UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Recoil effects in bound muon decay Brookfield, Gary John 1981

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1981_A6_7 B76.pdf [ 3.12MB ]
Metadata
JSON: 831-1.0095209.json
JSON-LD: 831-1.0095209-ld.json
RDF/XML (Pretty): 831-1.0095209-rdf.xml
RDF/JSON: 831-1.0095209-rdf.json
Turtle: 831-1.0095209-turtle.txt
N-Triples: 831-1.0095209-rdf-ntriples.txt
Original Record: 831-1.0095209-source.json
Full Text
831-1.0095209-fulltext.txt
Citation
831-1.0095209.ris

Full Text

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 i n THE FACULTY OF GRADUATE STUDIES ( Department of Physics ) We accept t h i s t h e s i s as conforming to the required 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 t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the requirements f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be granted by the head o f my department o r by h i s o r her r e p r e s e n t a t i v e s . I t i s understood t h a t c o p y i n g or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a llowed without my w r i t t e n p e r m i s s i o n . Department of PHYSICS The U n i v e r s i t y of B r i t i s h Columbia 20 75 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date Apr. 2S , iqBI DE-6 (2/79) ABSTRACT An unbound muon at r e s t decays i n t o two n e u t r i n o s and an e l e c t r o n . Momentum c o n s e r v a t i o n f o r b i d s the e l e c t r o n to have energy g r e a t e r than one h a l f the muon mass. However, i f the muon i s bound to a nucleus, i t s o r b i t a l motion -and nu c l e a r r e c o i l make i t p o s s i b l e f o r the decay e l e c t r o n energy to approach the muon mass. These hi g h energy e l e c t r o n s are an important background e f f e c t i n the search f o r muon to e l e c t r o n c o n v e r s i o n . T h i s t h e s i s i n v e s t i g a t e s r e c o i l c o r r e c t i o n s to the e l e c t r o n spectrum and asymmetry from bound muon decay u s i n g an e f f e c t i v e p o t e n t i a l approach. T h i s approach s i m p l i f i e s the equations d e s c r i b i n g a f i n i t e l y massive nucleus and e l e c t r o n (or muon) i n t e r a c t i n g e l e c t r o m a g n e t i c a l l y , to a one p a r t i c l e D i r a c equation f o r a p o t e n t i a l w e l l . For l a r g e Z, c a l c u l a t i o n of the e l e c t r o n spectrum r e q u i r e s i n c l u d i n g the e f f e c t s of f i n i t e n u c l e a r s i z e and e l e c t r o n wave f u n c t i o n d i s t o r t i o n . Such a c a l c u l a t i o n would i n v o l v e n u m e r i c a l l y generated wave f u n c t i o n s and numerical i n t e g r a t i o n s , and i s not done here, though the a p p l i c a b l e methods and ge n e r a l formulas are presented. For small Z the c a l c u l a t i o n can proceed a n a l y t i c a l l y and an ex p r e s s i o n has been d e r i v e d f o r the spectrum and asymmetry i n c l u d i n g r e c o i l . i i i 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 Center Of Mass Transformation 14 2.3 Bound States 22 2.4 S c a t t e r i n g States 28 3 BOUND MUON DECAY 33 3.1 The Wave Functions 33 3.2 The Decay Rate 36 3.3 The Phase Space 40 4 PLANE WAVE BORN APPROXIMATION 43 4.1 The Approximate Wave Functions 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 Conclusions 61 i v FIGURES F i g . 1. The I n t e g r a t i o n Region 42 F i g . 2. The I n t e g r a t i o n Region For F i x e d pfc 53 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. Recoil/no 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 Parts 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 i n t e r a c t i o n s has l e d to a renewed i n t e r e s t i n the search for 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 lepton number. In p a r t i c u l a r , f o r a greater understanding of the lepton-guark generation problem, experiments have been proposed to search f o r decays v i o l a t i n g muon number con s e r v a t i o n , such as , fk -»3e , and n e u t r i n o l e s s e l e c t r o n to muon conversion. In order to detect these very rare events one has to know the backgrounds very a c c u r a t e l y . One of the most important of the backgrounds to e l e c t r o n to muon conversion i s the conventional decay of negative muons from the ground s t a t e of atoms. These mesonic —lO atoms form almost immediately (. < 10 s ) when a muon beam enters matter. U n l i k e the. decay of free muons, the decay of bound muons can produce a small f r a c t i o n of e l e c t r o n s i n the energy range |;rfy» < E e K m/» • This i s because the nucleus can take up a l a r g e r e c o i l momentum with only a small f r a c t i o n of the a v a i l a b l e energy. At the high energy end of t h i s range the background becomes small enough that one might be able to detect the ~ 100 Mev e l e c t r o n s expected from muon to e l e c t r o n conversion. The high energy end of the spectrum i s a l s o the part most a f f e c t e d by nuclear r e c o i l . The e l e c t r o n spectrum from bound muon decay was f i r s t c a l c u l a t e d by Porter and P r i m a k o f f 1 i n 1951. Later work by G i l i n s k i and Mathews 2, U b e r a l l 3 , and Huff* i n 1960-61 improved the approximations used for 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 of the q u a l i t a t i v e changes to the muon decay r a t e caused by the atomic b i n d i n g . G i l i n s k i and Mathews c a l c u l a t e d the asymmetry of the decay f o r the f i r s t time. L a t e r , Hanggi, V i o l l i e r , R a f f , and A l d e r 5 " ' repeated the e l e c t r o n spectrum c a l c u l a t i o n u s i n g n u m e r i c a l l y generated wave f u n c t i o n s f o r the e l e c t r o n and muon, i n c l u d i n g the e f f e c t s of vacuum p o l a r i z a t i o n , the f i n i t e s i z e of the nucleus and some of the r e c o i l c o r r e c t i o n s t o be d i s c u s s e d below. Herzog and A l d e r 7 extend the above work to i n c l u d e the p o s s i b l i t y of a small admixture of s c a l a r , pseudoscalar and tensor c o u p l i n g s i n a d d i t i o n to the standard V-A weak i n t e r a c t i o n , and a l s o d i s c u s s the e f f e c t of e l e c t r o n bremstrahlung on the spectrum. Other e f f e c t s not c a l c u l a t e d i n these papers are the e l e c t r o m a g n e t i c vertex c o r r e c t i o n s to the weak decay v e r t e x , the e f f e c t s of a nuclear magnetic moment, and the r e c o i l e f f e c t s . The l a t t e r of these i s the s u b j e c t of t h i s t h e s i s . To c a l c u l a t e the e f f e c t s of r e c o i l on the wave f u n c t i o n s , b i n d i n g e n e r g i e s , decay r a t e s e t c . of bound muon decay, we use an e f f e c t i v e p o t e n t i a l method. In the l i m i t of i n f i n i t e n u c l e a r mass, the p o t e n t i a l one would use to c a l c u l a t e the muon and e l e c t r o n wave f u n c t i o n s would, of course, be the Coulomb p o t e n t i a l , V = ~2<x/r , with perhaps some m o d i f i c a t i o n due to nucle a r s i z e . I f the nu c l e a r mass i s not i n f i n i t e then the p o t e n t i a l must be modified to i n c l u d e the e f f e c t s of nucl e a r motion. To f i r s t order i n the nu c l e a r v e l o c i t y , V N , the m o d i f i c a t i o n s are caused by the presence of a magnetic f i e l d around the nucleus i n a d d i t i o n to the e l e c t r i c f i e l d . A p p l i e d to a system with a s p i n 1 / 2 , p o i n t nucleus t h i s approach g i v e s the 3 B r e i t i n t e r a c t i o n 8 . The e f f e c t i v e p o t e n t i a l we w i l l d e r i v e here i s a p p l i c a b l e to s p i n l e s s n u c l e i with s p h e r i c a l l y symmetric charge d i s t r i b u t i o n s . Even though the e l e c t r o n c o u l d be extremely u l t r a - r e l a t i v i s t i c , i t all o w s us to d e f i n e a c e n t e r of mass and a reduced mass i n a s i m i l a r way to the completely non-r e l a t i v i s t i c approach. Besides the reduced mass, the equations of motion i n the cente r of mass i n v o l v e a m o d i f i e d p o t e n t i a l and a "reduced" energy. Aspects of the e f f e c t i v e p o t e n t i a l approach have been a p p l i e d by Grotch and Yennie' to the energy l e v e l s of hydrogen, by F r i a r 1 0 ~ l x to e l e c t r o n s c a t t e r i n g from n u c l e i , and by B a r r e t t et a l 1 2 to the energy l e v e l s of muonic atoms. They d i d t h e i r c a l c u l a t i o n s i n the center of mass frame and d e r i v e d t h e i r equations i n a d i f f e r e n t way than i n t h i s t h e s i s . In the bound muon decay c a l c u l a t i o n the p o s i t i o n of the cente r of mass i s more important because i t i s not the same i n the muon-nucleus system and the e l e c t r o n - n u c l e u s system. The r e c o i l c o r r e c t i o n s of Hanggi et a l and Herzog et a l i n c l u d e only the e f f e c t s of the change i n phase space allowed t o the outgoing p a r t i c l e s by the f i n i t e mass of the nucleus, and none of the reduced mass, e f f e c t i v e p o t e n t i a l and m o d i f i e d wave f u n c t i o n changes made here. Sec t i on / 2 of t h i s t h e s i s develops the e f f e c t i v e p o t e n t i a l method and c o n s i d e r s i t s a p p l i c a t i o n to the s c a t t e r i n g and bound s t a t e s of e l e c t r o n s (muons) and n u c l e i . S e c t i o n 3 a p p l i e s the r e s u l t s of s e c t i o n 2 to bound muon decay. The i n t e g r a l s over n e u t r i n o momentum are done and the c a l c u l a t i o n proceeds as f a r as p o s s i b l e without the e l e c t r o n and muon wave f u n c t i o n s a c t u a l l y being s p e c i f i e d . 