u r e m e n t s i n weak i s described later  in  9  The  three  experiments;  limits  quoted  above  were  n e u t r i n o s were n o t d e t e c t e d .  conceptually  the  simplest  to  obtain  set  The  limit  i s that  n e u t r i n o ' s mass. The muon momentum f r o m t h e p i o n TI was  measured p r e c i s e l y  momentum  The mass not will  =  ™4  in  this  +  *"£ '  h  decay  at  SIN  [ 2 4 ] . Energy-  decay  yields  ^ir (  +  )^ of  of  neutrino  the  pion  result  does  mixing.  on t h e mass o f t h e t a u n e u t r i n o was o b t a i n e d  ~C' neutrino  *  mass w i l l  spectrum; the endpoint  will  ^~  ^  ^  (2.1.6)  T  change t h e endpoint be l o w e r e d  from t h e  of the e l e c t r o n value  Finally, measurement  to set this  was n e g l e c t e d . L e s s t h a n  neutrino 600 e v e n t s  limit.  t h e e l e c t r o n n e u t r i n o mass l i m i t of  expected  The l i m i t q u o t e d a b o v e was s e t  a s s u m i n g a V-A weak i n t e r a c t i o n a n d v a n i s h i n g e l e c t r o n neutrino mixing  from  decay  the massless neutrinos case.  were o b t a i n e d  We  later.  a measurement o f t h e t h r e e - b o d y  mass. A g a i n ,  the  (2.1.5)  quoted above. Note t h a t t h i s  return to this point  Nonzero  muon  (2.1.4)  two-body  i n t o account the p o s s i b i l i t y  limit  the  is  f o r t h e n e u t r i n o mass.  l e d to the l i m i t  The  for  on  p r e c i s e m e a s u r e m e n t s o f t h e muon momentum a n d  take  which  ~*r  i n an e x p e r i m e n t  conservation  following value ^  *  /+  i n weak d e c a y  the high-energy  was s e t t h r o u g h  a  end of t h e e l e c t r o n spectrum i n  10  another three-body H Nuclear  decays  decay He  3  which  is  corrected factors  V  roughly  versus  linear. from  generally  the  the  represented  X  square  root  (F),  energy  /  of  t h e number o f e v e n t s ,  divided  by  of the decay  some  electron.  mass.  For  zero  In e q u a t i o n  f o r the  above.  The  v  i n v e s t i g a t e d to set  the  limit  experiment claims a rather high r e s o l u t i o n  g i v e n . However, t h e r e a r e o t h e r problems  this  limit.  First,  the o u t g o i n g e l e c t r o n can s c a t t e r  i n the decay  giving  some u n c e r t a i n t y  for  issue  i s the u n c e r t a i n t y  to  in  neutrino of the  fH  i n the energy the  The  l e v e l s of t h e energy  levels  i s performed  analysis  was  done  2  neglecting  the  of  more  mass a w a i t s r e p e t i t i o n o f t h e e x p e r i m e n t . The  1 1  Acceptance  radiate  nonzero  5  N0 ).  or  the  spectrum  (C H  2  establishing  the f r e e decay; however, the a c t u a l experiment in Valine  (45 e v / c  i t s energy.  s t a t e . C a l c u l a t i o n s h a v e been made o f  with t r i t i u m  quoted  which i s necessary to set the  limit  important  deviates  It i s basically  y  e  FWHM a t t h e end o f t h e yfl s p e c t r u m )  case  n e u t r i n o mass, t h e p l o t i s  d r o p p i n g t o z e r o a t E =E^°' ' -m .  t h i s d e v i a t i o n w h i c h was  final  kinematic  F o r n o n v a n i s h i n g n e u t r i n o mass, t h e K u r i e p l o t  linearity  a  (2.1.8)  4  E™"'* i s t h e maximum e l e c t r o n e n e r g y  neutrino  by  of  f o r Coulomb e f f e c t s  zero  (2.1.7)  [E:^-£j H(Er -Eef-^T  (2.1.8) above, of  ^  such as t h i s a r e  Kurie plot. This i s a plot  *  €  +  possibility  of  11  neutrino mixing, A  more r e c e n t  which complicates  measurement o f t h e yS e n e r g y s p e c t r u m of  U n i v e r s i t y of G u e l p h [27] a best from  value that  detector  of  20  of  s e t s a mass l i m i t  e V / c . The  the. p r e v i o u s l y  worse  (^300  experiment. Another experiment We  return  weak  of  these  has  been a n a l y z e d  secondary  parameters. reanalyzed  The  SIN  decay  a linear  for  the  example of the 14 M e v / c the  t a u and  the  conclusions  could occur  2  t y p e , we  We  The  the  was  neutrino  also  This  height  the P o n t e c o r v o  1% p r o b a b i l i t y . As  of n e u t r i n o m i x i n g .  If  mixing  recently [ 3 2 ] , As  of  mass  6-  complicated  include this  an  n o t e d above by  possibility  a n o t h e r example of the n e u t r i n o e x p e r i m e n t s of the mention no-neutrino  been  [33,34]. (/3/3-v) 0  w i t h about  [29-31],  of n e u t r i n o m i x i n g a  described  spectrum.  +  experiment  reached,  as  [28].  a n a l y s i s o f muon d e c a y .  As  long  mentioned  (2.1.4).  i n t h e yx  t r i t i u m decay e x p e r i m e n t s are  possibility  i n our  possibility  final  s u p e r p o s i t i o n o f mass  theoretically  decay  was  e f f e c t s and  previously  p e a k s i s g o v e r n e d by pion  with  2  resolution  of n e u t r i n o m i x i n g  are  the  (Si(Li))  i s proposed at Chalk R i v e r  e i g e n s t a t e s , then secondary peaks occur possibility  the  source  the pion  eigenstates  at  different  experiment  from the  t o the q u e s t i o n  I n t r o d u c t i o n . Consider  neutrino  The  2  i n t e r a c t i o n s are d i f f e r e n t  in the  method was  mentioned  eV/c ).  H  Ve  experimental  2  3  [26].  of m < 6 5 e V / c  a s o p p o s e d t o /S s p e c t r o m e t e r ) and  considerably state  the matter c o n s i d e r a b l y  recognized  If  neutrinos  decay  takes  as are  d o u b l e yS d e c a y a  signature massive  p l a c e . On  the  (fifiov),  f o r Majorana  Majorana other  first  which  has  neutrinos  particles,  then  hand, the e x i s t e n c e  of  12  (^Pov) d e c a y Majorana  does not n e c e s s a r i l y  mass  experiments  (^  o v  i t strongly  suggests  neutrinos this.  a r e s e n s i t i v e t o n e u t r i n o mass i n  range. There from  but  imply that  i s no u n d i s p u t e d e v i d e n c e  )  ($3 v) 0  the  f o r nonzero  have  a  decay  10-100 e V / c  2  n e u t r i n o mass  experiments.  As an e x a m p l e o f t h e s e c o n d experiment,  t y p e o f n e u t r i n o mass  we d i s c u s s n e u t r i n o o s c i l l a t i o n s  sensitive  [5,11,35],  Consider  a p u r e weak e i g e n s t a t e , 1/^ , o f momentum p, a t t i m e t = 0 . A f t e r a time  t,  this  state  evolves to the following,  i n terms  of t h e  mass e i g e n s t a t e s , 1pi ,  = E  K<u  where E =Jlpi m \ Then 2+  the  2  L  eigenstate  ^  b  e  f- b i < x . u 4 a  is  evolving  a  probability  o f momentum p a f t e r  'J  There  (2.1.9)  t E t t  nonzero  of  finding  the  weak  a t i m e t i s g i v e n by  ^  a  f l K  u  b l <  probability  e  (2.1.10)  of  one  weak n e u t r i n o s t a t e  i n t o a n o t h e r p r o v i d i n g t h e n e u t r i n o masses a r e n o t z e r o  or degenerate.  The mass d i f f e r e n c e a n y e x p e r i m e n t  is  sensitive  13  t o d e p e n d s on t h e o s c i l l a t i o n Solar  neutrino  experiments are s e n s i t i v e  d i f f e r e n c e s of the order solar rate this  neutrino capture  of 10~  discrepancy.  2  to neutrino  [5,36,37].  The  mass  observed  r a t e [ 3 8 ] i s a b o u t 30% o f t h e c a l c u l a t e d  Earth  based  experiments)  differences  experiment  of  are  a  few  experiments  typically eV/c  2  (reactor  sensitive  [5,36,37].  and  to neutrino  The  reactor  of Reines e t a l . [ 4 0 ] , i n which the r a t i o of charged  to n e u t r a l current  reactions  r\  was  eV/c  6  energy.  [ 3 9 ] . O s c i l l a t i o n s h a v e n o t been r u l e d o u t a s a s o l u t i o n t o  accelerator mass  l e n g t h and t h e n e u t r i n o  measured,claims  0.7< A m < 1 . O e V / c . 2  2  experiment  This  a  -v-  n  + e  (2.1.11)  mass-squared  difference  c l a i m h a s been c h a l l e n g e d  by a  of similar  [ 4 1 ] w h i c h i s c o n s i s t e n t w i t h no o s c i l l a t i o n s .  Bubble  chamber a n d e m u l s i o n e x p e r i m e n t s a t CERN and FNAL [ 4 2 , 4 3 ] and an oscillation  search  oscillations.  The  a t Los Alamos [44,45] f i n d recent  CERN  no  evidence  beam dump e x p e r i m e n t s  for  [46] a r e  a l s o c o n s i s t e n t w i t h no o s c i l l a t i o n s [ 3 6 ] . Obviously, mass  is  very  the experimental  situation  u n c l e a r ; no f i r m c o n c l u s i o n s  m a g n i t u d e o f t h e n e u t r i n o mass h a v e been  regarding  neutrino  on t h e e x i s t e n c e o r  reached.  14  2.2 The G e n e r a t i o n  The  of Neutrino  Masses  SU(2)XU(1)  gauge  standard  interactions  [7-9] w i l l  theory  be r e v i e w e d  here.  c o n c e p t o f s p o n t a n e o u s symmetry b r e a k i n g is  used,  for  t h e v e c t o r bosons and f e r m i o n s .  of  electroweak  In t h i s theory, the  [3,47,48] i s v i t a l  v i a t h e H i g g s mechanism [ 3 , 4 9 - 5 2 ] , These  t o generate  ideas  are  and  masses  described  below. In  the  minimal  S U ( 2 ) X U ( 1 ) gauge t h e o r y ,  m a s s l e s s by c o n s t r u c t i o n . We d i s p l a y t h i s ways  of  extending  of  later.  Some  possible  t h i s model t o i n c l u d e m a s s i v e n e u t r i n o s a r e  a l s o g i v e n . We c o n s i d e r t h e s e than  the neutrinos are  extensions  of  SU(2)XU(1)  rather  d i s c u s s i n g a grand u n i f i e d model s i n c e t h e b a s i c mechanisms neutrino  mass  generation  are  i l l u s t r a t e d adequately.  p r o b l e m o f t h e m a g n i t u d e o f n e u t r i n o masses w i l l the c o n t e x t SU(2)XU(1) The write  a  of grand  unified  theories,  as  The  be d i s c u s s e d i n  well  as  f o r the  extensions.  basic  i d e a i n f o r m u l a t i n g any gauge t h e o r y  "gauge  [3,53] i s t o  i n v a r i a n t " d e s c r i p t i o n of t h e p h y s i c a l system.  The  s y s t e m i s assumed t o p o s s e s s a symmetry  described  Lie  g r o u p . A t r a n s f o r m a t i o n u n d e r t h e g r o u p i s g i v e n by  by  some  (2.2.1) where satisfy  the T  a  (a= 1 ,2  N) a r e t h e g e n e r a t o r s  the f o l l o w i n g commutation  of t h e group.They  relations. (2.2.2)  15  a be The f of  are called  matrices,  generators  the s t r u c t u r e constants  L , which s a t i s f y  of the group.  t h e same a l g e b r a  a  A  set  (2.2.2) as t h e  i s s a i d t o form a r e p r e s e n t a t i o n of t h e  group.  That  is, [ L \ Then  a  set  according  of  L  w  n fields  L  b  (2.2.3)  c  <P- ( i = 1 , 2 , . . . , n) i s s a i d t o t r a n s f o r m  t o t h e r e p r e s e n t a t i o n L i f they  commutation  satisfy  the  following  relations. [ T % ( p  Thus,  l = i £*"  under  L  ^ >  -  <P.M  a gauge t r a n s f o r m a t i o n ,  (2.2.4)  the f i e l d s  Q  transform  as  follows.  (p.Jx)  *  U(QUS) -  Notice  that  the  coordinates. local  transformation  (2.2.5a)  L  Le  A;- W. (%) J I J  parameter  If this  gauge  (p (*) U"' (BUS)  e  may  i s the case,  transformation;  (2.2.5b)  depend  on  is  space-time  the transformation  i f 6  is  i s called a  independent  of  x the  i s s a i d t o be g l o b a l .  From t h e f i e l d s o f t h e s y s t e m , a L a g r a n g i a n which  the  is  i n v a r i a n t u n d e r t h e gauge t r a n s f o r m a t i o n  constructed (2.2.1).  The  i d e a o f d e s c r i b i n g i n t e r a c t i o n s by means o f a gauge  theory  has  been  theoretical  r a t h e r s u c c e s s f u l and i s c e r t a i n l y  the primary  strategy at present. The  kinetic  energy  terms  of  a  Lagrangian  contain  16  derivatives.  Such  transformation. "vector  gauge  corresponding 1  from is  If  terms  a r e not i n v a r i a n t under a l o c a l  local  gauge  fields",  to  — -  =  iy  preserve  transform  —  Therefore, existence  gauge  (2.2.7))  photon  a  each ranges  below.  "  L  new  The d e r i v a t i v e  defined  (2.2.6)  experimentally.  the  new  In  f i e l d s A ^ must  ±(^U)IL  ]  the requirement of l o c a l  ]  (2.2.7)  gauge i n v a r i a n c e i m p l i e s t h e  o f N gauge f i e l d s A^L(x) a n d a l s o d i c t a t e s t h e f o r m  most  familiar  1x1  matrices.  (x)  the  coupling  form  gauge t h e o r y  =  g  i s quantum  U(1)  which  case,  equation  is  electrodynamics represented  (2.2.1)  (2.2.8)  i s the e l e c t r i c  c h a r g e a n d t h e gauge  t h e p h o t o n . A mass t e r m f o r gauge  m A . ( x ) A ^ x ) i s n o t gauge i n v a r i a n t . ( s e e 2  / U  i s desirable since a  needed t o m e d i a t e t h e l o n g  requirement of l o c a l  by  becomes  e  In t h e case of qed, t h i s is  of  fields.  For t h i s  A^(x) represents  the  index.  C ^U)--U(9U))CK^U)VL' -  U  qed,  A  A  J  invariance,  H e r e t h e gauge g r o u p i s  unitary  field  L  introduced,  superscript  g i s t o be d e t e r m i n e d  i n t e r a c t i o n s of t h e s e The  of  V r  required,  as f o l l o w s .  CA^U)  In  The  is  be  by t h e c o v a r i a n t d e r i v a t i v e  The c o u p l i n g c o n s t a n t  (qed).  must  t o a group generator.  y  the  A^(x),  t h r o u g h N a n d jx. i s t h e L o r e n t z  replaced  order  invariance  gauge  fields equation massless  range f o r c e . I n f a c t ,  gauge i n v a r i a n c e c a n be t a k e n  as a  the  "reason"  17  for  the  photon's  interactions,  masslessness.  which are short  In  the  case  range, t h i s  to  mediate  vector  is  not a l l o w e d .  violates  mass  local  term  gauge  renormalizability  invariance  of  the  the  not  gain  spontaneous  the  symmetry b r e a k i n g ,  r e s o l u t i o n of both The a  solution  also  therefore,  may  possess  i n the ground  example  often  the  abandoning  the  interactions  symmetry,  c o n c e p t of provides  some  symmetry  (vacuum)  which  state.  used t o i l l u s t r a t e  this  The idea  (f)  simple  classical  [3,53] i s t h a t of a  (2.2.9) under  of t h e system 2  vacuum.  Rather,  there  are  s y s t e m must c h o o s e  transformation (2.2.10)  i s Figure  1. The  ground  state  or  i s a t t h e p o t e n t i a l minimum w h i c h , f o r t h e (p=0.  Of c o u r s e ,  transformation  the  - jP  potential is illustrated  c a s e o f jjl <§, i s a t  the  that  f i e l d , (p, w i t h t h e f o l l o w i n g p o t e n t i a l .  T h i s p o t e n t i a l i s symmetric  the  a  i s n o t , however,  V l f l - ' l f  vacuum  only  problems.  manifest  The  a  destroys  t h i s p r o b l e m . The  or h i d d e n  yet  c o n c e p t o f h i d d e n symmetry i n v o l v e s t h e h y p o t h e s i s  system  scalar  to  Massive  Such a term not  r e q u i r e m e n t o f l o c a l gauge i n v a r i a n c e f o r t h e weak does  weak  interaction  but  theory;  the  i s a problem.  gauge f i e l d s a r e r e q u i r e d boson  of  The  (2.2.10)  f o r y/- >0, V ( ^ ) 2  (2.2.10) two  symmetry  but  symmetric  (p =0  i s manifest  is still is  minima  not at  symmetric the  in under  minimum.  p) - j 6yU. /A\ The  one o f t h e m i n i m a shown i n F i g u r e  2  1  as  its  18  ground s t a t e and then  t h e symmetry i s no l o n g e r m a n i f e s t  s t a t e . The s y m m e t r y i s s a i d t o be h i d d e n  Fig.  2  (2.2.9)  JJL?>Q,  the s c a l a r theory  contains  potential  as  Choosing  t h e vacuum  v a l u e o f t h e quantum f i e l d  in  quantity i s called  t h i s case,  described  by  one o f  s t a t e o f the  t h e minima  system, the  t h e vacuum e x p e c t a t i o n v a l u e  We d e f i n e a new f i e l d , (j)' , w h i c h h a s  n  (vev)  and,  (2.2.11) zero vev,  f ' = <p -0L =.  nonzero.  i s g i v e n by  0  0  of t h e  expectation  i n t h e vacuum i s s e e n t o be  io I (p |o> = <<p> = fb/^/X  T h u s <^')  the potential  no mass t e r m s . T h e f o l l o w i n g d i s c u s s i o n shows  a mass t e r m a r i s e s .  This  or broken.  The S c a l a r P o t e n t i a l V((p) f o r j x ? < § a n d /c >0  1:  With  how  in that  (2.2.12)  I n t e r m s o f t h i s new f i e l d ,  \J({)') = If* -  as follows.  3  +t *f X  the p o t e n t i a l i s  * - A * *  (2.2.13)  19  The the  new  field  original  (2.2.10)  h a s a c q u i r e d an o r d i n a r y mass J 3 A a ~* a l t h o u g h  field  of  was  massless.  the potential  Also  the o r i g i n a l  i snot apparent  i n this  symmetry  form; i t has  been h i d d e n . T h i s s i m p l e example i l l u s t r a t e s t h e concept o f symmetry  breaking  an o r i g i n a l l y In the  o r h i d d e n s y m m e t r y . A mass i s g e n e r a t e d  massless  gauge  fields  to  acquire  i t i s necessary f o r a m a s s . The f o l l o w i n g  e x t e n s i o n o f t h e s i m p l e example j u s t g i v e n w i l l  Upon  i l l u s t r a t e how a  c a n a c q u i r e a mass. T h i s i s t h e H i g g s  Consider a charged field.  from  massless theory. .  t h e c a s e o f t h e weak i n t e r a c t i o n s ,  gauge f i e l d  spontaneous  imposing  scalar  field,  (p ,  gauge  invariance  s y m m e t r y , t h e L a g r a n g i a n c a n be w r i t t e n  + y  u>  f(j)~  mechanism.  coupled under  to a  a  gauge  local  U(1)  i n the following  ((p (p^ f  A  form.  (2.2.14)  where JL >0 a n d \>0. I n t h e a b o v e , 2  F^v For  <^A -  E  (2.2.15)  V  t h e c a s e o f y<- <0, t h e f i e l d  the  2  gauge  field  yW- >0, t h e f i e l d 2  remains  (p h a s an o r d i n a r y mass t e r m  and  m a s s l e s s . However, f o r o u r c h o i c e o f  ^ can d e v e l o p a vev, as i n t h e p r e v i o u s  example.  That i s , < 0 I (p \0) = C(p)  0  where  = v/f£  (2.2.16)  20  V The  following  interpretation  ^  /A  (2.2.17)  p a r a m e t r i z a t i o n of  (p [ 5 2 ] ,  allows  for a  clear  of the r e s u l t s which a r e o b t a i n e d .  (p = Here,  =  x  e  and  •  5  are  hermitian  fields.  (2.2.18) Under  t h e U ( 1 ) gauge  transformat ion IL ~ the f i e l d s  Z,  transform  as  (2.2.19) follows  (see  equations  (2.2.1)  and  (2.2.7)). (p  <P' =  ^  ( i r + ^  (2.2.20)  (2.2.21 ) In  terms of t h e t r a n s f o r m e d  * ±  Py_ k' y[(xv + y[) r  - \xr The  scalar  boson  is  fields,  '  +  Tj  the Lagrangian i s  -  ( y ^ *• 3 A i r ) 1  constant  (2.2.22)  field  h a s a c q u i r e d a mass \| 2y/. ; t h i s m a s s i v e  called  the  2  from the L a g r a n g i a n ;  H i g g s b o s o n . The f i e l d  i t h a s been  "eaten"  by  scalar  ^ has d i s a p p e a r e d the  gauge  field,  21  which  s u b s e q u e n t l y p i c k s up a mass, e^. T h i s e x a m p l e  how a gauge f i e l d c a n a c q u i r e mass t h r o u g h Having  surveyed  the  concepts  b r e a k i n g and t h e H i g g s mechanism, SU(2)XU(1)  group  SU(2)XU(1).  determinant of  we  theory of electroweak  that the electroweak  SU(2)  the Higgs  spontaneous  now  briefly  interactions.  interactions are  mechanism.  the  fe^bc'  I t i s postulated  described  the t o t a l l y  The  particle  content  fermions  are  assigned  to  group.  m o d e l . The g e n e r a t o r o f and  is  by  the  gauge  is  antisymmetric as  follows.  constants Levi-Civita Left-handed  d o u b l e t r e p r e s e n t a t i o n s o f t h e weak  Right-handed  Right-handed  hypercharge  the  S U ( 2 ) i s t h e g r o u p o f 2X2 u n i t a r y m a t r i c e s o f  tensors.  i s o s p i n SU(2)  symmetry  review  1. I t h a s t h r e e g e n e r a t o r s . The s t r u c t u r e are  singlets.  of  illustrates  fermions  neutrinos  are  the  group,  U(1)  are  excluded Y,  assigned  to  i n the minimal  is  called  weak  a s s i g n e d t o each p a r t i c l e a c c o r d i n g t o the  r u l e below. That i s , Q=  T  3  Here, Q i s the e l e c t r i c weak  isospin.  