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Recoil effects in bound muon decay Brookfield, Gary John 1981

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RECOIL EFFECTS IN BOUND MUON DECAY by GARY JOHN BROOKFIELD .Sc., The U n i v e r s i t y of B r i t i s h Columbia, 1978  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES ( Department of P h y s i c s )  We a c c e p t t h i s t h e s i s as conforming t o the r e q u i r e d  standard  THE UNIVERSITY OF BRITISH COLUMBIA March, 1981  (5) Gary John Brookf i e l d , 1981  In p r e s e n t i n g requirements  this thesis f o r an  of  British  it  freely available  agree for  that  understood  that  Library  shall  for reference  and  study.  I  for extensive  copying  be  her or  shall  publication  not  be  Date  DE-6  (2/79)  Apr.  2S  , iqBI  of  Columbia  make  further this  thesis  head o f  this  It  my  is  thesis  a l l o w e d w i t h o u t my  PHYSICS  The U n i v e r s i t y o f B r i t i s h 20 75 Wesbrook P l a c e Vancouver, Canada V6T 1W5  the  representatives.  permission.  Department of  copying of  g r a n t e d by  the  University  the  h i s or  f i n a n c i a l gain  the  that  p u r p o s e s may by  f u l f i l m e n t of  degree at  I agree  permission  department or  for  advanced  Columbia,  scholarly  in partial  written  ABSTRACT  An  unbound muon a t r e s t d e c a y s  electron.  Momentum  energy g r e a t e r is  t h a n one h a l f  bound t o a n u c l e u s ,  make  i t possible  muon  mass.  This  thesis  spectrum  a  interacting  energy  massive  generated  corrections  asymmetry  important conversion.  t o the e l e c t r o n  using  an e f f e c t i v e equations  n u c l e u s and e l e c t r o n  ( o r muon)  a  one  particle  Dirac  well.  Such  nuclear a  a n d an e x p r e s s i o n recoil.  spectrum  s i z e and  calculation  wave f u n c t i o n s  including  recoil  the  to  a r e presented. For small  analytically  an  simplifies  requires  electron would  and n u m e r i c a l  i s n o t done h e r e , t h o u g h t h e a p p l i c a b l e  formulas  i f t h e muon  t o approach the  are  Z, c a l c u l a t i o n o f t h e e l e c t r o n  distortion.  an  t o have  nuclear  f o r muon t o e l e c t r o n  electromagnetically,  numerically  -and  electrons  approach  the e f f e c t s of f i n i t e  function  motion  recoil  and  the electron  f r o m bound muon d e c a y  This  finitely  large  including  and  investigates  fora potential  For  energy  i n the search  approach.  describing  equation  high  neutrinos  t h e muon mass. However,  its orbital  a n d asymmetry  potential  forbids  f o r t h e decay e l e c t r o n  These  background e f f e c t  and  conservation  i n t o two  wave involve  integrations,  methods a n d  general  Z the c a l c u l a t i o n can proceed  h a s been d e r i v e d  f o r the spectrum  iii  CONTENTS  1 INTRODUCTION  1  2 THE EFFECTIVE POTENTIAL METHOD  5  2.1 The E f f e c t i v e P o t e n t i a l .'  5  2.2 The C e n t e r Of Mass T r a n s f o r m a t i o n  14  2.3 Bound S t a t e s  22  2.4 S c a t t e r i n g S t a t e s  28  3 BOUND MUON DECAY  33  3.1 The Wave F u n c t i o n s  33  3.2 The Decay Rate  36  3.3 The Phase Space  40  4 PLANE WAVE BORN APPROXIMATION  43  4.1 The Approximate Wave F u n c t i o n s  43  4.2 The Decay Rate  47  4.3 The Phase Space I n t e g r a t i o n  52  4.4 The End Of The Spectrum  58  4.5 C a l c u l a t i o n s , R e s u l t s And C o n c l u s i o n s  61  iv  FIGURES  F i g . 1. The I n t e g r a t i o n Region F i g . 2. The I n t e g r a t i o n R e g i o n For F i x e d  42 p  53  fc  F i g . 3. The E l e c t r o n Spectrum From Bound Muon Decay  65  F i g . 4. Log Graph Of The T a i l Of The Spectrum  66 9  F i g . 5. R e c o i l / n o R e c o i l Decay R a t i o  67  F i g . 6. The Asymmetry  68  F i g . 7. The End Of The Spectrum  69  F i g . 8. The Three P a r t s Of The R e c o i l C o r r e c t i o n  70  1  1 INTRODUCTION  The advent of gauge t h e o r i e s of p a r t i c l e led  interactions  has  t o a renewed i n t e r e s t i n the s e a r c h f o r e x o t i c i n t e r a c t i o n s  which v i o l a t e baryon number or l e p t o n number. In p a r t i c u l a r , f o r a greater understanding  of the l e p t o n - g u a r k  experiments  have  muon number  conservation,  neutrinoless  been  electron  One  electron  to  negative  muons  of  muon  to  the  as  muon  c o n v e r s i o n . In o r d e r t o d e t e c t to  most  the  , fk -»3e  such  know  the  ,  and  backgrounds  very  important of the backgrounds t o  conversion  from  problem,  proposed t o s e a r c h f o r decays v i o l a t i n g  t h e s e v e r y r a r e events one has accurately.  generation  is  the  conventional  decay  of  ground s t a t e of atoms. These mesonic —lO  (. < 10 s ) when a muon beam  atoms form almost immediately  enters  m a t t e r . U n l i k e the. decay of f r e e muons, the decay of bound muons can  produce  |;rfy» < E recoil  K e  a  s m a l l f r a c t i o n of e l e c t r o n s i n the energy range  /» • T h i s i s because the n u c l e u s can take  m  momentum  with  only  a  Mev high  energy  by n u c l e a r  of  from muon  this  range  the  background  to  electron  conversion.  100 The  end of the spectrum i s a l s o the p a r t most a f f e c t e d  recoil.  e l e c t r o n spectrum  calculated  by  Porter  and  from  bound  Primakoff  1  muon in  G i l i n s k i and Mathews , U b e r a l l , and H u f f * i n 2  the  large  s m a l l enough t h a t one might be a b l e t o d e t e c t the ~  e l e c t r o n s expected  The  a  s m a l l f r a c t i o n of the a v a i l a b l e  energy. At the h i g h energy end becomes  up  approximations  3  used  for  decay 1951.  was  first  L a t e r work by  1960-61  improved  the wave f u n c t i o n s . The  paper by  2  Huff  i n c l u d e s a good d i s c u s s i o n o f t h e  the  muon  decay  r a t e c a u s e d by t h e a t o m i c  Mathews c a l c u l a t e d t h e asymmetry  of  time.  Raff,  Later,  electron  Hanggi, V i o l l i e r ,  for  the  recoil  admixture addition the  of  scalar,  effect  of  electron  and  nuclear  m a g n e t i c moment, and t h e r e c o i l  binding an  calculate energies,  effective  the  t h e weak d e c a y v e r t e x ,  rates etc.  electron  of  course,  nuclear potential motion.  size. must To  If  system w i t h  order  perhaps  nuclear  some  mass  Other  latter  of  to include in  the  the be  the  muon  the  effects  nuclear  nucleus  this  a  due  to  then the  of  nuclear N  magnetic field.  and  Coulomb  velocity, V  of  use  nuclear  modification  i n a d d i t i o n to the e l e c t r i c  a spin 1 / 2 , point  functions,  i s not i n f i n i t e  a r e c a u s e d by t h e p r e s e n c e  around the nucleus a  the  be m o d i f i e d  first  modifications  would,  , with  The  of i n f i n i t e  calculate  V = ~2<x/r  discuss  o f bound muon d e c a y , we  to  potential,  in  electromagnetic  on t h e wave  mass, t h e p o t e n t i a l one would u s e functions  couplings  thesis.  p o t e n t i a l method. I n t h e l i m i t  wave  7  the e f f e c t s of a  effects.  the e f f e c t s of r e c o i l decay  of  small  on t h e s p e c t r u m .  corrections  To  wave  a  i n t e r a c t i o n , and a l s o  bremstrahlung  of t h i s  of  tensor  vertex  i s the subject  the  generated  possiblity  not c a l c u l a t e d i n these papers a r e  these  repeated  below. Herzog and A l d e r  effects  to  first  i n c l u d i n g the e f f e c t s of  the  V-A weak  the  o f t h e n u c l e u s a n d some  pseudoscalar  t o the standard  for  numerically  size  include  decay  to  G i l i n s k i and  5  c o r r e c t i o n s t o be d i s c u s s e d  e x t e n d t h e a b o v e work t o  binding.  changes  and A l d e r " '  e l e c t r o n a n d muon,  vacuum p o l a r i z a t i o n , t h e f i n i t e the  the  spectrum c a l c u l a t i o n using  functions  qualitative  , the field  Applied to  approach gives the  3  Breit is  interaction .  The e f f e c t i v e  8  applicable  charge  spinless  distributions.  extremely mass  to  Even  ultra-relativistic,  potential with  spherically  though  the  electron  way  motion  a  "reduced" energy. Aspects of the e f f e c t i v e  by F r i a r  by  et a l  Barrett  did  their  their bound  o f mass i n v o l v e  1  1  0  ~  l  to electron  x  t o the energy  2  calculations  equations  in a different  i s more i m p o r t a n t  include only the  because  /2  potential  and  approach  levels  of  from n u c l e i ,  and  muonic  atoms.  They  in this thesis.  In  of the center  the  o f mass  system.  of Hanggi  et a l  by t h e f i n i t e  of t h i s  method and c o n s i d e r s of e l e c t r o n s  Section decay.  potential  and  Herzog  et  al  i n phase space a l l o w e d t o mass o f t h e n u c l e u s , a n d  potential  and m o d i f i e d  wave  c h a n g e s made h e r e .  