4 In s e c t i o n 4 , crude approximations are made f o r these wave f u n c t i o n s which enable the remaining i n t e g r a l s to be done a n a l y t i c a l l y and the e f f e c t s of r e c o i l on the e l e c t r o n spectrum to be seen f o r small 2 n u c l e i . L e f t f o r f u t u r e work i s the exact c a l c u l a t i o n u s i n g n u m e r i c a l l y generated muon and e l e c t r o n wave f u n c t i o n s , i n c l u d i n g the e f f e c t of f i n i t e n u c l e a r s i z e . N o t a t i o n : We w i l l use the conventions of Bjorken and D r e l l 1 3 u n l e s s otherwise noted. One exce p t i o n i s that we normalize our s p i n o r s so t h a t U*U=I i n s t e a d of U U = I . In p a r t i c u l a r we use u n i t s so that C = f\-\ . F o u r - v e c t o r s w i l l always have a Greek index, p/* . Thr e e - v e c t o r s w i l l be w r i t t e n as. p , thus p / , = ( E p ' i . The f o u r - v e c t o r s c a l a r product i s w r i t t e n as p A-p f t = p^p^ . The t h r e e - v e c t o r s c a l a r product i s w r i t t e n pAfo . So we get P A P B = E ^ B " P A * Pa • P 1 S t h e magnitude of the t h r e e - v e c t o r p , and p l^s the u n i t v e c t o r i n the d i r e c t i o n of p , that i s , p = p / p 5 2 THE EFFECTIVE POTENTIAL METHOD 2 . 1 The E f f e c t i v e P o t e n t i a l The e f f e c t i v e p o t e n t i a l method w i l l enable us to c a l c u l a t e the e f f e c t s of n u c l e a r motion on the wave f u n c t i o n s d e s c r i b i n g the e l e c t r o m a g n e t i c i n t e r a c t i o n of n u c l e i and e l e c t r o n s (or muons) . We assume that the n u c l e a r mass , M , i s much l a r g e r than a l l other energy v a r i a b l e s i n the problem and so we n e g l e c t terms of order M or s m a l l e r i n e v e r y t h i n g t h a t f o l l o w s . T h i s i m p l i e s that the nucleus moves n o n - r e l a t i v i s t i c a l l y and we can z drop terms of order VN or s m a l l e r , where VM i s the n u c l e a r v e l o c i t y . The nucleus i s a l s o assumed to have zero s p i n and a s p h e r i c a l l y symmetric charge d i s t r i b u t i o n . The e l e c t r o n (or muon) i s t r e a t e d as a D i r a c p a r t i c l e whose motion may w e l l be r e l a t i v i s t i c but whose energy must be s m a l l e r than M f o r our approximations to be v a l i d . C l a s s i c a l e l e c t r o d y n a m i c s p r o v i d e s the model f o r our development of an e f f e c t i v e p o t e n t i a l . For an e l e c t r o n of mass f\ and a nucleus of mass M , we get the p o t e n t i a l v=-£tf./r only i n the l i m i t M -» oo . Here zl i s the nuclear charge number, r i s the s e p a r a t i o n of the charges and <x = ftV+ir i s the f i n e s t r u c t u r e c o n s t a n t . If M i s not i n f i n i t e then the nucleus w i l l move and i t s motion w i l l e f f e c t the p o t e n t i a l i n two ways. 1 ) The nucleus produces a magnetic f i e l d around i t p r o p o r t i o n a l to V* which propagates and causes a f o r c e on the 6 e l e c t r o n p r o p o r t i o n a l to V e . In t h i s way the e f f e c t i v e p o t e n t i a l w i l l get a term p r o p o r t i o n a l to V NV t . 2 ) The f i n i t e time of pr o p a g a t i o n of the f i e l d w i l l cause the e l e c t r o n to "see" the p o s i t i o n of the nucleus as i t was some time b e f o r e . T h i s would cause terms of order VM to appear i n the e f f e c t i v e p o t e n t i a l but we have choosen to drop terms t h i s s m a l l . C a l c u l a t i o n of the e f f e c t i v e p o t e n t i a l to f i r s t order i n V* g i v e s then the Darwin p o t e n t i a l 1 4 Ve» = ^ L l - i ( v v „ + v . - r t f . - r M ] (2 . 1 ) T h i s p o t e n t i a l i s gauge i n v a r i a n t and w i l l g i v e r e c o i l c o r r e c t i o n s to order VM ( o r M 1 ) . Extending t h i s to the case of an e l e c t r o n and a s p h e r i c a l l y symmetric nucleus we get where p(r') i s the charge d i s t r i b u t i o n of the nucleus, normalized so that P -»« The i n t e g r a t i o n v a r i a b l e r i s the c o o r d i n a t e measured from the c e n t e r of the nucleus, and R = r - r . Note that both the charge d e n s i t y and the shape of the nucleus depend on V« because of the F i t z g e r a l d c o n t r a c t i o n , but only to order V^. so we can ignore these e f f e c t s . To c a l c u l a t e Ve^ . we choose 7 r = r lo ,o , i) r'= r'CsinS co5<t>,sm0 sin^cojB) R = C-r'smBcos^-r'sinesm^, r-r'cos9) R 2 = rl+ r' 1 -2rr'cos6 d 3 r ' * r , 2 d r d(cose)d4> and we d e f i n e = 4-ir \Vdr-pcr') (-2oc) e where we have used j r > r1 f r < r' VcrO i s the p o t e n t i a l one gets when the nucleus i s f i x e d . Using the above d e f i n i t i o n of V we get V^Cft = V(r)[ I - H-v,] - j^r 'pcr ' ) ^ ^ . R ^ . R The remaining i n t e g r a l must have the f o l l o w i n g form ( ? r>r' (-p r < r ' dicosB) _ f 'd ( : 8 If we pick V6 = V N ="(o, 1,0) we get £(n s^ d^ dKcosejr^ dr'pcr')1!^  r,xsirfe 5 i n l $ =+F$Vdr'pert ( - Z o O \ 3,r* 0 ( TP r * r ' where we used 5in z e d(wB) _ ) 3r* r < r 1 S 3r" It i s useful to express j - in terms of V and W , where \ r - J F » r > r ' = -p,$V<r')r'Mr' We get j- = V - U .To calculate 9 we choose V ^ a ( 0 ( i , 0 and V6=(0,-|,i) so that VV r t = 0 V ? V e r = I . This gives 9 g(r-j s \d4>dccosB)r,1dr,pcrM -^5 = V-3J =3U-2V u s i n g P u t t i n g these r e s u l t s together we get For p o i n t charges we have U=V and we r e t r i e v e (2.1). So f a r we are s t i l l i n the realm of c l a s s i c a l e l e c t r o d y n a m i c s . To go to quantum mechanics we r e p l a c e VN and V e by the e l e c t r o n and n u c l e a r v e l o c i t y o p e r a t o r s . T h i s g i v e s Ve&Cr) = V-^(2V-U)6tp,-^(3U-2V)ocr rp„ (2.2) We have chosen the order of the f a c t o r s i n Ve$ so that i t i s He r m i t i a n . Other o r d e r i n g s which have the same c l a s s i c a l analog are e i t h e r i d e n t i c a l to the above or are not He r m i t i a n . For example, i s e q u i v a l e n t to (2.2) because 10 f . f ^ j v ? - ^ U c t p . = * p . U - i 6 c r f £ and i s not Hermitian 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 ( 2V -3U0 f Care must be taken to d i s t i g u i s h PM = " ' v r M from p = - i v r where _ > - > - > _> r" = re " r* .We w i l l see i n the next s e c t i o n that p„ can be expressed as p M = ( | - j ^ P T - p where P T i s an operator which commutes with r and a l l f u n c t i o n s of . r such as V and U , whereas, of course, Er-, p-] = . T h i s makes i t e a s i e r t o d e r i v e the above r e l a t i o n s . E quation (2.2) g i v e s the e f f e c t i v e p o t e n t i a l i n the form which we w i l l use. The e f f e c t i v e p o t e n t i a l can a l s o be w r i t t e n i n the form v e > } ( n=v - 2^ { £ P . , v} - ^ ra • p., c pi, v ] 3 where W(r) =^dV'pcn(-BoOR = - - ^ V v c r ) d r ' + r'V(r")dr' and s a t i s f i e s 11 V*W=2V §7 = r U Here we use [pJ.W] =-Z(V + rU-|p) T h i s i s s i m i l a r to the form f o r Vejj. that Grotch and Yennie' use though they d e r i v e i t f o r the c e n t e r of mass frame and f o r p o i n t n u c l e i o n l y . Using t h i s r e p r e s e n t a t i o n of Ve# we can . c a l c u l a t e ^ P N ' Vett ' Pn y where p N and p j are the incoming and outgoing n u c l e a r momenta. I f we use t e m p o r a r i l y the n o r m a l i z a t i o n ^ _ i r x • p J < r « l p M ) = 6 then < ft.' i v« i A, > « e!1* t - ^ * . ( a ,4«^ v«^  where - J - » i _ i V(Q^ ) = <p*'l V I p„> = 6 H Vcr) wc<p = < ft! I w i p„> = JjdV e ,r'* wen V(o^ and W(cp can most e a s i l y be expressed i n terms of F(cp the form f a c t o r . 12 ? t % t = -Jeff e * * pen We then get < P « ' I V ^ ! p w > = = c - 2 e o e - ^ [ l - « . . l % ^ ] where <*x=* ^ -The second term i n <Pi!i I Veyj. I g i v e s the i n t e r a c t i o n of the two c u r r e n t s . The matrix element can be w r i t t e n as <&'i V f r > = e 5 * * is. £ j „ - 1 : ? - 1 where J M i s the n o n - r e l a t i v i s t i c r e d u c t i o n of the c u r r e n t f o r a s p i n zero p a r t i c l e given by J / C ^ = 2 F ( ^ ( l , - % p - ) To f i n d the Born amplitude f o r e l e c t r o n - n u c l e u s s c a t t e r i n g we c a l c u l a t e the matrix element of V^ JJ. between e i g e n s t a t e s of e l e c t r o n momentum. T h i s g i v e s 13 <P« Pt I V ^ I P N Pe> — UV?