adopted  The  +  Y/X  charge  (2.2.23-) and T  following  3  i s t h e t h i r d component  of  n o t a t i o n o f Cheng a n d L i [ 5 4 ] i s  here.  (0 -aJ)  (2.2.24)  1 The numbers  i n parentheses  under  SU(2)  isospin  the and  and  give  U(1)  hypercharge,  the  transformation properties  groups;  t h a t i s , t h e y a r e t h e weak  respectively.  The  subscript  OL  22  represents  flavour  subscripts  L  and  and R  i  is  the  correspond  SU(2)  to  index  left-  ( i = H / 2 ) . The  and  right-handed  p r o j e c t i o n s , r e s p e c t i v e l y . F o r some f i e l d 1j)  r  (2.2.25) l  M  The m a t r i x In  K  5  i  (  n  -  (  2  .  2  .  2  6  )  i s d e f i n e d i n A p p e n d i x A.  o r d e r t o w r i t e an S U ( 2 ) X U ( 1 )  gauge f i e l d s must  be  introduced,  invariant Lagrangian,  corresponding  to  the  four four  g e n e r a t o r s o f t h e g r o u p . The t h r e e gauge f i e l d s c o r r e s p o n d i n g t o the  SU(2) generators w i l l  U(1)  gauge f i e l d  a=1,2,3 also  is  the  be d e n o t e d by A^_(x) a n d we d e n o t e t h e  a s B^_(x). H e r e , yt<-  is  SU(2) index. F i n a l l y ,  the  Lorentz  index  and  a Higgs SU(2) doublet i s  introduced. <t'  <  M  (2.2.27)  The S U ( 2 ) a n d U ( 1 ) r e p r e s e n t a t i o n s a r e i n d i c a t e d The g e n e r a l S U ( 2 ) \ U ( 1 ) i n v a r i a n t L a g r a n g i a n constructed  from these elements i s g i v e n  below.  i n parentheses. which  can  be  23  ^  <  f l ^  t i ,  ( p _ ^)  L  (2.2.28)  R  where ^  = ^ A * =  In  the  ^ B  equations  V  y  - ^ A % - ^  M  € ^  C  A ^  A'  (2.2.29)  B ^  V  above,  (2.2.30) g  and  g'  are  t h e gauge c o u p l i n g s  a s s o c i a t e d w i t h S U ( 2 ) a n d U ( 1 ) , r e s p e c t i v e l y . The equation  (2.2.28)  interaction. the  gives  the  pure  gauge  N e x t t h e gauge i n v a r i a n t  first  fields  line  of  p a r t of the  k i n e t i c energy  terms  for  f e r m i o n s a r e g i v e n . The H i g g s p a r t o f t h e L a g r a n g i a n a n d t h e  Yukawa  couplings  of  the  Higgs  and  fermions  L a g r a n g i a n . T h e r e a r e no gauge f i e l d mass t e r m s o r terms  included  symmetry.  since  these would v i o l a t e  complete lepton  the l o c a l  the mass  SU(2)XU(1)  24  The n e u t r a l component o f t h e H i g g s f i e l d  for  gets  a nonzero vev  t h e c a s e o f ju- <0 2  0' W o ' / J I ! ]  (2.2.31)  where V  x  = yu!~ /X  -  T h i s breaks the SU(2)xU(1) symmetries as Higgs doublet  c a n be p a r a m e t r i z e d  (2.2.32) described  above.  The  as f o l l o w s .  (2.2.33)  I  a. The  T*-  (a=1,2,3)  transformation  are  the  SU(2)  generators.  U n d e r an S U ( 2 )  with (2.2.34)  the  fields  transform  (p'  (qr°pU^  (f) —  t - a ; —  =  as f o l l o w s .  r A / - - u © [ T ^ ; - ^ u "  B^ — " i ^  (2.2.35)  ,  ^ u l u  ,  r a  ( 2  .2. 7) 3  =  (-2.2.38)  =  (2.2.39)  The r e s u l t s a r e t h a t t h e s c a l a r H i g g s f i e l d  gets a  mass  ^-2jj}\  The f o l l o w i n g c o m b i n a t i o n s o f gauge f i e l d s d i a g o n a l i z e s t h e mass  25  matrix  o f t h e gauge b o s o n s . +  (2.2.40)  7  and  yield  (2.2.41)  7^  masses (2.2.42) (2.2.43)  respectively.  The f o u r t h gauge b o s o n  (2.2.44)  remains  massless  and  corresponds  fermions  a l s o a c q u i r e mass. The mass  upon d i a g o n a l i z i n g t h e m a t r i x only is  one  lepton  t h e p h o t o n . The  eigenstates  are  charged obtained  . F o r i n s t a n c e , f o r t h e c a s e of  family, this  t h e mass e i g e n s t a t e w i t h  to  is trivial  and t h e s i n g l e SL f i e l d  eigenvalue (2.2.45)  The a c q u i s i t i o n illustrated neutrino basically  symbolically  remains built  been e x c l u d e d term  of a fermion  massless  mass v i a  in  Figure  in  mechanism  is  i t i s c l e a r why t h e  SU(2)XU(1).  The r i g h t - h a n d e d  content  f o r the neutrinos. Therefore,  Higgs  2. Now  minimal  i n t o the theory.  from i t s p a r t i c l e  the  so t h e r e  This  neutrino is  no  is has  Yukawa  the neutrinos cannot pick  up  26  «v\  9  Fig.  a  2:  F e r m i o n Mass C o n t r i b u t i o n due t o H i g g s  Doublet  mass t e r m v i a t h e H i g g s m e c h a n i s m t h e way t h e c h a r g e d  leptons  do. It theory  i s a l s o e a s y t o s e e , a t t h i s p o i n t , how reduces  energy  corresponding  The  b o s o n mass e i g e n s t a t e s ,  terms  contain,  for  the o r i g i n a l example,  fermion  one  part  t o t h e weak i n t e r a c t i o n .  c h a r g e d weak c u r r e n t  f o r the leptonic sector i s  ^ ^ i  The  SU(2)xu(1)  t o an e f f e c t i v e V-A weak i n t e r a c t i o n t h e o r y . I n  terms of the v e c t o r kinetic  the  neutral current  (2.2.47)  f o r this sector i s  (2.2.48)  For  momenta  clearly  much  smaller  than  M  w  reduce t o e f f e c t i v e four-fermion  (M ), x  these i n t e r a c t i o n s  current interactions.  27  i  f  ( 3 ^  -  ^  T  t  (2.2.49)  -z-  (2.2.50)  rP\^  (2.2.51)  where 7X Here,  G  F  is  the  f e r m i o r weak i n t e r a c t i o n  boson m e d i a t e d muon d e c a y , boson  reduces  than M , w  Qj^/M^J ,  to  and t h i s  the propogator  reduction  ^  o r  m o m e n t  of  um  constant. the  In the W  intermediate  t r a n s f e r much  smaller  follows.  We t u r n now t o t h e q u e s t i o n  o f n e u t r i n o mass.  The  general  f o r m o f a n e u t r i n o mass t e r m i s £  = - i  M  Here v  c  ^  + &  u  i s the f i e l d  V~i f c +" < ^ ^ r V  + U-C^  (2.2.52)  c o n j u g a t e t o v. T h a t i s , C T v *  (2.2.53)  where C i s t h e c h a r g e c o n j u g a t i o n in  Appendix  B;.  The  first  operator.  two t e r m s  Details are supplied  i n equation  (2.2.52) a r e  M a j o r a n a mass t e r m s ; i f y they v i o l a t e  C  = V  l e p t o n number c o n s e r v a t i o n  (2.2.54) by two u n i t s .  The  last  t e r m i s an o r d i n a r y D i r a c mass t e r m s u c h a s t h a t f o r t h e c h a r g e d  28  leptons.  In  does not the  minimal SU(2)XU(1)  above the  neutrino  p i c k up a D i r a c mass t e r m v i a t h e H i g g s m e c h a n i s m  right-handed  Since  as d e s c r i b e d  component o f t h e n e u t r i n o  SU(2)XU(1)  l e p t o n number forbidden  possesses  Cheng  and  minimal  handed  two  mass  term  is  be  which  can  enlarged  accommodate  a  of  enlargement  the  neutrino  only occur field  is  i n the Higgs s e c t o r alone  if  the  neutrino  i n the Higgs s e c t o r or  A D i r a c mass t e r m can of  to also  b a s i c ways o f a c h i e v i n g a n e u t r i n o mass.  can  component  Therefore,  a Majorana  L i [ 5 4 ] h a v e i n v e s t i g a t e d some e x t e n s i o n s  theory  fermion>sector.  excluded.  theory.  minimal SU(2)XU(1) theory, mass. T h e r e a r e  is  a g l o b a l symmetry c o r r e s p o n d i n g  (L) c o n s e r v a t i o n ,  i n the minimal  field  since  in  the  not  The the  rightexcluded.  will  generate  o n l y M a j o r a n a mass t e r m s . Thus t h i s e n l a r g e m e n t must n e c e s s a r i l y violate  lepton  enlargement  number  i n the Higgs s e c t o r  In the H i g g s triplet  H,  singlet H Higgs  + +  conservation.  sector,  a  singly  can  result  the  charged  We  discuss  e x a m p l e s of  first. addition  of  either  an  s i n g l e t ~L* , o r a d o u b l y  i n a mass t e r m f o r t h e  SU(2) charged  n e u t r i n o s . For  the  triplet,  (2.2.55)  the  f o l l o w i n g a d d i t i o n a l Yukawa c o u p l i n g o f t h e  leptons  is  Higgs  and  the  allowed.  (2.2.56)  29  Here,  f^  is  Higgs t r i p l e t  <T  a n t i s y m m e t r i c . When t h e n e u t r a l component o f t h e  develops  • W>  a vev such  =' W„  0  that  O j  (2.2.57)  t h e Yukawa H i g g s - l e p t o n c o u p l i n g ( 2 . 2 . 5 6 ) r e s u l t s mass t e r m  in a  Majorana  f o r t h e n e u t r i n o s g i v e n by  d . t  ^ 1  \  L  (2.2.58)  T^^b-  (2.2.59)  where =  If  a  minimal  singly  charged  Higgs  singlet  i s added t o t h e  SU(2)x>U(1) m o d e l , t h e a l l o w e d Yukawa c o u p l i n g  is  given  by  <£ where Zee  -  y  f^  b  iac ^bj ^£ L  i t  generate  J  is  the s i n g l e t  charged.  (2.2.60)  +in.c.  Higgs  Therefore,  T h i s m o d e l was s t u d i e d by cannot  develop  a  fm.  the following t r i l i n e a r <P.L  7 T e  C  j  vev  s o m e t h i n g more i s n e e d e d t o  a mass f o r t h e n e u t r i n o s . I f a s e c o n d H i g g s d o u b l e t  added, then  where  L  a n d e.- a r e a n t i s y m m e t r i c .  [ 5 5 ] , Of c o u r s e ,  since  L  is  scalar coupling i s allowed. (2.2.61)  (p, a n d (p a r e t h e two H i g g s d o u b l e t s . T h i s c o u p l i n g , when 2  30  taken  with  the Higgs-lepton  l e p t o n number. T h e r e f o r e ,  Yukawa c o u p l i n g , d o e s n o t c o n s e r v e  n e u t r i n o s can a c q u i r e a Majorana  t h r o u g h a one l o o p d i a g r a m s u c h a s t h a t shown i n F i g u r e  mass  3.  Q«P>o  >  v.  Fig.  3:  1  «—  /  0  Neutrino Majorana S i n g l e t jv*  ii**  singlet  cannot a c q u i r e  ^—<— +  Mass  J u s t as i n the l a s t case, Higgs  «  (  v  c  Contribution  due  to  the a d d i t i o n of a doubly  Higgs  charged  i s n o t enough t o g e n e r a t e a n e u t r i n o mass; i t  a v e v . H o w e v e r , a n e u t r i n o mass c a n be  generated  through loop diagrams i f the Higgs s e c t o r i s f u r t h e r enlarged i n such  a  way  violates  as  to  a l l o w some t r i l i n e a r  l e p t o n number c o n s e r v a t i o n  scalar coupling  which  when j o i n e d w i t h t h e l e p t o n -  H i g g s Yukawa c o u p l i n g . The  simplest  expanding  the  right-handed Majorana  way  fermion  neutrino  to  generate  neutrino  mass  terms  by  s e c t o r of m i n i m a l SU(2)XU(1) i s t o add a  singlet  for  each  family.  Now,  a  bare  mass t e r m i s n o t r u l e d o u t by t h e S U ( 2 ) * U ( 1 ) s y m m e t r y .  Also, the neutrinos  p i c k up a D i r a c  doublet  j u s t as t h e charged l e p t o n s do. That i s , t h e  mechanism  Lagrangian  mass  term  v i a the  Higgs  31  (2.2.62)  becomes, upon t h e H i g g s d e v e l o p i n g  6 ^  where  <£h,  symbolically  = V ( i A [ 2 ) f O^b * i n Figure  A combination lead  to  equation  a  neutrino  field  ^  + V. c_ .  The  Dirac  mass t e r m i s i l l u s t r a t e d  possibilities  mass  term  of  a  neutral  described  the general  Also, the a d d i t i o n  and  (2.2.63)  2.  of the  (2.2.52).  neutrino  <  a v e v , ir /fai  of  Higgs  both  above  form g i v e n i n  a  singlet  will  right-handed  fu°  leads to a  M a j o r a n a mass t e r m when t h e H i g g s p i c k s up a v e v . In  grand u n i f i e d  direct  product  such  s t r o n g SU(3) t h e o r y  t h e o r i e s , a s i n g l e gauge as  group  G  (or a  GxG) d e s c r i b e s t h e u n i f i c a t i o n  and e l e c t r o w e a k  of the  SU(2) U ( 1 ) . (2.2.64)  T h e r e i s o n l y one gauge c o u p l i n g . The c o u p l i n g s three  low  energy  grand u n i f i c a t i o n choosing can  symmetry  groups  f o r each of  are related  g r o u p G. T h e r e i s a g r e a t d e a l  t o t h a t of the of  leeway  t h e H i g g s s e c t o r o f GUTS m o d e l s . T h a t i s , H i g g s  be i n t r o d u c e d w h i c h t r a n s f o r m  representation  of  Majorana  terms  mass  G  according  t o some  the  neutrinos.  Also,  in  fields  complicated  and, thus, can almost c e r t a i n l y for  the  introduce  some  (eg.SO(lO)) [56,57] n a t u r a l l y accomodate a r i g h t - h a n d e d  groups neutrino  32  field,  allowing  a  Dirac  mass t e r m as w e l l  as a bare Majorana  mass t e r m . H o w e v e r , t h e m i n i m a l SU(5) room  in  the  neutrino. Also, number B=+1  (B-L)  i s conserved.  Similarly,  a n t i l e p t o n s and mass  term  Thus, i n course,  for  the  for other  and  Clearly,  SU(5),  (which  the  i t can  be  into  be  the  SU(2)XU(1) model, the charged i t s extensions,  terms  are  SU(5)  the  as an  GUT  fermion  t h e m a g n i t u d e s of t h e The  yields  terms f o r the  for  a  for  Majorana  i s forbidden.  the  Of  a d d i t i o n of singlet  or  incorporated  a of  into  the  neutrino  masses.  masses a r e resulting  arbitrary.  n e u t r i n o mass the  masses. as  S U ( 2 ) A S U ( 2 ) K U(1) U  n e u t r i n o can  instance,  the unacceptable  u q u a r k s and  L=-1  same i s t r u e f o r t h e c a s e of  symmetric models such  SO(10),  have  o f t h e m a g n i t u d e o f t h e mass. I n  SO(10) a r i g h t - h a n d e d In  assigned  massless.  SU(5) be  lepton  particles  remain  e x t e n d e d by  extended to include neutrino  naturally. typically  the q u e s t i o n  also arbitrary.  In l e f t - r i g h t  minus  B-L)  for generating  next  or  violates  have  SU(2)xu(1).  mechanisms e x i s t  discuss  not  right-handed  Therefore,  neutrinos  We  GUT  a l l . other  neutrino which transforms  j u s t as  a  number  H i g g s f i e l d s . T h u s , a n e u t r i n o mass c a n  SU(5)  In  for  baryon  particles.)  m i n i m a l model can  right-handed  does  c a r r y l e p t o n number L o f +1,  the n e u t r i n o  minimal  simply  (Baryons are c o n v e n t i o n a l l y  h a v e B=-1  leptons L=0  [1-3]  representations  in minimal SU(5),  while antibaryons  B=0.  new  particle  GUT  the  situation  neutrinos  accommodated  Higgs of e q u a l  [58].  ways o f a c h i e v i n g s m a l l p h y s i c a l m a s s e s . One  be  R  There  mechanism Dirac  mass  are  some  possibility  i s that  33  the  H i g g s v e v t h a t y i e l d s t h e D i r a c n e u t r i n o mass i s v e r y  [58].  T h i s can occur  different  component  masses. ( T h i s occurs rather  i f the neutrino  artificial  of  the Higgs f i e l d  and L a n g a c k e r  for  matrix mass for  their  of a very  situation  means  of  final  To u n d e r s t a n d  c o r r e c t i o n s ' [58].  l a r g e Majorana  of a very  explain  this  generation estimate  Other  f o r the  Majorana  small  value  mechanism i n  f o r t h e p h y s i c a l n e u t r i n o mass.  t h i s mechanism of e n s u r i n g  a  small  physical  mass, we must r e t u r n t o t h e g e n e r a l n e u t r i n o mass t e r m  content.  proceed i n terms of a s i n g l e  We w i l l  simplicity.  I t i s important  t o note the  L  i/£ a r e weak s i n g l e t s .  doublets  T h u s , t h e D i r a c mass t e r m  s i n g l e t a n d a l e f t - h a n d e d member  as f o r charged fermions. right-handed  fermion  lepton family f o r  B o t h i / a n d v£ a r e members o f S U ( 2 )  a right-handed  the  is  o f t h e l a r g e M a j o r a n a mass a n d ,  (2.2.52).  just  mass  large right-handed  of t h e L a g r a n g i a n  and  fermion  b e l o w b u t f i r s t n o t e t h a t t h e v a r i o u s schemes d i f f e r  in their  neutrino  a  a s m a l l p h y s i c a l n e u t r i n o mass a r e  t h e p h y s i c a l n e u t r i n o mass. We  thus,  the other  a t y p i c a l D i r a c mass y i e l d s an a c c e p t a b l y  more d e t a i l in  than  by  weak s i n g l e t n e u t r i n o . D i a g o n a l i z a t i o n o f t h e mass  for this and  generated  [ 3 ] p o i n t s o u t t h a t t h e scheme  radiative  ensuring  b a s e d on t h e g e n e r a t i o n right-handed  is  f o r t h e (120) of Higgs i n SO(10).) T h i s  c o u l d be u p s e t by d i v e r g e n t possibilities  mass  small  of  while connects  a  doublet,  The t e r m i n 6, w h i c h we w i l l  M a j o r a n a mass t e r m , c o n n e c t s  left  and  call  right-  h a n d e d s i n g l e t s . We saw t h a t t h i s t e r m c a n a p p e a r a s a b a r e mass term  i f B-L i s n o t i m p o s e d . The t e r m i n (X ( l e f t - h a n d e d M a j o r a n a  mass t e r m ) c o n n e c t s  left-  and  right-handed  members  of  weak  34  doublets.  These  terms  occur,  as  we  have shown, f o r c e r t a i n  e x p a n s i o n s i n t h e H i g g s s e c t o r . The most g e n e r a l mass m a t r i x f o r one  family i s  (2.2.64)  If is  the Higgs sector  i s not extended so t h a t ( l i s  a t y p i c a l D i r a c mass t e r m  z e r o and  (^ 1 G e v / c ) w h i l e  if $  then the  2  (ft /&). 2  e i g e n v a l u e s o f t h e mass m a t r i x a r e , a p p r o x i m a t e l y , & a n d The  latter  (small) eigenvalue corresponds to  ordinary physical Gell-Mann,  the  mass  of  neutrino. Ramond, a n d S l a n s k y [ 5 9 ] p r o p o s e d i n t r o d u c i n g a  H i g g s r e p r e s e n t a t i o n w h i c h b r e a k s B-L ( a ( 1 2 6 ) i n S O ( 1 0 ) ) large  mass  scale  M a j o r a n a mass masses  of  scheme o f  in  term.  the  a  symmetry  GUT,  Their  order  representation  of  scheme 5  2  in  a  SO(10)  of  His  at  a  large right-handed  typically  1 0 " eV/c .  breaking  f o r which  allowing  yields  neutrino  Witten [ 6 0 ] suggested a via a  spinor  Higgs  t h e r e i s no t r e e l e v e l M a j o r a n a  b u t a l a r g e r i g h t - h a n d e d M a j o r a n a mass o c c u r s a t level.  the  the  two  mass loop  scheme y i e l d s p h y s i c a l n e u t r i n o m a s s e s o f t h e o r d e r  eV/c . 2  Mohapatra the  order  of  SU(2) A  SU(2)K  which  are  L  and S e n j a n o v i c [ 6 1 , 6 2 ] MeV/c  U( 1  in  2  left-right  symmetric  ) . They i n t r o d u c e two H i g g s f i e l d s  triplets  under  w e l l a s an S U ( 2 ) X S U ( 2 ) L  their  o b t a i n n e u t r i n o masses  K  SU(2)  U  and S U ( 2 ) ,  H i g g s d o u b l e t . They  R  A  L  of  theory and  r e s p e c t i v e l y , as  break  the  parity  35  and  B-L  symmetries  by l e t t i n g  p i c k up a n o n z e r o v e v o f t h e  o r d e r o f t h e mass o f t h e r i g h t - h a n d e d gauge b o s o n bound  on  this  experiments light one  mass f r o m c h a r g e d  A  lower  [63] and n e u t r a l [64] c u r r e n t  i s a b o u t 300 G e V / c . Then t h e mass o f 2  neutrinos  W^.  the  ordinary  i s i n v e r s e l y p r o p o r t i o n a l t o t h i s mass a n d , i n  c l a s s of models  which  Mohapatra  and  Senjanovic  discuss,  t a k e s the form Tn.y o<. -m^ L  This  gives,  for  / -vrb ^  (2.2.65)  w  example,  a t a u n e u t r i n o mass o f t h e o r d e r o f  Mev/c . • 2  These  theoretical  estimates  for  the  n e u t r i n o mass a r e v e r y l o o s e , r a n g i n g f r o m The  question  of  10"  5  eV/c  f e r m i o n masses i s not a t a l l w e l l  Thus, i t i s important accessible  magnitude  situations.  t o s t u d y n e u t r i n o mass  in  2  of  the  t o Mev/c . 2  understood.  experimentally  36  2.3 The E f f e c t i v e F o u r - F e r m i o n T h e o r y o f Weak I n t e r a c t i o n s  We  saw  i n the last  of e l e c t r o w e a k transfer  s e c t i o n how t h e S U ( 2 ) A U ( 1 )  i n t e r a c t i o n s reduced,  much  smaller  than  weak  e f f e c t i v e V-A f o u r - f e r m i o n  theory  not  the  the h i s t o r i c  Originally, analogy  route  Fermi  i  current  (j )^.  The  w  electromagnetic  n  vector  boson  mass,  effective  t h e qed L a g r a n g i a n role  process  of  vector  invariant  the  was g i v e n  Lorentz  four-fermion  He  %i  f  1jJ a r e t h e f i e l d in  terms  A p p e n d i x A. The g extends  over  was  theory.  f o r ^6-decay i n  replaced  the  vector  ( « j a ^ A ) by a weak v e c t o r emitted  photon  in  an  t o the e l e c t r o n - a n t i n e u t r i n o (point) four-  such Lagrangian  transformations  &°tl\v,y U%ri -^ )^-]  given  t o an  interaction.  