Sec t i on  states  equations  o f mass frame a n d d e r i v e d  the p o s i t i o n  none o f t h e r e d u c e d mass, e f f e c t i v e function  of  o f t h e change  particles  be  i t i s n o t t h e same i n t h e m u o n - n u c l e u s  corrections  the e f f e c t s  outgoing  levels  way t h a n  s y s t e m and t h e e l e c t r o n - n u c l e u s recoil  the  a modified  scattering  i n the center  muon d e c a y c a l c u l a t i o n  The  could  by G r o t c h a n d Y e n n i e ' t o t h e e n e r g y  hydrogen,  symmetric  t o t h e c o m p l e t e l y non-  of  have been a p p l i e d  here  i t a l l o w s us t o d e f i n e a c e n t e r o f  a p p r o a c h . B e s i d e s t h e r e d u c e d mass,  i n the center  derive  nuclei  and a r e d u c e d mass i n a s i m i l a r  relativistic  we w i l l  The  calculation  thesis  develops the e f f e c t i v e  its application (muons) a n d  3 applies integrals  of s e c t i o n  2  to  bound  muon  n e u t r i n o momentum a r e done a n d t h e  p r o c e e d s as f a r as p o s s i b l e  muon wave f u n c t i o n s a c t u a l l y  and bound  nuclei.  the r e s u l t s over  to the s c a t t e r i n g  potential  being  without  specified.  the electron  and  4  In  4,  section  functions  which  analytically t o be s e e n  for  2  is  generated  muon  and  the e f f e c t of f i n i t e  We  unless  will  particular  we u s e u n i t s  as.  p  have  as  written  p fo  of  A  spinors  a Greek  A  ft  direction  exact  U*U=I  instead  C = f\-\  .  three-vector  = B  , and  is  E ^ B " P * A  p  is,p = p/p  Pa  Four-vectors  l^s t h e  scalar  •  that  UU=I  of  . Three-vectors w i l l  The P  B j o r k e n and  that  = p^p^ . A  of  exception  p/*  P  functions,  One  index,  using  wave  conventions  four-vector  , that  spectrum  size.  The  p  wave  be done  calculation  electron  the  that  . So we g e t  of p  the  .  the three-vector  to  on t h e e l e c t r o n  nuclear  so  /  p -p  integrals  noted.  so  p ,=(Ep'i  , thus  written  use  otherwise  our  f o r these  nuclei. work  normalize  always  remaining  future  Notation: 1 3  the  and t h e e f f e c t s of r e c o i l  numerically  Drell  enable  f o r small  Left  including  c r u d e a p p r o x i m a t i o n s a r e made  P  unit  1  S  t  h  . In will  be w r i t t e n product  scalar  we  is  product i s e  magnitude  vector  i n the  5  2 THE  2.1  The  Effective Potential  The the  e f f e c t i v e p o t e n t i a l method w i l l  e f f e c t s of  the  E F F E C T I V E POTENTIAL METHOD  nuclear  electromagnetic  muons) . We  of  implies  that  the  M  the  or  wave  of  nuclear  energy v a r i a b l e s  order  the  interaction  assume t h a t  than a l l o t h e r terms  m o t i o n on  to  functions  nuclei  mass , M  i n the  smaller  e n a b l e us  and ,  describing  electrons  is  much  p r o b l e m and  in everything  calculate  larger  so we  that  (or  neglect  follows.  n u c l e u s moves n o n - r e l a t i v i s t i c a l l y  and  This  we  can  z  drop  terms  velocity.  of  The  spherically The  nucleus  symmetric  electron  m o t i o n may than  order  M  well  f\ only  and  the  the  r  fine  two  an  ways.  proportional  to  The V*  M  separation  but  as  a Dirac  nuclear  spin  and  a  Here  of  the  the an  the zl  M  model  i s the  i s not effect  and  of  the  causes a  our mass  v=-£tf./r  nuclear  =  charge  ftV+ir  is  then  the  potential  in  <x  infinite  nucleus produces a magnetic w h i c h p r o p a g a t e s and  for  electron  potential  charges  i t s motion w i l l  smaller  valid.  get  .  If  p a r t i c l e whose  whose e n e r g y must be  provides  , we  constant.  move and 1)  mass M -» oo  structure  i s the  charge d i s t r i b u t i o n .  e f f e c t i v e p o t e n t i a l . For  limit  VM  where  assumed t o have z e r o  electrodynamics  i s the  nucleus w i l l  is also  smaller,  relativistic  a n u c l e u s of  in  number,  of  or  a p p r o x i m a t i o n s t o be  Classical development  N  ( o r muon) i s t r e a t e d  be  f o r our  V  field force  around i t on  the  6  electron  proportional  potential  will  time  of  propagation  "see"  the p o s i t i o n  g e t a term  T h i s would c a u s e potential  but  Calculation gives  then  to V  .  e  way N  of  the  field  have  choosen  of the e f f e c t i v e  will  to  potential  some  drop  potential  to  effective  . 2 ) The  t  t o appear  M  the  time  terms first  this order  corrections  to order  Extending symmetric  where  is  this  nucleus  gauge V  t o the case  in  V*  (2.1)  1  and  will  give  recoil  ).  o f an e l e c t r o n  and a  spherically  we g e t  p(r') i s t h e c h a r g e  so t h a t  invariant  ( o r M  M  small.  1 4  +  potential  before.  i n the e f f e c t i v e  Ve» = ^ L l - i ( v v „ v . - r t f . - r M ] This  finite  cause the e l e c t r o n t o  a s i t was V  terms o f o r d e r  t h e Darwin  this  proportional to V V  of the nucleus  we  In  distribution  of the n u c l e u s ,  normalized  P  -»«  The the  integration  variable  r  c e n t e r of the n u c l e u s ,  charge  d e n s i t y and the  To  calculate  of  e  we  the  r  measured  choose  from  . Note t h a t both the nucleus  depend  c o n t r a c t i o n , but o n l y t o o r d e r  effects.  V ^.  coordinate  R= r -  and  shape  because of the F i t z g e r a l d we c a n i g n o r e t h e s e  i s the  on  V«  V^. so  7  r = r lo,o, i)  r'= r'CsinS co5<t>,sm0 sin^cojB) R = C-r'smBcos^-r'sinesm^, r-r'cos9) R = + r' -2rr'cos6 2  1  l  r  d r' * r 3  and  , 2  dr  d(cose)d4>  we d e f i n e  = 4-ir  \Vdr-pcr')  (? (-2oc)  e  where we have  (-p  is  the p o t e n t i a l  the above d e f i n i t i o n  <  '  r  integral  j  r >r  f  r < r'  1  one g e t s when t h e n u c l e u s  of  V^Cft = V(r)[ I The r e m a i n i n g  r  used  _ f ' ddicosB) (co:  VcrO  r>r'  V  i s fixed.  we g e t  H-v,]  - j^r'pcr') ^  must have t h e f o l l o w i n g  form  ^.R^.R  Using  8  If we p i c k  £(n  V = V ="(o, 1,0) 6  we get  N  s^d^dKcosejr^dr'pcr')!^ r sirfe 5 i n $ 1  ,x  l  =+F$Vdr'pert (-ZoO \ ,* 3 r  ( TP  0  r  * ' r  where we used  S  5in z ed(wB) _ ) 3r* r < r  3r" It i s u s e f u l t o express  j-  1  i n terms of V  and  \r-JF»  W , where  r  >  r  '  = -p,$V<r')r'Mr' We get  j- = V - U  V =(0,-|,i) so that 6  .To calculate VV  rt  =0  V?V  e  9  r = I  we choose  V ^ ( 0 i , 0 and  . This gives  a  (  9  g(r-j  \d4>dccosB)r drpcrM ,1  s  = V-3J  -^5  ,  =3U-2V  using  Putting  For  these  results  point charges So  far  we  electrodynamics. V  together  we have are To  we g e t  U=V  still  in  the  This  realm  of  classical  go t o quantum m e c h a n i c s we r e p l a c e  by t h e e l e c t r o n and n u c l e a r v e l o c i t y  e  (2.1).  a n d we r e t r i e v e  V  N  and  operators.  gives  V Cr) = V - ^ ( 2 V - U ) 6 t p , - ^ ( 3 U - 2 V ) o c r rp„ (2.2) e&  We  have c h o s e n  Hermitian. are e i t h e r  the order  of the f a c t o r s  so  that  O t h e r o r d e r i n g s w h i c h have t h e same c l a s s i c a l identical  t o t h e above  example,  is  i n Ve$  e q u i v a l e n t t o (2.2)  because  or  are  not  i t is analog  Hermitian.  For  10  f.f^jv?-^  Uctp.=*p.U-i6crf£  and  is  not H e r m i t i a n s i n c e  ( V c t f r f =a-p„V = V c c - p > i a - r ^ ( U a . p / = Uwp\, + i d - r ( 2 V - 3 U 0 f Care  must  be  taken to d i s t i g u i s h  PM = " ' v  r M  from  _ > - > - >  .We  r  expressed commutes  as  above  in  see i n the next  p =(| - j ^ P M  r  T  - p  where  and a l l f u n c t i o n s  o f c o u r s e , Er-, p-] =  section P  is  T  of . r  that an  such as  . T h i s makes  we  can  e>}  be  o p e r a t o r which V  and  i t easier  U  ,  to derive  (2.2) gives  will  the e f f e c t i v e  u s e . The e f f e c t i v e  potential  potential  in  can a l s o  the be  the form  v  (n=v - 2^ {  £  P . , v}  - ^ r a • p., c pi, v ] 3  where  W(r) =^dV'pcn(-BoOR = --^Vvcr)dr'+ and  p„  relations.  Equation which  will  with  whereas, the  where  r  _>  r" = e " * r  p=-iv  satisfies  r'V(r")dr'  form  written  11  §7  V*W=2V  =  r  U  H e r e we use  [pJ.W] =-Z(V + rU-|p)  This  i s similar  though they nuclei  t o t h e form  derive  f o r Vejj.  i t f o r the center  that Grotch  and Y e n n i e ' use  o f mass frame and f o r p o i n t  only.  Using  this  ^ P N ' Vett ' Pn y nuclear  representation  where  momenta.  ^  <r«lp )= M  _ ix  6  •p  r  J  p  If  N  we  and use  of  Ve#  we  can  .calculate  p j a r e t h e i n c o m i n g and temporarily  the  outgoing  normalization  then  <ft.'i v« i A, > « e!* t - ^ * . ( a,4«^ v«^ 1  where - J  - » i  _ i  V(Q^) = <p*'l V I p„> =  6  H  Vcr)  wc<p = <ft!I w i p„> = JjdV e '* wen ,r  V(o^  a n d W(cp  form f a c t o r .  c a n most e a s i l y  be e x p r e s s e d  i n terms of  F(cp t h e  12  ? t % t = -Jeff e * *  We  pen  then g e t  <P«'IV^!p > = w  =c- eoe-^[l-«..l%^] 2  where  <*x=* The  second  term  i n <Pi!i  ^-  I V yj. I  two c u r r e n t s . The m a t r i x  gives  e  element  <&'i V f r > = e * *  spin  J  M  zero  i s the n o n - r e l a t i v i s t i c particle  given  J/C^ To  find  calculate electron  the  the  reduction  -1:?-1  of the c u r r e n t  for a  by  =2F(^(l,-%p-)  Born a m p l i t u d e  the matrix  of  c a n be w r i t t e n a s  is. £ j „  5  where  the i n t e r a c t i o n  element  momentum. T h i s  gives  for electron-nucleus of  V^JJ.  between  scattering  eigenstates  we of  13  <P« Pt I V ^ I P N P > e  —  UV?