& P H + Pt - pi - Pi ^ J / Ztc where and we used Je = 0 and <£o = 0(M ). T h i s i s e x a c t l y what one would expect from QED. I t i s important to note t h a t any term of the form with toc. (4>J=0 , can be added to V«j£ without a l t e r i n g the Born amplitude. However, such a change would a f f e c t the wave f u n c t i o n s , b i n d i n g e n e r g i e s and matrix elements of s t a t e s i n our c a l c u l a t i o n . So i t i s not p o s s i b l e to d e r i v e ( 2 . 2 ) from the Born amplitude a l o n e . For a d i s c u s s i o n of t h i s p o i n t see R o s e 1 5 , Grotch and Yennie' or F r i a r 1 1 . 14 2 . 2 The Center Of Mass Transformation The wave equation f o r the n u c l e u s - e l e c t r o n system i s given by - 2j 5 i(3U-2V)oc-rr.p l (3 T We have i n c l u d e d the k i n e t i c but not the r e s t energy of the nucleus i n the Hamiltonian and i n E T , the t o t a l energy. ff\ i s the e l e c t r o n mass. The wave f u n c t i o n has four components and i s a f u n c t i o n of both r e and rN . If t h i s were a completely n o n - r e l a t i v i s t i c Hamiltonian, we would tr a n s f o r m to the c e n t e r of mass (CM) c o o r d i n a t e s and separate out the CM motion from the r e l a t i v e motion. Whether t h i s can be done f o r the r e l a t i v i s t i c case i s not c l e a r . For guidance, we t u r n again to c l a s s i c a l mechanics and c o n s i d e r f i r s t the case of the nucleus and e l e c t r o n without i n t e r a c t i o n . For t h i s system we can d e f i n e the c e n t e r of mass frame to be the frame i n which the sum of the t h r e e momenta of the p a r t i c l e s i s ze r o . The v e l o c i t y of t h i s frame w i l l be ( p N t Pe)/( E N t £t). T h i s does not d e f i n e a CM p o s i t i o n however. A d e f i n i t i o n of the CM which has the c o r r e c t v e l o c i t y i s 15 R T = ^  + E w f > (2.3) where the e n e r g i e s of the p a r t i c l e s , E e and E„ , i n c l u d e the r e s t masses. T h i s i s not the only p o s s i b l e d e f i n i t i o n s a t i s f y i n g our requirements, and i t i s not even r e l a t i v i s t i c a l l y i n v a r i a n t because of the way that the energie s appear i n the d e f i n i t i o n and because measuring *e and r„ sim u l t a n e o u s l y i n a p a r t i c u l a r frame i s not an i n v a r i a n t procedure. I t i s p o s s i b l e to d e f i n e a CM which e l i m i n a t e s these p r o b l e m s 1 7 but i f one of the p a r t i c l e s i s n o n - r e l a t i v i s t i c , which i s our circumstance, i t w i l l reduce to the above form anyway. T h i s suggests that RT would be a u s e f u l CM c o o r d i n a t e f o r use i n s i m p l i f y i n g the wave equ a t i o n . The e l e c t r o n and nucleus i n our problem, however, are not f r e e and so t h e i r e n e r g i e s and momenta are not c o n s t a n t . A l s o the f i e l d i t s e l f w i l l have energy and momentum which must be taken i n t o account. As a crude approximation we can assume , f o r the purposes of c a l c u l a t i n g RT , that the e l e c t r o n and nucleus get one h a l f of the f i e l d energy each. T h i s approximation w i l l have to be c o r r e c t only t o z e r o t h order i n M ' to be c o n s i s t e n t with our other approximations. Thus the t o t a l energy i s d i v i d e d i n t o two p a r t s E„-r E* + V = (M + £ V ) H E . + jrV)+0(M-') So the CM c o o r d i n a t e i s 16 a l | . i ^ ) ^ + E ^ f % + o ( M - ) I f we were to apply t h i s r e s u l t to quantum mechanics we would expect that the wave f u n c t i o n of the CM of the system should be p r o p o r t i o n a l to o , at l e a s t f o r l a r g e s e p a r a t i o n s where V _» i s s m a l l . Here R- i s the t o t a l momentum of the system, From a mathematical p o i n t of view i t i s d i f f i c u l t to use RT as a CM c o o r d i n a t e because i t i s not a simple l i n e a r combination of I and I'M , but i t depends a l s o on I r t-r r t| through V and E e • So i n s t e a d we make the c o n v e n t i o n a l non-r e l a t i v i s t i c CM t r a n s f o r m a t i o n which i s , i n t h i s circumstance, j u s t a mathematical convenience of no p h y s i c a l meaning. The new c o o r d i n a t e s would be r = rc - r N and the conjugate momentum o p e r a t o r s would be PT = Pc + P* With these s u b s t i t u t i o n s the Hamiltonian becomes (2.5) 17 ( 2 . 6 ) Hr = [ap rpm-r^ +V t jJ3f(2V-U)at.p + ^ (3U-2V)ar rp ] - ^ ( 3 U - 2 V ) c t r r . f h = Hrft| + HcM where H T i s s p l i t up i n t o H ^ i and H t n a c c o r d i n g to the square b r a c k e t s . T h i s Hamiltonian i s not separable because of -» A II II the appearance of P and r i n H c» . However, H h t l depends on *~ and p , i t s conjugate momentum, o n l y . Let T£ 6 I be a s o l u t i o n of then i t can be shown that -i - i ¥ < r R ) = $|+^rS-?i+2i(E-m--jrU)r ^ 3 | Y r t i ( r ) 6 , T ( 2 . 7 ) s a t i s f i e s H T ¥ = ( E + - ^ ) f = E T * ( 2 . 8 ) (Note that we are us i n g PT f o r both the t o t a l momentum operator and i t s e i g e n v a l u e . The c o r r e c t meaning should be c l e a r from context.) P R has the obvious i n t e r p r e t a t i o n as the t o t a l momentum of the system while ET = i s the t o t a l energy. and E i s the i n t e r n a l energy. T h i s way of approaching the CM 18 t r a n s f o r m a t i o n i s s i m i l a r t o that of Grotch and Kashuba 1*, though they use i t f o r a p o i n t , s p i n 1/2 nucleus. We can now compare the wave f u n c t i o n of the CM given above wit h the one i n v o l v i n g R r which we had a n t i c i p a t e d . Using (2.3) and (2.4) we get i s i n c e , to z e r o t h order i n M~* , the i n t e r n a l energy E i s the e l e c t r o n .energy p l u s the p o t e n t i a l , that i s , E = E e + V+OCW'). Thus we get = L l + | j [ ( E - m - i v ) g r i J e , i l C T h i s e x p l a i n s the r- "r term i n (2.7) s i n c e f o r l a r g e f , U=V+0(r~*) . A l s o t h i s i n d i c a t e s that f o r l a r g e -V we should make the replacement L I + -J r (E-m4uirP T ] e i ' T ^—* e''** i n (2.7). T h i s w i l l be important i n d i s c u s s i n g the asymptotic form of s c a t t e r i n g s t a t e s . The r e l a t i v e ( i n t e r n a l ) wave equation i s 19 Lot £+£m + j$ +V+ 2l5[(2V-U)i-p The form of t h i s equation does not depend on the c h o i c e of the « -» -» CM d e f i n i t i o n , but i s a r e s u l t of d e f i n i n g r r - P w + pe , so we expect % e\ to be a p h y s i c a l l y meaningful i n t e r n a l wave f u n c t i o n . A s o l u t i o n of the i n t e r n a l equation i s XdCft = | l+2H(V + pm+rU|-r)j Y.(h where X s a t i s f i e s (ap+pm.+Vo^Yo - E o X ( 2. 9) and V. = V - ^ ( V l + Ur|^) ( 2 . H , Note that f o r the Coulomb p o t e n t i a l V0=»V . The wave equation s a t i s f i e d by Yc i s simply the D i r a c equation with a p o t e n t i a l , and can be s o l v e d by e x i s t i n g techniques n u m e r i c a l l y and a n a l y t i c a l l y . The wave f u n c t i o n Y0 has no p h y s i c a l i n t e r p r e t a t i o n and i s j u s t a computational convenience. If we now put the two t r a n s f o r m a t i o n s together we get (except f o r n o r m a l i z a t i o n ) 0 20 +V+pm + r U ^ ] ( Y0ch eiPr' (2.12) R The H e r m i t i a n conjugate of t h i s i s - V + p m . - r u £ ] j where we have used LrU^:]1" = - L f U ^ + 2V] T h i s g i v e s us the u s e f u l r e s u l t t h a t - Y . f [ l+fi(5c-P T +pm)] Y 0 (2.13) The term i n 3^ has no p h y s i c a l i n t e r p r e t a t i o n being p u r e l y a mathematical consequence of the r e l a t i o n s h i p between % and Xti . The term i n «.PT , however, has an i n t e r p r e t a t i o n because i t i s p a r t of the t r a n s f o r m a t i o n of X a to Y which are both p h y s i c a l l y meaningful. I t g i v e s the change i n e l e c t r o n d e n s i t y due to the F i t z g e r a l d c o n t r a c t i o n when going from the CM frame to frames i n which the CM i s moving. There i s no s i m i l a r change to the nu c l e a r -density because the nucleus i s n o n - r e l a t i v i s t i c in a l l frames we w i l l c o n s i d e r . Because of t h i s , the e l e c t r o n c u r r e n t 21 i s a Lorentz four v e c t o r . In the CM frame we get J«r = ( Y-*i %ti , Y-J o c Yrfct ) I f we now i n t r o d u c e a n o n - r e l a t i v i s t i c CM motion V T = Pr/M the c u r r e n t w i l l undergo a Lo r e n t z t r a n s f o r m a t i o n , which to f i r s t order i n M , i s given by In p a r t i c u l a r , f o r the e l e c t r o n d e n s i t y we get J.' = ¥ f ¥ = Yrt, Y^ + Y*. a v T Y r e l T h i s i s the same as (2.13) 22 2.3 Bound States For bound s t a t e s we can r e w r i t e (2.10) i n terms of the binding energies B - m - E Bo = m 0 - E0 This gives B = B 0 - ^ As an example we c a l c u l a t e the r e c o i l c o r r e c t i o n to the binding energy of the Is l e v e l of hydrogenic atoms. We have V0=V=~£oc/r . S o l v i n g (2.9) by the standard methods we g e t 1 5 Eo = m„V I - (Zoo*' and using (2.10) = m (I - ) - m ( I ~ ) F i n a l l y we get where Thus we see that to order (Zee} 1 the binding energy i s wha-t we would expect i f we t r e a t e d the e l e c t r o n and the nuclear r e c o i l n o n - r e l a t i v i s t i c a l l y by using a reduced mass i n a Schroedinger 23 e q u a t i o n . The l e a d i n g r e l a t i v i s t i c c o r r e c t i o n to the b i n d i n g energy i s of order and the l e a d i n g r e l a t i v i s t i c r e c o i l c o r r e c t i o n i s of order to • We now c o n s i d e r the n o r m a l i z a t i o n of the wave f u n c t i o n given by ( 2 . 