under proper  r  momentum  o f weak i n t e r a c t i o n s . T h i s  F o r muon d e c a y , t h e more g e n e r a l  +  operators  scalar  , .3.,, 2  p a r t i c l e s and  of c r e a t i o n and a n n i h i l a t i o n  sixteen linearly  ( V ) ,tensor  (P) c o u p l i n g s . These  operators i n summation  (T), axial  are  i n d e p e n d e n t 4x4 m a t r i c e s  is  below.  u.c.  f o r the i n t e r a c t i n g  ( S ) ,vector  which  i s given  a n d g' a r e c o u p l i n g c o n s t a n t s . The  (A), and p s e u d o s c a l a r the  of  o f t h e weak d e c a y . Thus he w r o t e down a l o c a l  fermion  are  limit  w i t h quantum e l e c t r o d y n a m i c s . (je.™^.  The  i n the  [ 1 4 ] w r o t e down h i s L a g r a n g i a n  current  pair  to  gauge m o d e l  vector  represented  listed  below.  by  37  P  In  the  Lagrangian  of L o r e n t z merely  ( 2 . 3 . 1 ) , t h e u s u a l c o n t r a c t i o n and  indices  convention  i s t o be assumed  t o w r i t e the c o u p l i n g index  That these o p e r a t o r s a r e indeed i n t h e f o l l o w i n g way. C o n s i d e r s i x t e e n o p e r a t o r s such  Now the  f o r the m a t r i c e s  linearly  some l i n e a r  ("^ . I t  i as a  independent  summation is  subscript. c a n be  seen  s u p e r p o s i t i o n of  the  that  m u l t i p l y i n g w i t h one o f t h e s i x t e e n m a t r i c e s P  and t a k i n g  trace,  L  ^ T  ( P  r  m  =  o  i- i The s i x t e e n m a t r i c e s the  identity,  P  .  m a t r i c e s f o r which i=j .  Thus  Since  P P  a l l the  =1  P  a l l have v a n i s h i n g  In f a c t ,  trace,  the only products  the t r a c e i s not zero  are  o f two o f  those  ( i = 1 ,...,16), the l a s t equation  coefficients  c  v a n i s h and t h e  except  P  for  for these which  becomes  are, indeed,  38  linearly  independent.  The  Lagrangian  (2.3.1)  is  in  current"  f o r m . The p a r t i c u l a r  a product  o f two n e u t r a l c u r r e n t s ; t h i s  retention  form g i v e n  form of the f o u r - f e r m i o n  We w i l l  show l a t e r  (charge  exchange) and t h e l i n e a r  important Some  in  of  the  electrodynamics, That  Gamow a n d T e l l e r should  also  i s , only  interaction  [66] noted  be  Lagrangian.  form  couplings g a  The  g  be  included g  v  to  weak  1932  describe-decay In analogy  only  the  was n o n v a n i s h i n g .  (W)  in  with vector  T h i s was t h e  Lagrangian.  In  1936,  in At  a  more g e n e r a l  this  point,  f o r m o f t h e weak  the  Lagrangian  was  ( 2 . 3 . 1 ) by Feynman a n d  such  as  described  by  Gell-Mann  [67].  i n t e r a c t i o n s and  parity  Fermi's  conserving  original  weak  e i t h e r a l l o f t h e g- o r a l l o f t h e g j must v a n i s h . L  parameters. parameters  form  will  neutrino  (2.3.1).  t h u s , no l o n g e r p a r i t y c o n s e r v i n g . F o r a  Lagrangian,  another  a n d g' i s i n o r d e r .  now i n c l u d e s , f o r i n s t a n c e , VA t y p e  interaction,  Lagrangian.  The weak i n t e r a c t i o n was f u r t h e r g e n e r a l i z e d  t h e form of e q u a t i o n  This  in  charge  t h a t SS, AA, TT, a n d PP i n t e r a c t i o n s  included  parity-conserving.  (2.3.1) i s  independence of the H  he  vector current-vector current  is,  rewritten  [ 1 4 ] w r o t e down h i s L a g r a n g i a n  interaction.  to  weak i n t e r a c t i o n  be  "current-  i s the s o - c a l l e d  t h e c u r r e n t - c u r r e n t form of e q u a t i o n  quantum  i n equation  had p r o p o s e d t h e e x i s t e n c e of  Fermi  so-called  t h e two f o r m s .  discussion  After Pauli [65],  that i t can  in relating  the  a n d g ^ , b e i n g c o m p l e x numbers, f o r m a s e t o f 20 r e a l One to  overall  phase  is  arbitrary,  be d e t e r m i n e d by e x p e r i m e n t .  leaving  The e n e r g y  19  spectrum  39  f r o m yti*  of t h e p o s i t r o n s calculated expressed  f o r the  decay  case  (Michel  spectrum)  V  A  * lo  A V  been  of m a s s l e s s n e u t r i n o s p r e v i o u s l y and  i n terms of t h e f o l l o w i n g parameters  $ = ~ ^ (3( b  has  [68,69].  + 6 b ")  ^  TT  where  K=a„ + a The The  t  k ? p  parameters  H-(a „ + a v  symmetries  in  the  appear energy  A  W  above  T T  the  Michel  parameters.  f o r them were o b t a i n e d w i t h o u t i m p o s i n g  on t h e L a g r a n g i a n . The ^ a n d 7^ p a r a m e t e r s  isotropic  part  of  of  the positron  e n d . ^ , a s an o v e r a l l  factor  appear  t h e M i c h e l spectrum w h i l e ^ and S  i n t h e a n g u l a r dependence. end  6 a  ^>,7^,^, a n d 6 a r e c a l l e d  expressions  any  A  ^ is  sensitive  to  the  high  spectrum as ^ i s t o t h e low energy multiplying  the  angle-dependent  p a r t o f t h e s p e c t r u m , g i v e s an e n e r g y - a v e r a g e d a s y m m e t r y w h i l e 6 g i v e s t h e energy dependence of t h e asymmetry. Experiments  on  measurement o f t h e  muon  Michel  decay  have  parameters.  been m a i n l y f o c u s s e d on There  are  experiments  40  under  way a t t h e meson f a c t o r i e s t o m e a s u r e p r e c i s e l y  ^parameters  [ 7 0 , 7 1 ] . However, t h e f o u r  sufficiently  complicated that  a g i v e n measurement i m p l i e s constants.  There  are  i tisdifficult  in  not  Michel  terms  enough  of  t h e ^ and  parameters  are  to interpret  what  the  basic  measurable  coupling  quantities  d e t e r m i n e a l l 19 r e a l p a r a m e t e r s , anyway. The s i t u t a t i o n c a n somewhat  simplified  by  a l l of  the  coupling  invariance  t o be d e t e r m i n e d . A l s o ,  rather a r b i t r a r y assumptions  ^ [ ^ e Y  ?  ( l ^ W p C ^ r O - Y  We saw i n t h e l a s t the l i m i t V-A  This  ^ J  form.  (2.3.2)  s e c t i o n how t h e S U ( 2 ) * U ( 1 )  happened  s t a t e s . For the  M i c h e l parameters  because  theory reduced, i n  case  t h e l e p t o n s were a s s i g n e d t o  of  take the values  this  this  section  use  the  V-A  interaction,  the  below.  The c a l c u l a t i o n s o f muon d e c a y follow  i f the  o f momenta much s m a l l e r t h a n t h e weak b o s o n mass, t o a  form.  helicity  s  real,  a r e imposed  t h e n t h e L a g r a n g i a n t a k e s on t h e f a m i l i a r V-A  £=  implies  c o n s t a n t s c a n be t a k e n t o be  l e a v i n g only t e n r e a l parameters following  be  i m p o s i n g some symmetry r e s t r i c t i o n s on  the L a g r a n g i a n . F o r i n s t a n c e , time r e v e r s a l that  to  V-A  w i t h massive n e u t r i n o s which form  of t h e L a g r a n g i a n t o  41  d e s c r i b e t h e weak i n t e r a c t i o n . renormalizable. interactions, resolves local  w h i c h was r e v i e w e d  theory, but by  Chapter where  SU(2)X"U(1)  rather  the  intermediate  briefly  weak  energies  Lagrangian  theory in  of  the  i s not  electroweak last  section  That theory  i s not a  interactions  are  therein  v e c t o r b o s o n s . As was p o i n t e d o u t i n  1, f o r t h e c a s e o f muon d e c a y w i t h a l l the  massless  neutrinos,  o f t h e p a r t i c l e s a r e much s m a l l e r  than  i n t e r m e d i a t e v e c t o r boson masses, t h e d i f f e r e n c e between t h e  results  o f an e f f e c t i v e V-A c a l c u l a t i o n  calculation result  i s expected  the e f f e c t s rather  adopted  full  to  SU(2)XU(1)  i n s i g n i f i c a n t . This  t o be t r u e f o r t h e c a s e o f m a s s i v e n e u t r i n o s , i n Chapter  1, t h e a i m h e r e i s  probe  the  t h e e f f e c t i v e V-A  of  weak  study  on muon d e c a y  s t r u c t u r e o f t h e weak theory  to  interaction.  interactions  is  henceforth.  It  was  mentioned e a r l i e r i n t h i s s e c t i o n that the general  weak L a g r a n g i a n so-called  £ =% [ %  ( 2 . 3 . 1 ) c o u l d be r e w r i t t e n i n a n o t h e r  charge exchange form i s g i v e n  r  r  t  i s obtained  ^ ] [ f  e  r  (5.-3/ v > % 3  upon p e r f o r m i n g  the charge r e t e n t i o n form of interchanging linear  a  o f massive n e u t r i n o s and t h e i r m i x i n g s  than  Therefore,  and  were f o u n d t o be e x p e r i m e n t a l l y  as w e l l . A l s o , as noted  It  gauge  t h e problem of r e n o r m a l i z a b i l i t y .  mediated  the  The  This effective  e  a Fierz the  This  below.  vwc.  (2.3.3)  transformation  Lagrangian.  two o f t h e i n t e r a c t i n g  form.  This  [ 1 5 , 7 2 ] on refers  to  p a r t i c l e s a n d amounts t o a  t r a n s f o r m a t i o n of the c o u p l i n g c o n s t a n t s , which i s  given  42  by  hi  H  6  H  \  -a.  0  X  -\  0  -a.  o  \  1  0  -oL  -\  -4.  fo  -4  I /  where i , J? = S,V,T,A,P ( i n t h a t o r d e r ) The  transformations  They  are  obtained  Lagrangian  above as  are  follows.  known a s t h e F i e r z Consider  one  term  identities. of  the  (2.3.1)  where  Now  the  c a n be r e g a r d e d a s s i x t e e n  components <*•% , M ^ y j  are  , whose  l a b e l l e d by t h e i n d i c e s /5 a n d % . T h e n , f o r e a c h  c a n be e x p r e s s e d  independent  4x.4 m a t r i c e s  matrices  P J  J  in  terms  as f o l l o w s .  of  the  sixteen  linearly  43  Then  the above term  o f t h e A<  of the L a g r a n g i a n  can  be  r e w r i t t e n i n terms  as  J  The  minus  sign  anticommuting. i n the  appears Upon  Fierz-rearranged  that,  since  the  Lagrangian c o u p l i n g s g.  the  index  j  summation i s i m p l i e d . M u l t i p l i n g t a k i n g the t r a c e  field  operators  comparison w i t h the the c o r r e s p o n d i n g  i d e n t i f i e d w i t h t h e new  Note  since  is  (2.3.3),  the  A  are term  can  be  index,  no  i n t h i s manner  not  a  by one  Lorentz of  the  V  K  and  yields  0  The  last  products  More  step  follows  of the m a t r i c e s  explicitly,  from V  the  p r o p e r t i e s of t h e t r a c e s o f  . Rearranging  yields  44  Substituting  S,V,T,A,  transformation  matrix  i d e n t i t y and t h e second For  the  V-A  and  P  for  k  results  in  the  Fierz  g i v e n above. T h i s proves the f i r s t  Fierz  identity i s obtained s i m i l a r l y .  assumptions,  the  results  of  a  Fierz  transformation are  1  The  (~ % x  + 2-3  A)  p r i m e d c o n s t a n t s t r a n s f o r m i n t h e same way. T h u s , t h e c h a r g e  e x c h a n g e f o r m o f t h e V-A L a g r a n g i a n i s  (2.3.4)  Choice of e i t h e r the  Lagrangian  convenience.  t h e c h a r g e r e t e n t i o n o r charge exchange form of for  calculating  i s merely  a  matter  of  45  I I I . MUON DECAY WITH MASSIVE  NEUTRINOS  I n t h i s C h a p t e r , we make t h e a s s u m p t i o n t h a t n e u t r i n o s massive  and  related  by  that a  Introduction.  the  unitary  mass  a n d weak n e u t r i n o e i g e n s t a t e s a r e  transformation,  as  decay  (1.2)  for  n e u t r i n o s . The s p e c t r a are  the  two  cases  g i v e n . We s p e c i a l i z e  of  t o the three-neutrinos retain only  f o r n e u t r i n o mass i n t h e M e v / c  with the experimental also  limits.  or Majorana  of  2  the  world  muons  and, w i t h  one  neutrino  Numerical  results  range, i n accordance  Electron-neutrino correlations are  i n v e s t i g a t e d a s a means o f  are D i r a c  spectrum  D i r a c and Majorana  as m a s s i v e enough t o c o n t r i b u t e s i g n i f i c a n t l y . given  i n the  f o r b o t h p o l a r i z e d and u n p o l a r i z e d  the assumption of h i e r a r c h i a l m i x i n g ,  are  described  Our a s s u m p t i o n o f n e u t r i n o mass i s i n d e p e n d e n t o f  any p a r t i c u l a r m o d e l . We c a l c u l a t e t h e M i c h e l muon  are  particles.  determining  whether  neutrinos  46  3.1  The C a s e o f D i r a c  We (1.2)  Neutrinos  calculate, i n this section, the rate  f o r the case of massive  neutrinos  and a n t i n e u t r i n o s  Dirac  quantum  Conventionally, antineutrino  number  the  neutrinos.  lepton  i s assigned  the This  number,  L=1  while  a L.  the  h a s L=-1. section  s e r v e a s a model  c a l c u l a t i o n s and, thus,  are presented  first  case of N lepton  take the general rate  i s calculated  oj. = J  <f I£  where t h e i n i t i a l (1.2).  i s ,  them d i f f e r e n t v a l u e s o f  called  neutrino  The c a l c u l a t i o n s o f t h i s  transition  That  are distinguishable particles.  f a c t c a n be r e p r e s e n t e d by a s s i g n i n g conserved  f o r t h e muon d e c a y  and f i n a l  detail.  We  f a m i l i e s . The f o l l o w i n g  f o r muon d e c a y  M  W  i n considerable  for later  a x I L>  (3.1.1)  4  states are  those  of  the  reaction  That i s ,  (3.1.2)  (3.1.3) The  operators  particles Appendix m  L  are  b'  and A.  and  d  1  are creation operators f o r physical  antiparticles,  For instance,  w i t h momentum p  u  respectively,  bV(p, ,s, ) c r e a t e s  a n d s p i n s . The s t a t e s  p h y s i c a l mass e i g e n s t a t e s .  L  as  presented  a neutrino (3.1.2)  in  o f mass  and(3.1.3)  The weak L a g r a n g i a n , oh t h e o t h e r  47  hand,  i s written  created of  at  i n t e r m s o f t h e weak e i g e n s t a t e s ,  a weak i n t e r a c t i o n v e r t e x .  t h i s Lagrangian  instance,  the  is  given  electron  r e l a t e d t o the physical  in  The c h a r g e e x c h a n g e  equation  neutrino  (2.3.4)  field  neutrino  the p a r t i c l e s  (weak  fields  by  form  where,  for  eigenstate)  the  NXN  is  unitary  transformat ion N (3.1 .4) Here  the U^  which s a t i s f y  (a=e,yU,T, ...) a r e t h e P o n t e c o r v o m i x i n g p a r a m e t e r s equations  physical eigenstates,  (1.4) and ( 1 . 5 ) . Thus,  v  transition  are  given  the  i n equation  initial  integration,  •j-  rate  and  of  l l  t'('-^l^-«--  (3.1.5)  ( 3 . 1 . 1 ) c a n now be c a l c u l a t e d . The (A.6).  Upon o p e r a t i n g  final  states  and  with  doing  the the  fields  fields  on  coordinate  the result i s  (3.1.6)  L  where,  the  t h e weak L a g r a n g i a n c a n be w r i t t e n a s  I^"..L^.tV«0-V* ]tlfc  The  i n terms  including a l l possible  physical  neutrino'states,  (3.1.7)  48  Here, u  and  v  are  the  four-component  particle  and a n t i p a r t i c l e s o l u t i o n s ,  equation.  (See A p p e n d i x  In g e n e r a l ,  respectively,  representing of the  Dirac  A.)  the process -  is  spinors  "  <*  ^  +  .. .  +  (3.1.8)  r e p r e s e n t e d by t h e t r a n s i t i o n  (3.1.9)  Here the s u b s c r i p t s  "i"  initial  and  per  time i s given  unit  final,  and  "f"  on  respectively.  the  four-momenta  The d i f f e r e n t i a l  denote  probability  by  (3.1.10) L  Thus,  R =  f o r the process  1-  ^  (1.2),  the decay r a t e i s  , T ^ ^ t e ( ^ f ^ ^  (  ^ - F ^ I pvyi  (3.1.1D  49  N o t i c e t h a t t h e n e u t r i n o s p i n s a r e summed o v e r not  detected  equation  i n experiments.  The s q u a r e  (3.1.7).  calculated  in detail  7VL - Z  \7VL \  The m a t r i x of  this  since  element matrix  they a r e  ^~ i s g i v e n i n ic  element  i s  i n A p p e n d i x A. The r e s u l t i s  4  - (^)U^)\a^\\x^  (p . e  ?  J  y ;.p  (3.1.12)  ?  wherein  '  We  (  p \ "  f x  rewrite equation  W  S  (3.1.13)  ^  (3.1.12) f o r convenience a s  i .IU^IU^TVUJ  n*(%y Also,  +  the Lorentz  s e e n more e a s i l y functions.  rate  4  i n v a r i a n c e of the expression  after  using the following  formal  ( 3 . 1 . 1 1 ) c a n be trick  with  6-  Since  .*(£-n^ = the  ,3.,., )  ( 3 . 1 . 1 1 ) c a n be w r i t t e n a s  (3.1.15)  50  21  W J  IU,.I  1  (3.1.16)  X  where Q = jy- ft The  (3.1.17)  e-functions  particle  ensure  i s positive. This  function given  that  the  energy  of each f i n a l  f o l l o w s from t h e d e f i n i t i o n  state  o f t h e 0-  below. (3.1.18)  From e q u a t i o n incoherent  sum o f N  combinations In  muon  are  integrated  method o f d o i n g integration  to  rates, corresponding  experiments  detected;  hence  i n the c a l c u l a t i o n  performed their  performed  to  momenta  the  N  2  date, the must  be  o f t h e muon d e c a y r a t e . The  so f o r t h i s c a s e i s g i v e n h e r e . be  to  state neutrinos.  decay  not  over  we s e e t h a t t h e r a t e i s a c t u a l l y an  different  2  of f i n a l  the  neutrinos  (3.1.16),  involves  the  Notice that  the  following Lorentz  structure.  = AQ^Q  V +  & Q V  V  (3.1.19)  51  Using  the o v e r a l l  i n t e g r a t e over  1^--  energy-momentum  one o f t h e n e u t r i n o ' s  the 6-function  6-function  four-momentum, we  Q^'^^yi^-y  j d ^ S(^-mfH((Q-f^-vH^e(^  Using  conservation  property  =  ^ 0 The  (3.1.21)  i s t h e root of f ( x ) , t h e i n t e g r a t i o n over  0  0 ( Q.  i 0 p t w^- ^ L  0-function  positive values  6(p ) L  of  has  been  I  0  A a n d B.  0  will  (3.1.22)  t  used  t o ensure that only the i n order  t o solve f o r the  t h e two  be c a l c u l a t e d . E a c h o f t h e s e  V  - (A*  integrals  ij?  y i e l d s an e q u a t i o n i n  WGf  L  Since any  X p r (Q- p O "  (3.1.19),  *(Q -aQ-fL + ^ < W ^ f J  in  i  gives  First  = ^ I "  1/  p  c o n t r i b u t e . Now,  p a r a m e t e r s A and B o f e q u a t i o n and  obtain (3.1.20)  %[tt%S\ ]W\^ where x  to  i s a Lorentz reference  (jVQ-™ °)  (3.1.23)  c  i n v a r i a n t q u a n t i t y , i t can  frame.  We c h o o s e t o e v a l u a t e  f r a m e o f t h e four-momentum Q. I n t h i s  frame,  be  evaluated  i t i n the rest  equation  (3.1.23)  becomes  Lh^$)0t-  Applying  ^Q  equation  i B  -iQ E t<-wv^e(Q -E VE Q ->i^ b  c  (3.1.21)  0  to  L  L  a  (3.1.24)  t h e 6 - f u n c t i o n and w r i t i n g t h e  52  measure as d f 3  L  where d £ l Av4P>=  = i-jE-t-v^  E dE aii , L  t  (3.1.25)  p  represents the angular  PL  TT HQ  integration,  the result i s  0 ( Q M ^ W ^  A '"(^rH^vn^(Q"m^-Mrt ,  (3.1.26)  4  where  M  M.* ~-  f •  V  ^  v  ^  '  ^  v  The p h a s e s p a c e c o n s i d e r a t i o n s w h i c h dealt  0  0  give  w i t h i n A p p e n d i x C. A s e c o n d e q u a t i o n  by c a l c u l a t i n g I  I  ( 3  0  the  - -  ©-function  1  2 7 )  are  i n A and B i s f o u n d  .  0  - (A-*- B") s( Q*~-2Q-^i  =  Evaluating this  integral i n the  following equation AtB= Solving results  %  X  , / i  +y*?-nj] 6 (Q- p ) t  rest  frame  £ ( Q*- ^  of  Q  yields  the  i n A a n d B.  (q>^ Mf-VQ*+vv^^ynVVQ -vv^^^^^(q^U mrt  equations  (3.1.  L  t  |  :  (3.1.29)  (3.1.26) and (3.1.29) f o r A and B y i e l d s t h e  below.  A- J L x W q ^ ^ w - ' U Q V Q ^ ^ t ^ V i ^ b ^ S l  (3.1.30)  53  Of c o u r s e , reduce  f o r t h e case of  t o the usual values A =.  ir/ia.  Substituting  X  " *  =  ^  [  v  +  these  tt/a4 (3.1.19)  (3.1.32)  yields  Q^t™?) i « - ^ r t erg* T  +[Q- I Q U ^ > W W - - * ^ H  results  [15] of  into equation  > ^  q  neutrinos,  B=  back  i ^  massless  I  (3.1.33)  Therefore,  •Q "(f -p A 2  t  For relations  A  e(Q '- ^ i t w ^ l  muons  (3.1.34)  a  decaying  at  rest,  the  following  kinematic  are useful.  Gf= ^ h - X t S e ^  (3.1.35)  (^-QV  (3.1.36)  V a - S ^  ( f ^ Q V ^ ( l - X ' A )  (3.1.37)  54  (3.1.38) (3.1.39) where (3.1.40) (3.1.41) (3.1.42) and p  e  i s a unit vector  momentum,"p . I n t h i s £  are  i n the d i r e c t i o n of the p o s i t r o n ' s  notation,  the nine  differential  decay  three rates  55  ^  - ^ ^ ^ M ^ V \ ^ i e ^ S p l ] e ^ - X ^ M ^ ^  N e g l e c t i n g t h e e l e c t r o n mass and Michel  summing o v e r  s p e c t r u m f o r p o l a r i z e d j**  1- i t l - K^- A( I- ^ ^ ^ f  +(6^-^  a :  (3.1.43)  electron spin,  is  l c _ ^ ^_ ^ - ^ © f X - C ^ t : ^ ^ l  the  x  (3.1.44)  where ^For  COSG  ( 3 . 