&  PH  + Pt - pi  - Pi  ^  J/  Z  tc  where  and would  we  expect It  with  used  Je from  =  0  and  <£o = 0(M  toc. 4>J=0 , c a n be added  to  (  functions,  However, binding  calculation.  So  such  a  any  V«j£  i s exactly  term of the  what  one  would  affect  e n e r g i e s and m a t r i x e l e m e n t s  i t i s not p o s s i b l e  Y e n n i e ' or  Friar  1 1  .  form  without a l t e r i n g  change  amplitude a l o n e . For a d i s c u s s i o n G r o t c h and  This  QED.  i s important to note that  amplitude.  ).  the  Born  the  wave  of s t a t e s  i n our  to derive  ( 2 . 2 ) from t h e  of  point  this  see  Born  Rose  1 5  ,  14  2.2  The  Center  The  Of  Mass  Transformation  wave e q u a t i o n  f o r the  n u c l e u s - e l e c t r o n system  i s given  by  - j (3U-2V)oc-rr.p 3 2  We  have  i n c l u d e d the  nucleus  kinetic  i n the Hamiltonian  t h e e l e c t r o n mass. The a  r  f u n c t i o n of both If  separate t h i s can guidance,  For  but and  be  the  done we  and  e  CM  f o r the  turn of  r  N  not  w h i c h has  velocity define  can  , the  energy  of  t o t a l energy.  the  ff\  f o u r components a n d  n o n - r e l a t i v i s t i c Hamiltonian,  motion  from  mass the  relativistic to  of the  this  a CM  (CM)  coordinates  and  case  is  not  electron  and  clear.  without  t h r e e momenta o f  position is  we  r e l a t i v e m o t i o n . Whether  c l a s s i c a l m e c h a n i c s and  frame w i l l  velocity  is is  For  consider  interaction.  d e f i n e t h e c e n t e r o f mass frame t o be  sum  the c o r r e c t  T  rest  T  .  of  again  of  the  E  in  center  the n u c l e u s  s y s t e m we  not  wave f u n c t i o n has  to the  i n which the  z e r o . The does  out  the case  this  frame  l(  t h i s were a c o m p l e t e l y  would t r a n s f o r m  first  5i  be  (p t N  the  the p a r t i c l e s  Pe)/( E t £ ). N  however. A d e f i n i t i o n  t  is  This  of t h e  CM  15  R  T  = ^  +  E w f  >  (2.3)  where t h e e n e r g i e s o f t h e p a r t i c l e s ,  E  e  a n d E„  , include  rest  masses. T h i s i s n o t t h e o n l y p o s s i b l e d e f i n i t i o n  our  requirements,  because and  because  frame  measuring  t h e e n e r g i e s appear  *e  a n d r„  i s n o t an i n v a r i a n t  non-relativistic,  to  the  useful  above  the  of  so  field  a  the  correct  definition  to define a  b u t i f one o f t h e p a r t i c l e s  anyway. T h i s s u g g e s t s  that  i t will R  T  w o u l d be a  t h e wave e q u a t i o n .  i n our problem,  however,  a r e not  e n e r g i e s a n d momenta a r e n o t c o n s t a n t . A l s o  will  have e n e r g y  and  momentum  which  must  crude  field  approximation  RT , t h a t energy  we c a n assume  be  the electron each.  only t o zeroth order  , f o r the purposes  and n u c l e u s get  This approximation  will  one  i n M ' t o be c o n s i s t e n t  a p p r o x i m a t i o n s . Thus t h e t o t a l  energy  i s divided  E„-r E * + V = (M + £ V ) H E . + jrV)+0(M-') t h e CM c o o r d i n a t e i s  half  have t o be  parts  So  reduce  account.  calculating  other  1 7  i s our circumstance,  and n u c l e u s  their  itself  into As  of  electron  and  taken  form  the  It i s possible  CM c o o r d i n a t e f o r u s e i n s i m p l i f y i n g  The free  which  in  invariant  simultaneously i n a particular  procedure.  CM w h i c h e l i m i n a t e s t h e s e p r o b l e m s is  satisfying  a n d i t i s n o t even r e l a t i v i s t i c a l l y  o f t h e way t h a t  the  with  our  i n t o two  16  a  If  l|.i^)^ E^f +  we were t o a p p l y  expect  this  result  o(M-)  % +  t o quantum  mechanics  we  would  t h a t t h e wave f u n c t i o n o f t h e CM o f t h e s y s t e m s h o u l d be  proportional  to o  , at least  f o r l a r g e s e p a r a t i o n s where  V  _»  is  small.  From R  T  a  coordinate  combination V  of  I  and  E  relativistic just  is  the  mathematical  a s a CM  through  R-  Here  CM  momentum  p o i n t o f view  because  and e  total  I'M ,  i t  is  but  the  t o use  not  linear  a  simple  i t depends a l s o on  t r a n s f o r m a t i o n which  system,  i tisdifficult  • So i n s t e a d we make t h e  a mathematical convenience  of  I r -r | t  conventional  i s , in this  r t  non-  circumstance,  o f no p h y s i c a l m e a n i n g . The  new  c o o r d i n a t e s w o u l d be  r =r - r c  and  the conjugate  N  momentum o p e r a t o r s  w o u l d be  (2.5) PT With these  =  Pc + P * s u b s t i t u t i o n s the Hamiltonian  becomes  17  H = [ a p r p m - r ^ +V t jJ3f(2V-U)at.p r  +^(3U-2V)arrp ] (2.6)  -^(3U-2V)ctrr.fh H | + Hc  =  rft  H  where square  is  T  split  appearance of P and p Let  then  up  H^i  into  brackets. This Hamiltonian -»  the *~  M  r  , i t s conjugate T£ I  in H » c  momentum,  be a s o l u t i o n  6  i s not II  A  and  and H  t n  according to the  separable II  . However, H  htl  because  of  d e p e n d s on  only.  of  i t c a n be shown t h a t  -i - i  ¥ < r R ) = $|+^rS-?i+2i(E-m--jrU)r ^ 3 | Y r t i ( r ) 6  , T (2  . ) 7  satisfies  H ¥ T  (Note and  = ( E + - ^ ) f  t h a t we a r e u s i n g  PT  f o r both  i t s e i g e n v a l u e . The c o r r e c t  context.)  P  R  has  the  E  i s the i n t e r n a l  ET =  energy.  (2.8)  the t o t a l  meaning  obvious  momentum o f t h e s y s t e m w h i l e and  = ET*  should  momentum be  interpretation  operator  clear  from  as the t o t a l  i s the t o t a l T h i s way o f a p p r o a c h i n g  energy. t h e CM  18  transformation  i s similar  to  that  though they use i t f o r a p o i n t , We  Grotch  1/2  nucleus.  c a n now compare t h e wave f u n c t i o n  w i t h t h e one i n v o l v i n g and  spin  of  R  and  Kashuba *, 1  o f t h e CM g i v e n  above  w h i c h we h a d a n t i c i p a t e d . U s i n g (2.3)  r  ( 2 . 4 ) we g e t  i  since,  to zeroth  order  i n M~*  e l e c t r o n .energy  plus  the  , the i n t e r n a l energy p o t e n t i a l , that  is,  E  E =  i s the  E + V+OCW'). e  Thus we g e t  = Ll+ j (E-m-iv)gr Je i  |  This  explains  t h e r- "  U=V+0(r~*) . A l s o the  this  r  , i l C  [  term  in  (2.7)  indicates that  since  f o r large  -V  large  we s h o u l d  f  ,  make  replacement  L I +-J (E-m4uirP ]e ' ^—* i  r  in  for  (2.7).  This  form of s c a t t e r i n g The  relative  T  T  will  be i m p o r t a n t  e''**  in discussing  states. ( i n t e r n a l ) wave e q u a t i o n i s  the asymptotic  19  Lot £+£m + j$ +V+ 2l5[(2V-U)i-p  The  form o f t h i s  equation  d o e s n o t d e p e n d on t h e c h o i c e «  CM d e f i n i t i o n , expect  % \  but i s a r e s u l t  t o be a p h y s i c a l l y  e  A solution  of d e f i n i n g meaningful  of the i n t e r n a l  -»  of  the  -»  r r - P w + pe  internal  , so we  wave f u n c t i o n .  equation i s  XdCft = | l+2H(V + pm+rU|-)j Y.(h r  where  X  satisfies  (ap+pm.+Vo^Yo - E o X  ( 2  . ) 9  and  V. = V - ^ ( V Note  that  by  and  be  Y  solved  by  The  interpretation  and i s j u s t  (except  we  now  existing  the  f o r normalization)  (2.H, V=»V  . The wave  0  the Dirac equation  wave  put  Ur|^) potential  i s simply  c  analytically.  If  +  f o r t h e Coulomb  satisfied can  l  techniques  Y  function  0  a computational two  with  potential,  numerically  has  no  and  physical  convenience.  transformations 0  a  equation  together  we g e t  20  (2.12)  +V+pm + r U ^ ] ( Y ch e ' iPr  R  0  The H e r m i t i a n c o n j u g a t e o f t h i s i s  -V+pm.-ru£]j where we have  used  LrU^:] " = - L f U ^ + 2V] 1  This  gives  us t h e u s e f u l r e s u l t  that  (2.13)  The  Y . [ l+fi(5c-P f  t e r m i n ^3  it  .  The  i s part  physically  term  +pm)] Y  h a s no p h y s i c a l  a mathematical consequence Xti  T  i n «.P  T  i n t e r p r e t a t i o n being  o f t h e r e l a t i o n s h i p between  of X a  I t gives  to Y  t h e change  due  t o t h e F i t z g e r a l d c o n t r a c t i o n when g o i n g  to  frames  i n w h i c h t h e CM i s m o v i n g .  to the nuclear in  a l l frames  current  There  -density because t h e n u c l e u s we w i l l  purely  %  and  , however, h a s an i n t e r p r e t a t i o n b e c a u s e  of the transformation meaningful.  0  consider.  Because  which  are  in electron from t h e  CM  both  density frame  i s no s i m i l a r c h a n g e is  non-relativistic  of t h i s ,  the e l e c t r o n  21  is  a Lorentz  four J«r  If  we  now  current order  vector. ( Y-*i  =  introduce will  in M  , Y-J  frame we  , i s given  rfc  a Lorentz  CM  by  f o r the e l e c t r o n d e n s i t y  we  f  r  i s t h e same a s  motion  transformation,  J.' = ¥ ¥ = Y t, Y ^ + Y*. a v  This  get  oc Y t )  a non-relativistic  undergo  In p a r t i c u l a r ,  %ti  In t h e CM  (2.13)  T  get  Y  rel  V = Pr/M  the  which t o  first  T  22  2.3 Bound S t a t e s  For bound s t a t e s we can r e w r i t e binding  i n terms  of the  energies  BThis  (2.10)  m-  E  Bo = m  0  -  E  0  gives  B=B -^ 0  As  an example we c a l c u l a t e t h e r e c o i l c o r r e c t i o n t o t h e b i n d i n g  energy  of  the  Is  level  of  hydrogenic  atoms.  