1 2 ) . We use box n o r m a l i z a t i o n f o r s i m p l i c i t y of i n t e r p r e t a t i o n . The two p a r t i c l e wave f u n c t i o n must s a t i s f y If we t r a n s f o r m c o o r d i n a t e s to r and K we get 1 = S 5 ¥ f ( r r l ) T ( r R ) d*r d JR = V e ( ^ f 5 Y 0 rcr)Ll + ^ ( ^ P T t | 5 m ) 3 Y 0 ( h d 3 r where we used ( 2.13) and i n c l u d e d an e x t r a f a c t o r as a n o r m a l i z a t i o n c o n s t a n t . N i s given by ||*= ^ Y 0 ( r)LI+ 1Jf(5cPT + 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 i s + V + jSm + rU^] I Y.(r) e '^S;" 1 I t i s convenient to w r i t e Y(rR) = Y(r)-%=r ( 2 . 1 5 ) where 24 Y ^ = N | l+2i5j[aPr + Z i ( E - m - x U ) r P T tV + pm + r U ^ : ] \ ToCn (2.16) and s a t i s f i e s s v» Now we c o n s i d e r the angular momentum p r o p e r t i e s of the wave f u n c t i o n s . The t o t a l angular momentum of the e l e c t r o n - n u c l e u s system i s J T = J „ f J t = rN x p„ + n x pe + j; °» Changing the c o o r d i n a t e s to r and K. g i v e s J T = R x P - j - r x p + j C T t = C + J L i s the angular momentum due to the motion of the ce n t e r of -» mass, j i s the angular momentum i n the CM frame. _> In the CM frame ( P r a O ) we get Y = N j l+^[V + £m + rU^] j Y. I v . , (2.17) Since j commutes with [V + Bm + r U ^ J , any To which i s an e i g e n s t a t e of j g i v e s an e i g e n s t a t e of or with the same 25 quantum numbers. That i s , i f Y 0 i s a s o l u t i o n of ( 2 . 9 ) with '^ = \ ro* ^2. , f o r example, then Y as given above i s the wave f u n c t i o n of the two p a r t i c l e s t a t e with t o t a l angular momentum quantum numbers TT=3/x M T = , / i • The s o l u t i o n s of ( 2 . 9 ) s a t i s f i e d by Ho are commonly l a b e l e d by the quantum numbers H , X , /* where ft i s a r a d i a l quantum number, X = U l + l ) + 0 ~ 'A-( i s the o r b i t a l quantum number of the upper two components of Y, ), and /A i s the' eigenvalue of j a . These quantum numbers are more f u l l y d e s c r i b e d i n R o s e 1 5 . Thus we get The s o l u t i o n s YC^*/*! are orthogonal and w i l l g i v e a set of orthogonal s t a t e s of the CM system, . The n o r m a l i z a t i o n constant i s then given by ( 2 . 1 4 ) . YoLnKyui (I Xlt\KMl = 6MA«. V/ ( Nunx/of ( 2 . 1 8 ) These s i m p l i f i c a t i o n s occur i n the c e n t e r of mass system only . If we have PT T" 0 , that i s , we are not i n the CM frame,then the angular momentum e i g e n s t a t e s , g i v e wave _» f u n c t i o n s f o r the two p a r t i c l e system, , which are not e i g e n s t a t e s of the t o t a l angular momentum, TT . T h i s i s because the terms i n co Pr and r-Pr i n the Y ~* Y t r a n s f o r m a t i o n do not commute with j or T T , and a l s o JV now c o n t a i n s the angular momentum of the motion of the center of mass, L . i n s p i t e of t h i s i t i s s t i l l u s u a l l y convenient to l a b e l the two p a r t i c l e s t a t e s as Y[PT HK/A] , because of the way which we so l v e ( 2 . 9 ) f o r Y 0 . Since n , X , and yw are not "good" quantum numbers f o r 26 the system we have no guarantee t h a t the s t a t e s rl?y nKywl are o r t h o g o n a l so t h i s must be checked. The matrix element between such s t a t e s i s 5 $ Y r[ PT n x/0 T [ PT' n' x>'3 dV d*R V» Vg (2.19) = 6S( PT - fr) NN'^ Y.^hxiO [ I + + (6c-PT + £rrO] YJnYjuj d3r v* I f we do the angular i n t e g r a t i o n f i r s t , we f i n d t h a t the term i n P does not couple d i f f e r e n t angular momentum s t a t e s because p commutes with both j and J £ . However, the operator oc PT i s a v e c t o r operator (with r e s p e c t to r o t a t i o n s of the CM only) and so i t can couple s t a t e s with j ' = j , j + I , i. = 1"! , t + l , and fk - KK - I, JA , }\+ I . C o upling between s t a t e s with 1 = I i s f o r b i d d e n by p a r i t y . The r e s t r i c t i o n s on ,j and I correspond to X' = - K ^ K - l j V + i . Thus the operator fV i s capable of d e s t r o y i n g the o r t h o g o n a l i t y of the s t a t e s YlPy . The d i f f e r e n t i a l equation s a t i s f i e d by Y C P T H K / A ] i s given by (2.6), (2.8), and (2.15) HY-Lfc-p +0m + 2 ] 3 j p 2 T V + ^ H ( 2 V - U ) o : . p + z L ( 3 U - 2 V ) 5 c . f r p t ^ P r - R P P r - 2 J ^ ( 2 V - U ) ^ P r - 2 H ( 3 u " 2 v ) i Y = E Y Here Pr i s a constant v e c t o r . For Pr 10 t h i s equation i s not s p h e r i c a l l y symmetric and so we expect that the energy 27 ft e i g e n s t a t e s w i l l not be angular momentum e i g e n s t a t e s . T h i s i s e x a c t l y what we get when we use (2.16) to get Y from Y 0 However, we would s t i l l expect t h a t energy e i g e n s t a t e s of Y having d i s t i n c t e i g envalues would be o r t h o g o n a l s i n c e H i s H e r m i t i a n . The energy e i g e n v a l u e s c o r r e s p o n d i n g to HoLnV/*! would be the same f o r s t a t e s with the same M and but d i f f e r e n t yu , so we w r i t e them as E 0 Cn* ]. These ei g e n v a l u e s w i l l i n g e n e r a l be d i s t i n c t . I f t h i s i s the case then the eigenvalues c o r r e s p o n d i n g to YCPy ttK/A^. , namely (us i n g (2.10)) ELn*] = EoLnKl E«CnK] + m1) w i l l a l s o be d i s t i n c t and we get = 6„n« St*. \ a function o£ yu '^ PT PT' ] and v. ve Hn' I d funct ion oj- /A/A' J i f we use (2.15) and (2.19). (For the Coulomb p o t e n t i a l we have the degeneracy E«tn*] = E 0 Cn-K] and so to prove the above equations we have to use the e x p l i c i t s o l u t i o n s f o r . % .) The operator <X- P r cannot couple s t a t e s with X = K' and so i t can now be droppped from (2.19). T h i s e l i m i n a t e s any f u r t h e r p o s s i b l e c o u p l i n g between s t a t e s with d i f f e r e n t yU and we get f i n a l l y 28 s b[?T~fr) Kn' Our n o r m a l i z a t i o n c o n d i t i o n (2.14) again reduces to 2.4 S c a t t e r i n g S t a t e s The s c a t t e r i n g s o l u t i o n s of (2.9) are l a b e l l e d by t h e i r asymptotic momentum p 0 and h e l i c i t y A0 . That i s , f o r l a r g e r Y,t.f.A.] ^ e , p , ' r m & x o The n o r m a l i z a t i o n and o r t h o g o n a l i t y c o n d i t i o n s depend on these asymptotic forms and are not a f f e c t e d by the wave f u n c t i o n d i s t o r t i o n near the nucleus. We have chosen Y0 so that i t i s u n i t d e n s i t y normalized 29 1 2 E . Eo+m. ucp.x^uup.x'O * ' l Eo • Po + Wo1 X\ op 0 i s a two component s p i n o r s a t i s f y i n g To normalize the e l e c t r o n - n u c l e u s wave f u n c t i o n and to check f o r o r t h o n o r m a l i t y of the d i f f e r e n t s t a t e s we c a l c u l a t e Y L P r P o M T L P ^ A ' j t f r d ^ = V 8 S'( PT - Pi) J Y^C Po A,] r I C oc- PT f fm ) ] Yo L Po' Ai] d 3r = 8*( J - PT') 6*( Po " p, ) N N' lApo XO L I + ^ ( « ' ? r * p,1 Ai ) We have i n c l u d e d the e x t r a f a c t o r N/V& as a n o r m a l i z a t i o n c o n s t a n t . 30 _ EQT- ftu r / , X 0& Po \ -v* -vy s i n c e I f we d e f i n e M _ i - _ L ( J £ + E d i i N = 1 2M E E ' we get where ¥LP Tp 0Ao3 = N j I +JMLK?T + 2i (E-m-JrU)r .P T C ^ - * ( 2 , 2 1 > Once again i t i s convenient to w r i t e 31 P i P r ' * Y C P T P O A O 3 = Y £ P T P O M - ^ -where Y[P T PoA.] =| | T^LocPr +2i(E-m-|;U) rfV * x D P -> ( 2 ' 2 2 ) The wave f u n c t i o n s TlR-p^Ao! t h e r e f o r e form a u s e f u l set of c o r r e c t l y normalized, orthogonal wavefunctions. Now we need to f i n d the asymptotic momenta of the e l e c t r o n and the nucleus which correspond to Pr and p 0 . The asymptotic form of Y i s Y C P T P O X . 3 = { lT^C^FT+2i ( E " m ) r P T T ^ m -|- l -^3^ , i P * P T J1 2 R ' - T P E " ~ E ~ i N o»P.