1.45)  e  the case of m a s s l e s s  neutrinos,  g i v e s the u s u a l M i c h e l spectrum The many  parameters  specialize world  general  N to  of  (3.1.44)  obtained here c o n t a i n s  meaningfully.  to the phenomenologically  (N=3). We  reduction  [15,16],  neutrino result analyze  a  Therefore,  interesting  assume t h a t t h e m a s s e s m>  are  three  too we  neutrino  i n ascending  order  56  of  i/..  that  T h a t i s , m <m <m . C u r r e n t e x p e r i m e n t a l mass 1  14<m <46 e v / c 2  [25]. I f v  &  families,  then  This assumption  i/,,  •m  [ 2 3 , 2 4 ] , and  similarly  to choosing  matrix  to  for and  7  be  is  6,  largest.  supported  on m i x i n g a n g l e s and  6  2  can  be  are  m <250  the  v  other  6 <4.8X10~ . 3  2  the d i a g o n a l elements  i n the l e p t o n s e c t o r but  t r u e i n many GUTS m o d e l s and  need t o be  and  2  1  verified  Under t h i s a s s u m p t i o n , 3  v  e t c . so t h a t 6 < 4 X 1 0 "  1 f  mixing  experimental evidence  6  m <.5 M e v / c  [21,22],  i s mostly  m_£m  limits  3  corresponds  Pontecorvo  experimentally be  2  v  Mev/c  the  2  It  by  analogy  not to  with  sector [73],  n e g l e c t e d . Only can  is  i s expected  i n the quark  k e p t . Then t h e M i c h e l s p e c t r u m  of  terms i n  be w r i t t e n  as  3  t ( 3 . 1 .46) where (3.1.47)  M( l-x Y  5  (3.1.48)  57  + a - 0 ( l - X - « H f > ( »-^-"|tVl e C i - x - H S ^ The  factor  neutrinos. 6 -*0.  R  It i s easily  Then,  3  gives the standard  0  by  the  parameters, equation  (3.1.49)  M i c h e l spectrum w i t h  seen t h a t R  and R  3  orthonormality  (3.1.46) reduces  3 3  massless  reduce t o  R  when  0  c o n d i t i o n s of the mixing to  the  standard  Michel  spectrum. In  general,  the  Pontecorvo  undetermined. Corresponding  elements of the mixing can  be  estimated  angles,  i n analogy  hierarchial  are  mixing;  I n GUTS, t h e  mixings  t o t h e e s t i m a t i o n of quark particular the  e  t o assume t h a t t h e d i a g o n a l  matrix are largest.  i n terms of masses. T h i s  called  parameters  t o t h e a s s u m p t i o n made a b o v e t h a t i /  v,, e t c . , i t i s r e a s o n a b l e  i s mainly  mixing  values  estimate of  the  mixing  will  be  neccessary  parameters a r e  HAtJ 1  lUUil  1  ^ t / i y =.00^  - "^e />vL-c =-0003  l U ^ P ^ Therefore,  (3.1.50a)  ^ / Y H  X  -  (3.1.50b)  .0^  (3.1.50c)  f r o m t h e o r t h o n o r m a l i t y c o n d i t i o n ( 1 . 5 ) , we g e t lU^f'-.SH  Numerically,  R  3  and  R  (3.1.51)  3 3  are  comparable;  however,  the  58  contribution strongly  of  the  The  by  to the Michel  spectrum  s u p p r e s s e d by t h e f a c t o r s o f m i x i n g  case of h i e r a r c h i a l  very  latter  parameters f o r the  mixing.  hierarchial  mixings  are really  o n l y a g u e s s . Some n o t  s t r i n g e n t c o n s t r a i n t s c a n be p u t on t h e looking  at  for m  3  the  Pontecorvo  (3.1.46) i s  p i o n d e c a y s . The a n a l y s e s  i n t h e Mev/c  2  mixing  o f TT 2 d e c a y s  Ue3 . F o r m  l e s s than  3  [30,74]  e  r a n g e , i n d i c a t e an u p p e r l i m i t  parameter  parameters  of  .02 f o r  .5 M e v / c , no 2  good c o n s t r a i n t s e x i s t . A l s o , t h e d e c a y Tf—y>i_i/ h a s been and  h i e r a r c h i a l mixings v.  neutrino being the  muon  give a very  small  The most r e c e n t  3  spectrum  performed  search  at  SIN  probability  o f f i n d i n g a 6-11 M e v / c  less  1%.  than  T h u s , m <6 M e v / c 3  v  are  2  2  hierarchial  i f lu  l >.06 2  /L3  probability  sets  4  a  neutrino  mixing,  displayed relative  and  limit  in this  on t h e decay  one e x p e c t s  of  a b o u t 6%.  [32].  t o the Michel  d e c a y w i t h m a s s l e s s n e u t r i n o s . The m i x i n g s Figure  of t h e  f o r secondary peaks i n  4 [ 7 5 ] , t h e e f f e c t s o f a heavy v  In F i g u r e and  For  2  analyzed  3  mixing  into  v,  s p e c t r u m f o r muon  are  hierarchial  in  t h e muon i s u n p o l a r i z e d . The d i f f e r e n c e between  m a s s i v e and m a s s l e s s c a s e s i s g r e a t e s t a t t h e h i g h e n e r g y end o f the p o s i t r o n spectrum. T h i s h i g h energy r e g i o n Figure are  5  [ 7 5 ] . The e f f e c t s f o r s e v e r a l d i f f e r e n t  shown; f o r a 1 M e v / c  shows  up  i s magnified  2  neutrino,  f o r instance,  in  values  of m  the  effect  3  only  after  x>.99. T h a t t h e l a r g e s t e f f e c t o f m a s s i v e  neutrinos  should  occur  at the high  spectrum  i s expected.  neutrinos  predicts  that  The  energy  standard  the Michel  V-A  end  of  theory  spectrum  the  Michel  for massless  peak  at  x=1  59  (neglecting  electron  conservation  makes i t most f a v o u r a b l e  out  with  depicted as  maximum  This  available  i n Figure  massive  mass).  energy.  should  energy from t h e p o s i t r o n t h a t from  the  on t h e m i x i n g  3  to  come  v i o l a t i n g mechanism  and, indeed,  such  d o e s t a k e away some  otherwise  spectrum  2  3  momentum  These c o n s i d e r a t i o n s a r e  i t would  Michel  d e p e n d s on ( m / n y , ) , where m and  f o r the positron  6 [76], A c h i r a l i t y  neutrinos  difference  i s because angular  obtain.  The  with massless  neutrinos  i s t h e mass o f t h e h e a v y  neutrino,  p a r a m e t e r s b e t w e e n t h e h e a v y a n d t h e two l i g h t  neutrinos. In F i g u r e the  7 [ 7 5 ] , t h e dependence of t h e M i c h e l  Pontecorvo mixing  Here,  U  3  l u ^ l =.2  and  2  3  U^  0 f o ra l l curves.  M e  illustrates  very  the  effects  of  The h i e r a r c h i a l  are d i s p l a y e d along Evidently  decay  seems  at the from  of  positron these  virtual  e 3  corrections  values  U^  3  from  [75]  on t h e M i c h e l  value  massless  8  of  U^  3  is  D*e3 = U/x3  neutrinos  case.  strong. unpolarized  muon  t o be a way o f d e t e c t i n g t h e  However,  electromagnetic  radiative  Figure  and t h e c h o i c e  spectrum  results  no f i r m c o n c l u s i o n s  4.  x>.95.  Ue.3. i s n o t v e r y  Therefore,  Chapter  U  and t h e  of the spectrum t o  t h e parameter  of  e f f e c t s of a massive n e u t r i n o . spectrum  until  with the standard  t h e d e p e n d e n c e on  Looking  2  3  the h i e r a r c h i a l  value  on  i s shown f o r m = 5 M e v / c .  yLL3  H i e r a r c h i a l values  pronounced  spectrum. In a l l the curves, taken.  U  .5 a r e shown. The s e n s i t i v i t y  i s not  3  parameter  spectrum  the  effects  corrections  c a n made f r o m t h e s e  to the Michel  on  are  this large.  results.  The  spectrum a r e c a l c u l a t e d i n  60  Fig.  4:  The M i c h e l s p e c t r u m f o r u n p o l a r i z e d /•* d e c a y . The s o l i d c u r v e i s f o r m a s s l e s s n e u t r i n o s . The d a s h e d curve i s for m =l0 MeV/c ; the dash-line curve i s for m =20 M e V / c . 2  3  2  3  61  0.26  0.25  0.24 X JO GC  0.23  Wi — • - T«t = 10  0.22  i  0.21 0.82 Fig.  _L  0.9 X  0.95  1.0  5: The h i g h - e n e r g y end of t h e M i c h e l spectrum with hierarchial m i x i n g . The s o l i d curve i s f o r massless n e u t r i n o ; t h e d a s h - d o t c u r v e i s f o r m =3 M e V / c ; t h e dashed curve i s f o r m = 5 MeV/c ; t h e s q u a r e - l i n e curve i s f o r m =l0 MeV/c ; the t r i a n g l e - l i n e curve i s for m =20 M e V / c . 2  3  2  3  2  3  2  3  62  (a)  Allowed  —_ e  Fig.  (b)  —  «  6:  v  *- v  —  Forbidden  z - T  S  Helicity considerations f o r t h e M i c h e l s p e c t r u m o f jx* decay with massless neutrinos. Angular momentum conservation allows t h e c o n f i g u r a t i o n (a) and f o r b i d s (b).  Another case of r e l e v a n c e i s t h a t of t h e decay muons w i t h t h e p o s i t r o n e m i t t e d a n t i p a r a l l e l (^.•p =-l).  This  e  high precision this  [70,71],  case  experiments c u r r e n t l y For  deviations  from t h i s  Mev/c  3  i /  2  the  case  from  for  the  way  dR/(x dxd^ ) 2  t  (1-x) f a l l o f f  each  under  of p o l a r i z e d muon  spin  t h e r e a r e two which  measure  o f m a s s l e s s n e u t r i n o s , one w o u l d  other  at  z^ =-1.  The  e  a r e shown i n F i g u r e 9 f o r a 5  f o r a couple of d i f f e r e n t  displaced  to  i s c o n s i d e r e d here because  o b t a i n a (1-x) d i s t r i b u t i o n  are  »  so  m i x i n g s c h e m e s . The c u r v e s as  to  illustrate  their  d i f f e r e n c e s . O n l y t h e c a s e o f m a x i m a l m i x i n g ( l u ^ l = I U^u. \ = 1 / 3 ) 2  2  3  could  be  distinguished  from  kinks which a r e d i s c e r n a b l e to  the  kiriematical  t h e m a s s l e s s n e u t r i n o s c a s e . The  i n t h e maximal m i x i n g case  turning off of R  3 3  a t x=1-46  2  a r e due  and of R  x = 1 - 6 . F o r m =3 M e v / c , w h i c h i s n o t d i s p l a y e d , t h e f i r s t 2  2  3  kink occurs  would  be  measurable  only  3  at  such  f o rl a r g e mixing and t h e second  ( a t x=.9992) o u t s i d e t h e r a n g e  planned  in  the  present  63  0.26  Fig.  7:  Sensitivity of the M i c h e l spectrum to the mixing p a r a m e t e r U,^ f o r m = 5 M e V / c . The d a s h e d c u r v e i s f o r IU/ASI = . 0 5 9 ; the s o l i d curve i s f o r I U \ =.2; the dash-line curve i s f o r \ U ^ r = . 5 . U ~0 f o r a l l the curves. 2  3  2  2  /  3  F C 3  3  64  Sensitivity of t h e M i c h e l spectrum t o the mixing parameter U f o r m = 5 MeV/c and l U ^ t =.059. The s o l i d c u r v e i s f o r m a s s l e s s n e u t r i n o s . The d a s h e d c u r v e is f o r | U«,3l = 3 x 10" *; t h e d a s h - l i n e curve i s for l u l = l U > \ = .059. 2  e ?  3  2  2  e 3  2  3  2  65  experiments.  noted  earlier, 3  massive  interactions deviate  was  f o r an i n t e r m e d i a t e mass v .  expected that  As  will  neutrinos cause  with  a  large  Therefore, i t i s predominantly  t h e measurement o f t h e  from t h e (1-x) d i s t r i b u t i o n t o w i t h i n  currently  planned  experiments.  mixing  ^  i s not unlikely  left-handed parameter  to  t h e s e n s i t i v i t y of  66  Fig.  9:  H i g h - e n e r g y end of M i c h e l spectrum f o r spin o f /a* antiparallel to positron momentum. The s o l i d l i n e i s the (1-x) behaviour of massless neutrinos. The dashline curve i s f o r I U/u l =lu«. \ =.1 m i x i n g and t h e d a s h e d c u r v e i s f o r l u - l = l U \ = l / 3 . The mass o f m i s 5 MeV/c . 2  2  3  3  2  3  2  r  2  e3  3  67  3.2 The C a s e o f M a j o r a n a  The  case of  interacting considered can  Neutrinos  neutrinos  via  a  in this  be d i f f e r e n t  being  massive  predominantly  Majorana  left-handed  coupling  s e c t i o n . The p h y s i c a l mass e i g e n s t a t e s , from t h e c o r r e s p o n d i n g  . ( i = 1 , . . . , N ; a = e ,J*,Z, . . .) F o r t h e  of  families,  lepton  unitary  these  transformation.  emphasize  that  Dirac case. expressed  ,3.2.,)  i t can  be  The f i e l d s ^,  mixing  different  mass  or  This  decomposition  contains  only  type  of  one  of  a  can  as i n  Majorana  field  f i e l d . This  i s a r e f l e c t i o n of  of the p a r t i c l e s . (3.2.2)  t=  W i t h t h e above phase, a Majorana p a r t i c l e and  are  indistinguishable.  This  First,  two  i n the c a l c u l a t i o n  distinct  types  i t s antiparticle  indistinguishability o f muon d e c a y ,  f o r the case of D i r a c n e u t r i n o s now  be  operators  1/> ^  complications  to  creation/destruction operator, i n  t o the case of the D i r a c  the Majorana p r o p e r t y  here  eigenstates,  i n terms of c r e a t i o n and a n n i h i l a t i o n (3.1.11).  contrast  matrix  from t h e m a t r i x U of t h e  weak  equation  arise  case  two b a s e s a r e r e l a t e d by an NXN  symbol V i s used f o r t h e P o n t e c o r v o  are  ifj,.,  general  1^--r. v . ^ The  is  e i g e n s t a t e s o f t h e weak  interaction, N  particles  of f i n a l  t h e two p h y s i c a l n e u t r i n o s  in  i n the l a s t  presents  which  d i d not  s e c t i o n . There  s t a t e i n t h e decay the  final  some  state  (1.2). may  be  68  distinct.  That  i s , the f i n a l  state i s  deUp^sO o t - H f e , ^ eta ( ^ ^ . M c ^ The k  operator  ay (k , s ) c r e a t e s x  , and s p i n s . T h i s  i s the  K  Appendix  B.  On  the other  C^j '  a neutrino  notation  (3.2.3)  o f mass m ,  momentum  K  we  have  presented  h a n d , t h e two n e u t r i n o s  in  i n the f i n a l  s t a t e c o u l d a l s o be i n d i s t i n g u i s h a b l e . T h e n ,  =d (p Se)av-.("k,*^ aX  (3.2.4)  +  £  This  e )  L  l a t t e r p o s s i b i l i t y does n o t occur  neutrinos  since  corresponding As handed  a  Dirac  effective  contribution neglected  case  of  Dirac  i s d i s t i n g u i s h a b l e from i t s  antiparticle.  i n the Dirac  right-handed  particle  f o r the  case,  here.  choose  weak L a g r a n g i a n  current of  we  the  predominantly  to describe  c a n n o t be r u l e d o u t b u t  the  order  The L a g r a n g i a n  of  only  6X10~  left-  t h e muon d e c a y . A i t would  make  [77] and, so,  5  i s w r i t t e n i n terms  of  a is  Majorana  mass e i g e n s t a t e s a s f o l l o w s . + V\.e. Symbolically, transition following  the  muon  d e c a y c a n be r e p r e s e n t e d  r a t e s t o each of the N  2  final  states.  (3.2.5)  a s t h e sum o f That  i s ,  the  i s calculated.  x H l C f c - \  A  +  ^ X - \Cfj,.^  (3.2.6)  69  In  the above, (3.2.7) (3.2.8)  The  initial  factor to  o f 1/2  the  I I) i s given  state  i n equation  necessary  Majorana  o f 1/2  the  limit  yields  the standard  into  of  result,  Upon s u b s t i t u t i o n  first  no m i x i n g  particles  is a statistical  because of t h e i d e n t i c a l p a r t i c l e s  In  Lagrangian  of  The s e c o n d f a c t o r  |f'^.  ( 3 . 1 . 2 ) . The  (3.2.6) remedies a d o u b l e c o u n t i n g  indistinguishability  antiparticles.  i n equation  due and  factor  i n the f i n a l  state  and m a s s l e s s n e u t r i n o s  this  as r e q u i r e d .  of t h e i n i t i a l  the expressions  and f i n a l  f o r o£.  and  states  jf^,- , one  and  the  obtains  (3.2.9)  where  (3.2.1Oa)  and  (3.2.10b)  Also,  70  j  - c*ir)U*(p*-p.-i-i)(ynA-yn£)  0.2.10  where  n^-^oS/S*^*^  <-3  2  12)  and  t ^ f . , 0 V ^ ^ H v ^ ^ f c j i , ^ Vf(  T»£ " ^  In  t h e above  expressions,  conjugate neutrino  the spinor  (3.2.13)  lT( ^ )\ ?  r  corresponding  satisfies  U = c  C S  (3.2.14)  where C i s t h e c h a r g e c o n j u g a t i o n  matrix,  O  (3.2.15)  Evaluation  of t h e r a t e f o r t h e decay  the  of  case  Dirac  neutrinos,  s e c t i o n . The t r a c e s e v a l u a t e d that  case  (1.2) proceeds j u s t  as i n  w h i c h was d e t a i l e d i n t h e l a s t  h e r e a r e somewhat  different  from  s i n c e they i n v o l v e t h e conjugate s p i n o r s . D e t a i l s of  the e v a l u a t i o n of m a t r i x The  t o the  more g e n e r a l  elements a r e presented  presentation  i n Appendix  of Majorana p r o p e r t i e s  D.  i s given i n  A p p e n d i x C. The be  Michel  s p e c t r u m , n e g l e c t i n g e l e c t r o n mass, i s f o u n d t o  71  xl  (3.2.16) The  notation  reduces  to the usual  (The D i r a c massless As  i s t h e same a s t h a t o f  and  Majorana  3.1.  This  f o r t h e c a s e of m a s s l e s s  results  coincide  for  result  neutrinos.  the  case  of  consider  only  the  m,  kept  as  neutrinos.) for  the  three-neutrinos significant. is  result  Section  In  c a s e o f D i r a c n e u t r i n o s , we  world.  A g a i n , o n l y one  mass,  3  is  t h i s c a s e , t h e r a t e f o r u n p o l a r i z e d muon  decay  72  V^IM-V^I* R ; ^  + i  0.2.17)  where Ro-.TT ^ - ^ - x V e ( l - 0  (3.2.18)  (3.2.19)  -  R  0  z:  and  R  R  s  3  x  -  ( i - x - 4 ^ v  given  in  are  the  respectively,  /  (3.2.20)  A  (3.2.18)  equations  and  (3.2.19),  same a s e q u a t i o n s ( 3 . 1 . 4 7 ) a n d ( 3 . 1 . 4 8 )  summed a n d a v e r a g e d o v e r muon s p i n . T h u s ,  the  between  neutrinos l i e s  the  cases  o f D i r a c and Majorana  term R .  The d i f f e r e n c e b e t w e e n R  for  Majorana  3 3  the  case  i s displayed  proportional  to 6 . Also,  because  the  it,  the R  of 3 3  2  recall  term gave a n e g l i g i b l e  small  difference  terms.  here  The M i c h e l  i nthe  f o r t h e D i r a c case and i n equation ( 3 . 2 . 2 0 )  that,  f o r the  Dirac  f a c t o r s of small mixing parameters  spectrum. Thus, the s i t u a t i o n between  3 3  only  contribution  to  i s t h a t of a s m a l l spectrum  R  3 3  and i s case,  multiplying the  Michel  difference  i s , t h e r e f o r e , not a  73  good p l a c e t o t r y t o d e t e r m i n e w h e t h e r n e u t r i n o s a r e M a j o r a n a o r Dirac particles, and  i t s mixings  unlikely.  u n l e s s t h e mass o f v with the l i g h t  2  i s i n the tens of  neutrinos are large. This  Mev/c  2  seems  74  3.3. E l e c t r o n - N e u t r i n o C o r r e l a t i o n s  I t was shown i n t h e l a s t not  section that the  probably  does  neutrinos  are D i r a c or Majorana p a r t i c l e s .  electron-neutrino differentiating correlation  decay  a  correlations between  Dirac  measurements  means o f s t u d y i n g For  provide  have  means  to  of  spectrum  determining We  test  and  Michel  now  their  Majorana  previously  investigate  usefulness neutrinos.  been  Such  t h e s t r u c t u r e o f t h e weak i n t e r a c t i o n s [ 1 9 ] .  (1.2) and t h e n e u t r i n o i/  e  a r e t o be d e t e c t e d .  from  the  The n e u t r i n o  v i a t h e i n v e r s e /3-decay p r o c e s s  ^e. +  ^  —*"  eT  +•  p  (3.3.1)  i n a l a r g e - m a s s d e t e c t o r . The r e a c t i o n ( 3 . 3 . 1 ) i s c h o s e n neutrino  in  suggested as a  a c o r r e l a t i o n measurement, b o t h t h e p o s i t r o n  c o u l d be d e t e c t e d  whether  detection since i t selects a particular  f o r the  neutrino  type,  . The r e a c t i o n 1-  n  — -  yu-'  \  y  w o u l d n o t go f o r t h e l o w - e n e r g y n e u t r i n o s If  the  Pontecorvo then the  assumption  mixing dominant  that  matrices branches  the  (3.3.2) o f muon d e c a y a t r e s t .  diagonal  elements  (U o r V) a r e d o m i n a n t f o r muon  decay  of  i s again  into  the made,  Dirac  and  Majorana neutrinos a r e , r e s p e c t i v e l y , yU ^  +  e.* -V, C  ^  (3.3.3) Vl  (3.3.4)  75  By  the  same  participant the  reasoning,  i n the r e a c t i o n (3.3.1).  neutrinos  take  the neutrino  Of c o u r s e ,  a r e not a l l degenerate,  place.  