We  have  V =V=~£oc/r . S o l v i n g (2.9) by t h e s t a n d a r d methods we g e t  1 5  0  Eo = m„V I - (Zoo*' and u s i n g  (2.10)  = m (I -  )-  m( I  ~  )  F i n a l l y we g e t  where  Thus  we see t h a t t o o r d e r ( Z e e }  1  t h e b i n d i n g energy i s wha-t we  would expect i f we t r e a t e d t h e e l e c t r o n and t h e non-relativistically  by  nuclear  u s i n g a reduced mass i n a  recoil  Schroedinger  23  equation. energy  The  is  correction We given  of  relativistic  order  i s of  now by  leading  and  order  consider ( 2 . 1 2 ) .  use  two  particle  If  coordinates  1 =  = where we  used  normalization  box  to  r  and  K  0  included  an  i s given  (r)LI+ Jf(5cPT 1  recoil  the  wave  we  function  for simplicity  of  satisfy  get  J  e  ||*= ^ Y  binding  dR  V (^f5 N  of  wave f u n c t i o n must  f  constant.  the  relativistic  normalization  S 5 ¥ ( r r l ) T ( r R ) d*r  ( 2 . 1 3 ) and  to  •  We  The  transform  leading  normalization  interpretation.  we  the  to  the  correction  Y cr)Ll + ^ ( ^ P t | 5 m ) 3 Y ( h r  0  extra  T  0  factor  as  a  by  + P»n)]  Y o C h d V  (2.14)  v* and  the  c o r r e c t l y normalized  wave f u n c t i o n  is  + V + jSm + r U ^ ] I Y.(r) e ' ^ S ; " It  i s convenient  to  write  Y(rR) = Y(r)-%=r where  1  (2.15)  24  ^  Y  =  N  | l+2i5j[aPr + Z i ( E - m - x  U)rP  T  (2.16)  t V + pm + r U ^ : ] \ ToCn and  satisfies  s  v» Now we c o n s i d e r functions.  The  the angular  total  angular  momentum p r o p e r t i e s o f t h e wave momentum o f t h e e l e c t r o n - n u c l e u s  system i s J  T  = J „  = r  f  N  Changing  J t  x p„ +  the coordinates  to  r  n x pe + j; °» and  K.  gives  J =RxP-j-rxp+jCTt T  = C+ J L  i s t h e a n g u l a r momentum due t o t h e m o t i o n o f t h e -» mass, j i s t h e a n g u l a r momentum i n t h e CM f r a m e . In  (  t h e CM frame  _>  Pr O a  center  ) we g e t  Y = N j l + ^ [ V + £m + r U ^ ] j Y. I  Since  j  eigenstate  commutes of  j  of  (2.17)  v.,  with  g i v e s an  [ V + Bm + r U ^ eigenstate  J of  , any To or  with  w h i c h i s an the  same  25  quantum  numbers.  '^ \ ro* ^2. , of  Ho  /*  where  (  is  of  Y,  T  = /x  ft  M  3  T  a r e commonly  labeled  the o r b i t a l  ) , a n d /A  solutions  orthogonal  1 5  the  i s then  angular  occur ,  T  .  given  the angular  ,  X  ,  + 0 ~ 'Atwo components numbers  T h u s we g e t  will  give  a  s e t of  system,  .  The  by ( 2 . 1 4 ) .  V / ( Nunx/of  = 6MA«.  i n the center that  satisfied H  numbers  wave  momentum  o f j a . T h e s e quantum  CM  Xlt\KMl  P T" 0  i s the  quantum number, X = U l + l )  of  (I  have  frame,then  total  YC^*/*! a r e o r t h o g o n a l a n d  These s i m p l i f i c a t i o n s we  with  by t h e quantum  i s the' e i g e n v a l u e  constant  of ( 2 . 9 ) with  a s g i v e n above  state  described in Rose  YoLnKyui  If  Y  quantum number o f t h e upper  states  normalization  i s a solution  0  • The s o l u t i o n s o f ( 2 . 9 )  i  = , / T  i s a radial  a r e more f u l l y  The  then  t h e two p a r t i c l e  quantum numbers by  i s , i fY  f o r example,  =  function  That  (2.18)  o f mass s y s t e m o n l y .  i s , we  are  not  in  momentum e i g e n s t a t e s ,  the  CM  g i v e wave  _»  functions  f o r t h e two p a r t i c l e  eigenstates  of the t o t a l  the  terms  not  commute w i t h  angular spite  i n co Pr j  momentum  of t h i s  particle solve  and  of  , X  r-Pr T  T  0  , which a r e not  momentum,  TT  . This  i n t h e Y ~* Y  , and a l s o  usually  a s Y[PT HK/A]  states  n  angular  t h e motion  i tisstill  ( 2 . 9 ) forY Since  or  system,  JV  t r a n s f o r m a t i o n do now  of the center convenient ,  because  i s because  to  contains  o f mass, L label  the  the .i n two  o f t h e way w h i c h we  . , a n d yw  a r e n o t "good" quantum numbers  for  26  the  system  orthogonal such  we  have no  guarantee  so t h i s must  be  t h a t the  checked.  The  rl?y nKywl  states  matrix  element  are  between  states i s  5 $ Y [ P n x/0 T [ P ' n' x>'3 r  T  T  dV  d*R  V» Vg  (2.19)  = 6 ( P - fr) NN'^ Y.^hxiO [ I + + (6c-P + £rrO] YJnYjuj d r v* S  3  T  If  we  do  P  T  the angular  does  not  integration  couple  p  commutes w i t h  is  a vector operator  and  so  i t can  forbidden  by  parity.  The given  by  j  and  J  angular  .  Coupling  The  between  the  ( 2 . 8 ) , and  +  z  term  in  ,j  and fV  1"!  =  satisfied  by  T  only)  , t +l 1 =I  , is  I  correspond  is  capable  YlPy  states  oc P  t h e CM  states with  operator  equation  of , i.  j ,j+ I  r e s t r i c t i o n s on  Thus  differential  H Y - L f c - p +0m  t h a t the  momentum s t a t e s b e c a u s e  j' =  the o r t h o g o n a l i t y of the  (2.6),  find  . However, t h e o p e r a t o r  £  s t a t e s with  X' = - K ^ K - l j V + i .  destroying  different  we  (with respect to r o t a t i o n s  couple  fk - KK - I, JA , }\+ I  and  to  both  first,  of  . YCPTHK/A]  is  (2.15)  +2]3j p T V + ^H(2V-U)o:.p 2  L(3U-2V)5c.frp t ^ P r- R P P r  -2J^(2V-U)^P -2H r  (  3  u  "  2  v  )  i  Y  =E Y Here  Pr  spherically  i s a constant symmetric  v e c t o r . For and  so  we  Pr 10 expect  this that  equation the  i s not energy  27  ft eigenstates exactly  what we  However, having  we  would be different  not  angular  still  use  (2.16) t o  expect  energy  ,  so  i n g e n e r a l be  eigenvalues  we  If  corresponding  to  e i g e n s t a t e s of  this  YCPy  M  same  is  and  but  eigenvalues  case ,  ttK/A^.  Y  HoLnV/*!  0  the  0  to  E C n * ] . These is  H  since  corresponding  the  Y  from  orthogonal  w r i t e them a s  distinct.  Y  get  t h a t energy  eigenvalues  same f o r s t a t e s w i t h  yu  momentum e i g e n s t a t e s . T h i s i s  e i g e n v a l u e s w o u l d be  The  the  be  g e t when we  would  distinct  Hermitian.  will  will  then  namely  the  (using  (2.10))  ELn*] = will  also  be  distinct  EoLnKl and  we  E«CnK]  + m) 1  get  = 6„ « St*. \ a function o£ yu^' P P ' ] n  T  T  and  v. v  e  Hn' if the  we  use  (2.15) and  degeneracy  equations  we  operator  <X- P  now  be  possible finally  (2.19). 0  r  droppped coupling  cannot from  the  and  explicit  couple (2.19).  between  d  function oj- /A/A' J  ( F o r t h e Coulomb p o t e n t i a l  E«tn*] = E C n - K ]  have t o use  I  so  prove  solutions  states with This  to  so  any yU  above .)  and  eliminates  have  the  for. %  X = K'  states with d i f f e r e n t  we  and  The  i t can further we  get  28  s  Our  normalization  2.4  Scattering  The  condition  momentum  normalization  asymptotic  p  and  0  and  f o r m s and  distortion density  reduces  to  States  Y,t.f.A.] ^  unit  (2.14) a g a i n  s c a t t e r i n g s o l u t i o n s of  asymptotic  The  Kn'  b[?T~fr)  near  the  are  helicity  A  are 0  labelled  . That  by  their  i s , for large  r  e ' m&xo , p ,  r  orthogonality not  nucleus.  normalized  (2.9)  conditions  affected We  by  have chosen  d e p e n d on  the  Y  0  wave so  these  function that  it is  29  1 2E.  Eo+m.  ucp.x^uup.x'O * 'l  Eo • Po + Wo  1  X\ p o  0  i s a two component  To n o r m a l i z e  spinor  satisfying  the e l e c t r o n - n u c l e u s  orthonormality  of the d i f f e r e n t  wave f u n c t i o n and t o c h e c k f o r  s t a t e s we c a l c u l a t e  YLPrPoMTLP^A'jtfrd^  = V  8  S'( PT - Pi)  J  Y^C Po A,]  r  I  C oc- PT f fm ) ] Yo L Po' Ai] d r 3  = 8*( J - P ') 6*( Po " p, ) N N' lApo XO L I + ^ ( « ' r * ?  T  We  have i n c l u d e d  constant.  the  extra  factor  N/V&  as  a  p, Ai ) 1  normalization  30  _ EQT- ftu r  /  X & Po \ -v*  ,  0  -vy  since  If  we  define  M_i N  =  1  -_L(J£+ 2M E  Edii E  '  we g e t  where  ¥LP p Ao3 = N j I T  0  C  Once a g a i n  +JMLK?T  + 2i(E-m-JrU)r.P  ^  i t i s convenient  -  to write  T  *  (  2  ,  2  1  >  31  P  i P r  '*  Y C P T P O A O 3 = Y £ P T P O M - ^ where  Y[P PoA.] = | | T ^ L o c P r + 2 i ( E - m - | ; U ) rfV T  *  The  wave  functions  correctly Now and  normalized, we n e e d  TlR-p^Ao!  x  therefore  ->  ( 2  form a u s e f u l  to find  t h e a s y m p t o t i c momenta o f t h e and p  0  electron  . The a s y m p t o t i c  l  T  J  1  2 R ' i  T  P * P T  P E "  ~ E ~  N o»P.-r*irV-it*i(E-i«lr-'T/l1 1  U(p Xo)C  y"  o  we have  w h i c h we have  get  s e t of  = {lT^C^FT+2i(E"m)rP T^m-|- -^3^  ,i  2.2.  2 2 )  is  YCPTPOX.3  Here  '  orthogonal wavefunctions.  t h e n u c l e u s w h i c h c o r r e s p o n d t o Pr  form o f Y  D P  used  shown t o be a p p r o p r i a t e  f o r large  I f we now make t h e t r a n s f o r m a t i o n  r  in  section  ( 2 . 4 ) on t h e e x p o n e n t  we  32  P«-r + P -Rt^(E-m:>r-P ^^ T  So  ^  T  i s a s y m p t o t i c a l l y an e i g e n s t a t e o f p  e  and  p  rt  with  eigenvalues  Pea= P o - r ^ P r  PH« = ( | - | [ ) P T - P O Inverting  this  gives /1  P.  =  E \ •»  U - - j 5 [ ) P t a --pf Pn*  Pr = Pe* t i s easy  t  T  H  to confirm  tot  2  (2.10) a n d ( 2 . 