-r*irV-it*i(E-i«lr-'T/l1 1 U(p oXo)C y" Here we have used which we have shown to be a p p r o p r i a t e f o r l a r g e r i n s e c t i o n 2.2. I f we now make the t r a n s f o r m a t i o n (2.4) on the exponent we get 32 P«-r + PT-Rt^(E-m:>r-PT^ ^ So ^ 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 of p e and p r t with e i g e n v a l u e s Pea= Po-r^Pr PH« = ( | - | [ )PT- PO I n v e r t i n g t h i s g i v e s / 1 E \ •» E -* P. = U - - j 5 [ ) P t a --pf Pn* Pr = Pe* t pH. Note the d i f f e r e n c e between these formulas and (2.5). I t i s easy to c o n f i r m that t T t o t 2 M (2.23) — Eea + E H a u s i n g (2.10) and (2.8). 33 3 BOUND MUON DECAY 3.1 The Wave F u n c t i o n s We are now i n a p o s i t i o n to c a l c u l a t e the decay of bound muons i n c l u d i n g r e c o i l e f f e c t s . The r e a c t i o n i s In the i n t i a l s t a t e the muon i s bound i n the Is ground s t a t e of the muon-nucleus system which i s at r e s t i n the l a b . We d e s c r i b e t h i s system by the wave f u n c t i o n ^Cr^r^,) which has fr/*!*® , J = T T = , / 2 , M T = V = , 4 ' . Thus us i n g (2.17). N i s given by (2.18) and YM0 i s the Is s o l u t i o n of The t o t a l energy of the muon-nucleus system ( e x c l u d i n g the r e s t 34 mass of the nucleus) i s T h i s can a l s o be expressed as where i s the muon b i n d i n g energy. These equations come from (2.9) and (2.10), and Vo i s given by (2.11). The t o t a l energy, ET/A , i s the same as the i n t e r n a l energy, , because the CM i s not moving. The e l e c t r o n and the nucleus form a s c a t t e r i n g s t a t e so we can d e s c r i b e them by the wave f u n c t i o n ^eCrervi) . The cente r of mass of t h i s system i s not at r e s t i n the l a b frame s i n c e the n e u t r i n o s can c a r r y away some momentum. Thus, u s i n g (2.21) and (2.22), Y. = \ I +^[ocPref 2i(E t-me-yU)r-PTe = i t 4 where P-P. - fc IU--^r, « -» -> -> / • Ea v Et -» Prft= P*. Po =(l+"pf)Pe.-^ P M a Pea and p^ are the asymptotic momenta of the e l e c t r o n and nucleus i n the l a b frame. E 4 i s the i n t e r n a l energy of the e l e c t r o n - n u c l e u s system, though i n the above equations i t c o u l d 35 be r e p l a c e d by ETC or E«,t s i n c e they are a l l the same to z e r o t h order i n M~' . Y e i s a u n i t d e n s i t y normalized s c a t t e r i n g s o l u t i o n of (ap +|Sm.TVo) Yo. = E0. Y.c with asymptotic momentum p 0 and h e l i c i t y A 0 . The t o t a l energy of the e l e c t r o n - n u c l e u s system i s T h i s can a l s o be expressed as 2M p* f t O.D Ere H P i + W.1 t o W PK1" = E. a + E To c a l c u l a t e the decay r a t e we w i l l only need the wave f u n c t i o n s j _> , . . . evalu a t e d a t r«, = H* a n d so there w i l l be no ambiguity i n r> -» - J _> -> - i r = r e - r w and r = t > - r M . The n e u t r i n o wave f u n c t i o n s w i l l be box normalized plane waves where U + ( P T ^ > ) U I ^ M = 1 v f ( p ^ A > ) v ( p 9 X y ) = I T h e i r e n e r g i e s are E v = p v a n c * Ej/ = P-y • 36 3.2 The Decay Rate A l l the wave f u n c t i o n s i n the problem are energy e i g e n s t a t e s so we use the f o l l o w i n g formula f o r the t r a n s i t i o n r a t e , U)y> , from i n i t i a l s t a t e l i > to f i n a l s t a t e l£> The H|MT we use i s the standard V-A p o i n t weak i n t e r a c t i o n vertex f o r yw-*ev"v Since t h i s i s a p o i n t i n t e r a c t i o n we must e v a l u a t e H ) M T f o r -» -* -» -» -> -» l~e = f> = r v = rv — X" +• Y~u . T h i s g i v e s <i-1HIHTI »> - j 3 c • tu ,T( t ( l -»0%][ ; tT t l l -WV,]d l fd , r 1 i The i n t e g r a l should be over dreQ r» but we have used a r o Tu which i s e q u i v a l e n t . The exponent i s .#•-* -* ~s N -? . .- i -» frit, o o . *-i(p> + Pv + PU+P».Virt - i(P*+-pv+-j^Pe*+P*»)rr s i n c e K e - ' • * + • ]5j « The i n t e g r a l over rw i s now t r i v i a l and g i v e s momentum c o n s e r v a t i o n 37 I t i s u s e f u l to w r i t e H = p , + P f H o - E - y + E * (3.2) - " ft* " P»»« = ET/A ~ E Te Htt i s a f o u r - v e c t o r s i n c e i t equals the sum of the f o u r -momenta of the n e u t r i n o s . T h i s g i v e s We now use the F i e r t z t r a n s f o r m a t i o n 1 8 [ A Y r ( l - * 5 ) B 3 L C M I - Y s ) D ] = [X* r ( l - t f s )D3i :c i n which A , B , C , D are a r b i t r a r y D i r a c s p i n o r s . T h i s a l l o w s us to combine the n e u t r i n o s p i n o r s i n one f a c t o r . 38 5 e , ( , " i i ) H ' R Y.WI-where = i( H-pv-%) IE, »*( I - W.3 Nc We now square the matrix element • 15„ 6r( I- *sl V* 1L \/„ I - V «vl The only p a r t depending on n e u t r i n o s p i n i s i n the square b r a c k e t s , so to sum over the n e u t r i n o s p i n s we need E t M e( I - V vf ][ v* I'( l-^)u T] 2K r < r T h e r e f o r e i<si H l l l r i i>il=-^Jp iiH-p»-Piii^; N * The t r a n s i t i o n r a t e i s then 39 So f a r we have the t r a n s i t i o n r a t e t o a d i s c r e t e f i n a l s t a t e . For decay i n t o a range of f i n a l s t a t e s we must m u l t i p l y by the d e n s i t y 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 r a t e . ^(H-p^-p^^^^NrN^dp^d'p^cfpyd'py We can now i n t e g r a t e over the n e u t r i n o momenta us i n g ^ = S S ^ H - P > - ^ ^ V P : - % % i s the ste p f u n c t i o n . T h i s g i v e s The l a s t term i s zero from symmetry and we get d ' r ^ ^ g ^ N ^ N j L ^ H ' - H - H ^ i e C H - H O d ^ d ' K ( 3 . 4 ) We have made the change pMft-» H i n the i n t e g r a t i o n v a r i a b l e s . At t h i s p o i n t we haven't summed/averaged over electron/muon s p i n s . Now . we want to reexpress the t r a n s f o r m a t i o n from ^eo to T e , (2.22), i n terms of H and pe» i n s t e a d of p N » and p e« usi n g (3.2). 40 Y € = jl+2]5jE-dL-H-2i(E.- mt-j\X)rH The asymptotic momentum of V o e i s p 0 given by (2.23) T h i s i s as f a r as i s p o s s i b l e t o go i n the c a l c u l a t i o n without a c t u a l l y s p e c i f y i n g the wave f u n c t i o n s Yoe and Y>/» . 3.3 The Phase Space The phase space l i m i t s f o r the i n t e g r a t i o n over H and paa are set by p* > 0 H 0 > H > 0 The l a s t i n e q u a l i t y i s the e f f e c t of the s t e p f u n c t i o n We t h e r e f o r e c o n s i d e r the boundary given by Ho . I t i s simpler to use a r e l a t i v i s t i c form f o r E N a i n e x p r e s s i n g Ho EM«I = YFF+pJv - M T h i s change would g i v e the exact phase space to a l l orders i n M 1 i f we knew ETy* to the same degree of accuracy. Using (3.1) and (3.2) we get 41 H * H 0 ET/A ~ Efta " EHa = ET>*- E ^ - j M l T p i + 2 H p t o X T H l +M where A A X - H • Pe* Thus .. 2M(ErM-Eea)T(E T »-E t e ) 1 -P i Er^-Ee.+ Xpea') ( 3 > 5) = ET/> - E e . - ^ [ ( E t * ~ E ^ + p J * ] ~ M ( E t " ~ E J x + 0 ( M " t ) The maximum value of E f t a occurs when H- 0 c 2 M E T » + E t * + me t M Ax- 2(rt+ E T A K) = E v - 2 k ( E v - » n + 0 (1*0 0.6) and i s independent of X- . The i n t e g r a t i o n r e gion i s graphed i n F i g . 1. 42 F i g . 1. The i n t e g r a t i o n r e g i o n given by (3.5). Without r e c o i l the upper s u r f a c e would be f l a t . The d o t t e d l i n e on the 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 the 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 the peak leaves the i n t e g r a t i o n r e g i o n , g i v e s 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 APPROXIMATION 4.1 The Approximate Wave F u n c t i o n s The i n t e g r a l s over H i n (3.4) and over r i n (3.3) w i l l , i n g e n e r a l , have to be done n u m e r i c a l l y . A u s e f u l approximation f o r which these i n t e g r a l s can be done a n a l y t i c a l l y i s the plane wave Born approximation ( PWBA ). The approximations made are 1) the muon wave f u n c t i o n i s giv e n by the n o n - r e l a t i v i s t i c s o l u t i o n f o r a Is s t a t e i n the f i e l d of a p o i n t charge, m u l t i p l i e d by a D i r a c s p i n o r with zero lower components; 2) the e l e c t r o n wave f u n c t i o n i s u n d i s t o r t e d by the nu c l e a r charge. We w i l l show that these approximations are q u i t e good f o r n u c l e i with small 2 , which are a l s o the ones f o r which r e c o i l e f f e c t s are most important. F i r s t we c o n s i d e r the e f f e c t of f i n i t e n u c l e a r s i z e on the muon wave f u n c t i o n . Since the nucleus i s s m a l l e r than the Bohr r a d i u s f o r smal l 2 ,we would expect that the n u c l e a r s i z e c o r r e c t i o n would be of order R M / T B , where Rw i s the nucl e a r r a d i u s and fn, i s the Bohr r a d i u s of the muon o r b i t . For the Is s t a t e the Bohr r a d i u s i s (2ocm^ so the nu c l e a r s i z e c o r r e c t i o n would be of order 2(Xm^Ru . T h i s i s around .004- f o r hydrogen and .1 f o r oxygen. So f o r s u f f i c i e n t l y s m a l l 2 we can use a Coulomb p o t e n t i a l . To f i n d the muon wave f u n c t i o n we must s o l v e (2.9) usi n g the p o t e n t i a l VQ given by (2.11). For the Coulomb p o t e n t i a l we 44 get V0=V= ~2«x/r . So (2.9) becomes the equation of a s i n g l e p a r t i c l e i n a Coulomb f i e l d . T h i s equation