For instance, after  i / , should  then  t h e decay  be t h e d o m i n a n t i f t h e masses o f  oscillations  (3.3.3) t h e n e u t r i n o  v, c o u l d o s c i l l a t e i n t o  another n e u t r i n o and, thus,  likely  in  to  occurence  take  part  would  nondominant  be  not  greatly  (3.3.3). be  alter  For  the  However,  neutrinos  i n r e a c t i o n (3.3.1). the flux  from  For Majorana  neutrinos  should  the  decay  oscillations  of  type  take  (3.3.5)  L  p l a c e . In t h i s case,  the Pontecorvo matrix  an  enlargement  in  the  V c o u l d be  "*—*" take  Dirac case j u s t We  due  oscillations"  (^V  place. In t h i s case,  are  Higgs s e c t o r o n l y , as d e s c r i b e d i n  C h a p t e r 2, t h e n o n l y t h e " D i r a c t y p e  will  the  v, a n d  Thus o s c i l l a t i o n s  o f v, some d i s t a n c e  as this  in  r e c t a n g u l a r . I f i t i s assumed t h a t t h e M a j o r a n a m a s s e s to  be  the case of Majorana n e u t r i n o s , the s i t u a t i o n can  v can  by  not  o f t h e muon d e c a y o s c i l l a t i n g i n t o  somewhat d i f f e r e n t .  the  r e a c t i o n (3.3.1).  compensated  branches  then p a r t i c i p a t i n g  the  could  (3.3.6) the situation  i s similar  t o the  discussed.  calculate  two  kinds  of c o r r e l a t i o n s  b e l o w . The  type  i s where t h e e n e r g y o f t h e p o s i t r o n a n d t h e  the  positron  and  the  detected  angle  first  between  n e u t r i n o a r e measured. I n t h e  76  second type  discussed,  the energies  of both t h e p o s i t r o n and t h e  n e u t r i n o a r e measured. The  energy-angle c o r r e l a t i o n -  e  where t h e n e u t r i n o s a r e equation  (3.2.11)  undetected the For  by  +  f o r t h e dominant decay  "V, 57  Dirac  (3.3.7)  particles,  integrating  over  a n t i n e u t r i n o and e x p r e s s i n g  opening  angle  muons d e c a y i n g  i s calculated the  momentum  the result  in  from of t h e  terms  between t h e p o s i t r o n and t h e n e u t r i n o ,  of 0 i• e  a t r e s t , t h e r a t e i s g i v e n by  (3.3.8)  The  masses o f t h e p o s i t r o n and o f  neglected; summed  neutrino  v,  have  been  t h e s p i n s o f t h e p o s i t r o n a n d t h e n e u t r i n o s h a v e been  over.  Integrating azimuthal  the  The  over  angle  rate the  (3.3.8)  neutrino  is  f o r unpolarized  momentum  and  the  muons. electron  yields the following d o u b l e - d i f f e r e n t i a lrate.  77  1- 21 I U  •I ^ x  This expression  [ (\->0) siv, Oev/a - i c o ^ e  /a]  1  i s good t o o r d e r  6.  The  2  t v  (3.3.9)  property  IJUoJ - = \  (3.3.10)  3  has  been  effect  2  the  i s m a s s i v e and  3  3  in  first  t e r m . The  of m a s s i v e D i r a c n e u t r i n o s . For  only v \ U^ I  used  6  ,  2  mixings  the  i s n e g l i g i b l e . The  i s o b t a i n e d by  For  second term,  the case  limit  setting  of  second term g i v e s  the t h r e e n e u t r i n o being  world,  proportional  to  of m a s s l e s s n e u t r i n o s and  no  IU l = 1 and  6 = 0.  2  2  e i  Majorana  the  neutrinos,  the  rate  for  the  process ^ is both  calculated  *  of the n e u t r i n o s v  + il^J |^'[(p. ,Vp i  ?  first  term  -u  is  L  (3.3.11)  L  similarly.  terms w i t h a f a c t o r of  The  a?  For  the dominant branches, e i t h e r  , v. c o u l d be i / and, choosing •J \v \ , the d i f f e r e n t i a l r a t e i s 1  only  2  e 1  'V( ,. .Vp ..p^^(Q. ,- .) r 1  A ?  or  F  p  /  (1-|v^,\ ) 2  times  ?  p  the r e s u l t  (3.3.12)  f o r the  Dirac  78  n e u t r i n o case w h i l e t h e second outgoing in  neutrinos  the following  over  f o r t h e case  both  ( 3 . 3 . 1 2 ) c a n be s p l i t up  * d^'+ d £ "  muon . s p i n  just  (3.3.13) as  i n the Dirac  case.  and i n t e g r a t i n g over t h e undetected  n e u t r i n o ' s momentum, t h e r e s u l t s a r e a s f o l l o w s at  of  manner.  dR' a n d dR" a r e c a l c u l a t e d  Averaging  is  b e i n g v,. E q u a t i o n  d£ = f l - I V ^ d ^ Here,  term  f o r muon  decay  rest.  ^(•YK^ -  <2.->vyE^- olTvyE, + 4 E^E, Sxv^&^/x)  (3.3.14)  d£" =frpWe.^W/^^p^p, E, ( l y E e - ^ e ^ ^ ^ Q . v M ^ " )  (air*) Integrating  (3.3.15)  caEe")(aE0 over  the final  neutrino's  momentum a n d o v e r one  a n g l e , one o b t a i n s  dxdco=>£tv  d£_!  =  (i-X^m e v/a^ i  t  I V ^ M ^ ^  X^Cl-^) - CQ^6ev/2 2  Combining  these expressions with the r e s u l t  the f i n a l  result i s  (3.3.16)  (3.3.17)  f o rthe Dirac  case,  79  \V^, ^ L K (a,-*) 5'sv) Gey-h <L0S& 1 +  (3.3.18)  H  £V  Neglecting  neutrino  mass,  (3.3.17) f o r t h e D i r a c and difference  is  comparing Majorana  proportional  to  equations  cases  two  the  double  differential  \V^i\.  We  2  distribution  illustrate  I U ^ J = .1,  as  2  u s e d , t h e two t y p e s correlation The  distribution  branch  momentum  t o x=.5 a n d t h e  (e /Jl) . F o r t h e m i x i n g ev  o f n e u t r i n o s c o u l d be d i s t i n g u i s h e d i n a  type  of  correlation  we c o n s i d e r  5%  here i s the  i n p o s i t r o n and n e u t r i n o e n e r g i e s . A g a i n ,  r e a c t i o n i s that of  i s taken  calculated  2  10,  experiment.  second  detecting  a function of s i n  i n Figure  the  i s p l o t t e d f o r the  cases f o r t h e p o s i t r o n energy corresponding  mixing  and  ,respectively, the  d i f f e r e n c e between t h e D i r a c and M a j o r a n a c a s e s where  (3.3.9)  by and  to  contain  integrating  equation at  over  over a l l angles.  least the  (3.3.1), one  since the  the  v, .  The r a t e i s  undetected  For the case of D i r a c  dominant  neutrino's neutrinos,  t h i s proceeds as f o l l o w s . Summing o v e r a l l s p i n s a n d n e g l e c t i n g t h e m a s s e s o f v^ the  positron,  we  obtain,  upon  n e u t r i n o ' s momentum, f r o m e q u a t i o n b r a n c h i n t h e muon r e s t f r a m e  and  i n t e g r a t i o n of the undetected (3.2.11),  f o r the  dominant  80  Fig.  10: Energy-angle correlations f o r D i r a c (D) and M a j o r a n a (M) n e u t r i n o s . The v a l u e o f x = 1 / 2 i s chosen and the m i x i n g |Uu.,| = .1 i s u s e d . 2  81  AZ= GfjUeii " E^dE E,JE, dcosOe-v I I l u ^ 3  E, ( • ^ - • m . ? - . ^  1  t  S^-i^et-^-^^raEeE/i-co^eevV^)  We d e f i n e t h e f o l l o w i n g d i m e n s i o n l e s s  = in  analogy  = Q£TV£  lU.a  L  t^(l-i^V Z  t h e 6 f u n c t i o n t o do t h e  f o r the D i r a c case i s  (3.3.21)  l a ^ ^ ^ s j - ]  i s no x d e p e n d e n c e i n t h i s  distribution.  t h e case of Majorana n e u t r i n o s , the e q u i v a l e n t  the c a l c u l a t i o n a r e g i v e n  1^  The  (3.3.20)  A  integration, the result  that there  For  variable  w i t h t h e v a r i a b l e x. U s i n g  l a s t angular  Notice  2E,/m  final  below.  J  result i s  (3.3.19)  steps i n  82  ae  As  -  (^^Jv^^^(,-^-^\y^,\^L^a-^-^-v-4p(\-y^vs^S\  i n t h e e n e r g y - a n g l e c o r r e l a t i o n s , t h e d i f f e r e n c e between  Dirac  Dirac  distribution  neutrinos  neutrinos, We  difference  distribution The  is,  of  not vanish  i n x for  fixed  11. A t y p i c a l  point  are displayed.  Another  Majorana cases i s that t h e  at  energy  whether  The e x p e r i m e n t a l exceedingly  energies  must  y=1  while  task  the  difficult.  be  measured.  thus,  will  seems  are Dirac  of performing  Dirac  Both  promising  or  this  Majorana  measurement  the positron  and  For the neutrino,  this  of both t h e e l e c t r o n and proton  These e n e r g i e s  d e c a y i n g a t r e s t and, the rate  correlation  neutrinos  measuring t h e energies  reaction (3.3.1).  estimate  and  Majorana  does.  however,  involves  mixings  the Dirac  does  determining  neutrino  (3.3.22) i s quadratic  various  positron-neutrino  particles.  o f y. F o r  t h e s e two c a s e s i n F i g u r e  between rate  by ( 3 . 3 . 2 1 ) f o r t h e c a s e o f  given  f o r some f i x e d v a l u e  thedistribution  illustrate  Majorana  in x  i sflat  y=.25 was c h o s e n a n d  for  the  a n d M a j o r a n a d i s t r i b u t i o n s i s p r o p o r t i o n a l t o \u^\ *. The  y.  0.3.22)  be  will  be r a t h e r  difficult  l o w f o r muons  t o measure.  We  f o r t h e s e c o r r e l a t i o n measurements w i t h t h e  c u r r e n t l y a v a i l a b l e ji* f l u x a t t h e meson f a c t o r i e s a n d a 100 t o n 4TT w a t e r d e t e c t o r the  difference  t o be s e v e r a l t e n s between  proportional  t o \v^.,\ ,  information  on  2  Dirac  and Majorana  i t would  i t s magnitude  of events  be  valuable  i n order  per day.  Since  distributions i s to  have  some  t o help determine the  83  Fig.  11: E n e r g y - e n e r g y c o r r e l a t i o n s f o r y=1/4. The s o l i d l i n e i s for Dirac neutrinos; the rest are fdr Majorana neutrinos. The d a s h e d c u r v e i s f o r h i e r a r c h i a l m i x i n g , the dash-dot c u r v e i s f o r lvy,l =.067, and t h e dashc r o s s c u r v e i s f o r \Vu. \ =.1. 2  2  1  feasibility  of such c o r r e l a t i o n  experiments.  85  I V , THE INCLUSION OF RADIATIVE CORRECTIONS  The graph  r e s u l t s of t h e l a s t c h a p t e r  level  sensitive  the high-energy to  neutrino  indicate that at  end of t h e M i c h e l spectrum  mass.  However, t h o s e  compared w i t h e x p e r i m e n t a l d a t a s i n c e , as inclusion  of  radiative  M i c h e l spectrum difference,  corrections  case  of  i s well  [16].  When  differ  by  structure  Chapter  terms  of  i s g i v e n by t e r m s  constant  order  of  the  predictions  order  (amJ/M ),  where  2  w  a  [ 1 6 , 1 8 ] , The e f f e c t i v e t h e o r y  V-A  theory  diagrams f o r the f i r s t  f o r the  massive  of electroweak  (m^/M ^ ) 2  i s the  order v i r t u a l  i s expected t o  neutrinos  of  the  case  also.  3, u s i n g t h e  photonic c o r r e c t i o n s  which  i n F i g u r e s 12 a n d 13.  i n t e r f e r e n c e between t h e f r e e decay a m p l i t u d e amplitudes  fine  i n t e r a c t i o n s . The Feynman  must be i n c l u d e d i n t h i s c a s e a r e p i c t u r e d  the  2,  n e u t r i n o s , between t h e  T h u s , t h e c a l c u l a t i o n s a r e done h e r e , a s i n C h a p t e r  The  the  r a d i a t i v e c o r r e c t i o n s a r e i n c l u d e d t h e two t h e o r i e s  be a good a p p r o x i m a t i o n  effective  be  changes t h e  S U ( 2 ) X U ( 1 ) gauge t h e o r y a n d t h e e f f e c t i v e V-A t h e o r y f o r t h e p o s i t r o n spectrum  i s rather  known,  significantly  massless  tree  r e s u l t s cannot  [ 1 6 , 1 7 ] , As was p o i n t e d o u t i n  f o r the  the  a n d t h e sum o f  diagrams of F i g u r e 12(b,c,d) g i v e s t h e  radiative correction t o the  muon  decay  (1.2).  The  radiative  decay  juS  e,  +  v  iJ1  (4.1)  86  (a)  Fig.  (b)  12: Feynman d i a g r a m s f o r the free v i r t u a l photon c o r r e c t i o n s .  _ ^ » e  Fig.  (c)  =  r  ^  ~  l  (d)  muon  decay  K-  +  8m  13: S e l f - e n e r g y leptons.  correction  (a)  diagrams  f o r the  (b) y(k)  Fig.  and t h e  14: B r e m s s t r a h l u n g c o r r e c t i o n s  t o muon d e c a y .  charged  87  must not  a l s o be i n c l u d e d i n t h i s c o n s i d e r a t i o n s i n c e t h e p h o t o n i s detected  in  characteristics.  experiments  that  only  measure  The r e l e v a n t b r e m s s t r a h l u n g  the  e  +  diagrams are given  i n F i g u r e 14. It  i s the high-energy  sensitive  to  end of t h e M i c h e l  radiative  corrections  n e u t r i n o s . J u s t a s was t h e e f f e c t c o r r e c t i o n s tend the  value  to pull  which  f o r t h e case of massless  o f n e u t r i n o mass, t h e p h o t o n i c  down t h e s p e c t r u m t o z e r o a n d t o  t o be e q u a l l y i m p o r t a n t  lower  f o r the massive  case.  We c a l c u l a t e t h e r a d i a t i v e c o r r e c t i o n s o f F i g u r e s the case of D i r a c n e u t r i n o s o n l y . T h i s  12-14 f o r  i s because t h e d i f f e r e n c e  b e t w e e n D i r a c a n d M a j o r a n a n e u t r i n o s was shown t o be v e r y in  the  Michel  spectrum; t h i s  result  i s expected  when r a d i a t i v e c o r r e c t i o n s a r e i n c l u d e d . T h i s t h a t t h e d i f f e r e n c e b e t w e e n t h e two t y p e s pronounced  when  correlation  experiment discussed  Of  is  o f p o s i t r o n energy a t w h i c h t h e spectrum peaks. These  c o r r e c t i o n s are expected neutrinos  spectrum  course,  the  neutrinos  are  small  t o remain  true  i s due t o t h e f a c t  of n e u t r i n o s  detected,  is  most  such as i n the  i n C h a p t e r 3.  the diagrams of F i g u r e s  12-14 i n v o l v e  divergent  i n t e g r a l s . The c o n t r i b u t i o n t o t h e d e c a y r a t e o f e a c h d i a g r a m i s here  evaluated  cancel to order massless  explicitly c, l e a v i n g a  neutrinos  regularization integrals regularizing  and i t i s found t h a t t h e d i v e r g e n c e s  to  which  case.  We  handle occur  divergent  finite  answer,  just  use t h e technique the  [ 2 0 ] . The integrals  divergent idea is  to  of  of  as  i n the  dimensional  four-dimensional this  continue  method to  of some  88  arbitrary  dimension  convergent.  n  for  which  Then t h e n - d i m e n s i o n a l  the  cast  i n t h e form of p o l e s i n ( n - 4 ) . a  gauge  gauge  expanded  invariant  [78,79].. This  The  first  assumption  V  Further, assumed.  formulae  made h e r e  -W.^  i s , in  weak e i g e n s t a t e s v-  L  next  and  evaluated  regularization  technique  is  further  a r e g i v e n i n A p p e n d i x E. of N SU(2) l e p t o n d o u b l e t s .  t h a t t h e masses o f t h e N n e u t r i n o  such  <  that  yyl  nearest  (4.2)  H  neighbour  (HNN)  mixing  t h e NxN m i x i n g m a t r i x r e l a t i n g  the  neutrino  the d i a g o n a l elements  next dominant elements  are  taken  are those  ev  i s predominantly  Recall sum o f N  2  that  e  v,, i / ^ i s m a i n l y  the M i c h e l spectrum  v  2t  i s actually  to  be  immediately  t o t h e d i a g o n a l . F o r e x a m p l e , U >U >U ,. T h i s m i x i n g  means t h a t  is  , (where a=e,ytt, T , . . . ) a n d mass e i g e n s t a t e s  (where i = 1 , . . . , N ) ,  dominant  <  hierchial  That  Dimensional  regularization  mass e i g e n s t a t e s a r e o r d e r e d 7VL <  be  i n powers of ( n - 4 ) . D i v e r g e n c e s a r e  the g e n e r a l case  is  can  m e t h o d . We work i n t h e ' t H o o f t - F e y n m a n  d e s c r i b e d a n d many u s e f u l Consider  integrals are formally  integrals  and  is  results  the  scheme  etc. an i n c o h e r e n t  spectra. M dx  s r  dx  1  The  assumption  t h a t HNN m i x i n g h o l d s means t h a t  in  t h e same S U ( 2 ) d o u b l e t s a s JJ. o r e o r i n t h e d o u b l e t w h i c h i s  the nearest neighbour doublet)  will  t o that of  the  jx d o u b l e t  contribute significantly  only  neutrinos  ( i . e .the  to the Michel  T  spectrum.  89  Also,  since  (y,ytx')  the are,  respectively, and  m  neutrinos by  recent  as f o l l o w s  2  HNN  in  mixing,  < TVL, < 4 6  JL  the accuracy  t o z e r o . Thus, o n l y  neutrino  (v^'s nearest  results state  eV/c^  (4.4.4a)  KeV/c^  those  (4.4.4b)  spectra  having  virtual  corrections  nonzero  the  the  can  be  bremsstrahlung  mass.  Details  of  c o r r e c t i o n s are given  self-energy mass  these  by t h e sum o f t h e s e l f -  i n t e r f e r e d with the  amplitude  decay graph and i n t e g r a t e d over t h e o u t g o i n g  neutrino having  2  i n w h i c h a t most one  and  momenta. The a p p r o p r i a t e mass c o u n t e r t e r m from  m, a n d m  i n A p p e n d i c e s F and G.  energy and v e r t e x a m p l i t u d e s free  on m,  b e l o w f o r t h e c a s e o f o n l y one o f t h e f i n a l  c a l c u l a t i o n s are given  the  2  neighbour) i s massive c o n t r i b u t e t o the  virtual  given  neutrinos  The  v,  and  (4.3).  photonic are  and  e  predominantly  o f muon d e c a y e x p e r i m e n t s ,  set equal  The  ( v , e")  e x p e r i m e n t s c a n be u s e d t o s e t l i m i t s  TYL < O.S  sum i n e q u a t i o n  doublets  [21-24].  W  Within  the  correction. ,is  The  has  been  result,  for  neutrino subtracted  with  the i t h  90  t ( i - x M i - ^ M - ^ . ^  [ ( i - ^ ( i - x V ( i ^ ^ l ] F|  (4.5)  - i J ^ S e ^ a.  (4.6)  e  where  t  JU(l-y^ -  " " F x ] J ! ^ * - JL  and S The  a  = ^  Spence f u n c t i o n  /( \-%)  Li (x) 2  (4.7)  i s defined  as f o l l o w s .  X (4.8)  The  notation  i s that  o f C h a p t e r 3. T h a t i s ,  (4.9) where 6 i s t h e a n g l e b e t w e e n t h e p o s i t r o n momentum axis;  s^L i s t h e muon s p i n a n d p  of p o s i t r o n The  e  a unit vector  and  the ^-  i n the direction  momentum.  divergence appears as a pole  in  (n-4).  To  translate  91  this the  dimensional cutoff  regularization result  method  [ 1 7 ] one  simply  into that obtained makes  the  with  following  substitution h where  X  m l n  process given  — - J™. \  Kallen's result  double d i f f e r e n t i a l (4.1) w i t h  below. T h i s  integrated  one m a s s i v e n e u t r i n o  result  T h u s , we o b t a i n  out  i s reproduced.  decay r a t e f o r t h e  bremsstrahlung  i n the f i n a l  state i s  i s u s e f u l f o r muon d e c a y e x p e r i m e n t s i n  which t h e photon i s a l s o detected. been  (4.10)  m W N  i s t h e c u t o f f photon energy. Then, i n t h e l i m i t of  massless neutrinos, The  tJ/H^vFVT  The  neutrino  momenta  have  a n d t h e p h o t o n p o l a r i z a t i o n s summed  over.  92  (4.11)  where  3 P(3/a^  la)*  CQ^y  (4.12)  (4.13)  •f Y*  (4.14)  s  (4.15) Integration  over  the  result. _i  photon  momentum  yields  the  following  93  - 3 ^ j k ?  1  ^  i  0 - ^ ) 0 - ^ - ^  94  (4.16)  The  infrared  precisely given  divergence  cancels the  in  equation  6| o r h i g h e r  appears  divergence  of  (4.5). In these  powers of 6  e  as the  a pole  i n (n-4). I t  virtual  corrections  results,  t e r m s m u l t i p l i e d by  have been d r o p p e d b u t l n 6  terms  e  have  been r e t a i n e d . With Michel  these  results  f o r the  s p e c t r u m c a n now be o b t a i n e d .  neutrino world, predominantly  radiative  c o r r e c t i o n s , the  Specializing  to  u n d e r t h e a s s u m p t i o n o f HNN m i x i n g , the  neutrino  in  the  only massive neutrino t o c o n t r i b u t e for m  , i s the  decay.  Numerical  positron  s p e c t r u m f o r t h e d e c a y o f muons a t r e s t i s g i v e n by  this expression  3  i n the Mev/c  z~)  will  equation  given  muon  which i s  results  Interpreting  be  to  three  v, 3  (v  doublet  the  f o r massless  R  limit  for R  0  0  i s the usual  neutrinos  s p e c t r u m f o r w h i c h one o f t h e o u t g o i n g result  range. Thus, t h e  i n t e r m s o f t h e sum o f  ( 4 . 3 ) a n d t h e HNN m i x i n g ,  c o r r e c t e d spectrum  2  while  neutrinos  i s w e l l known a n d i s , o f c o u r s e ,  of massless neutrinos f o r the  spectra  of  radiatively R  3  i s the  i s m a s s i v e . The obtained  calculations  given  i n the above.  95  Both  R  piece  due t o r a d i a t i v e c o r r e c t i o n s .  Ro=  and  0  R  e  t ^<+x^]} 6 -  s p i n dependent  i s given =  i n t o a f r e e decay p i e c e  and a  (4.18a) -p x M l - a . x \ l  -aix  C  be d i v i d e d  Rcf + C  =  The  can  3  6(\-%)  p, R o  (4.18b)  (4.18c)  5  term of the r a d i a t i v e l y c o r r e c t e d  part  Ro^ ,  by  ^ ^ x^(\-^ O r M - i± J U - (All* E( I + Ht 34 X ^ + 3-Tic -  +  tL. C\-^ J^(i-%S]  * /*  t  \ 6(\-%)  (4.l8d)  In t h e above,  r M =  (Jk^ t  r)UJ*^]z]0 * ^  +3vv^-0  (J^*~"7  +0 (4.19)  The  s p e c t r u m f o r w h i c h one o f t h e n e u t r i n o s  i s massive i s given  below. %i  + ^  where t h e f r e e  part  (4.20a)  96  +  was  5  ( I - 0 [ ( I - <  >  I  - ( \ -  A  c a l c u l a t e d i n Chapter  ~  fc  2.  