8 ) .  these that  M  — Eea + E using  (2.23)  p.  N o t e t h e d i f f e r e n c e between It  E -*  H a  formulas  and ( 2 . 5 ) .  33  3 BOUND MUON DECAY  3.1 The Wave  We  are  Functions  now  muons i n c l u d i n g  In  the i n t i a l  the  ina position  recoil  state  J =TT  system = , /  using  2  by  , MT=V  (2.17). N  = ,  t h e muon  t h e wave 4'  the decay  o f bound  e f f e c t s . The r e a c t i o n i s  muon-nucleus system which  this  to calculate  i s bound  i n t h e I s ground  i sat rest function  i n t h e l a b . We  ^Cr^r^,)  state  of  describe  which has  fr/*!*® ,  . Thus  i s given  by (2.18)  and Y  M0  i s the Is  solution  (excluding  the rest  of  The  total  energy of t h e muon-nucleus system  34  mass o f t h e n u c l e u s ) i s  T h i s can a l s o  where  be e x p r e s s e d a s  i s t h e muon b i n d i n g e n e r g y . T h e s e  (2.9) a n d ( 2 . 1 0 ) , a n d Vo ET/A is  ,  is  e q u a t i o n s come  i s g i v e n by ( 2 . 1 1 ) . The t o t a l  t h e same a s t h e i n t e r n a l  energy,  from  energy,  , because  t h e CM  not moving. The  can  electron  describe  mass o f t h i s neutrinos  and t h e n u c l e u s form a s c a t t e r i n g  them by t h e wave f u n c t i o n system  can  i s not a t r e s t  state  so  we  ^eCrervi) . The c e n t e r o f  i n the l a b  frame  c a r r y away some momentum. T h u s ,  using  since  the  (2.21) and  (2.22),  Y. = \ I +^[ocP f 2i(E -m -yU)r-P re  t  e  Te  = it 4 where  P-P.-fc «  -»  Pr= ft  Pea  and p^  nucleus  in  IU--^r, ->  ->  P*.  Po =(l+"pf)Pe.-^ P  are the the  electron-nucleus  / •  E t -» M a  asymptotic  lab  Ea v  frame.  system,  E  though  momenta 4  of  the  i s the i n t e r n a l  electron energy  i n t h e above e q u a t i o n s i t  and  of the could  35  r e p l a c e d by ETC o r E«,t s i n c e t h e y  be  i n M~'  order  .  Ye  is a  unit  a r e a l l t h e same t o z e r o t h  density  normalized  scattering  s o l u t i o n of  (ap +|Sm.TVo) Yo. = E . Y. 0  with asymptotic of  momentum  the e l e c t r o n - n u c l e u s  p  = To  P i + W.  +  a  r  -»  1  p  O.D  ft  =r  e  -r  _>  w  The  and  ->  n  d  so t h e r e  will  .  .  be no a m b i g u i t y  . in  .  M  wave f u n c t i o n s w i l l  where  U (PT^>) U I ^ M = 1 +  v (p^A>)v(p Xy)= I f  9  energies  functions  - i  waves  Their  need t h e wave ,  a  r =t>-r  neutrino  only  _>  r«, = H*  at  -J  energy  E  j r>  . The t o t a l  as  c a l c u l a t e t h e d e c a y r a t e we w i l l  evaluated  0  to W PK " 2M *  1  E.  A  system i s  T h i s c a n a l s o be e x p r e s s e d  Ere H  and h e l i c i t y  0  c  are E  = v  p  a n c v  *  Ej/ = P-y •  be  box  normalized  plane  36  3.2 The Decay  All  the  eigenstates  , U)y>  rate  The  H|  vertex  Since  Rate  wave  functions  we  MT  use  is  i s a point  r  state  li>  for  to final  the standard  interaction . This  r  <i-1H I »> -  j  IHT  the  state  V-A p o i n t  are  weak  energy  transition l£>  interaction  integral  should  we  must  evaluate  H  )MT  for  gives  3 c •  which  problem  f o r yw-*ev"v  -» -* -» -» -> -» l~e = f> v = v — X" +• Y~u  The  the  s o we u s e t h e f o l l o w i n g f o r m u l a  , from i n i t i a l  this =  in  tu,T( (l-»0%][ tT ll-WV,]d fd r1i t  ;  l  ,  t  be o v e r d r e Q » r  b u t we have u s e d  a r o Tu  i s e q u i v a l e n t . The e x p o n e n t i s  .#•-*  -*  ~s  N  -?  ..-i  -»  frit, o  o  .  *-i(p> + Pv + PU+P».Virt - i(P*+-pv+-j^Pe*+P*»)rr since K  The  integral  conservation  over  e  - ' • * + • ]5j «  r  w  is  now  trivial  and  gives  momentum  37  It  i s useful to write  H = p ,  +  P f  H o - E - y + E *  - " ft* " P»»« H  is  tt  a  =  now  four-vector  use t h e F i e r t z  ET/A ~ E e T  momenta o f t h e n e u t r i n o s .  We  (3.2)  since This  i t e q u a l s t h e sum o f t h e f o u r -  gives  transformation  1 8  [AY (l-* )B3LCMI-Y )D] r  5  s  = [X* (l-tf )D3i:c r  s  in  which  allows  A  ,  B  , C  us t o combine  ,  D  are a r b i t r a r y  the n e u t r i n o  spinors  Dirac i n one  spinors. factor.  This  38  5e  =  , ( ,  "ii  '  ) H  R  Y.WI-  i( H-p -%) IE, »*( I - W . 3 N v  c  where  We now s q u a r e  the matrix  element  • 15„ 6 ( I- * l  V*  1L \/„  neutrino  spin  r  s  The  only  brackets,  part  depending  s o t o sum o v e r  on  t h e n e u t r i n o s p i n s we  I - V «vl i s i n the square need  E t M ( I - V v ][ v* I'( l - ^ ) u ] e  f  Therefore  i<si H The  2K  lllr  transition  i  T  r<r  i>i =-^Jp iiH-p»-Piii^; l  rate  i s then  N  *  39  So state.  f a r we h a v e t h e t r a n s i t i o n For  decay  by t h e d e n s i t y  rate  i n t o a range o f f i n a l  to  a  discrete  s t a t e s we must  of s t a t e s g i v i n g the d i f f e r e n t i a l  decay  final  multiply rate.  ^(H-p^-p^^^^NrN^dp^d'p^cfpyd'py We c a n now i n t e g r a t e  ^ = S S ^  is  The  last  the step  term  over the n e u t r i n o  momenta  using  - P > - ^ ^ V P : - % %  H  function. This  i s zero  gives  f r o m symmetry a n d we g e t  d'r^^g^N^NjL^H'-H-H^ieCH-HOd^d'K  We h a v e made t h e c h a n g e p -» H Mft  this  point  we h a v e n ' t  Now . we T  e  want  using  (3.2).  summed/averaged  to  , ( 2 . 2 2 ) , i n terms  i n the i n t e g r a t i o n  (3.4)  variables.  over electron/muon  reexpress the transformation  of H  a n d pe»  instead  of  spins.  from  p » N  At  ^eo t o  and p « e  40  Y = €  The  jl+2]5jE-dL-H-2i(E.- m -j\X)rH t  asymptotic  This  is  actually  are  given  0  by  (2.23)  without  s p e c i f y i n g t h e wave f u n c t i o n s Yoe a n d Y>/» .  phase  Space  space  limits  f o r the i n t e g r a t i o n over  H  and paa  s e t by  p* > 0 The  last  inequality We  It  is p  o e  a s f a r a s i s p o s s i b l e t o go i n t h e c a l c u l a t i o n  3.3 The Phase  The  momentum o f V  i s simpler  is  H > H >0 0  the  effect  therefore consider  of  the  step  t h e boundary g i v e n  t o use a r e l a t i v i s t i c  form  for E  N  a  in  function by  Ho  .  expressing  Ho EM«I =  This M  1  and  change  would  YFF+pJv  give  the exact  - M phase space t o a l l o r d e r s i n  i f we knew E y* t o t h e same d e g r e e o f a c c u r a c y . T  ( 3 . 2 ) we g e t  U s i n g (3.1)  41  H* H  ET/A ~ Efta " E a  0  H  = E >*- E ^ - j M T p i + 2 H p X T H l  T  +M  l  t o  where A  A  X - H • Pe* Thus  ..  2M(ErM-Eea)T(E »-E ) -Pi Er^-Ee.+ Xpea') 1  T  te  = E> - E . - ^ [ ( * ~ E t  T/  e  The maximum v a l u e  c  t AxM  Fig.  ^  +  p J  * ~ M ]  o c c u r s when  ( E t  "~  J  x  +  0  ( M  "  )  t )  H- 0  2(rt+  v  t  T A K  - 2 k ( E v - » n + 0(1*0  i s i n d e p e n d e n t o f X- . The i n t e g r a t i o n 1.  E  3>5  2 M E T » + E * + me E )  = E  and  of Efta  E  (  region  0.6)  i s graphed  in  42  F i g . 1. The i n t e g r a t i o n r e g i o n g i v e n by ( 3 . 5 ) . W i t h o u t recoil the upper surface would be f l a t . The d o t t e d l i n e on t h e x « - i plane i n d i c a t e s the l o c a t i o n of the peak i n t h e i n t e g r a n d d i s c u s s e d i n s e c t i o n 4.2. P o i n t A , where t h e peak leaves the integration region, gives the l o c a t i o n of the s t e p i n the e l e c t r o n energy spectrum.  43  4 PLANE WAVE BORN  4.1 The A p p r o x i m a t e Wave  The in  i n t e g r a l s over  APPROXIMATION  Functions  H  i n (3.4) and over  r  i n (3.3) w i l l ,  g e n e r a l , have t o be done n u m e r i c a l l y . A u s e f u l  for  which these  i n t e g r a l s c a n be done a n a l y t i c a l l y  wave B o r n a p p r o x i m a t i o n  for a Is state i n the f i e l d spinor  function these  with  which  zero  are also  made a r e 1)  by t h e n o n - r e l a t i v i s t i c s o l u t i o n  of a point charge, m u l t i p l i e d  by  a  l o w e r components; 2) t h e e l e c t r o n wave  i s u n d i s t o r t e d by t h e n u c l e a r  approximations  i s the plane  ( PWBA ) . The a p p r o x i m a t i o n s  t h e muon wave f u n c t i o n i s g i v e n  Dirac  approximation  a r e q u i t e good  t h e ones  c h a r g e . We w i l l  f o r n u c l e i with  f o r which  recoil  show t h a t  small  2  ,  e f f e c t s a r e most  important. First muon  we c o n s i d e r  wave  radius  function. Since  f o r small  correction  2  would  r a d i u s a n d fn, state  .1  To  expect  of order  RM/TB  nuclear  i s smaller that  size than  i s around  f o r o x y g e n . So f o r s u f f i c i e n t l y  t h e Bohr size  Rw i s t h e n u c l e a r  , where  so t h e n u c l e a r  . This  on t h e  the nuclear  r a d i u s o f t h e muon o r b i t .  i s (2ocm^  2(Xm^Ru  of f i n i t e  the nucleus  would  i s t h e Bohr  t h e Bohr r a d i u s  Coulomb  the  ,we be  w o u l d be o f o r d e r and  the e f f e c t  For the Is  size correction  .004- f o r h y d r o g e n  small  2  we c a n u s e a  solve  (2.9) u s i n g  potential. find  t h e muon wave f u n c t i o n we must  potential  V  Q  given  by ( 2 . 1 1 ) .  