has been s o l v e d and the Is s o l u t i o n i s given i n R o s e 1 5 . We w i l l not be usi n g the exact s o l u t i o n but i n s t e a d 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 Schroedinger equation with a Coulomb p o t e n t i a l , times a spinor with zero lower components. The lower components of the exact s o l u t i o n which we drop are of order . For hydrogen, ice .007 and f o r oxygen 2ot~.05 . F i n a l l y , we c o n s i d e r the e f f e c t of the Coulomb f i e l d on the wave f u n c t i o n of the e l e c t r o n . The exact s c a t t e r i n g s o l u t i o n s f o r the Coulomb p o t e n t i a l are a l s o i n R o s e 1 5 . From these s o l u t i o n s one f i n d s that the e l e c t r o n d e n s i t y a t the nucleus i s in c r e a s e d by a f a c t o r of order £a£ev/'p^ i f E e a » m e . So i f we use plane wave s o l u t i o n s i n s t e a d of the exact s o l u t i o n s , we are dropping terms of that o r d e r . Since the e l e c t r o n i s h i g h l y r e l a t i v i s t i c through most of the muon decay spectrum, t h i s approximation i s of order 2ot , the same as dropping the lower components of the muon wave f u n c t i o n . An exact c a l c u l a t i o n of the decay spectrum, i n c l u d i n g the three e f f e c t s d e s c r i b e d but no r e c o i l , shows t h a t PWBA i s a c t u a l l y more ac c u r a t e than we c o u l d expect based on the above a n a l y s i s . For example, Hanggi et a l 5 " ' f i n d that the sum of the three e f f e c t s i n c r e a s e s the decay r a t e by about four per cent i n s i l i c o n . So we have reason to hope that PWBA w i l l g i v e u s e f u l r e s u l t s up to around 2 ~ 15 . Note that the approximations we make i n usi n g PWBA do not i n v o l v e dropping any r e c o i l c o r r e c t i o n s . The muon wave f u n c t i o n i s then 45 where € = to^Ztx. and \X/* i s a s p i n o r with zero lower components s a t i s f y i n g U^U^»| , and E 0 / * = n y ( i - J F ( 2 c c ? ) Thus we get E T /* = [ I - jr( 2 * ? + j ^ ( 2a?] We have dropped terms of order (ICKY1" from the above. % i s given by (2.16) + J R t V t p r r y > + r u £ ] J Y . A where U=V = ~2cc / r using (2.11). The n o r m a l i z a t i o n constant i s given by (2.18) Tjr-SlCtl+SfpjY,.^  - H-K" So v -( I . " V C * « ? x { . 2oc ).£ p-€r T h i s i s now c o r r e c t l y normalized. Since the e l e c t r o n does not i n t e r a c t with the nucleus we know that the complete e l e c t r o n - n u c l e u s system wave f u n c t i o n i s 46 T h i s i s box normalized a c c o r d i n g to (2.20). Of course, t h i s r e s u l t c o u l d have been ob t a i n e d u s i n g the methods of s e c t i o n s 2.4 and 3.1 with zero p o t e n t i a l , but s i n c e we know the complete wave f u n c t i o n i n t h i s simple case, i t i s e a s i e r to s t a r t with i t r a t h e r than X e t o d e r i v e Y» . To get we need t o express the complete wave f u n c t i o n i n terms of r and R-e u s i n g (2.4).' Thus, u s i n g (2.22) • - t t.e''-**** We have dropped the s u b s c r i p t a , f o r asymptotic, from p e a and p ^ s i n c e i t i s not needed f o r plane wave s t a t e s . A l s o we are now l a b e l i n g the s c a t t e r i n g s t a t e s by. pe and A e i n s t e a d of p<, and X 0 but s i n c e there i s no i n t e r a c t i o n the n o r m a l i z a t i o n and o r t h o g o n a l i t y are the same. 47 4.2 The Decay Rate From (3.3) we get Note that p a e n t e r s the i n t e g r a l i n the same way as the momentum of the other n o n - i n t e r a c t i n g p a r t i c l e s , namely the , - * x n e u t r i n o s ( H = p > + p ^ ) . The i n t e g r a l i s of the form .-•rP,. - e r , 3 - ; _ Sire 2TT(2C*> Ce* , r ( i - -^ .^e ' 6 r a 3 r - - S + P 2 ) 2 M ( e x + P 2 ) Thus H U M 1 , €V* N r = C I + ^ ^ - ) ^ L u e l ( r ( l - ) f 5 ) u r 3 5 8 7 r £ 2ir(Zofl ? * H 6 l + P e l + H x + 2 p t r T r M ( e x T P e l T H 2 + 2 ^ - K y i We now e v a l u a t e the spinor p a r t , summing over e l e c t r o n s p i n but not muon s p i n s i n c e we w i l l want to c a l c u l a t e the asymmetry of the decay. 48 Ae Where P>»=(wyo) and 5^ i s the muon s p i n d i r e c t i o n . A f t e r some al g e b r a t h i s reduces to where Q = (I, - 3^) Hence M M * f l , ! V i i ^ \ 3 2 T r e 3 [p«QvrTpeo.QC - p«Q9oc- i€„Cf<rpeQPJ 2 e l e ( ^ o c ^ ? l + P t l + H l + £ p f F ! ) * S / ( e l + P e l + H 1 -r2p e R ) 4 " M ( e To c a l c u l a t e Nr N*(HrHff- H-H 9™ ) we need LPerQ^+Peo-Qc-pe 9^<rr - • e« c ,<r H 'H*" H H g " ) - HHpt-Q. + 2pe-H Q-H The decay r a t e i s then given by ( 3 . 4 ) 49 For s m a l l 2 the f u n c t i o n i n b r a c k e t s i s dominated by a sharp peak along the l i n e lp e+H| = 0 , or Pe»U and X = - l . Along t h a t 8 S 8 l i n e the denominator of the dominant term i s £ - tvy(2«5 so the i n t e g r a n d i s of order (£<x") l a r g e r on the peak than elsewhere i n the i n t e g r a t i o n r e g i o n . Around E e ^ i ^ the peak i s no longer w i t h i n the i n t e g r a t i o n r e g i o n and so the spectrum drops a b r u p t l y . See F i g . 1. A c a l c u l a t i o n u s i n g the r e s u l t s of of s e c t i o n 3.3 shows that the l i n e lp*+H|=»0 i n t e r s e c t s the i n t e g r a t i o n region boundary when p f t+E e=ET ><A. So the r e c o i l c o r r e c t i o n t o the p o s i t i o n of the s t e p i n the spectrum i s due to the r e c o i l c o r r e c t i o n of ET/» o n l y , and hence i s of order hn^/M . T h i s c o n t r a s t s with the r e c o i l c o r r e c t i o n to the end of the spectrum which i s of order flft*/M_. u s i n g (3.6). To proceed f u r t h e r we have to express H 0 i n terms of H and p e , the i n t e g r a t i o n v a r i a b l e s . Ho E-iy* " E F C - E M = ET,* " Ec - jrjyj p j = E ^ - E t - ^ ( p j T H z i - 2 p e H ) = H-^(HH2PeH) where H = ET/»"Ee~ft/2M. We have used here the n o n - r e l a t i v i s t i c e x p r e s s i o n f o r EM i n s t e a d of the r e l a t i v i s t i c e x p r e s s i o n used i n s e c t i o n 3.3. These two e x p r e s s i o n s are, of course, the same 50 to order M . The maximum e l e c t r o n energy p o s s i b l e , that i s , the end of the spectrum, i s g i v e n by H =0_ . Using the r e s u l t f o r Ho we get HHpe-Q t Zpe- HQH = A + B* H - C H L - D'-H jvH - E ' -H H 1 where A= r f ( 3 E t + P c X * ) B' = 2 H( Eet - Pe^  - 2H Pe(3E.+ p t v V M C = E * + P e t + H(3E e + PeVVM D ' = 2 t t 2 ( E e V - ^ V M E = ( E e t - P * V M T h e r e f o r e we get cf r = 6 V p N p , d X ( | + ) 9 ( H o . H ) n ' d H d | c l l M ^ •[At i'-H -CH 1 - D-HptH-E-HH 1] .5 J ! Ua+bHx + H1)* m /.M(o+bHx +H*l 5 where a = e a + p ^ , fc> = 2p« , and X = p e H . To do the i n t e g r a l over d<t>H we choose pt = 2. and H = HCsmBHt-oS^H4>H i 0 * ^ , B = &(sm8RtoS(i>BlSln&6Sl^<l>8,CosB0,), e t c . Terms which i n v o l v e a s i n g l e power of Sm^M or COST'H w i l l g i v e zero when i n t e g r a t e d whereas terms without such a f a c t o r w i l l be m u l t i p l i e d by 2lT Hence, f o r example, 51 ^ B-tid4>H= 2TrKB'cos9,co3eH In t h i s way we get 2 +> • f c [ A + B H x - C H * - D H l x * - E H s x ] i Ca+bHx+H 1) 4 " m^MCa+bHx+H1)51 (4.1) where D=2(pe-V+-j5j-(E ePet-pt a)) E =^(Ecfv V ~ pft") T h i s r e s u l t can be checked by t a k i n g the l i m i t £-» 0 which should give the f r e e decay r e s u l t . In t h i s l i m i t the peak i n the int e g r a n d a c t s l i k e a D i r a c d e l t a f u n c t i o n and so we get, f o r any f u n c t i o n Oj(H,>0 e -» A l s o O H 52 Using =CmJ+mt)3E»-4m /.E t 1-2m /»m l -(4Eerrv-3rnJ-rrv) V R we get d 3 r = ^ ^ ^ 0 ( m ^ E f c - P e ) •[SCrr^TmDEe-l-m^El-^m^mJ T h i s i s the same as the f r e e decay r e s u l t 1 ' . Note t h a t no terms of order IA ' appear and that pe=H x * - l i m p l i e s that p N 3 0 , that i s , the nucleus doesn't r e c o i l . These r e s u l t s are e x a c t l y what we expect. 4.3 The Phase Space I n t e g r a t i o n With p e f i x e d the i n t e g r a t i o n region i s the area a bed given i n F i g . 2. 53 e-H=H M F i g . 