The  v p ^ e ^ - x - ^ radiatively  (4.20b)  corrected  part  is  IT  [ i (\a + M S x - 11 x =0 +  X  -^fpff  )  X f ^  f-  3 x- ^ x ^ I \ x l k * / s j ] 5  ^  The  s p i n dependent p a r t  ^36  ~  +• <*_X  ^05  is  +  L X  t ( ' ^ - ^ x r 1^X^SWMx'Q J ) ^  _L  ^-nx^-SX^JvLX/^l  (4.20c)  97  U S - l o o * t^y^-H^x ") +- L ^ O i ^ + \\0-y^ 3  ^  J  H e r e r ( x ) i s g i v e n by e q u a t i o n ( 4 . 1 8 ) .  The the  d e p e n d e n c e o f t h e r a t e on 6, r a t h e r importance  of t h e p o s i t r o n The  of r a d i a t i v e  terms  6,  shows  clearly  c o r r e c t i o n s a t t h e h i g h - e n e r g y end  of  the  spectrum  1; t h e s e t e r m s a r e p r e s e n t i n R  a manifestation  of t h e s o - c a l l e d  reflect  a charged p a r t i c l e ' s  number  of  finite  than  that  spectrum.  ln(1-x)  approaches are  Recall  (4.20d)  very  resolution,  soft  0  singular  as  x  a s w e l l a s i n R, a n d  i n f r a r e d c a t a s t r o p h e . They  capability  photons.  are  of  emitting  S i n c e any r e a l  t h e s e t e r m s c a n be r e p l a c e d ,  a  large  experiment for x  has a  near  1  [ 1 6 ] , by — where m A x/2 i s t h e e n e r g y  JU(\-\+m) r e s o l u t i o n of the apparatus.  (4.22)  98  The  inclusion  o f a n o n z e r o n e u t r i n o mass h a s n o t r e s u l t e d  i n a n y a d d i t i o n a l mass s i n g u l a r i t i e s  i n t h e r a t e f o r muon d e c a y .  Wherever terms l o g a r i t h m i c i n 6 o c c u r , the  appropriate  factors  of  in  the  electron  form of l n 6  N a u e n b e r g - K i n o s h i t a theorem  U^.3.  15  The  for a  2  neutrino  spectrum  s p e c t r a l shape f o r t h e to  the  Michel  i s also  The  similar  singularities  massless  look at  f o r a 10 M e v / c  2  massive  case  with  2  due t o t h e k i n e m a t i c  the  massive  neutrino, this  neutrino.  now  that  spectrum s h i f t s case. this it  Figures shift  notice  t o a lower  is  plot  the  instance,  there  the  i s a kink  final  for a  state  5 Mev/c  2  a t t h e p o i n t x=.9998, w h i c h i s experimental  resolution.  The  i s t h a t t h e maximum o f t h e p o s i t r o n value  15 a n d 16 c l e a r l y  difficult  a  case.  case  o f x compared t o  the  show t h e d i f f i c u l t y  massless  of measuring  i n t h e p o s i t i o n o f t h e maximum. F o r s m a l l m i x i n g  i s very  neutrinos  to  as  the r a d i a t i v e  16, we  spectra. First  For  t u r n i n g - o f f occurs  feature  plotted  neutrinos  t u r n i n g o f f of  not measurable w i t h i n t h e p r e s e n t second  in  n e u t r i n o . T h e r e a r e two f e a t u r e s t o  f o r i n the massive n e u t r i n o  1-x-6  spectrum  t o the massless  c o r r e c t i o n s h a v e been a c c o u n t e d f o r . I n F i g u r e same s p e c t r a  are  n e u t r i n o mass a n d s e v e r a l v a l u e s o f  The s p e c t r a a r e a l l n o r m a l i z e d  very  of zero  terms, as expected from the Lee-  a  reference. overall  by  [80,81].  5 Mev/c  massless  multiplied  i n the l i m i t  mass  We d i s p l a y t h e h i g h - e n e r g y e n d o f t h e Figure  are  6 and v a n i s h  n e u t r i n o mass. O n l y t h e u s u a l present,  they  ,  t o d i s t i n g u i s h t h e m a s s l e s s and massive  spectra.  Figure  17  [76] g i v e s the values  of the mixing  U^a  allowed  99  100  Fig.  16:  The h i g h - e n e r g y end o f t h e r a d i a t i v e l y c o r r e c t e d M i c h e l spectrum for a 10 M e V / c neutrino w i t h >U \ =.059 (dashed l i n e ) and equal mixing of I U^. \ = \U \ =. 1 (dash-dot curve). The solid curve i s the standard M i c h e l spectrum. 2  2  A3  2  3  2  t3  101  1  3  5  10 nyMeV/c ) 8  Fig.  17: L i m i t s o f a l l o w e d m i x i n g o f l u ^ a l as a f u n c t i o n of neutrino mass. H i e r a r c h i a l - n e a r e s t - n e i g h b o u r m i x i n g i s d e p i c t e d i n ( a ) ; (b) g i v e s t h e c a s e o f e q u a l m i x i n g . 2  102  Fig  18: H i g h - e n e r g y end o f M i c h e l s p e c t r u m w i t h t h e s p i n o f /x * a n t i p a r a l l e l t o t h e p o s i t r o n momentum. F o r m =5 M e V / c , the d a s h - d o t c u r v e h a s m i x i n g \U^ \ = \ U c l =.1 and t h e d a s h c u r v e h a s I U ^ l =. 067. The m a s s l e s s n e u t r i n o with radiative corrections case i s d e p i c t e d by t h e s o l i d l i n e . The d a s h - c r o s s l i n e i s t h e ( 1 - x ) f a l l o f f f o r no r a d i a t i v e c o r r e c t i o n s and m a s s l e s s n e u t r i n o s . 2  3  2  3  2  2  3  103  by t h e c u r r e n t d a t a a s a f u n c t i o n mixing  assumption,  i s made  of  in  these  graphs,  n e u t r i n o o f mass 5 M e v / c mixing  mass.  The  17b.  As  f o r the  an  example  case  of the  o f HNN m i x i n g , a  i s n o t r u l e d o u t by p r e s e n t d a t a  2  HNN  i n F i g u r e 17a a n d t h e r e s u l t s f o r  e q u a l m i x i n g a r e shown i n F i g u r e information  neutrino  for a  I U ^ l < . 07. 2  F o r t h e d e c a y o f p o l a r i z e d muons, we c o n s i d e r t h e same c a s e as  i n Chapter  3. T h a t i s , t h e d e c a y w i t h t h e  antiparallel  to  the  muon  neutrinos, this particular  at  point  discussed to  x=1  i n Chapter  be l i n e a r  by  still a  massive  d e v i a t e s from  massless  rate  i n F i g u r e 18. O n l y  unrealistic, neutrinos case.  mixing  2  the i n c l u s i o n of r a d i a t i v e linearity  but  n e u t r i n o s . With  no  the r e s u l t s f o r a 5 Mev/c  the mixing  probably  the  forbidden  t o z e r o o f t h e r a t e was seen  f o r the free decay. With  neutrino  is  momentum c o n s i d e r a t i o n s , a s we  v a n i s h e s a t x=1 f o r m a s s l e s s  present of  angular  configuration  3. The f a l l o f f  corrections the f a l l o f f  momentum  s p i n i s c a l c u l a t e d . F o r t h e case of  massless the  positron  longer  the  t h e mixing of  v a n i s h e s a t x=1. We  neutrino fordifferent  f o r the  rate  case  of  values  large,  i s t h i s case d i s c e r n i b l e  and  from t h e  104  V. SUMMARY AND CONCLUSIONS  This t h e s i s describes a r e c a l c u l a t i o n of the rate  f o r the  o r d i n a r y muon d e c a y yn with  the  there  i s mixing  e+  assumptions  independent the masses  of  neutrinos  any s p e c i f i c the  c a n be m a s s i v e a n d t h a t  The  question was  situation  lack  theoretical  The c a l c u l a t i o n  model f o r n e u t r i n o m a s s e s .  neutrinos  neutrinos  a  (5.1)  ^  t  that  experimental of  v  among t h e n e u t r i n o f a m i l i e s .  of  calculations. Majorana  —*  +  are  input  parameters  of d i s t i n g u i s h i n g  also  explored.  regarding  principle  matter  of  the  nature  of  fundamental problem. In the s o l u t i o n clue  to  a deeper understanding  the  restricting  the  for this  problem  of electroweak  of  massive n e u t r i n o s  study.  important, lies  the  i n t e r a c t i o n s and  t h e means t o d i s t i n g u i s h b e t w e e n c o m p e t i n g t h e o r i e s . implications  uncertain  t h e mass o f t h e  n e u t r i n o i s an  to this  f o r the  of n e u t r i n o s and t h e  n e u t r i n o t o be z e r o h a v e s e r v e d a s m o t i v a t i o n The  Indeed,  between D i r a c and  Both  the nature  is  Thus,  the  i n experimentally accessible  s i t u a t i o n s must be e x p l o r e d . Muon d e c a y  is  one  such  relevant  situation toinvestigate. The  calculations  here  i n t e r a c t i o n s are predominantly fermion  V-A  theory  were  done  a s s u m i n g t h a t t h e weak  left-handed; the e f f e c t i v e  h a s been u s e d . Muon d e c a y i s m a i n l y  fourstudied  105  e x p e r i m e n t a l l y t o determine t h e form of t h e However,  h e r e we h a v e assumed a s p e c i f i c  and, t h u s , have i s o l a t e d  so t h e f i r s t  and t h e r a t e  interactions.  interaction  the e f f e c t s of massive  R a d i a t i v e c o r r e c t i o n s a r e known t o decay  weak  be  f o r m (V-A)  neutrinos.  important  in  muon  o r d e r p h o t o n i c c o r r e c t i o n s t o t h e decay (5.1)  f o r the r a d i a t i v e  decay (5.2)  have  been  calculated.  The b r e m s s t r a h l u n g s p e c t r u m  w e l l as the usual M i c h e l spectrum. T h i s r e s u l t event  that  detected  a  muon  (5.1).  an  The  analysis.  which t h e photon i s  s p e c t r u m was o b t a i n e d f o r t h e g e n e r a l  families with  Thus,  calculations.  we  of  sum  arbitrary  of  N  of  our  application  to  spectra.  specialized  three  lepton  with  the  t  to  i t i s ,i n  T h e r e a r e t o o many  meaningful  numerical  the phenomenologically  families  f o r the  remaining  assumption  f a m i l y , t o be m a s s i v e . T h i s i s a  of  experimental  muons  decaying  n e u t r i n o masses i n t h e M e v / c 4,5,7-9).  hierarchial limits  these assumptions, the p o s i t r o n unpolarized  2  mixings;  F u r t h e r , we a l l o w o n l y one o f t h e t h r e e n e u t r i n o s ,  associated  Figures  i n the  f o r the uncorrected  i n t h i s g e n e r a l c a s e t o make a  important case  result  Michel  incoherent  parameters  that  in  3, r e s u l t s were p r e s e n t e d  case of N lepton fact,  experiment  i s useful  i s performed.  In Chapter decay  decay  i s given as  on  spectra  mixing  and  i t s  n e u t r i n o masses. W i t h  f o r both  polarized  and  a t r e s t were p r e s e n t e d f o r s e v e r a l 2  range  The s p e c t r a  and  several  f o r t h e decay  mixings  (see  o f p o l a r i z e d muons  106  are  rather  for  insensitive to neutrino  unpolarized  muon  decay  mass.  exhibit  n e u t r i n o mass a t t h e h i g h e n e r g y e n d . made  regarding  point  since  question  limits  conclude  distinguish  between  the  spectrum.  Michel  information order  corrections  the Majorana or Dirac  s t u d i e d . We  that  strong  No  Dirac  are nature  large.  could  However,  of neutrinos not  Experiments  in  i s obtained  which  more  of n e u t r i n o s .  Along t h i s  difference  between  i n positron  Dirac  and  line,  we  and  neutrino  exhibiting a  energies  on  the p o s i t r o n energy i f n e u t r i n o s  p a r t i c l e s and  is  independent  of  detected.  M a j o r a n a n e u t r i n o s . The  quadratically  positron  depends  a r e Majorana  energy  for  Dirac  The d i f f e r e n c e b e t w e e n t h e D i r a c a n d M a j o r a n a r e s u l t s  is  proportional  to  its  nearest  n e i g h b o u r , v,.  light  the mixing  between t h e y x - t y p e n e u t r i n o and Of  o f s u c h a measurement  w o u l d be d e t e c t e d +• n  -»  course,  the  i s questionable.  calculated  an  event  experimental The  outgoing  v i a t h e i n v e r s e yS decay p r o c e s s -  eT +  p  (5.3)  W i t h t h e p r e s e n t l y a v a i l a b l e muon f l u x a t t h e meson we  direct  e n e r g y - a n g l e a n d e n e r g y - e n e r g y c o r r e l a t i o n s f o r muon  distribution  neutrino  to  seem t o be n e c e s s a r y i n  The e n e r g y - e n e r g y c o r r e l a t i o n s l o o k most p r o m i s i n g ,  feasibility  was a l s o  by m e a s u r i n g  decay i n which both t h e p o s i t r o n and t h e n e u t r i n o a r e  neutrinos.  be  the  possible  and Majorana n e u t r i n o s  t o determine the nature  marked  spectra  sensitivity to  statements  i t i s probably  on t h e n e u t r i n o  calculated  a  the  on t h e n e u t r i n o mass o r m i x i n g s a t t h i s  radiative  of  However,  rate  of  several  tens  facilities, p e r day f o r a  1 07  100 t o n 4Tf  water d e t e c t o r .  detecting  both  the  However,  the  e l e c t r o n and p r o t o n  difficulty  t o d e t e r m i n e t h e n e u t r i n o e n e r g y . Muon  flight  be  c o n s i d e r . One with  a  means o f p r o b i n g  s u c h as c o r r e l a t i o n s . The Majorana  tantalizing  results  s t a t e m e n t t h a t we a r e  Michel  also  encouraging  rather a  at  energy-energy  spectrum at  displayed first  i t s endpoint.  in fact,  corrections decay  to  reducing  the  (5.2) a r e g i v e n  that  the  Michel  three neutrino world  made. The  is  a  decay  It  the  is  well  spectrum  from  from  maximum  first  is  been  of h i e r c h i a l nearest  largely  radiative  was  insensitive  neutrinos. Again,  neighbour  and, (HNN)  dimensional  done f o r Chapter to the only  further, mixing  its  photonic  established in  considered  gauge i n v a r i a n t method o f  at  order  calculation  the  radiative  maximum  ( 5 . 1 ) and t h e r a t e f o r t h e  spectrum  the  t h e peak i n t h e  known t h a t  i t t o z e r o . The  i n C h a p t e r 4. The  has  uncorrected  Qualitatively,  reducing  d i f f e r e n c e b e t w e e n D i r a c and M a j o r a n a  assumption  Dirac  i n F i g u r e s 4 and 5 a r e  sight.  t h e c a s e o f D i r a c n e u t r i n o s s i n c e i t was 2  and  nature,  correlations  n o n z e r o n e u t r i n o mass i s t o s h i f t  c o r r e c t i o n s a l s o decrease endpoint,  Majorana  f a c e t of the n e u t r i n o ' s  spectrum t o lower p o s i t r o n e n e r g i e s , spectrum  left  s t r i k i n g d i f f e r e n c e between t h e for  to  difficult  t h e e f f e c t s o f n e u t r i n o mass on t h e  unpolarized  of  experiment  i s , we must l o o k t o this  in  i f not q u i t e a c c e s s i b l e encouragement.  Of c o u r s e ,  effect  correlation  decay  t o determine the fundamental D i r a c or  of the n e u t r i n o . That  ingenious  and  feasible  m i g h t make t h e g e n e r a l  no e a s y way  nature  more  in  i n t h e i n v e r s e /3 - d e c a y  (5.3) i n o r d e r might  lies  has  the the been  regularization  108  was  used  to  handle  the  divergences  which  occur  c o r r e c t i o n s . The c o n t r i b u t i o n o f e a c h c o r r e c t i o n and  the  finite  divergences  were f o u n d t o c a n c e l  was  to order  answer, j u s t as i n the case of massless  in  these  evaluated a,leaving a  neutrinos.  The p o s i t r o n s p e c t r a a r e d i s p l a y e d f o r b o t h p o l a r i z e d Figure  18) and u n p o l a r i z e d  (see F i g u r e s  I n t h e c a s e o f p o l a r i z e d muon neutrinos probably  could  be  detected  unrealistic  masses i n t h e Mev/c mixing.  For  2  neutrino  unpolarized  muon  muon  weak  doublets  15 a n d 16) muon d e c a y s .  the  effects  experimentally masses  range and m i x i n g  neutrino with small mixings and  decay,  and  only  massive  f o r what a r e  mixings,  angles  decay,  of  neutrino  l a r g e r than  the e f f e c t  i n t o the neutrinos  in  Cabibbo  of a massive the  i s s m a l l . The s e n s i t i v i t y  electron  to neutrino  mass a t t h e h i g h e n e r g y end o f t h e p o s i t r o n s p e c t r u m i s washed  out  with  now q u a l i t a t i v e l y  neutrinos in Figure massive  .07.  The r e s u l t s o f t h e c a l c u l a t i o n  i n t o t h e ji-type  by t h e p r e s e n t  neutrino 2  data  are  shapes  i s allowed  a r e summarized of  the  n e u t r i n o weak e i g e n s t a t e ,  ,  given  of t h e  as  mass. F o r i n s t a n c e , u n d e r t h e HNN neutrino  spectral  t h e same f o r b o t h t h e m a s s l e s s a n d m a s s i v e  17. 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C o n v e n t i o n s a n d N o t a t i o n  The  properties  of D i r a c  been u s e d e x t e n s i v e l y i n presented those  in  this  properties,  the  s p i n o r s and  matrices  calculations  of  w h i c h have  this  work  are  A p p e n d i x . A l s o , a s an e x a m p l e o f t h e u s e o f a  relevant  squared  matrix  element  is  calculated. The and  conventions  Drell  e m p l o y e d i n t h i s work a r e t h o s e  [ 8 2 ] . The m e t r i c  g^v i s g i v e n  of  below.  I I  0 0 0\ o -\ 0 0 \ 0 0 -\ 0  V o The  s c a l a r product  o  o  Bjorken  ,  <A  1)  -i /  o f two f o u r - v e c t o r s , a a n d b, i s  defined  as  follows. OL- b - a^.VT = (X^ ky«J * >  The  free Dirac  equation  V  =  a  o b ~ a. • Y>  (A. 2)  0  describing a spin  1/2 p a r t i c l e  o f mass m  is i r r f  where  the  ^  anticommutation  I  — yvL  are  — 0  4x4  matrices  ( A < 3 )  which  the  following  relations.  i s t h e 4x4 u n i t m a t r i x . A n o t h e r m a t r i x  matrices i s  obey  constructed  from the  114  r-trrr.r The the  <A.S>  f o u r component s p i n o r 7^ w h i c h i s a g e n e r a l  free  Dirac equation  s i n g l e - p a r t i c l e plane  c a n be e x p r e s s e d  as a s u p e r p o s i t i o n of  wave s o l u t i o n s a s f o l l o w s .  V i s a n o r m a l i z a t i o n v o l u m e a n d t h e e n e r g y , E, obeying  the  Dirac  solution to  equation  i s given  for a  particle  i n terms of i t s t h r e e -  momentum a n d mass a s E= J l j ^ t ^ The  operators  b ( p , s ) and d ( p , s ) a r e t h e  f o r a p a r t i c l e and a n t i p a r t i c l e , and  d^(p,s)  operators.  (A.7)  are  the  respectively. Likewise,  particle  and  The c r e a t i o n a n d a n n i h i l a t i o n  usual anticommutation r e l a t i o n s as given l  annihilation  antiparticle operators  operators b^(p,s) creation  satisfy  the  below.  [V> (p,s\WfVs' )l -  ^''h(f-f')  (A.8)  U%s)  w U f - p ^  ( .9)  V  ,  ^( <y)\= f  -{ W p ^ ^ c K p ' . ^ V ( ^ ( - p . S ^ c K - p ^ S ' ^ ^0  A  (A.10)  H e r e p i s t h e four-momentum a n d s i s t h e s p i n - p o l a r i z a t i o n f o u r v e c t o r g i v e n by  1 15  S ^S°, ?V-^"g_5  ^  equation  u ( p , s ) i s a f o u r component  In  represent  a  antiparticle satisfy  (A.6),  particle  £ V (  solution  solution.  the following  f  The  p \  while  particle  (A. 1 1 )  ir(p,s)  spinor taken t o represents  and a n t i p a r t i c l e  an  spinors  equations.  u.{^-yvO -o  (A.12) (A.13)  where ^- " The  so-called  slash  U  1  6  (A.14)  notation  used  above  has  the  following  meaning.  / The  3  Oy.  (A. 15)  f o l l o w i n g p r o p e r t i e s of the s p i n o r s a r e very u s e f u l .  utp,,^ T-  TUD,^  =  (^J-TVQ (  ^ ( p ^ i U p , ^  I+h  i)  - jp'+i^-  (A.16a) (A.16b)  W j ^ s W t p ^ - (jzC-wv ) ( i + ^ y ) a.  (A. 17a)  "!L T H p / O V-(j?,«S> ^ j^-Tw.  (A.17b)  >  Extensive matrices.  u s e i s made o f t h e f o l l o w i n g p r o p e r t i e s o f t h e D i r a c ^  1 16  Yf- YJ^  ( A . 18)  0  p =-V*^ R  (A.19)  i ^ r - ^n'-^f For  four-vectors  ^*r-  cc^ h\ 6y  (A.20)  a , b, a n d c , t h e f o l l o w i n g r e l a t i o n s  hold.  ^W=-U^i-a,CL-b  (A.21 )  V  (A. 22)  = - 2 ?t  JZC  A  X 0CWtf = A  4flC-b  A  X  *yu* ^  (A.23)  =  A  ^ products of t  The t r a c e s o f v a r i o u s Tr ^  - Tr V"  (A.24) matrices are  = 0  s  (A.25)  Tr y Y  - H  (A.26)  T r ( odd  #  (A.27)  v  T r r n f In  o"f ^ ma-irices^ = 0  f  r ^  = ^  t h e above, ^/ ? u>  tr  L  ^  V  f  u.29)  r  i s the completely  antisymmetric  tensor  such  that t°  yZi  Equation of  a  =+l  (A.30)  (A.20) i s p a r t i c u l a r l y  product  of  several  useful  Dirac  incalculating  V matrices,  the  trace  as i t e f f e c t i v e l y  117  reduces the  number of m a t r i c e s  i n the product.  