F o r t h e Coulomb p o t e n t i a l we  44  get  V =V= ~2«x/r  particle the  in  with  zero  solution  in Rose  but instead  equation lower  2ot~.05  Finally,  we c o n s i d e r  the  solutions  e q u a t i o n h a s been s o l v e d a n d  . We w i l l  not  be  of order  potential  the electron  Rose  density  £a£ /'p^ i f E ev  instead  order. most  in  scattering .  »m  .  e  components o f t h e muon wave  t h e muon  effects described  but  no  more  analysis.  F o r example, H a n g g i e t a l  e f f e c t s increases  results  in  using  t h a n we c o u l d  t h e decay  have r e a s o n  up t o a r o u n d  do  muon wave f u n c t i o n  shows  expect  " ' find  i f we  is  highly  the  this lower  not  that involve  i s then  including the that  PWBA  is  b a s e d on t h e above  that  by a b o u t  t o hope t h a t  correct ions. The  5  rate  2 ~ 15 . N o t e  PWBA  these  decay spectrum,  spectrum,  recoil,  actually  make  accurate  on t h e  function.  c a l c u l a t i o n of t h e decay  we  .007  solutions  So  the electron  i s o f o r d e r 2 o t , t h e same a s d r o p p i n g  So  ice  s o l u t i o n s , we a r e  approximation  silicon.  spinor  From  through  three  the  a t the nucleus i s  of the exact  Since  of  e a  1 5  relativistic  three  a  . For hydrogen,  are also  by a f a c t o r o f o r d e r  exact  using  the n o n - r e l a t i v i s i c s o l u t i o n of the  t h e e l e c t r o n . The e x a c t  terms o f t h a t  An  single  t h e e f f e c t o f t h e Coulomb f i e l d  p l a n e wave s o l u t i o n s  dropping  a  .  one f i n d s t h a t  increased use  of  Coulomb  of  c o m p o n e n t s . The l o w e r components o f t h e e x a c t  w h i c h we d r o p a r e  function  1 5  equation  w i t h a Coulomb p o t e n t i a l , t i m e s  f o r oxygen  wave for  i s given  solution  Schroedinger  becomes t h e  a Coulomb f i e l d . T h i s  Is solution  exact  and  . So (2.9)  0  t h e sum o f t h e four  PWBA w i l l  per cent i n give  useful  the approximations dropping  any  we  recoil  45  where € =  to^Ztx.  satisfying  \X/* i s a s p i n o r  and  with  zero  lower  components  U ^ U ^ » | , and  E *  =  0 /  ny(i-JF(2cc?)  T h u s we g e t  E * =  [ I - jr( 2 * ? + j ^ ( 2 a ? ]  T/  We  have dropped  given  by  given  U=V by  1  from t h e above.  %  is  (2.16)  +  where  (ICKY "  terms o f o r d e r  J R  tVtprry +ru£]JY. >  = ~2cc/r u s i n g  (2.11).  A  The n o r m a l i z a t i o n  constant  is  (2.18)  Tjr-SlCtl+SfpjY,.^ - H-K" So v  This  -(  I . "VC*«?x { .  i s now c o r r e c t l y Since  know t h a t  the  2oc  ).£  p  -€r  normalized.  electron  does not i n t e r a c t  the complete e l e c t r o n - n u c l e u s  with  the nucleus  we  s y s t e m wave f u n c t i o n i s  46  This  i s box n o r m a l i z e d a c c o r d i n g  result  could  have  to  been o b t a i n e d  (2.20).  using  Of  course,  t h e methods o f s e c t i o n s  2.4 a n d 3.1 w i t h z e r o  p o t e n t i a l , b u t s i n c e we know t h e  wave f u n c t i o n  simple case,  rather  in this  than X e  t  derive  o  c o m p l e t e wave f u n c t i o n  Thus, u s i n g  Y»  i n terms o f  -tt  with i t  we need t o e x p r e s s t h e a n d R-e  using  (2.4).'  .e''-****  We  have  dropped the s u b s c r i p t  p^  since  i t i s n o t needed  labeling  the  p<,  and X  and  orthogonality  0  r  to start  complete  (2.22)  •  now  i t i s easier  . To g e t  this  a , f o r asymptotic,  f o r p l a n e wave s t a t e s .  s c a t t e r i n g s t a t e s b y . pe  but s i n c e  there  and  i s no i n t e r a c t i o n t h e  a r e t h e same.  from p Also  A  e  e a  we  and are  instead of  normalization  47  4.2  The Decay  From  Note  Rate  (3.3) we  that  p  get  enters  a  momentum o f t h e  other  (  integral  in  non-interacting  -*  ,  neutrinos  the  the  same way  particles,  as the  namely  the  x  H=p +p^).  The  >  integral  i s of the form  Sire C e. -*• r P ,(.i - - ^ . ^ e '- ear , 3r- ; _- - S ,rP  6r  2TT(2C*>  3  + P ) 2  M(e +P )  2  x  2  Thus  H U M , €* N =CI+^^-)^Lu l( (l-)f )u 3 1  V  r  e  5  r  8  7  5  r  r  2ir(Zofl  £  *H6 +Pe + H +2ptrTr l  We  now  evaluate  not  muon s p i n  the  decay.  l  x  the spinor  since  we  will  part,  M(e  x  Pe TH +2^-Ky l  T  ?  2  summing o v e r e l e c t r o n  want t o c a l c u l a t e t h e  spin  asymmetry  i  but of  48  Ae  Where P>»=(wyo) and algebra  where  5^  i s t h e muon s p i n  direction. After  some  t h i s reduces to  Q = (I, - 3^)  Hence  MM*  f l , ! V i i ^ \  32Tre  3  [p«QvrTp .Q - p«Q9oc- i€„f<rpeQJ P  eo  2e /( e + Pe +H l  To c a l c u l a t e  C  C  e(^oc^  l  l  1  -r2p  e  R )  N N*(H H - H-H 9™ r  ff  r  LPerQ^+Peo-Qc-pe  ^9<rr  " M ( el + P t l + H  4  )  - • e« ,<r c  - H H p t - Q . + 2pe-H Q-H The d e c a y r a t e  i s then given  by ( 3 . 4 )  we  l  ?  +£pfF!)*S  need  H'H*" H H g " )  49  For  small  peak  2  along  the the  function in brackets  line  lp +H| = 0  , or  e  i s d o m i n a t e d by  Pe»U  and  X =-l  8 line  the  d e n o m i n a t o r of  integrand the  i s of  the  abruptly. section  See 3.3  correction  the To p  e  F i g . 1. shows region  t o the  recoil hn^/M  and  o r d e r (£<x")  region  that  the  line when  p o s i t i o n of  the  . This contrasts with  , the  the  i s of o r d e r  f u r t h e r we  the  spectrum  using  the  results  lp*+H|=»0  So  >  i n the  is  no  drops of  the  spectrum  recoil  c o r r e c t i o n to the  flft*/M_.  using 0  the  i s due  and  H  of  recoil  only,  have t o e x p r e s s  hence  in  longer  intersects  p +E =ET <A. e  the  peak t h a n e l s e w h e r e  so  step  ET/»  of  so  peak  ft  that  8  S  - tvy(2«5  is £  sharp  . Along  the  and  A calculation  correction  proceed  the  E e ^ i ^  boundary  spectrum which  term  l a r g e r on  integration  integration  of  dominant  i n t e g r a t i o n r e g i o n . Around  within  the  the  a  i s of  to  order end  (3.6). i n t e r m s of  H  integration variables.  Ho  E-iy* " E - E = ET,* " Ec - jrjyj p j FC  =  M  E^-E -^(pjTH i-2peH) z  t  = H-^(HH2PeH) where H = ET/»"Ee~ft/2M. We expression in  for  s e c t i o n 3.3.  EM  have u s e d  i n s t e a d of  T h e s e two  the  here  the  non-relativistic  r e l a t i v i s t i c expression  expressions  are,  of c o u r s e ,  the  used same  50  to  order  M  the  end o f the spectrum,  Ho  for  .  The maximum e l e c t r o n i s given  energy p o s s i b l e ,  by H = 0 _  .  Using  that i s ,  the  result  we g e t  HHpe-Q t Zp - HQH e  = A + B* H -  CH  - D'-H  L  jvH -  E'-H H  1  where  A= r f ( 3 E + P c X * ) B' = 2 H( E e t - Pe^ - 2 H Pe(3E.+ p v V M t  t  C =E*+Pet+H(3E + PeVVM e  D'=2tt 2(EeV-^V E Therefore  cf  M  =(E t-P*VM e  we g e t  r =6 V p N p , d X  .  (|+ ) 9 ( H o  H )  n'dHd|cllM^  •[At i'-H - C H - D-HptH-E-HH ] 1  .5  1  J  !  U a + b H x + H )*  m .M(o+bHx + H * l  1  where a = e + p ^  ,fc>= 2p«  over  choose  a  d<t>  we  H  p = 2.  B = &(sm8 toS i>B Sln& Sl^<l> ,CosB ), (  R  ,  l  6  power o f Sm^M  8  or  ,  0  COST'H  t  and X = p  for  example,  e  H  . To do t h e  integral  and H = HCsmBHt-oS^H4>  e t c . Terms will  whereas terms w i t h o u t such a f a c t o r Hence,  5  /  give will  which zero  involve when  be m u l t i p l i e d  H  i  0  * ^,  a single integrated by  2lT  51  ^ B-tid4> = 2TrKB'cos9,co3eH H  In  t h i s way we g e t 2  •  f  +>  c  (4.1)  [A+BHx-CH*-DH x*-EH x] l  s  i Ca+bHx+H ) " m^MCa+bHx+H ) 1 1  4  1  5  where  D=2(pe-V+-j5j-(E Pet-pt )) a  e  E = ^ ( E c f v V ~ p ") ft  This should  result  give  integrand any  t h e f r e e decay r e s u l t . acts  function  O  H  e -» Also  c a n be c h e c k e d by t a k i n g  like  Oj(H,>0  a Dirac  In t h i s  the l i m i t limit  £-» 0  t h e peak  which i n the  d e l t a f u n c t i o n a n d s o we g e t ,  for  52  Using  =CmJ+mt)3E»-4m .E -2m »m 1  /  t  l  /  -(4Eerrv-3rnJ-rrv)  V R  we g e t  d r=^^^0(m^E 3  f  c  -Pe)  •[SCrr^TmDEe-l-m^El-^m^mJ  This of  i s t h e same a s t h e f r e e d e c a y order  that  IA '  1  a p p e a r a n d t h a t p =H e  i s , the nucleus doesn't r e c o i l .  what we  Note  that  x * - l implies These  results  no  terms  that  p  are  exactly  3 N  0  expect.  4.3 The Phase  With given  result '.  p  Space  e  i n F i g . 2.  Integration  f i x e d the i n t e g r a t i o n region  i s the area  abed  ,  53  e-H=H M  Fig. H  2. The i n t e g r a t i o n  i s given  by (3.5) w i t h  region  for fixed  p  e  X=-l .  H = E M-Et~2M"^ E X M Ec" pt) -  1  TY  To  do t h e i n t e g r a l  up  into  analytically  an i n t e g r a l  integral  over  over  i t i s best  the rectangular  the t r i a n g l e  at<X  t o break  region  the i n t e g r a l  abet  and  an  .  $I(H,x}dHdx = $I(H,x)dHdx - ^ K M d H d x abet  abed  ade  H +i  +i H  = $ $I(H,x)dHdx-$ $ItH,x)dHdx - I  ©  Because  of  integral  will  the  the  way  we have s p l i t  be o f o r d e r  integrand.  I(Hy) = I(H;0 i n t h i s  This  M '  allows  region.  Thus  the i n t e g r a l  in spite us  to  up, t h e s e c o n d  of the sharp  make  the  peak  in  approximation  54  ^I(H,X1 dHdx = $$HH,x1 d x d H - ™ " ( E  Itx) '  E t m  I{R  X>>d  *  a n d we g e t  * *'  2  I  •(A+BHx-CH -DH x -EH x)H dxdH 2  1  l  5  z  2(E»-Ec)ft C, y \ AtBHx-CFf-DHV J +l  Y  -t  These  integrals  sketch The  here only  other  simple  but  the c a l c u l a t i o n  integrals  O - l  are evaluated  J L(a bHx+HM^  ( A  +  tedious  evaluate.  