2. The i n t e g r a t i o n r e gion f o r f i x e d p e H i s given by (3.5) with X=-l . H = ETYM-Et~2M"^ EXM - Ec" pt)1 To do the i n t e g r a l a n a l y t i c a l l y i t i s best to break the i n t e g r a l up i n t o an i n t e g r a l over the r e c t a n g u l a r region a b e t and an i n t e g r a l over the t r i a n g l e at<X . $I(H,x}dHdx = $I(H,x)dHdx - ^ K M d H d x abed abet H +i ade +i H = $ $I(H,x)dHdx-$ $ItH,x)dHdx © - I Because of the way we have s p l i t the i n t e g r a l up, the second i n t e g r a l w i l l be of order M ' i n s p i t e of the sharp peak i n the i n t e g r a n d . T h i s a l l o w s us to make the approximation I(Hy) = I(H;0 i n t h i s r e g i o n . Thus 54 I^(H,X1 dHdx = $$HH,x1 d x d H - ( E ™ " E t m Itx) I { R'X > > d* and we get * *' 2 I •(A+BHx-CH2-DH1xl-EH5x)HzdxdH 2(E»-Ec)ft C,l + y \ AtBHx-CFf-DHV J Y -t These i n t e g r a l s are now simple but t e d i o u s t o e v a l u a t e . We sketch here only the c a l c u l a t i o n of the most important i n t e g r a l . The other i n t e g r a l s are ev a l u a t e d i n a s i m i l a r way. J J TaTbHxTW H d x d H O - l = J L ( a + b H x + H M ^ ( A " G H 1 ) ^ + ( B H " E H ^ ( ^ + l b ^ ) <4-3> o D | [ t / x 2 (a + H^x . ( a t H Y O l , , ^ , , I f SHi-THVUK 3+3Vbrr-TVH 5 , u -H where R » a + bH + H 1 S=2baA+abB-2cfD T=3baB-6abD U = bB-2btC-abE-^aD-6b2D V=-b£-2D T h i s i n t e g r a l can now be s i m p l i f i e d by us i n g 55 The i n t e g r a l i n (4.3) becomes vj|dH+[-^-J-(u-av+ab^f -fl "H | r B [ S - rQ ( U -QVT2b 1 V ) 3 H T L T T 6 a b V ] H 1 d | i -fl = [ v R ~ ^ - ( a - a V + 2 b l V ) ^ z -[S+a(U-aV+Zb lV)]Rz+LT+6abV]RJ The v a r i a b l e s R , , Rz and R j are d e f i n e d i n (4.5). With the other i n t e g r a l s i n (4.2) done i n the same way we get the f i n a l r e s u l t 56 dT _ 32 e 5P e /. , nutZgf \ -fS+a(U-aV+2b 1 V))Ri + (T+6abV)R J ^ I C3DH1" bDH* y , f T » 3 a b D i P (4-4) ~2BHw£ C2R~ "~~2R~ ~ 5 R + + { , + z  }K* +(a-6aD + bxD)R, +2D(^RIogR-p t L(H+p t)logR-H + 2e1t])^ ^ ^ ^ L ^ [ ( A _CH l )R 6 + BH R7 + DH*R,] + ^ b 9 R + f 3 [ ( A - C H l ) ^ + B H ^ - P H % ] ^ ^ H We have expressed the r e s u l t i n terms of d E e i n s t e a d of dpe , and we d i v i d e d by the f r e e decay r a t e ( 6 ZW^/l92Tr 5 ) and m u l t i p l i e d by the muon mass t o g i v e a di m e n s i o n l e s s q u a n t i t y . C o l l e c t e d below are the d e f i n i t i o n s of a l l the v a r i a b l e s needed to e v a l u a t e (4.4). m r = fry (I - j ^ + l ^ a a 1 ) e = m0/»2rx A =H 1 (3E e +rt '^^ B = 2 H ( E e H - f t - K ^ + - P e ^ E ^ ( E c M ^ - f t V H =ETy.-Ee-zK(E^-Ee-pe)1 57 R= a+bHtH* S=2blA+abB-2a*D T=3baB-6abD U = bB-2tfC-abE-4-aD-6b*D V = -bE-2D 5 =b'>\+abB+3alD T=2baB+4abD D =-blC+bB+6aD A=4e l t ^ i t a n " 1 ^ D I I o . qiga-ba) + b(3a-tf)H b(6a-b*) ^ R, = x lo 9 R + 2TR 2A t D Za + bH | 3b(b-r2H) . 3b K l " 2AR2 * 2 AaR A*^ D _ ab-r(b*-2a)H . (2a + b»)(b+2H) , 2a -rb*. * 3 ~ 2ARZ "** 2 A*R A* x R+= A R +-2 " t (4.5) o . gb-f(b*-2aOH , 2QO. R 5 " AR + I t n _ I , a + H2 2bH * 6blHl R ' ~ bfl * baH2 + 3baH* D 3(a-rH*) , q(a+Ha)* . IKq + H*  K 8 = tfH* 2b*H3 "*! 6b+H+ The d i f f e r e n t i a l decay r a t e (4.4) i s a f u n c t i o n of the momentum, A J Ffe , and pe- S*. i n the form so that i f we i n t e g r a t e over e l e c t r o n d i r e c t i o n , J \ e , the term i n Y Z w i l l g i v e zero. That i s 58 v. 4r = ^ Y , ( P O = ^ 3 ^ ( ^ , 0 ) = A - n - v / r>.A — A i r — <*E. The asymmetry parameter, 0CA i 1 S d e f i n e d as dUc T J l e , d t * d i U (4.6) = Y*(PO These are the f u n c t i o n s we w i l l use to present the r e s u l t s of the c a l c u l a t i o n s . 4.4 The End Of The Spectrum I t i s u s e f u l to have an approximate e x p r e s s i o n which g i v e s the form of the spectrum at the hig h energy end. I t w i l l be simpler to d e r i v e t h i s by redoing the i n t e g r a l i n (4.1) ra t h e r than t r y i n g t o make some expansion of (4.4). We use the f o l l o w i n g d e f i n i t i o n s <r = ny / M p i s t h e r e f o r e a measure of the d i s t a n c e from the end of the spectrum. We w i l l keep only the f i r s t s i g n i f i c a n t order i n p 59 (which w i l l be p 5 ), and the f i r s t order r e c o i l and b i n d i n g c o r r e c t i o n s to t h i s . In t h i s way we get rry = ny( I - <x) &r/» = M/» ( i ~ i z } u s i n g (4.5). We a l s o n e g l e c t the e l e c t r o n mass. Thus The i n t e g r a t i o n l i m i t on H i n (4.1) i s of order l*yp and so to make the p dependence more e x p l i c i t we d e f i n e H=wyprv , where now W i s of order one. T h i s g i v e s a + bHx + H 2 - e x + p e l + H^ZpcHx Z I _ 2 n , 5 ( a l + b H x + H ir ryMa+bHx+H*) 1 " n j ? u H = rryp ( 1 + c O C * m ^ ( l - ^ c r - | r ) ( l + piv) The term i n E i n (4.1) w i l l not c o n t r i b u t e to the order we are c a l c u l a t i n g . The i n t e g r a t i o n l i m i t s are 60 -I < x < 1 0 < h < l + a ( l - x } so we get, f o r example, ^ r ? x d h d X = - f cr Proceeding i n t h i s way we get the f i n a l r e s u l t ^ . g [ ( . | ? , . « » t c } . * i « W ] ( ^ u.7) T h i s has been made dim e n s i o n l e s s i n the same way as (4.4). As noted before, H and Mo are of order Wi^p at the end of the spectrum. T h i s i s because the phase space allowed to the n e u t r i n o s goes to zero f o r maximum e l e c t r o n energy. In (3.4) there are f i v e f a c t o r s of H or H 0 , three i n d H and two i n LHTH°"- HH g r f f ] . T h i s e x p l a i n s the dependence of the end of the spectrum, and shows that i t i s not a r e s u l t of the p a r t i c u l a r approximations we are making f o r the wave f u n c t i o n s , but i s a f e a t u r e of the exact c a l c u l a t i o n , with or without r e c o i l , f o r a l l 2" . The asymmetry parameter d e r i v e d from (4.7) us i n g (4.6) i s (4.8) 61 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 In t h i s s e c t i o n we present the r e s u l t s of the c a l c u l a t i o n of the e l e c t r o n spectrum and asymmetry of bound muon decay. For Ee l e s s than we have used (4.4) to c a l c u l a t e the decay r a t e . For E e g r e a t e r than t h i s i t i s d i f f i c u l t t o use f o r numerical reasons. The l a r g e r terms i n (4.4), such as the one i n € 5 R t , are of order one, but we know from s e c t i o n 4.4 that at the end of the spectrum, (4.4) should be of order (oZaCf . For t h i s to be p o s s i b l e there must be a c a n c e l a t i o n of a l l terms of order I , £ot , p , c^ioc , ... .(2:*} £ 5 e t c . I f we use a computer to e v a l u a t e (4.4), we would have t o c a l c u l a t e the term i n £ 5 Ri a c c u r a t e l y enough so that i t s (picc^ 5 dependence i s not l o s t i n round o f f e r r o r s . For i n s t a n c e , f o r a hydrogen nucleus and Ee = *V we would r e q u i r e a 20 decimal p l a c e machine f o r three d i g i t accuracy i n the result-. For t h i s reason, the h i g h energy end of the spectrum was c a l c u l a t e d by doing the i n t e g r a t i o n i n (4.1) n u m e r i c a l l y . T h i s i s r e l a t i v e l y easy because the sharp peak d i s c u s s e d i n s e c t i o n 4.2 i s not i n the i n t e g r a t i o n region f o r these e n e r g i e s . We remind the reader t h a t the r e c o i l e f f e c t s i n (4.4) d i s c u s s e d here and p l o t t e d i n the graphs, i n c l u d e the phase space changes due to nuclear motion, the e f f e c t s of us i n g the "reduced" mass, T(\0^ , and a l s o the wave f u n c t i o n m o d i f i c a t i o n s necessary i n the e f f e c t i v e p o t e n t i a l approach. These three p a r t s of the r e c o i l e f f e c t are separated only i n F i g . 8 and i t s a s s o c i a t e d d i s c u s s i o n . A l s o , when we t a l k of the spectrum 62 without r e c o i l we mean the spectrum i n the l i m i t . M -* Oo o n l y , without im p l y i n g t h a t £ -* O which would g i v e the f r e e decay spectrum. F i g . 3 shows the g e n e r a l f e a t u r e s of the bound muon decay spectrum i n c l u d i n g the s t e p i n the spectrum at ?jMju> d i s c u s s e d i n s e c t i o n 4.2. For comparison, the decay spectrum of an unbound muon i s 1 3 T h i s has been made di m e n s i o n l e s s i n the same way as (4.4) and n e g l e c t s the e l e c t r o n mass. A p l o t of t h i s spectrum on F i g . 3 would be almost i n d i s t i n g u i s h a b l e from the bound spectrum f o r £ = 1 . The changes i n the s p e c t r a due to r e c o i l are too sma l l to be seen on t h i s graph. These e f f e c t s can be seen i n F i g . 4, however, which i s a p l o t of the l o g of the decay r a t e versus the e l e c t r o n energy. F i g . 5 i s a p l o t of the r a t i o of the decay r a t e with r e c o i l to the decay r a t e without r e c o i l . T h i s shows c l e a r l y t h a t r e c o i l i s p r o p o r t i o n a l l y much more important i n the h i g h energy end of the spectrum. A simple c a l c u l a t i o n u s i n g order of magnitudes shows why t h i s i s so. R e c o i l changes i n g e n e r a l must go to zero both as M-» 00 and as £-*0 . Thus they must be of order 2£o;ryy/M or s m a l l e r . (An e x c e p t i o n to t h i s i s the r e c o i l change i n E*** which i s of order ry/IA . T h i s i s p o s s i b l e because EM** i s d i s c o n t i n u o u s as 2">0 .) In the low energy h a l f of the spectrum the decay r a t e without r e c o i l i s of order one, so the decay r a t i o i s of order 1+ 0(£<xm/»/fA>) . In the h i g h energy h a l f of the spectrum, however, 63 the decay r a t e goes to zero as £ 0 and so i t must be of order 2oc or s m a l l e r . The decay r a t i o i s then of order 1 + 0(wy/M) . Thus we see t h a t r e c o i l e f f e c t s are more important i n the high energy h a l f of the spectrum. A l l the p l o t s i n F i g . 