repeatedly  until  point  where  equations  (A.25-29) can  properties given  above  for  the  V  case i n Appendix  E.  a p p l i e d . The  the  extended to the n-dimensional As  an  example  of  we  present  the  element  for  muon  quantity  t o be  matrices,  n.  decay  calculated  -  E  the  manipulation  calculation with  of  massive  i s , according  It  can  Dirac  used  matrices  of the the  be  are  spinors  squared  and matrix  neutrinos.  to equation  be  The  (3.1.7)  im*-  (a.3D  V--SpiyrS  wherein the m a t r i x  element i s  (A.32)  H e r e , V" (p„ , s. ) i s t h e represents  spinor  representing  t h e p o s i t r o n ; -\J"( p., , s^)  t h e a n t i m u o n , i r ( p . ,s. ) t h e a n t i n e u t r i n o , and J  4.  the n e u t r i n o . M  i s most  components of the  easily  s p i n o r s and  u(p- ,s )  J  t  found  matrices  through as  labelling  L  the  follows.  'To?  Upon a p p l y i n g  the complex c o n j u g a t i o n  have  Using  equation  (A.17),  and  some r e a r r a n g e m e n t ,  we  1 18  The c o m m u t a t i o n p r o p e r t i e s o f t h e t m a t r i c e s y i e l d ,  finally,  ^ U ^ L u ^ ^ ^ Now  (A-.33)  t h e s q u a r e d m a t r i x e l e m e n t , 7^1, c a n be c a l c u l a t e d .  Labelling  t h e m a t r i c e s and r e a r r a n g i n g ,  Therefore, (A.34) where  T> Tr [ ( ^ ~ Yy V  Calculation equation  of  H-  Y ftjY ( i- ^ (ft - ^ •) Y ? (I - Y)1 A  y  the f i r s t  y  t r a c e , T,,  (A.18) and t h e f a c t  the t r a c e s i m p l i f i e s  to  that  proceeds as f o l l o w s .  Using  119  7> T r  1+  V ' - V l  S i n c e t h e t r a c e o f an odd number o f K reduces  Using  matrices  vanishes,  this  to  equations  (A.27) a n d  ( A . 2 8 ) , we  find  where t h e n o t a t i o n  h a s been u s e d .  Therefore, ^ where  U  the  Similarly,  the squared M  U  ^  I  following  t e n s o r h a s been  used.  m a t r i x element i s 1  ^ ( p ^ . p ^ C ^ p - ) property  of t h e c o m p l e t e l y  (A.35) antisymmetric  1 20  B. M a j o r a n a  We  States  g i v e h e r e some o f t h e c o n v e n t i o n s i n t h e d e s c r i p t i o n  Majorana  particles.  Case [ 4 ] has f o r m u l a t e d  o f n e u t r i n o s . The f i e l d  the Majorana  conjugate to a f i e l d  where t h e c h a r g e c o n j u g a t i o n  ^  theory  i s d e f i n e d as  o p e r a t o r , C, i s  O L ^ T Therefore,  equation  field  (B.2)  (B.1) c a n be r e w r i t t e n a s  it* A Majorana  of  f  satisfies  (B.3) the f o l l o w i n g c o n d i t i o n . (B.4)  That  i s , a Majorana p a r t i c l e  antiparticle.  This  is  condition  not  distinguishable  to  a  description  T h i s becomes c l e a r component  satisfying  the  Dirac  i n t e r m s o f two-component s p i n o r s .  upon w r i t i n g  the f i e l d  ij/ i n t e r m s o f two two-  fields.  and a p p l y i n g t h e  condition  (B.4).  discussion,  following  f  Dirac  its  a l l o w s a r e d u c t i o n from the four  component s p i n o r d e s c r i p t i o n o f a p a r t i c l e equation  from  the  / matrices  i s used.  5  For  diagonal  the  purpose  of  this  r e p r e s e n t a t i o n of the  121  1  o/  1  U  0  I 0 0 -I J The  result  i sthat ^  (B.6)  c a n be w r i t t e n i n t e r m s o f a  component s p i n o r , (f), i n t h e f o l l o w i n g  f--  (.  single  two-  manner.  '*)  (B.7)  i  Since  if  satisfies  1  following  the Dirac  free particle  The each  equation  must s a t i s f y t h e  =  .  0  (B  +G.V(<r(n+)-K(p=o  equations other.  (p  equations.  >JP **-1<p-~^<f>* -l[tr±(P*)  equation,  ( 9) B  (B.8) and (B.9) above a r e complex In  terms  of  8)  the original  f o r a Majorana p a r t i c l e  conjugates of  field,  c a n be w r i t t e n t o  ijj , t h e D i r a c reflect  the  Majorana property as  i>-nV This  i s  known  as  f* = o the  decomposition of t h e f i e l d  Majorana i s g i v e n by  ( B  equation.  The  ,  1 0 )  Fourier  1 22  In  contrast t o the decomposition  equation  (A.6),  equation  c r e a t i o n and d e s t r u c t i o n commutation  (B.11)  contains  operator  a(p,s)  [a (k '~\ - i a  =o  (B.i2b)  the Dirac  elements.  the  + )  p r o p e r t y of p a r t i c l e and a n t i p a r t i c l e  particle  However,  decomposition  cp(t)~fcZ  case,  j^U  ?  i ti s interesting  AK(  )  f  i  AV"  t  ?  work  decomposition  in a  L  and u ( p A )  are  u ( p, A\ = L >L  (E-x-y)  - i r ^ A ^ = -L TVL ^  equations  two-component  i t i s simply a matter of  basis.  (D y i e l d s t h e f o l l o w i n g  ( E - ?•  (B.13)  x  F  helicity  Substitution  of t h e  (B.8) f o r t h e  equations.  * ( p , \)  u  matrix  to also write the Fourier  (B.13) i n t o t h e M a j o r a n a e q u a t i o n  two-component f i e l d  of  field  s p i n o r s . The s y m b o l A d e n o t e s h e l i c i t y ; to  calculation  \an >U(y,AW ?- l  ( B . 1 3 ) , u.(p,\)  being  (B.11) c a n be u s e d , j u s t a s  f o r the  o f t h e two-component  Now, i n e q u a t i o n  (B. 14)  * (p^\\  (B. 15)  demonstrate t h a t t h e p a r t i c l e and a n t i p a r t i c l e  s p i n o r s , u and ir, a r e n o t independent. since  following  (B.12a)  i n d i s t i n g u i s h a b l e . The d e c o m p o s i t i o n  These  the  type of  $>(f-f)  reflects  choice  with  one  in  U (pA<xtY,vA = W t  in  only  given  relations.  +  This  f o r a Dirac p a r t i c l e  i ti sa reflection  This result  is  expected  of the i n d i s t i n g u i s h a b i l i t y of p a r t i c l e  1 23  and  antiparticle,  the  Majorana  c o n d i t i o n . The s p i n o r s c a n be  expressed  i n t e r m s o f an o r t h o n o r m a l  helicity.  T h a t i s , t h e s p i n o r s * ( p ) a n d /3 (P)  basis satisfy  b a s i s of s t a t e s of d e f i n i t e of  the  helicity  the f o l l o w i n g r e l a t i o n s . (B.16) (B.17)  (B.18)  Therefore  the  ot (p)  spinor  (yS(p))  (negative) h e l i c i t y  state. Therefore i f  component o f u ( p , X )  i s taken  represents the  a  positive  positive  helicity  t o be  u ( ^ ^ = J l f t T j F i oi ( p ^ then  the p o s i t i v e h e l i c i t y  component o f t h e a n t i p a r t i c l e  is given, according to equation  ir(p.i)"\  -cm  p h a s e o f ot a n d p **( J)  spinor  by  CE+a-p' )jEi-lp\ c^  (B.20)  N  Since the P a u l i matrix  Therefore,  (B.15),  (E^-lplO  positive helicity  (B.19)  r  c  t o one  2  i s o f f - d i a g o n a l , i t changes a s t a t e of of  negative  helicity.  The  relative  are chosen such t h a t = -I^/30p>  equation  ( B . 2 1 ) becomes  (B.21)  124  tf(p,0  = -^  where t h e f a c t Then  a  (B.22)  t h a t /3(p) i s a h e l i c i t y  s o l u t i o n f o r the negative  eigenstate  helicity  h a s been u s e d .  part  of the spinor  is  It  follows  from e q u a t i o n  u(p,a )=  (B.14)  that  (B.23)  18 ( " 3 ^  >  The n o r m a l i z a t i o n  h a s been c h o s e n s u c h  that  I L u " ( ^ x W p ^ = X . i r ^ l ^ \ W ( | ) , ^ - a.E  (B.24)  1  M o h a p a t r a and S e n j a n o v i c  [60] p o i n t  c a s e , p a r t i c l e and a n t i p a r t i c l e the  decomposition  (p(x> ^  In  out t h a t , f o r  order in  j  t  >  )  to a  present  Lagrangian,  the  following  The c h a r g e c o n j u g a t e f i e l d s a r e d e f i n e d ^  '  ^  ^  , t  l  since  (B.25)  t h e form o f Majorana and D i r a c  p r o j e c t i o n s of a four-component f i e l d  - (IV  massless  (B.13) becomes l  a  the  states are d i s t i n g u i s h a b l e  j^[ (p ^^(p ,V- "P-\oi (^ir(p,x )e^"  mass t e r m s  t  e  ' ("TV  left  and  type right  1|* a r e d e f i n e d .  as (B.27)  125  =  1 ^  - (tf V  (B.28)  C  The mass t e r m s o f t h e L a g r a n g i a n c a n be w r i t t e n a s f o l l o w s  = " ( A l l  +  ^  f  (B.29)  where X - fyu + ^ \ ^ Obviously,  ^  s  l  R  t  (B.30a)  + ^R t  C  R  (B.30b) (B.30c)  "X a n d ^ r e p r e s e n t M a j o r a n a p a r t i c l e s . The t e r m i n D  i s t h e u s u a l D i r a c mass t e r m .  126  C. P h a s e S p a c e C o n s i d e r a t i o n s  and N o r m a l i z a t i o n  The p h a s e s p a c e f o r muon d e c a y a t r e s t  Using  the  three-dimensional  integrations,  to  do  one  of  the  t h i s becomes  J  y  6-function  i s g i v e n by  l E  ) i E,  e  '  where 'Ej " = \ |I +- pt \^ i  ^  3  = l ^  -  1  v \jp!W a\^M\pt\^obe - -v ^ e  L  (c.2)  The c o n d i t i o n -\ ^ c o s etc ^ \ gives  the  boundary  of  t h e D a l i t z p l o t . The e n e r g y 6 - f u n c t i o n  gives ^  -  (  y  ^  E  ^  E q u a t i n g t h e two e x p r e s s i o n s f o r E equation  i n cos ©^ 2  (C.3) 2  above y i e l d s t h e  following  127  =  y^t-  ^ >  ( E ^ E,V>  +- 4 (-vy ( E + E V E e E ^ t " ^ + e  i-v^--**^  L  For the case of massless n e u t r i n o s ,  R  [ ( T ^ ^V\ftV I-Yvy, ( E +  -14  m^,  t h e c o n d i t i o n on c o s e ^  ^ - i T v ( E SL E  Defining  this  e  e  (C.4)  reduces t o  + iEe'Eil  e  Neglecting  H E ^ E ^ & ^ E - ^ E ^  t E i H a E  t  (C.5)  gives  E ;  £\  (C.6)  El  the f o l l o w i n g v a r i a b l e s ,  X =  X, = 2_E_ yvy^.  this  becomes  This  result yields x ^ For the case  neglecting  (C.7)  L  Wy  = 1, a s e x p e c t e d .  i # j a n d m^O,  the equation i n c o s 9 2  e u  becomes,  t h e e l e c t r o n mass,  HE^E^co^G^ =[^-yH  i J  -2ry(EetE  T h u s t h e c o n d i t i o n on c o s 9 , ; 2  becomes  N L  )tiE E l e  L  : L  '  (C.8)  128  -I ^ "VH.^ ~™t~  klfA^iEe.+  El)  4-a.EeE, ^ \  (C.9)  I n t e r m s o f t h e v a r i a b l e s x , x. , a n d 6 = m / n u , t h e c o n d i t i o n i s J  L  This  L  »wa.y  The  last  Neglecting  - I- ^  case  (C.10)  which  must  be c o n s i d e r e d  e l e c t r o n mass, e q u a t i o n  = Iwfi - a - y y ( E condition  of cose-  0 - x - x- M  ,  J  t  .1  This  /  y i e l d -s x y- < I- "a"*" - ( X + X iY i O 0 x  The  O  j  ^  t  i s f o r m. =m*0.  (C.4) r e d u c e s t o  +E^ tXEeE' T  (C.I 1 )  t  i s , i n terms of t h e v a r i a b l e s x, x , and  ^  s ) xN  L  v4  1  reduces t o  [i-xVx'Li^-xHt-^/ail r [(l-xY ^*; '!^ 0 1  The  This  condition  that  reduces t o  t h i s e q u a t i o n have r e a l  1  roots i s  129  I- X- 4 ^ so x  = 1-46 Now  O  (C. 12)  for this  2  t h a t the  differential  >  case.  upper l i m i t  decay r a t e s can  normalization.  All  on be  spectra  x  0  J  been  established,  integrated to obtain  will  spectrum w i t h massless n e u t r i n o s .  l i n ^ a. MSTr*  has  be  normalized  the  the  overall  to the  usual  That i s ,  I,  4  0  =  For  the  neutrinos,  ( . C  case the  considered total  in  Section  (3.1),  that  of  1 3  )  Dirac  rate i s  ^ IUL^i* t [1U„1S I  (  i  - ^  ( c . 14)  130  where t h e f o l l o w i n g  r e s u l t s h a v e been  used.  \-HVL_  Thus,  the n o r m a l i z a t i o n f a c t o r N f o r equation  (3.1.44) i s g i v e n  by  which  i s approximately NT-  for  l U U ^  s m a l l \U 3!  equal t o  f l l l ^ N i ' ^ )  (c.15)  2  e  For Majorana n e u t r i n o s , the t o t a l  rate i s  o where R  is  v  integral  given  by  equation  i n the e q u a t i o n above, the r e s u l t ,  fc= ^-p - l U ^ l U ^ l S Thus,  (C.14).  the  difference  in  1  Upon  performing  the  to order 6 , i s 2  "  (C16)  n o r m a l i z a t i o n from  the D i r a c case i s  negligible. This  completes  the  phase  space  considerations  and  131  normalizations self-energy  f o r the  and  integration  vertex  are  as  case  of  t h e f r e e muon d e c a y . F o r t h e  corrections  given  above.  the The  bremsstrahlung photon energy i s d e r i v e d The p h a s e  2fr  d  to  limits upper  on  the  x-  on  the  limit  below.  s p a c e f o r muons r a d i a t i v e l y d e c a y i n g a t r e s t i s  to  to  ^ (  4!£?fdHf4iEi  sU  , i l ^ V  ?  / A  ?  J  H U y -  E - w-E'-fc.^ e  -E -u3-E--E ^ e  (C17)  J  where E ^ = (-vy-  E  €  - u> -  E ^  and 5.  8) where G i s t h e a n g l e between j?  k a n d p^ a n d (p i s t h e a n g l e  between  a n d p ^ +k*. A l s o ,  E q u a t i n g t h e two e x p r e s s i o n s f o r E. , r e a r r a n g i n g , 2  yields  and  squaring  1 32  = \ U j + w * - 3 r y ( E v c o + E O * 1 E e / c o - v E ^ E - ^ u A ^ l cose) j (C. 1 9) e  Since  L  0^cos <p<1, t h e t e r m  i n square b r a c k e t s i s g r e a t e r than or  2  equal t o t h e term  [ For t h e case  i n curly  1  - {  o f o n l y v.  CL E^~ +• b E  brackets.  L  V  }  h a v i n g mass m , (m =0) t h i s r e d u c e s t o L  C  £  0  (C.20)  where  O b v i o u s l y , c^O. R e a r r a n g i n g  (C.20) a n d w r i t i n g  b i n terms of  a,  we h a v e ( a - M ™ . V ( E - v o + E e - Y y H H w j - E ^ - _c_ x  t  Since E  L  V-  v E^ - w y . ^ 0 e  by e n e r g y c o n s e r v a t i o n a n d  4 ^  E  L  -<^/E- < 0  (C2i)  133  it  follows  that  Substituting  i n the expression  f o r a , we  obtain  Therefore,  *  W  For  YVL^( i - x + ^ - S j " )  ^ = co"b9  t h e .case o f m a s s l e s s n e u t r i n o s , The  overall  normalization  the reduction  for  the  case  i s obvious. when r a d i a t i v e  c o r r e c t i o n s a r e i n c l u d e d r e m a i n s t o be d o n e . F i r s t the  spectrum which contains  only massless outgoing  i n t e g r a t e d o v e r x. The r e s u l t o f  integrating  (C22)  the  part  of  neutrinos i s  the  self-energy,  v e r t e x , and b r e m s s t r a h l u n g terms i s  {<>.E.  (dx  i - V. +  ^ k>^>  f^\{-^^l^\  (C.23)  0  The  integrals  Integrating  involved are l i s t e d  the r a d i a t i v e c o r r e c t i o n s f o r the case of  massive outgoing  neutrino  JjS.E.-t-V. v B . V  U  a t t h e end of t h i s  G-F^V  Appendix. a  single  yields this result  /«_ Xt-TT -^ ^ + QC^)\ 2  (C.24)  0  Some of  of  the 6  listed  t h e i n t e g r a l s were done n u m e r i c a l l y 2  term i s not given  at  the  so t h e c o e f f i c i e n t  h e r e . The r e s t o f t h e i n t e g r a l s  end of t h i s A p p e n d i x . I t i s important  are  to notice  134  that  no  ln6  or l n 6  normalization  +  The  m  ^ l  /  i  [ l U  terms  2  occur  in  the  rate.  The  overall  i s , thus,  e  \ S \ U e , \  l  a  ] [ ( ^ ^ V  +  f  f o l l o w i n g v a l u e s of the Spence f u n c t i o n  t  SC^"]]  were  (C.25)  used. (C.26) (C.27a)  U ^ O - s ^ ~ Tr*/& Equations  ^ - ^ J U ^  (C.26) a r e o b t a i n e d by T a y l o r e x p a n s i o n .  i n t e g r a l s are  listed  x  below.  = -  2. (dx J  X lwX -  *! 3 w V 3  3. U x  X Jkx  X  x  3  =  4  j K y  4. \-x  5.  3£  - Jf2 C|  -  *1 It  (c.27b) The  relevant  7. [d*  9.pX  -d-x)[t-(i-xU  x J k ( ^ =  ^0  3 ( I - V ^ ] A ( I ^  - ( l - ^ [ l - i ( l - ^ j U ( l ^ + ( l - ^ [ | - T i ( l  136  12.  13.  i L^U\  =  x?  14. U *  ^  i  J^V  l  15. $ a x  -s  - -  X  X  + 4 ^  xs - - - S * 2  1  17. ] dx  %*JUx  -- - s  1  ir -/^ 3  18  9. j  Jx  s ^ J U x  -~  ^ d r . - ^  JXvd-x)  ~^ =  20. j d *  21  S1  dx  - S'  22. ] dx  ^  o  X- =  H  3  23. j d x  =  I  24. ^dx  SX  = i S*  25. ^ d x  <S V  =  t  26. J d x  W x  1-^ _ 27. \ dx S X 4  28. y d K  ^  =  JkX  -S  = 6  i X^Jkx -  0  4  3-  29. (dx  S X ^ JlwX b  0-x^  30. \ d x  0  "b^Jl/KK  =  0  138  31 . j d x  % i jt^t L  32. ^ d x S o  fe  33.  £ ^ x JUK(\-*)~  Jdx  34.\dx  X* J k x  =0  - 0  $\JU(j-0=  35.Jdx  ' I L ^ J ^ ^  ^ J ^ ^ H  +£  x  1  S ^ I U l - X ^  36,  37. f<U  38, J d x  S JLO-xV H  S  f e  J*(i-*>  W l  + ~ L ^  ^ ( i J k ^ ^ )  39,  40,. ( d x X ^ i - O - x V - " l i  41. W x  x J^Cl~x-^) -  +  a  A  S  S * - ^ J " ^  IS  139  °  0-K v  43  44.  S (  d<  l  45. S  dx  l  - ^ J ^ S *  =  X  J^X  S  -  1- H  0 ^ 46. £ M  I  d-X o  47.  1  3  -  ^  O-xV  \ dx  48. • d<  X  $  x  - O  X JLx  - 0  1  $  5  49  - i s  1  50  51.  52,  \ dx  s  fc  * -^x 3  =  t  AO  fe  3 6  X j  1 40  53. f d x x J U ( \ - V - s M = 3  o r ^ 55. ) c\x S  , X^A/l-^-S^--  4,  o  56. U x  57.  \ dx  58.  J dx 0  X^Jw x j n  l ^ J v ^ - i )  - - X . ( l - 3 ^ ^ ) + 21 - 5 s  - ~ T t ^ ( | - 4 S ^ 4-  X J k xJ 3  ' ^ ^  U J x )  = i r  2  - ^  -  1  141  D. D e t a i l s o f M a j o r a n a C a s e C a l c u l a t i o n  I n t h i s A p p e n d i x , most o f t h e d e t a i l s o f t h e d e r i v a t i o n equation  (3.2.16)  neutrinos  are  properties  for  the  presented.  are  useful  charge conjugation  Michel  The in  spectrum  following  manipulating  with  of  Majorana  definitions  and  the Dirac algebra.  The  matrix i s  c- L ^ r  (D.D  I t has the f o l l o w i n g p r o p e r t i e s .  c y  c  (D.2)  The c o n j u g a t e ^  =  L  U  spinor  i s defined  - CV°  (D.3a)  = uC  1  (D.3b)  The f o l l o w i n g p r o p e r t y involving Dirac U It  c  U  1  is  very  useful  in  calculating  traces  spinors.  =( CU  follows  equations  from  V^C^l = the  C ( UU.)  C -  properties  (D.4)  yf-r^  of  the  \  matrices  and t h e  (D.2) a n d (D.3) t h a t  L v ^ V i - O V * ^ and  by  tu^)5 (l-^Wj>eYl e  (D.5a)  142  V I I-  [ UV  Then t h e m a t r i x in  the f i n a l  U  y i l* = [V ( y  element  1 ( t-V> U v. ( ^ ] ?  (D. 5b)  C  f o r t h e c a s e o f two d i f f e r e n t  neutrinos  state i s  where k ( q ) i s t h e four-momentum  of i /  (v  L  ) and (D.7)  Equation  (D.4) was u s e d t o o b t a i n t h i s  Similarly,  result.  f o r t h e case of i d e n t i c a l  neutrinos  i n the  final  state,  and  PJTYi^^, ^ U s I l ^ \ M ^ f ( p V i - ^ V  where  the  latter  result  is  obtained  from  i n t e r c h a n g i n g t h e momenta k a n d q. The i d e n t i c a l s t a t e c r o s s terms a r e  (D.9)  A  the  former  neutrinos  by  final  143  .(£  where  We^W  m-  is  straightforward  ^ ^ ^ P e p  the to  mass  of  obtain,  (D.10)  v- . L  involves  than t h e o t h e r s so i t s d e r i v a t i o n Using  In  equation  This  last  result,  r a t h e r more  i s given  although  manipulation  in detail  below.  (D.5),  t e r m s o f c o m p o n e n t s , we h a v e  v-  The s p i n o r  subscripts  have  Rearranging  and u s i n g e q u a t i o n s  been  dropped  in  (D.2-4)  C-- i C x i i K ^ l u C ^ L t V ^ e Y . d - ^ l ^ U ^ V • [c v l  w  U ^ l [ v •] K  K $  [v  (, -  this  step.  1 44  \  --i l\i  (i-u  <-». o]„„L c Y ^ r X  Y;>C£  u  (f+« •)  NOW  = - [c (i- o £]' [ c r cllc % cl [ c ? c] T  SO  =  T r [-Y  Obviously,  e  £  since  1! X ^ e  A  this  K ( iv V i + 5 Y, ^ x  result  is  ^ Y ' ( i Y,\] =  i n d e p e n d e n t o f k and q,  o t h e r c r o s s t e r m i s t h e same. With these r e s u l t s ,  +  the c r o s s s e c t i o n i s  the  145  - 1< ( p -  (D. 1 1 )  e  F o r muons d e c a y i n g a t r e s t ,  v  IV .\Mv J\[al^ (^V(p ) -^X ( .^l] e  r  y  a  v  ; l  p t  ( D  .  1 2 )  where  (D.13)  and  (D.14)  with (D.15)  146  I\. same  was  c a l c u l a t e d i n A p p e n d i x C.  I  v  i s derived  in exactly  the  way. It  is  only  a m a t t e r of a l g e b r a i c m a n i p u l a t i o n  r a t e d e r i v e d here i n t o the  form of e q u a t i o n  (3.2.16).  t o put  the  147  E. D i m e n s i o n a l  Reqularization  Dimensional r e g u l a r i z a t i o n divergent  Feynman  attractions, u s e . The gauge  amplitudes  in a practical  dimensional  invariance.  i s a technique finite  The  [20].  sense, l i e s  regularization  used One  to  of i t s major  i n the s i m p l i c i t y  procedure  also  Ward i d e n t i t i e s a r e e x p l i c i t l y  u s i n g t h i s method p r o v i d i n g t h e y do  not  depend  render  on  of i t s  preserves satisfied quantities  which are unambiguously d e f i n e d o n l y f o r four d i m e n s i o n a l space. This  is  the case because the b a s i c  dimensional space-time One  regularization  f e a t u r e of t h e t e c h n i q u e of  i s a continuation  in  the  number  of  dimensions.  