important  We  integral.  i n a s i m i l a r way.  H  "  to  o f t h e most  TaTbHxTW  JJ =  a r e now  G H 1 )  ^  d x d H  + ( B H  "  E H  ^ ( ^ l b ^ ) <-> +  4  o  t / x (a + H^x . ( a t H Y O l , , ^ , , 2  D|[  I  f  SHi-THVUK +3Vbrr-TVH , 5  3  u  -H where  R » a + bH + H  1  T=3b B-6abD a  S=2b A+abB-2cfD a  U = bB-2b C-abE-^aD-6b D t  V=-b£-2D This  i n t e g r a l c a n now be s i m p l i f i e d  by u s i n g  2  3  55  The  integral  i n (4.3) becomes  vj|dH [-^--(u-av ab^f J  +  +  "  -fl  r [S-rQ(U-QVT2b V)3H LTT6abV]H B  |  1  T  H  1 d | i  -fl =[vR~^-(a-aV+2b V)^z l  -[S+a(U-aV+Zb V)]R +LT+6abV]RJ l  z  The v a r i a b l e s With get  R, ,  the  the f i n a l  R  z  other result  and  Rj  integrals  are defined  i n (4.5).  i n ( 4 . 2 ) done i n t h e same way we  56  dT  _  32 e P /. , nutZgf \ 5  e  -fS+a(U-aV+2b V))Ri + ( T + 6 a b V ) R ^ 1  J  I  C3DH " bDH* 1  ~2BHw£ C2R~ "~~2R~  y  ~  5  , f T » R  +  +  {  ,  3  a  b  D  i P  (4-4)  z * }K  +  +(a-6aD + b D)R, x  +2D(^RIogR-p L(H+p )logR-H + 2e t])^ 1  t  t  ^  ^  ^  L  [ _ C H ) R + BH R + DH*R,] + ^ b R  ^  l  (A  6  7  9  f [(A-CH )^+BH^-PH%]^^ l  +  3  We have e x p r e s s e d  t h e r e s u l t i n terms of d E  and  by  we  divided  the  free  decay  o f dpe  ( 6 W^/l92Tr Z  multiplied  by t h e muon mass t o g i v e  Collected  below a r e t h e d e f i n i t i o n s o f a l l t h e v a r i a b l e s  to evaluate  m  dimensionless  5  ,  ) and  quantity. needed  (4.4).  = fry (I - j ^ + l ^ a a ) 1  r  a  rate  instead  e  H  e = m »2rx 0/  A =H (3E +rt'^^  B =2H(EeH-ft-K^+-Pe^  E  H  1  e  ^(EcM^-ftV  =E .-Ee-zK(E^-Ee-p )  1  Ty  e  57  R= a+bHtH*  S=2bA+abB-2a*D l  T=3b B-6abD  U = bB-2tfC-abE-4-aD-6b*D  V = -bE-2D  5 =b'>\+abB+3aD  T=2b B+4abD  D =-bC+bB+6aD  a  l  a  A=4e  l  l  t ^ i t a n "  D I I o . qiga-b ) + b(3a-tf)H R, = l o R + 2TR a  x  9  1  ^  b(6a-b*) ^ 2A t  Za + bH | 3b(b-r2H) . 3b " 2AR * 2 A R A*^ _ ab-r(b*-2a)H . (2a + b»)(b+2H) , 2a -rb*. *~ 2AR "** 2 A*R A* D K l  D  2  a  3  Z  R+=  A  x  +-2"t  R  (4.5)  o . gb-f(b*-2aOH , 2 Q O . 5" AR I +  R  n  R  _ I , a+ H 2bH * 6b H  t  2  l  l  ' ~ bfl * b H 3b H* 3(a-rH*) , q(a+H )* . IKq + H* 8 tfH* 2b*H "*! 6bH+ a  2 +  a  a  D K  The  3  =  differential A  Ffe  in Y  e  Z  rate  (4.4) i s a f u n c t i o n  o f t h e momentum,  J  , a n d p - S*.  so t h a t  decay  +  i n t h e form  i f we i n t e g r a t e will  give  zero.  over e l e c t r o n That i s  direction, J \  e  ,  the  term  58  v. =  A-n-  v / r>.A — A i r  4r <*E. = ^ Y , ( P O The  asymmetry  =  p a r a m e t e r , 0C  A  ^  i  1  —  3 S  the  the  0  )  (4.6)  use t o p r e s e n t  the r e s u l t s of  calculations.  It  Spectrum  i s u s e f u l t o have an a p p r o x i m a t e  form  simpler than  ,  d t * d i U  f u n c t i o n s we w i l l  4.4 The End Of The  the  ^  Y*(PO  =  are  (  d e f i n e d as  dUcTJle,  These  ^  of  the  to derive  trying  following  to  spectrum  at the high  t h i s by r e d o i n g make  some  expression  gives  energy end. I t w i l l  the i n t e g r a l  expansion  which  of  in  (4.1)  (4.4).  We  be  rather use t h e  definitions  <r = ny / M  p  i s t h e r e f o r e a measure o f t h e d i s t a n c e  spectrum.  We  will  keep o n l y  the f i r s t  from t h e e n d  s i g n i f i c a n t order  of in  the p  59  (which w i l l  be  p  ), and  corrections  to t h i s .  5  In  the  this  first way  order  we  recoil  and  binding  get  rry = ny( I - <x)  &r/»  using  The to  ( 4 . 5 ) . We  also neglect  integration make t h e  where now  p  W  M/» ( i ~ i  =  limit  on  z  the  H  i s of  a + bHx + H  order  -  2  one.  e +p x  e l e c t r o n mass. Thus  in  d e p e n d e n c e more  l e  }  (4.1)  i s of  explicit This  we  i  define  and  H=wyprv  so ,  gives  I  l  l*yp  + H^ZpcHx  Z (a +bHx+H r  order  _ 2  ryMa+bHx+H*)  1  ,5  n  "nj?  u  H = rryp ( 1 + c O  C * m ^ ( l - ^ c r - | r ) ( l + piv)  The  term  in  calculating.  E The  in  (4.1)  will  not  contribute  integration limits  are  to the  order  we  are  60  -I < x < 1 so we g e t ,  0 < h < l+ a(l-x}  f o r example,  ^ Proceeding  in this  r ? x d h d X = - f cr way we g e t t h e f i n a l  result  ^ . g [ ( . | ? , . This  has  been  noted  before,  spectrum.  This  neutrinos there  are five  T  rff  ]  spectrum,  particular but  is  recoil, The  and is  a  Mo  i n t h e same way a s ( 4 . 4 ) . As  a r e of order  because  goes t o z e r o  LH H°"- H H g the  H  made d i m e n s i o n l e s s  the  phase  f o r maximum  f a c t o r s of H  shows  or H  approximations feature  of  that  space  , three  0  allowed  energy. in d H  i t  is  exact  not  a  calculation,  with  (3.4)  a n d two i n  result  f o r t h e wave  the  to the  In  dependence o f t h e  we a r e making the  a t t h e end of  electron  . This explains the and  Wi^p  end  of  of the  functions, or without  f o r a l l 2" . asymmetry p a r a m e t e r d e r i v e d  from  (4.7) u s i n g  (4.6)  is  (4.8)  61  4.5 C a l c u l a t i o n s , R e s u l t s  In  this  And  Conclusions  s e c t i o n we p r e s e n t  the r e s u l t s  of  the  calculation  of  t h e e l e c t r o n s p e c t r u m a n d asymmetry o f bound muon d e c a y . F o r  Ee  less  rate.  than  numerical € R  t  reasons.  to evaluate accurately round  digit end  of  s u c h a s t h e one i n  .(2:*} £  i t s (picc^  dependence  5  instance,  discussed these  i s relatively  necessary  here  and  that  plotted  mass, T(\ ^ , 0  i s not l o s t  in  p l a c e machine f o r t h r e e  reason,  easy  the  high  energy  the integration i n because  effect  are  discussion.  the in  recoil  the  the graphs,  motion,  effects include  the e f f e c t s  of  and a l s o t h e wave f u n c t i o n  i n the e f f e c t i v e  the r e c o i l  associated  5  sharp  energies.  s p a c e c h a n g e s due t o n u c l e a r "reduced"  i n £ Ri  i n s e c t i o n 4.2 i s n o t i n t h e i n t e g r a t i o n r e g i o n  We r e m i n d t h e r e a d e r discussed  . For  f o r a h y d r o g e n n u c l e u s and  r e q u i r e a 20 d e c i m a l  This  at  e t c . I f we u s e a computer  5  t h e s p e c t r u m was c a l c u l a t e d by d o i n g  peak  (oZaCf  be o f o r d e r  i n the result-. For t h i s  numerically.  that  must be a c a n c e l a t i o n o f a l l t e r m s o f , ...  For  we w o u l d  (4.1)  of  i n (4.4),  decay  t o use f o r  we would have t o c a l c u l a t e t h e t e r m  errors.  accuracy  the  i tisdifficult  (4.4) s h o u l d  , c^ioc  (4.4),  this  calculate  one, b u t we know from s e c t i o n 4.4  enough so t h a t  off *V  =  p  I , £ot ,  (4.4) t o  The l a r g e r t e r m s  t o be p o s s i b l e t h e r e  order  for  than  end of t h e spectrum,  this  Ee  greater  e  , a r e of order  5  the  E  For  we h a v e u s e d  i n (4.4) the phase using  modifications  p o t e n t i a l a p p r o a c h . These t h r e e separated  Also,  when  only we  in talk  Fig. 8 of  the  parts  and i t s  the spectrum  62  without  recoil  without  we  mean t h e s p e c t r u m  implying  that  £ -* O  M -* Oo  i n the l i m i t .  which  would  give  only,  the free  decay  spectrum. Fig. spectrum in  3 shows t h e g e n e r a l including  section  muon i s  This  1  4.2.  has  £ =1  . The  almost  s e e n on  electron  this  g r a p h . These  i s a plot  o f an  as  spectrum  unbound  (4.4)  on  and  Fig.  3  f r o m t h e bound s p e c t r u m f o r due  to r e c o i l  effects  c a n be  are too seen  of the l o g of the decay  spectrum.  as  simple  small  i n F i g . 4,  r a t e versus the  £-*0  ry/IA  recoil >  /  recoil  shows c l e a r l y  that  recoil  i n the h i g h  calculation  i n g e n e r a l must go  to this . This  1+ 0(£<xm»/fA )  with  e n e r g y end  u s i n g o r d e r of  to zero  i s the r e c o i l  i s possible energy  half  change  because  of  magnitudes  EM**  or  M-»  00  smaller. is  of  i s discontinuous  so the decay  . In t h e h i g h e n e r g y h a l f  as  E*** w h i c h  in  of the spectrum  i s o f o r d e r one,  both  2£o;ryy/M  . Thus t h e y must be o f o r d e r  .) I n t h e low  without  This  rate  i s so.  changes  (An e x c e p t i o n  of the decay  much more i m p o r t a n t  A  this  Recoil  of the r a t i o  rate without r e c o i l .  proportionally  shows why  2">0  i n the spectra  5 i s a plot  to the decay  order  decay  energy.  Fig.  and  of t h i s  indistinguishable  changes  however, w h i c h  the  spectrum  made d i m e n s i o n l e s s i n t h e same way  t h e e l e c t r o n mass. A p l o t  be  is  the decay  muon  ?jMju> d i s c u s s e d  i n the spectrum at  For comparison,  been  would  be  o f t h e bound  3  neglects  to  the s t e p  features  the ratio  decay  as  rate  i s of o r d e r  of the spectrum,  however,  63  the decay  rate  2oc  order  goes  or  1 +0(wy/M)  .  