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UBC Theses and Dissertations

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

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THE EFFECTS OF MASSIVE NEUTRINOS AND THEIR MIXINGS ON MUON DECAY PATRICIA ANN KALYNIAK M . S c , U n i v e r s i t y of B r i t i s h Columbia, 1 9 7 8 B . S c , U n i v e r s i t y of C a l g a r y , 1977 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF PHYSICS We a c c e p t t h i s t h e s i s as co n f o r m i n g t o the r e q u i r e d s t a n d a r d THE UNIVERSITY OF J u l y P a t r i c i a Ann BRITISH COLUMBIA 1 982 K a l y n i a k , 1982. I n 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 o f t h e r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e 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 s t u d y . 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 c o p y i n g o f 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 g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r 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 n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 )E-6 n/R-n i i ABSTRACT T h i s t h e s i s c o n t a i n s a study of the e f f e c t s of massive D i r a c and Majorana n e u t r i n o s and t h e i r m i x i n g s i n the e + spectrum f o r the muon decay jx*~*e* v e v^Kv^. ). The s p e c t r a f o r both p o l a r i z e d and u n p o l a r i z e d muons a r e g i v e n f o r the t h r e e -n e u t r i n o w o r l d w i t h a s i n g l e n e u t r i n o of mass i n the MeV/c 2 range. E l e c t r o n - n e u t r i n o c o r r e l a t i o n s are c a l c u l a t e d and proposed as a p o s s i b l e s i g n a t u r e of Majorana n e u t r i n o s . The f i r s t - o r d e r r a d i a t i v e c o r r e c t i o n s t o the muon decay and the r a d i a t i v e decay jm*-*e* i / e are i n c l u d e d i n t h i s a n a l y s i s f o r the case of a s i n g l e D i r a c n e u t r i n o of mass i n the MeV/c 2 range m i x i n g i n t o the t h r e e - n e u t r i n o w o r l d . The method of d i m e n s i o n a l r e g u l a r i z a t i o n i s used f o r the c a l c u l a t i o n of the r a d i a t i v e c o r r e c t i o n s . i i i TABLE OF CONTENTS page A b s t r a c t i i L i s t of F i g u r e s v Acknowledgements v i i Ch a p t e r s I I n t r o d u c t i o n 1 I I Background and M o t i v a t i o n 7 1. E x p e r i m e n t a l S i t u a t i o n 8 2. The G e n e r a t i o n of N e u t r i n o Masses 14 3. The E f f e c t i v e F o u r - f e r m i o n Theory of Weak I n t e r a c t i o n s 36 I I I Muon Decay W i t h M a s s i v e N e u t r i n o s 45 1 . The Case of D i r a c N e u t r i n o s 46 2. The Case of Majorana N e u t r i n o s 67 3. E l e c t r o n - n e u t r i n o C o r r e l a t i o n s 74 IV The I n c l u s i o n of R a d i a t i v e C o r r e c t i o n s 85 V Summary and C o n c l u s i o n s 104 B i b l i o g r a p h y 109 Appendices 113 A C o n v e n t i o n s and N o t a t i o n 113 B Majorana S t a t e s 120 C Phase Space C o n s i d e r a t i o n s 126 D D e t a i l s of Majorana Case C a l c u l a t i o n 141 E D i m e n s i o n a l R e g u l a r i z a t i o n 147 F S e l f - e n e r g y and V e r t e x C o r r e c t i o n s 154 G C a l c u l a t i o n of the R a d i a t i v e Decay 166 V LIST OF FIGURES page F i g . 1 The s c a l a r p o t e n t i a l V(<p) f o r ^ _2<0 andyu?>0 18 F i g . 2 Fermion mass c o n t r i b u t i o n due t o Higgs d o u b l e t . . 26 F i g . 3 N e u t r i n o Majorana mass c o n t r i b u t i o n t o Hi g g s s i n g l e t -L* 30 F i g . 4 The M i c h e l spectrum f o r u n p o l a r i z e d jx* decay. ... 60 F i g . 5 The h i g h - e n e r g y end of the M i c h e l spectrum w i t h h i e r a r c h i a l m i x i n g 61 F i g . 6 H e l i c i t y c o n s i d e r a t i o n s f o r the M i c h e l spectrum of JA* decay w i t h m a s s l e s s n e u t r i n o s 62 F i g . 7 S e n s i t i v i t y of the M i c h e l spectrum t o the m i x i n g parameter Uy.3 f o r m3 = 5 MeV/c 2 63 F i g . 8 S e n s i t i v i t y of the M i c h e l spectrum t o the m i x i n g parameter U e3 f o r m3=5 MeV/c 2 and I U ^ l 2 = .059 64 F i g . 