e n c o u n t e r s i n t e g r a l s of t h e  form  (E. 1 )  which  are  divergent  analogous dimension  integral,  for  some  values  analytically  of  OL .  continued  However, to  the  space-time  n  (E.2)  is  convergent  for  some  value  o f n. H e r e we  i n t e r n a l momentum p = ( p , , . . . , p ^ ) w h i l e t h e  h a v e n-component  external  momentum  k  remains a f o u r - v e c t o r . Rearranging,  (i)  148  where is  B ( k ) = m + k . W e make a c h a n g e o f v a r i a b l e s 2  legitimate  2  since  the i n t e g r a l i s convergent  t o p'=p+k,  which  f o r some n,  (ii)  J ( n Tr\  P e r f o r m i n g a Wick r o t a t i o n such t h a t  po-* i p o a n d  defining  K  2  as = - PI " the  ~~ H  1  i n t e g r a l becomes  i [ ^ L  _1  Rewriting  = i.(-iY<C^w  t h e dummy v a r i a b l e  t(-iyfd\  The  ( i i i )  integrand  split  (iv)  .  ( ) V  i s i n d e p e n d e n t o f a n g l e s s o t h e i n t e g r a t i o n c a n be  \ ±  = ^£l„S  where p=J^- ...- +  and  +  angular  _  k a s p, we h a v e  \  up i n t h e f o l l o w i n g  ^>  '  fashion.  ?  the  r ' generalization  (  vi)  of t h e n - d i m e n s i o n a l  integral i s =  r ^ -  s  l  ^  9  - A ^ ^  (vii)  149  The l a s t a n g u l a r I o  integral, written  i n t h e s t a n d a r d manner i s  j<f>  Using W a l l i s s i n e over  1  formula  f o r the i n t e g r a t i o n of the  mth  power  of  t h e i n t e r v a l 0 t o ft,  it  ^ S i ^ 8 6&  the a n g u l a r  -  J¥  r ( a ( ^ v i M  •  (  v  i  i  i  )  i n t e g r a l i s s e e n t o be  Then o u r i n t e g r a l becomes  which  W  where  i s e v a l u a t e d as  P""'  the  definition  change of v a r i a b l e s o f t h e gamma  = f V -  e  -M  S  w i t h S. = ( x + 1 ), we 2  x-= arc  follows = f a *  x=  x"-'  ?  <xi)  . h a s been u s e d .  Using the  function .  r  obtain  C"d*  V '  e"  i S U  i  i  )  150  A second change of v a r i a b l e s t o y=x  T=  *  Defining  [ i 5^ ' e -  s  2  U  yields  ^""^'^  U  i  v  )  t h e new v a r i a b l e ^ = y s , we h a v e  r Now  t h e s a n d ^ i n t e g r a t i o n s a r e s e p a r a t e d a n d c a n be  as d e f i n i n g gamma f u n c t i o n s  so  recognized  finally  (xvi )  Then o u r o r i g i n a l  integral  becomes  (E.3)  Further  integrals  differentiation example,  so  that  with  are  obtained  respect  to  from  this  one  upon  t h e e x t e r n a l momentum k. F o r  151  (E.4)  Other  integrals are obtained s i m i l a r l y .  For instance,  (E.5)  Clearly, of  o<.  the divergences and  i n these  the dimension  of  integrals  for  space-time  certain  values  continued to n=4 are  d i s p l a y e d a s p o l e s i n ( n - 4 ) i n t h e gamma f u n c t i o n s . D e f i n g £. =(4n), t h e gamma f u n c t i o n e x p a n s i o n i s  _1L where 1 i s t h e E u l e r - M a s c h e r o n i expand  constant.  a l l n-dependent q u a n t i t i e s  the l i m i t  of n equal  contributions. of  t- (SCO - V - Y It is  i n powers of  t o four i n order t o  In order t o i l l u s t r a t e  avoid  this,  important  to  before taking missing  finite  we t a k e t h e e x a m p l e  1 52  The  ln4lT  p a r t o f t h e r e s u l t w o u l d be m i s s e d  to  four  prematurely.  scaling the  This  point  of the coupling constant  electric  convenient  charge  e  i n n dimensions. For  to define a dimensionless e =  e  D  set  a l s o comes i n t o p l a y  has dimension  0  i fn is  (mass) " ' 2  coupling  u  i n the  instance,  so t h a t  1  equal  i t is  e by  C^)  where jx i s some mass s c a l e o f t h e p r o b l e m . We  list  b e l o w t h e Feynman p a r a m e t r i z a t i o n s  in putting integrals  into the general  form  which are u s e f u l  (E.I).  i  ^  I S c ^ o - ^ r  —T~- =  rU 0  ^  ( E  X"  -  6 )  (E 7)  L a x •¥ b ( l - K H  U-vV.  fdx,  c i ^  (E.8)  ( Z-a^ x X L  „ \ vn  Since  =  the  algebra,  '  «  evaluation the  Dirac  X  of rj  U-X )  Feynman  matrices  ,(E.9)  amplitudes must  be  involves generalized  Dirac to  n  d i m e n s i o n s . I t i s when one e n c o u n t e r s K , w h i c h  is  defined  terms  ^  ^ , that the  5  of  the  completely  antisymmetric  tensor  V o l  in  1 53  method o f d i m e n s i o n a l r e g u l a r i z a t i o n tensor  t^,^  dimensions i s as  V « X  f o r the Dirac  to arbitrary n matrices  in  n  (E.10)  r  (E.II)  ( a - ^ X ,  The p r e s c r i p t i o n  l^Y^  K  The  follows.  7 1  ^  ill-defined.  c a n n o t be u n a m b i g u o u s l y g e n e r a l i z e d  d i m e n s i o n s . The g e n e r a l i z a t i o n  ^V".  becomes  =0  we have u s e d f o r X5 i s  yu= 0^ . - . , n  (E.12)  1 54  F.  Self-Energy  We for  use  is  the  internal  The given  help  The  n  self-energy  ' t H o o f t - F e y n m a n gauge p r o p o g a t o r s  fermion  lines, respectively.  c o n t r i b u t i o n f o r a c h a r g e d l e p t o n of mass m  by  f 4 ^ (-tOrV-i  of e q u a t i o n  coupling  has  performing  ( E . 1 0 ) and  (E.11),  IvxV-ieoU the D i r a c a l g e b r a t h i s can  been s c a l e d a s d e s c r i b e d  i n the  y  \ t ( f  after  parametrization  denominator  Then  Corrections  f o l l o w i n g two  dimensions,  Feynman  Vertex  p h o t o n and  - L Z ( ^ =  In  and  (E.6)  integrand.  is Let  used  be  written  i n A p p e n d i x E. to  (  with  F  1)  the  as  Now  reexpress  the the  1 55  where  Upon a c h a n g e o f v a r i a b l e s , t h e s e l f - e n e r g y  The  n-dimensional  of e q u a t i o n s  c o n t r i b u t i o n becomes  i n t e g r a t i o n s a r e c a r r i e d o u t t h r o u g h t h e use  ( E . 3 ) a n d ( E . 4 ) . The r e s u l t i s  VW The l e p t o n  self-energy  c o r r e c t i o n c a n be e x p a n d e d i n  powers  of  '-^-m); t h a t i s , Z . ( ^  -  A  V  B ( j f < - > v O <- • • •  (F.5)  Therefore,  A  We  (F.6)  =  define  Then  L - 4- - W  (  F > 7 )  156  Expanding  i n p o w e r s o f <c,  ( 3 [ g_ - L  A = ^ _  This  term  counterterm.  is  + J w / H i r w M l + H\  eliminated  (See F i g u r e  by  the  (F.B)  subtraction  Starting  r e s p e c t t o p,  a  mass  13.)  The s e c o n d t e r m i n t h e e x p a n s i o n o f ^M^) follows.  of  is  evaluated  from e q u a t i o n ( F . 4 ) , and d i f f e r e n t i a t i n g  as with  we have  (F.9,  I n t e g r a t i n g and expanding  i n p o w e r s o f i.,  B = ^ - 5 [ | : - W ^ ) 1 - H 1  the r e s u l t i s  (  f  .  i  0  )  157  F o r t h e c a s e o f an a n t i l e p t o n  self-energy  term,  the  expansion  analogous t o (F.5) i s -  A't  B'tj/tw^  W o r k i n g t h r o u g h , one  (  the  term  A'  .  n  )  obtains ^  Again,  F  is  (F.12) eliminated  by t h e s u b t r a c t i o n  of t h e  amplitude i n the 't  Hooft-  a p p r o p r i a t e mass c o u n t e r t e r m . The d i v e r g e n t v e r t e x c o r r e c t i o n Feynman  gauge i s , i n n d i m e n s i o n s i s  The p a r t o f t h e • n u m e r a t o r  i n s q u a r e b r a c k e t s c a n be w r i t t e n a s a  sum o f t h e t e r m s N = <9(-n where  i -&  * $ l v )  (F.14)  158  00  Then  where  (F.16)  The  evaluation  of  these  three  U s i n g t h e Feynman p a r a m e t r i z a t i o n , can  be w r i t t e n  i n t e g r a l s proceeds as equation  (E.8),  follows.  the i n t e g r a l s  as f o l l o w s .  I  J  J  5Ttr[i -24( «y o-xo1] l  The n - . d i m e n s i o n a l i n t e g r a l s we u s e d a r e  ?e  M  ,(F  - ' ,  7)  159  where  ^ We  v ^y'  do t h e y i n t e g r a t i o n  (F.18)  1  n e x t . For t h e term of o r d e r  integral i s  •4 For  >  -  the term  The t e r m i n k  t  i n k, t h e y i n t e g r a l g i v e s  2  yields  o  Now  the x i n t e g r a t i o n s  Using  must be d o n e . T h e s e  are,first,  k°,  the  y  160  and  the r e s u l t i s  - iJ^Se  - 2  iWx j W l - x )  + AJUSTj  where x=2E /iry _ a n d t h e Spence f u n c t i o n e  A  a  The  integrals  The r e s u l t s  In  T u  f o r I and 1 1  2  are given  (F.21)  a t t h e end o f t h i s A p p e n d i x first.  o f 1 ° , i t was i m p o r t a n t t o e x p a n d R  of t because of the f a c t o r  be s e t e q u a l t o f o u r  t e r m s . The r e s u l t i s  is  ^ t-U  used above a r e l i s t e d  t h e x' i n t e g r a t i o n  powers can  J o  (F.20)  in  the  3  in  l/£. I n t h i s c a s e , h o w e v e r , n integral  without  losing  any  161  xO-x+s^ The f i n a l  Again,  n  1  integral  can  x  (F.22)  J  t o be done i s  be s e t e q u a l t o f o u r  i n the term w i t h the  P(3-n/2) b u t n o t i n t h e l a s t t e r m . I n t h e l a s t be e x p a n d e d a s (1 -(£/2)InR). The r e s u l t  term  R *- "  factor 2  of the i n t e g r a t i o n s  must is  ( x-a.) +  '  X ( I- X +  (F.23)  162  where  A=  X ~H^^  (F.24)  a  Contracting neglecting  the integrals with the  electron  mass  t h e i r a p p r o p r i a t e numerators and wherever  possible,  the  vertex  simplifies to  t-  1 L  >y  A  \-x  '  vy^  \-x  '1  '  5L (F.25)  The  x'  listed  3)\<U' ±  integrals  which  we u s e d  i n t h e above c a l c u l a t i o n s a r e  below.  - J±  -^$e -  !  ^X *  ^  (i^X-JUO  163  X ( I- X +  ^  1  j  a(v-x+  0  ^  x-a.- v!T  It  now  remains  corrections over  the  to  (S+V)  the  self-energy  w i t h the f r e e decay diagram  neutrino  contributions  interfere  momenta.  The  and  (M) and  self-energy  and  vertex integrate vertex  are (F.26)  where  (F.27)  (F.28) (F.29a)  6*  = ^ IbTf  -U l ^ -  L  yvL  ^  ( l t l£± l-X  JUxY" M  (F.29b)  164  i l  J _ ^ \ [ \ +- _X J U x l l-X \ -wy, J 1-X  X  L  J  Of c o u r s e t h e f r e e d e c a y a m p l i t u d e i s  So t h e v e r t e x  and s e l f - e n e r g y  c a n be w r i t t e n a s  where  Therefore,  the  n  f o r the i n t e r f e r e n c e  required  elements are  = ^(^.pjH^.po  (F.29c)  165  So  The  integration  over  t h e n e u t r i n o momenta i s , f o r t h e c a s e o f  o n l y one n e u t r i n o m a s s i v e enough t o c o n t r i b u t e  significantly,  where  Q = f ^ - pe  Putting  a l l of  equation  (4.5),  spectrum  with  considerat ion.  this  together  yields,  the photonic v i r t u a l only  one  neutrino  after  some  algebra,  c o r r e c t i o n s t o the Michel  being  massive  enough  for  166  G. C a l c u l a t i o n  In rate  of the R a d i a t i v e  this  Appendix,  f o r the r a d i a t i v e  Feynman  diagrams  the decay  Muon D e c a y  d e t a i l s of the c a l c u l a t i o n (1.6)  are  a r e shown i n F i g u r e  where ^  the  equation  i n t h e Feynman g a u g e .  (fr**t->Q V i - V s W ^  ?  ^ V Y S W (  P  (G.D  0  (6.2)  (G.1) a b o v e , £ i s t h e p o l a r i z a t i o n  p h o t o n . The f o l l o w i n g e^e^ L  properties  vector  of  of & a r e u s e f u l . (G.3)  = 0  (G.4)  I ^ l t K ^ - ^ Rearranging  four  = -l  Summing o v e r p h o t o n p o l a r i z a t i o n s  yields  relevant  i s t h e n e u t r i n o p a r t of t h e a m p l i t u d e . That i s , / ^ (  In  The  14. The a m p l i t u d e f o r t h e  sum o f t h o s e two d i a g r a m s i s t h e f o l l o w i n g  tu(pQ^  given.  of the  equation  y i e l d s , i n t h e Feynman gauge  ( C  .5,  (G.1) f o r t h e a m p l i t u d e a n d s q u a r i n g  167  (G.6) Summing o v e r e l e c t r o n the n e u t r i n o  and n e u t r i n o  spins  and  integrating  over  momenta,  Z  (G.7)  In t h e above, Q =  Y/*r per  £  (G.8)  and  (G.9)  168  where  V-s y\v\«,  ' ^  ^  '  f  i  ^  i  '  ^  -  f  f  M  s  V  i  t  -  ^  ^  l  ( G  .  1 0 )  Therefore,  Recall  that  the integral  U l £  was  calculated  only  one n e u t r i n o  coefficients  3  5-  S*(Q-  in  -  ? i  )  Section  ?  J  ft"  (G.12)  (3.2). For the present case,  i s massive  enough  Va  the  A and B r e d u c e t o  vis.)  u r  Q" (G.14)  x  was  f o u n d t o be s y m m e t r i c  the antisymmetric part of X labelled  where  for consideration,  TT " ( 0>  Also,  for  p i  (T)^  through  the charged lepton  part  will  (5)^ *  n  i n t h e i n d i c e s ^ and o so  not  contribute.  equation  The  terms  (G.7) a r e t h e t r a c e s  of the c u r r e n t . Only  the  parts  of  169  these  terms  symmetric  in  ^  and  OJ w i l l  contribute  bremsstrahlung  r a t e s i n c e t h e y a r e m u l t i p l i e d by N ^ ,  symmetric  ^  in  and u>. T h e s e s y m m e t r i c p a r t s  to the  which  is  of the t r a c e s a r e  l i s t e d below.  = M4^)[i (peLi-(f ) ^^cj p  e  e  e [ 0  (i.p )]  (G.16)  e  = ^ [ ( ^ ^ ^ - e H . - ^ - i l e J ) t(p U^,(p .-eV(p ..i)e^ e  - ^ u> ( ( • f e - - ^ J ( ^ - e ) ?  p^^pe^l  y  /  (G.17)  170  (G.18)  (G.19)  T ^ ^ Y  ?  f  e  ^ ^ L ( h ^ i  s  (G.20)  -- 4L(p>U £ (p^V ?  t-(  -fc^fe- pj- ( . fc.)0 ?e  1 72  (4f/>[$^jvO- ^ ( ^ K i ^ t  fc L(jfeL(£-^ e  t  i (pe' W  p*i>Vl  (i-peVie^^  ^ ^ ( ^ ^ [ ^ ( p e ^ - C i p e ^ . - ^ ^ p e . ^ ^  6  !  1  !  p  K e  ^ ^ l  <G.23)  where p  r  = p^- V V y . ^  The d o u b l e d i f f e r e n t i a l  (G.24) rate f o r the bremsstrahlung  therefore,  cPj> dH e  UE^Ca^^avy)  CaTr^  process i s ,  173  -u-^pe-^l^-^vi-avcp^-feife-an  -(fe • p t V ^ Q l L ^ . Q ) - ( p ^ - £ ) - ( ^ ^ \ C e - Q ^  At  this  point,  the  p h o t o n p o l a r i z a t i o n s have n o t been  o v e r . Of c o u r s e , t h i s c o u l d However,  the  neutrinos terms  for  have been done a t an e a r l i e r the  h a s been c a l c u l a t e d by  of  intermediate and  rate  Eckstein  the step  (G.25)  polarization  radiative Pratt vector,  decay and  Eckstein G.  i n our c a l c u l a t i o n s e r v e s as  choose  with  stage. massless [83]  Therefore, a  summed  check.  t h e gauge s u c h t h a t G - p ^ O . I n t h i s  in this  Pratt gauge  174  and  i n the l i m i t  of massless neutrinos  agreement w i t h t h e i r  equation  (G.25)  is in  result.  Summing o v e r p h o t o n p o l a r i z a t i o n s w i t h t h e h e l p o f e q u a t i o n (G.5)  yields  e q u a t i o n (4.11) f o r t h e d o u b l e d i f f e r e n t i a l  r a t e . To o b t a i n t h e e l e c t r o n  s p e c t r u m , t h e p h o t o n momentum  be  i n t e g r a t e d o v e r . Of c o u r s e , t h i s  the  integration  isfirst  w ^ i s t h e maximum p h o t o n  phase  space c o n s i d e r a t i o n s  i s a d i v e r g e n t p r o c e d u r e so  e n e r g y , w h i c h was c a l c u l a t e d  i n Appendix  * y 0- *  the ^ - a x i s .  from  C t o be  (G.27)  A l s o ^ = c o s 8 , where 9 i s t h e a n g l e b e t w e e n and  must  g e n e r a l i z e d t o n d i m e n s i o n s . That i s ,  Here,  <*\w =  decay  the  photon  momentum  To c a r r y o u t t h e k i n t e g r a t i o n , we d i v i d e t h e  r a t e up i n t h e f o l l o w i n g  manner.  175  \'2. -I  B Q  1  1'  A  ""5"  , l  "("^TTipJ t Q » p e - p > ( 4 - p^(pe-Q^  cfep^f  ^f^  [  +  x  A(>-^('(p -Q^3L  _ I  e  +  j v ^  d  ?  e  f  _ ape.n  ^.p^ct-^n \  5"  176  [<pe-P^V-fe-0>-^-ffeKp -6l)] lA  Defining  t h e | - a x i s t o be  momentum,  x  the  (G.28)  direction  of . t h e  electron  t h e f o l l o w i n g q u a n t i t i e s c a n be e x p r e s s e d i n t e r m s o f  o>, t h e p h o t o n Q  in  (Pz-QY  energy.  = y^y ( w - u ^ 7 a - x - v \P ^)  (G.29)  0  (G.31 )  where NT and  = J X^- 4 C '  >-  i s g i v e n by e q u a t i o n ( G . 2 7 ) .  Also,  C l - * * ^ a- X -V The s p i n - i n d e p e n d e n t p a r t This  part  (G.32)  (G.33)  of the r a t e w i l l  c o n s i s t s of t h e seven terms  be d e a l t w i t h  labelled  first.  " 1 " t h r o u g h "7"  i n e q u a t i o n (G.28). I n t e g r a t i n g over o, t h e s e terms  become  177  N  (G.34)  (G.35)  (G.36)  178  (G.37)  (G.38)  (G.39)  179  w -71 n  7  (G.40)  where  (G.41 )  (G.42!  180  0  and  iW^+  t  lZVl_  l  ^  (G.45)  to  (G.46)  where  Wo. - J doo I t * W ^ ) (LO -OJ^  0  0  Wo, - \ dto ( u W k ) ) co  (G.47)  and  -uX. =  w  -  (  G  .  4  8  )  181  Therefore,  Wo (G.49)  0^ =  (G.50)  (G.51 )  (G.52)  As an e x a m p l e o f t h e results the  most  equation  \d = -__L  Now,  above,  W,  calculations will  complicated  be d e r i v e d .  type.  After  involved  in  obtaining  the  This integral i s , i n fact, an  integration  by  parts  (G.42) becomes  l^w  vv 00  because  '  of  Lt  J -3(tO -to^(u3 -u>) + ZLuJo-utiLbX^-oo) m  the  overall  0  factor  of 1/(n-4),  expanded i n powers of (n-4) i n o r d e r t o c a l c u l a t e terms. That i s ,  (G.53)  * must be  a l l the f i n i t e  182  (G.54)  After  substituting  into equation  o b t a i n e d by s t r a i g h t f o r w a r d  (G.53),  the f o l l o w i n g  result i s  integration.  2. U)  0  COvw \  l/Oy (G.55)  Using (G.56) this  r e s u l t c a n be p u t i n t o t h e f o r m  derivation  of  integration  by p a r t s f o r t h e r e s t  overall  factor  immediately. by  UTi  of  similar  l/(n-4)  Then e q u a t i o n s  straightforward The a n g u l a r  factor  is  (1 - y )  2  so  to  of the  equation  (G.42).  above. In the  can  be  set  do  no  equal to four  (G.43-44) and (G.50-52) a r e o b t a i n e d  integration. integrations  remain  to  be  done.  Again,  the  must be e x p a n d e d a s  "MA, -X.  to  initial  of t h e i n t e g r a l s , t h e r e i s n  The  the i n t e g r a t i o n  (G.57) of terms c o n t a i n i n g  a factor  of  l/(n-4).  183  For  a l l the other terms,  integrals  n  can  be  w h i c h we u s e d a r e l i s t e d  The r e s u l t f o r t h i s w h i c h we o b t a i n The  i n equation  dependent  part  evaluated.  As seen from e q u a t i o n  terms  (s^-k)  in  and  in  contribution  way. T h i s  of  ^  However,  \ i i  the  £  The  Appendix.  bremmstrahlung  rate  remains  to  be  consists  of  The (s • k) t e r m s c o n t a i n in  their  t h e ( s y k) t e r m i s f o u n d  iHcocU.ae^  d  c a n a l s o be e x p r e s s e d  no  evaluation.  i n the f o l l o w i n g  V  l  W J ,  (G.58)  i n terms of t h e a v a i l a b l e  four  as f o l l o w s . T •= X  * ( p  e  ^  t /3 ( y Y  i s , ( s ^ ; k ) = c ( s ^ - p ), e  P - 3" =  t o (s^- p ) s i n c e e  (s^- p^)=0;  a n d o must be c a l c u l a t e d . We have  (^S v  (G.60)  fi(*\ ^  (G.61)  x  fc  p^-T  (G.59)  r  Then t h e ( k ) t e r m i s p r o p o r t i o n a l that  of t h e  (G.28), t h i s p a r t  ( s yp ) .  four.  t e r m c a n be w r i t t e n a s ( s ^ ) ^ ( J ) ^ where  =  vectors  of  to  (4.16).  d i v e r g e n c e s so n c a n be s e t e q u a l t o f o u r The  equal  a t t h e end of t h i s  spin-independent part  i s given  spin  set  e  -  y v  So  7-  Now  f * <W3V.^pe-7)] UC X\ '  (p^. J ) a n d (p^- J ) a r e c a l c u l a t e d by e x p l i c i t l y  (G.62)  doing  the  184  and  ^  integrations  p a r t . The  CJ i n t e g r a t i o n s  spin-independent at  {Sjj p )  have  so  The f i n a l  spin-independent  The  result  listed  Appendix. spin  dependent  part  i t i s evaluated i n n dimensions.  a l l been  calculation.  j u s t as f o r t h e  i n v o l v e d h a v e a l r e a d y been g i v e n i n t h e  term of the  t  divergence  n=4  p a r t a n d t h e ^ i n t e g r a l s a r e among t h o s e  t h e end o f t h i s The  with  done ^  in  the  contains  The  spin-independent  integrals  part  i n t e g r a l s a r e a t t h e end o f t h i s  of these  calculations  is  given  a  of  this  Appendix.  in  equation  were u s e d  i n these  (4.16). We  list  here  the angular  i n t e g r a l s which  calculat ions.  2  4,  " j U ^ ^ x r T  I  1  x+C  185  . i  =  10 \ A  11  12.  13  (dx  !  x5  =  C^sl)  /—V  (  -  186  15.  J d ^ J k ( V x - i - v P ^ = - 2.  x-  -I  + JUH-O-XV-S^ V ( a - x ^  Jw/a-x +  ^  (xU  V  L  1  31-X-vP  /  1-xtV  (x (X  (xX-vvP^f  (x-  a - x-\p  P  187  

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