t o z e r o as  decay  ratio  Thus  see  that  recoil  we  in  5  recoil  ends  Fig.  a t lower energy  6 i s a plot  of  negative s h i f t  1  ~'  3  the  7  discussion  is  of o r d e r are  more  of the spectrum. A l l the  plots  asymmetry aside  effects  the spectrum w i t h  parameter  recoil. defined  from the change  t h e asymmetry  a plot  of s e c t i o n  spectrum  decay  of  in E  in , is  M A X  o f t h e unbound  muon  by  decay  ~  Fig.  decay  then  be  o f t h e asymmetry a t h i g h e n e r g y p r e d i c t e d  T h i s assumes t h e e l e c t r o n  the  must  than the spectrum w i t h o u t  effect,  (4.8). For comparison, is  it is  t o z e r o a t h i g h energy because  ( 4 . 6 ) . The main r e c o i l the  so  The  i n the high energy h a l f go  and  smaller.  important Fig.  0  £  rate  goes  by  o f t h e end  4.4.  defined  rate divided  mass c a n be  o  in  ^  o f t h e s p e c t r u m b a s e d on  i s the d i s t a n c e  4.4.  The  SlZCZap^/Sl*  to _ I +  neglected.  1  ~*(r*.)  relative  . From at  from the decay  (4.7),  p= o  end  rate  the  the of  i s the  relative  . From t h e g r a p h i t  6  appears that quite  the  large  this plot  we  and  d e p e n d e n c e o f t h e end o f t h e i s independent of  2  spectrum  is  . U s i n g an e s t i m a t e f r o m  get  In F i g . 8 we 2 - I  p  decay  have d i v i d e d  spectrum to the  into  correction  due  modified  calculate  the m a t r i x element,  the three wave  recoil parts.  correction These  function,  2) t h e " r e d u c e d " mass,  to  are ,  1)  the  used  to  Tf\ * , 0  the  given  64  by  (2.10) and 3) t h e r e c o i l  ( 3 . 5 ) . The m o d i f i e d calculation neglected neglected  (4.4) t h i s  C  , 0  gives in  i  £  above  The  modified  so  is  ,  H  recoil  The p h a s e  0  terms  • Neglecting result  i n (3.5) and ( 3 . 6 ) .  a l l t h e M* a l l three by M~*  obtained  of the spectrum  . Of c o u r s e ,  the other  In terms A  the modified  wave  8 should  be compared w i t h recoil  F i g . 5 which  effects  ,B  ,  0  0  i n (4.4). P l o t t e d only  one  of  recoil.  (ZcOrvy/M  and  two c o r r e c t i o n s w h i c h a r e o f  p r o p o r t i o n a l l y much more i m p o r t a n t  a l l three  i n our  of these c o r r e c t i o n s  including  becomes  with  terms i n  wave f u n c t i o n c o r r e c t i o n i s o f o r d e r than  by  space c o r r e c t i o n i s  c o r r e c t i o n s , t o the spectrum without  much s m a l l e r  fty/M  Tt\ ^= fly .  t h e M~'  are the r a t i o s  the  order  i H  given  The " r e d u c e d " mass c o r r e c t i o n i s  w o u l d mean d r o p p i n g  the r e c o i l e s s  Fig.8  = Ho^».  choosing  by d r o p p i n g  of  i n t h e phase space  wave f u n c t i o n c o r r e c t i o n i s n e g l e c t e d  by s e t t i n g by  change  included.  function  correction  for larger  gives  the  £  decay  .  Fig. ratio  Fig. 3. The e l e c t r o n s p e c t r u m f r o m bound muon d e c a y . The d e c a y r a t e h a s been made dimensionless and i s given by ( 4 . 4 ) . The e l e c t r o n e n e r g y i s g i v e n i n t e r m s of t h e muon mass.  66  0.4  0.5  0.6  0.7  ELECTRON  0.8  ENERGY  0.9  1 0  F i g . 4. L o g g r a p h o f t h e t a i l o f the spectrum. This shows the recoil e f f e c t s i n t h e h i g h e n e r g y h a l f of the spectrum. A 2=6 w i t h r e c o i l p l o t on t h i s graph would o n l y j u s t be d i s t i n g u i s h a b l e from t h e no r e c o i l plot.  67  Fig.  5.  Recoil/no  recoil  decay  ratio.  68  F i g . 6 . The asymmetry. T h i s parameter d e f i n e d i n ( 4 . 6 ) .  i s a plot  o f t h e asymmetry  69  F i g . 7. The end o f t h e s p e c t r u m . This graph follows from the discussion in section 4.4. p is the d i s t a n c e f r o m t h e end o f t h e s p e c t r u m d e f i n e d i n 4.4. The relative d e c a y r a t e i s t h e d e c a y r a t e d i v i d e d by  70  \ co  \  o  \  Z= l -WRVE F U N C T . REDUCED MRSS •PHRSE S P R C E  ' ' o h—  CE CU  >CE LU  o'  O  0.0  i  0.2  i  i  i  0.4  ELECTRON  1 0.6  1  \ r ~  0 8  ENERGY  F i g . 8 . The t h r e e p a r t s o f t h e r e c o i l c o r r e c t i o n . T h i s shows a division of the t o t a l recoil effect for hydrogen into the three parts discussed i n the text, namely, t h e m o d i f i e d wave function correction, the "reduced" mass c o r r e c t i o n , a n d t h e c o r r e c t i o n due t o phase space. P l o t t e d here a r e t h e s p e c t r a w i t h each of the c o r r e c t i o n s a l o n e , divided by spectrum without recoil.  1.0  71  We  have  calculate  s e e n how t o a p p l y  the  muon d e c a y . section  The  2  as  functions, electron  the  recoil  potential  general  approach  energies,  and made  decay  rate  result  i s given  apparent  and  asymmetry  decreases  nuclei  the  change  in  decrease the  approximately The  calculations get  The us  is  Hy|»o  high  recoil  energies.  wave Born a p p r o x i m a t i o n reliable  electron  for  and  small  used  transformation  give  t o ^te.  a n <  t o do f o r n u m e r i c a l  mixes t h e a n g u l a r  Once g e n e r a t e d , however, Y Nt  i  a n <  3  are  most  for  larger  effect  i s the  This  is  to c a l c u l a t e  nuclei.  3  given  To  extend  these these  then  N  T  i s used  t/* 9 i v e n  ^3r  by  (2.16) and  , and because the  eigenstates e  w o u l d g i v e Veo  wave f u n c t i o n s b e c a u s e o f  presence of the d e r i v a t i v e operator,  .  the  muon wave f u n c t i o n s . T h e r e a r e many  The t r a n s f o r m a t i o n  (2.22) i s n o t e a s y the  wave Born  t o l a r g e r n u c l e i r e q u i r e s s o l v i n g (2.9) n u m e r i c a l l y  the  .  we  calculate  effects  c o m p u t e r p r o g r a m s a v a i l a b l e t o do t h a t , a n d t h e y and  3  of the spectrum. For hydrogen,  s m a l l e r . Another at  a  n u c l e i a n a l y t i c a l l y . The  ( 4 . 4 ) . The r e c o i l half  with  t h e form o f  plane  to  an  (4.8).  plane i s only  small  of  section  r a t e by a f a c t o r o f two;  asymmetry  by  In  in  o f wave  s p i n l e s s nucleus  l e a v i n g open  allowed  for  energy  the decay  derived  to the c a l c u l a t i o n  functions.  in section 4  i n the high  was  distribution.  wave  i n equation  c o r r e c t i o n s t o bound  e t c . of systems c o n s i s t i n g  charge  muon  p o t e n t i a l method t o  method  method t o bound muon d e c a y  approximation  to  a  symmetric  this  results  nuclear  o r muon and a n o n - r e l a t i v i s t i c  electron  recoil  order  effective  binding  spherically applied  first  the e f f e c t i v e  and  a  used r  e  i n (3.4) t o g i v e  used  to  express  i n (3.3) t o  the d i f f e r e n t i a l  72  decay  r a t e . • These  integral including  search  searches  ratios be of  H  over  is  are  done  straightforward.  to  give  the  we comment on t h e r e l e v a n c e  for -u  detected  these  e x o t i c decays w i l l  i n an e n e r g y  range  in this  near  We  muon  conventional corrections,  mass.  decay  spectrum  at least  taken  into  the r e s u l t s .  have is  shown subject  f o r the l i g h t e r account  results  process  in  thesis  end that to  of this  branching  can  designing  only  process  i s extremely the  spectrum  p a r t of t h e  substantial  nuclei.  to  and proposed  i n which the background  the extreme h i g h  analysing  these  t r y t o measure  s m a l l . T h i s means u s i n g  be  of  and beyond. E l e c t r o n s from t h i s  t h e c o n v e n t i o n a l decay c a l c u l a t e d  should  the  e l e c t r o n spectrum,  f o r muon t o e l e c t r o n c o n v e r s i o n . C u r r e n t  o f 10  the  Finally,  recoil.  Finally, the  steps  recoil  These c o r r e c t i o n s experiments  and  73  REFERENCES  1)  C.E. P o r t e r  and H. P r i m a k o f f ,  2)  V. G i l i n s k i  a n d J . Mathews, P h y s . Rev. 120  3)  H. U b e r a l l ,  P h y s . Rev. 119 (1960) 365  4)  R. W.  5)  P. H a n g g i , T h e s i s , U n i v e r s i t y  6)  P.  7)  (1960) 1450  H u f f , Ann. P h y s . 16 (1961) 288  Hanggi,  Lett.  P h y s . Rev. 83 (1951) 849  51B  R.  D.  of B a s e l ,  Viollier,  U. R a f f  (1973)  unpublished  a n d K. A l d e r ,  Phys.  (1974) 119  F. H e r z o g a n d K. A l d e r ,  Thesis,  University  of B a s e l ,  (1979)  unpublished 8)  G.Breit,  9)  H. G r o t c h a n d D. R. Y e n n i e , Rev. Mod. P h y s . 41  10)  J . L. F r i a r ,  Ann. P h y s . 81 (1973)  11)  J.,L. F r i a r ,  Ann. P h y s . 98 (1976) 490  12)  R. C. B a r r e t t , Lett.  13)  14)  J.  D.  P h y s . Rev. 34  47B  (1929) 553 (1969) 350  332  D. A. Owen, J . C a l m e t and H.  Grotch,  Phys.  (1973) 297  Bjorken  and  S.  D.  Drell,  M e c h a n i c s , New  York, McGraw-Hill,  J . D. J a c k s o n ,  Classical  Relativistic  Quantum  (1964)  E l e c t r o d y n a m i c s , New  York,  Wiley  (1962) 15)  M.  E.  Rose, R e l a t i v i s t i c  Electron  T h e o r y , New  York, Wiley  (1961) 16)  H. G r o t c h a n d R. K a s h u b a , P h y s . Rev. A 5 (1972) 527  74  17)  H. Osborn, Phys. Rev. 176 (1968)  18)  R. H. Good, Rev. Mod. Phys. 27  1514 (1955)  187;  K.  M.  Case,  Phys. Rev. 97 (1955) 810 19)  H.  M.  Pilkuhn,  Springer-Verlag,  Relativistic (1979)  P a r t i c l e P h y s i c s , New  York,  

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