5 go to zero at h i g h energy because the spectrum with r e c o i l ends a t lower energy than the spectrum without r e c o i l . F i g . 6 i s a p l o t of the asymmetry parameter d e f i n e d i n (4.6). The main r e c o i l e f f e c t , a s i d e from the change i n E M A X , i s the n e g a t i v e s h i f t of the asymmetry a t h i g h energy p r e d i c t e d by (4.8). For comparison, the asymmetry of the unbound muon decay i s 1 3 ~ ' ~ T h i s assumes the e l e c t r o n mass can be n e g l e c t e d . F i g . 7 i s a p l o t of the end of the spectrum based on the d i s c u s s i o n of s e c t i o n 4.4. o i s the d i s t a n c e from the end of the spectrum d e f i n e d i n 4.4. The r e l a t i v e decay r a t e i s the decay r a t e d i v i d e d by SlZCZap^/Sl* 1 . From (4.7), the r e l a t i v e decay r a t e goes to _ I + ^ ~*(r*.) at p= o . From the graph i t 6 appears that the p dependence of the end of the spectrum i s q u i t e l a r g e and i s independent of 2 . Using an estimate from t h i s p l o t we get In F i g . 8 we have d i v i d e d the r e c o i l c o r r e c t i o n to the 2 - I decay spectrum i n t o three p a r t s . These are 1) the c o r r e c t i o n due to the m o d i f i e d wave f u n c t i o n , , used to c a l c u l a t e the matrix element, 2) the "reduced" mass, Tf\0* , given 64 by (2.10) and 3) the r e c o i l change i n the phase space given by (3.5). The m o d i f i e d 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 i n our c a l c u l a t i o n by s e t t i n g = Ho^ ». The "reduced" mass c o r r e c t i o n i s n e g l e c t e d by choosing Tt\0^= fly . The phase space c o r r e c t i o n i s n e g l e c t e d by dropping the M~' terms i n (3.5) and (3.6). In terms of (4.4) t h i s would mean dropping a l l the M* terms i n A ,B , C , 0 i £ i H , H • N e g l e c t i n g a l l three of these c o r r e c t i o n s g i v e s the r e c o i l e s s r e s u l t o b t a i n e d by M~* 0 0 i n (4.4). P l o t t e d i n F ig.8 are the r a t i o s of the spectrum i n c l u d i n g o n l y one of the above r e c o i l c o r r e c t i o n s , t o the spectrum without r e c o i l . The m o d i f i e d wave f u n c t i o n c o r r e c t i o n i s of order (ZcOrvy/M and so i s much smal l e r than the other two c o r r e c t i o n s which are of order fty/M . Of course, the m o d i f i e d wave f u n c t i o n c o r r e c t i o n becomes p r o p o r t i o n a l l y much more important f o r l a r g e r £ . F i g . 8 should be compared with F i g . 5 which g i v e s the decay r a t i o with a l l three r e c o i l e f f e c t s i n c l u d e d . F i g . 3. The e l e c t r o n spectrum from bound muon decay. The decay r a t e has been made di m e n s i o n l e s s and i s given by (4.4). The e l e c t r o n energy i s given i n terms of the muon mass. 66 0.4 0.5 0.6 0.7 0.8 0.9 1 0 E L E C T R O N ENERGY F i g . 4. Log graph of the t a i l of the spectrum. T h i s shows the r e c o i l e f f e c t s i n the high energy h a l f of the spectrum. A 2=6 with r e c o i l p l o t on t h i s graph would only j u s t be d i s t i n g u i s h a b l e from the no r e c o i l p l o t . 6 7 F i g . 5. R e c o i l / n o r e c o i l decay r a t i o . 68 F i g . 6 . The asymmetry. T h i s i s a p l o t of the asymmetry parameter d e f i n e d i n ( 4 . 6 ) . 69 F i g . 7. The end of the spectrum. T h i s graph f o l l o w s from the d i s c u s s i o n i n s e c t i o n 4 . 4 . p i s the d i s t a n c e from the end of the spectrum d e f i n e d i n 4 . 4 . The r e l a t i v e decay r a t e i s the decay rate d i v i d e d by 7 0 co o ' ' o h— CE CU >-CE L U Z = l -WRVE FUNCT. REDUCED MRSS •PHRSE SPRCE \ \ \ o ' O 0 . 0 i i i i 1 1 r ~ 0 . 2 0 . 4 0 . 6 0 8 E L E C T R O N ENERGY \ 1.0 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 . T h i s shows a d i v i s i o n of the t o t a l r e c o i l e f f e c t f o r hydrogen i n t o the three p a r t s d i s c u s s e d i n the t e x t , namely, the m o d i f i e d wave f u n c t i o n c o r r e c t i o n , the "reduced" mass c o r r e c t i o n , and the c o r r e c t i o n due to phase space. P l o t t e d here are the s p e c t r a with each of the c o r r e c t i o n s alone, d i v i d e d by spectrum without r e c o i l . 71 We have seen how to apply the e f f e c t i v e p o t e n t i a l method to c a l c u l a t e the f i r s t order n u c l e a r r e c o i l c o r r e c t i o n s to bound muon decay. The e f f e c t i v e p o t e n t i a l method was d e r i v e d i n s e c t i o n 2 as a general approach to the c a l c u l a t i o n of wave f u n c t i o n s , b i n d i n g e n e r g i e s , e t c . of systems c o n s i s t i n g of an e l e c t r o n or muon and a n o n - r e l a t i v i s t i c s p i n l e s s nucleus with a s p h e r i c a l l y symmetric charge d i s t r i b u t i o n . In s e c t i o n 3 we a p p l i e d t h i s method to bound muon decay l e a v i n g open the form of the e l e c t r o n and muon wave f u n c t i o n s . The plane wave Born approximation made i n s e c t i o n 4 allowed us to c a l c u l a t e the decay r a t e and asymmetry f o r small n u c l e i a n a l y t i c a l l y . The r e s u l t i s given i n equation (4.4). The r e c o i l e f f e c t s are most apparent i n the high energy h a l f of the spectrum. For hydrogen, r e c o i l decreases the decay r a t e by a f a c t o r of two; f o r l a r g e r n u c l e i the decrease i s s m a l l e r . Another r e c o i l e f f e c t i s the change i n the asymmetry at high e n e r g i e s . T h i s i s given approximately by (4.8). The plane wave Born approximation used to c a l c u l a t e these r e s u l t s i s onl y r e l i a b l e f o r small n u c l e i . To extend these c a l c u l a t i o n s 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 to get the e l e c t r o n and muon wave f u n c t i o n s . There are many computer programs a v a i l a b l e to do t h a t , and they would g i v e Veo and Hy|»o . The t r a n s f o r m a t i o n to t^e. a n <3 t/* 9iven by (2.16) and (2.22) i s not easy to do f o r numerical wave f u n c t i o n s because of the presence of the d e r i v a t i v e o p e r a t o r , ^ 3 r , and because the t r a n s f o r m a t i o n mixes the angular e i g e n s t a t e s used t o express . Once generated, however, Y e and a r e used i n (3.3) to giv e Nt i a n <3 then N T i s used in (3.4) to gi v e the d i f f e r e n t i a l 72 decay r a t e . • These steps are s t r a i g h t f o r w a r d . F i n a l l y , the i n t e g r a l over H i s done to g i v e the e l e c t r o n spectrum, i n c l u d i n g r e c o i l . F i n a l l y , we comment on the re l e v a n c e of these r e s u l t s to the search f o r muon to e l e c t r o n c o n v e r s i o n . Current and proposed searches f o r these e x o t i c decays w i l l t r y to measure branching -u r a t i o s of 10 and beyond. E l e c t r o n s from t h i s process can only be d e t e c t e d i n an energy range i n which the background process of the c o n v e n t i o n a l decay c a l c u l a t e d i n t h i s t h e s i s i s extremely s m a l l . T h i s means using the extreme h i g h end of the spectrum near the muon mass. We have shown that t h i s p a r t of the c o n v e n t i o n a l decay spectrum i s s u b j e c t to s u b s t a n t i a l r e c o i l c o r r e c t i o n s , at l e a s t f o r the l i g h t e r n u c l e i . These c o r r e c t i o n s should be taken i n t o account i n d e s i g n i n g experiments and a n a l y s i n g the r e s u l t s . 73 REFERENCES 1) C.E. P o r t e r and H. Primakoff, Phys. Rev. 83 (1951) 849 2) V. G i l i n s k i and J . Mathews, Phys. Rev. 120 (1960) 1450 3) H. U b e r a l l , Phys. Rev. 119 (1960) 365 4) R. W. Huff, Ann. Phys. 16 (1961) 288 5) P. Hanggi, T h e s i s , U n i v e r s i t y of B a s e l , (1973) unpublished 6) P. Hanggi, R. D. V i o l l i e r , U. Raff and K. A l d e r , Phys. L e t t . 51B (1974) 119 7) F. Herzog and K. A l d e r , T h e s i s , U n i v e r s i t y of B a s e l , (1979) unpublished 8) G . B r e i t , Phys. Rev. 34 (1929) 553 9) H. Grotch and D. R. Yennie, Rev. Mod. Phys. 41 (1969) 350 10) J . L. F r i a r , Ann. Phys. 81 (1973) 332 11) J.,L. F r i a r , Ann. Phys. 98 (1976) 490 12) R. C. B a r r e t t , D. A. Owen, J . Calmet and H. Grotch, Phys. L e t t . 47B (1973) 297 13) J . D. Bjorken and S. D. D r e l l , R e l a t i v i s t i c Quantum Mechanics, New York, McGraw-Hill, (1964) 14) J . D. Jackson, C l a s s i c a l 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 E l e c t r o n Theory, New York, Wiley (1961) 16) H. Grotch and R. Kashuba, Phys. Rev. A 5 (1972) 527 74 17) H. Osborn, Phys. Rev. 176 (1968) 1514 18) R. H. Good, Rev. Mod. Phys. 27 (1955) 187; K. M. Case, Phys. Rev. 97 (1955) 810 19) H. M. P i l k u h n , R e l a t i v i s t i c P a r t i c l e P h y s i c s , New York, Sp r i n g e r - V e r l a g , (1979) 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0095209/manifest

Comment

Related Items