9 High-energy end of M i c h e l spectrum f o r s p i n of jjl* a n t i p a r a l l e l t o p o s i t r o n momentum 66 F i g . 10 E n e r g y - a n g l e c o r r e l a t i o n s f o r D i r a c (D) and Majorana (M) n e u t r i n o s 80 F i g . 11 Energy-energy c o r r e l a t i o n s f o r y=l/4 83 F i g . 12 Feynman diagrams f o r the f r e e muon decay and the v i r t u a l photon c o r r e c t i o n s 86 F i g . 13 S e l f - e n e r g y c o r r e c t i o n diagrams f o r the cha r g e d l e p t o n s 86 F i g . 14 B r e m s s t r a h l u n g c o r r e c t i o n s t o muon decay 86 F i g . 15 The h i g h - e n e r g y end of the r a d i a t i v e l y c o r r e c t e d M i c h e l spectrum f o r a 5 MeV/c 2 n e u t r i n o 99 F i g . 16 The h i g h - e n e r g y end of the r a d i a t i v e l y c o r r e c t e d M i c h e l spectrum f o r a 10 MeV/c 2 n e u t r i n o 100 F i g . 17 L i m i t s of a l l o w e d m i x i n g of IU/x3| 2 as a f u n c t i o n of n e u t r i n o mass 101 v i F i g . 18 High-energy end of the M i c h e l spectrum w i t h the s p i n of /u* a n t i p a r a l l e l t o p o s i t r o n momentum 102 v i i ACKNOWLEDGEMENT I w i s h t o thank my r e s e a r c h s u p e r v i s o r , John Ng, f o r h i s abundant h e l p and the time he spent w i t h me on t h i s work. I a l s o thank Doug Beder f o r a c t i n g as my o f f i c i a l s u p e r v i s o r . My t h a n k s , f i n a l l y , t o my husband, f o r h i s h e l p and encouragement. F i n a n c i a l a s s i s t a n c e from the N a t u r a l S c i e n c e s and E n g i n e e r i n g Research C o u n c i l and from the U n i v e r s i t y of B r i t i s h Columbia i s g r a t e f u l l y acknowledged. 1 I . INTRODUCTION The d e t e r m i n a t i o n of the n a t u r e of n e u t r i n o s has been of c o n s i d e r a b l e i n t e r e s t l a t e l y , p a r t i c u l a r l y i n the c o n t e x t of grand u n i f i e d t h e o r i e s [ 1 - 3 ] . Such t h e o r i e s attempt t o u n i f y s t r o n g and e l e c t r o w e a k i n t e r a c t i o n s . N e u t r i n o s have t r a d i t i o n a l l y been d e s c r i b e d as m a s s l e s s D i r a c p a r t i c l e s of d e f i n i t e ( l e f t - h a n d e d ) c h i r a l i t y . Many of the proposed u n i f i c a t i o n schemes, however, c o n t a i n massive Majorana n e u t r i n o s as p a r t of t h e i r p a r t i c l e c o n t e n t . A f r e e massive Majorana f i e l d i s d e f i n e d t o be an e i g e n s t a t e of charge c o n j u g a t i o n [ 4 - 6 ] . That i s , a Majorana p a r t i c l e and i t s a n t i p a r t i c l e a r e the same t o w i t h i n a phase. We choose t h i s phase such t h a t our Majorana c o n d i t i o n can be s t a t e d as f o l l o w s . Majorana p a r t i c l e and a n t i p a r t i c l e a r e i n d i s t i n g u i s h a b l e . In e q u a t i o n (1.1) above, IfJ i s a quantum f i e l d and the s u p e r s c r i p t " c " i n d i c a t e s o p e r a t i o n of charge c o n j u g a t i o n , which t r a n s f o r m s a p a r t i c l e i n t o i t s a n t i p a r t i c l e . Thus the d e t e r m i n a t i o n of whether or not n e u t r i n o s a r e massive and whether they a r e d e s c r i b e d by D i r a c or Majorana f i e l d s i s i m p o r t a n t i n d e c i d i n g w hich among competing t h e o r i e s a r e v a l i d . E x p e r i m e n t s have not so f a r d e t e r m i n e d whether n e u t r i n o s a r e D i r a c or Majorana p a r t i c l e s . Nor have they r u l e d out the 2 p o s s i b i l i t y of n e u t r i n o s b e i n g m a s s i v e . Thus, i t i s im p o r t a n t t o e x p l o r e t h e e f f e c t s of massive n e u t r i n o s and t o t r y t o f i n d a means of d i s t i n g u i s h i n g Majorana and D i r a c n e u t r i n o s by ex p e r i m e n t . Some of the e x p e r i m e n t s which a r e s e n s i t i v e t o n e u t r i n o mass or t o t h e i r Majorana n a t u r e a r e mentioned i n Chapter 2. A l s o i n t h a t C h a p t e r , we g i v e examples of the t h e o r e t i c a l schemes f o r n e u t r i n o mass g e n e r a t i o n . A b r i e f r e view of the SU(2)XU(1) gauge t h e o r y [7-9] of e l e c t r o w e a k i n t e r a c t i o n s i s a l s o g i v e n t o f a c i l i t a t e t h i s d i s c u s s i o n of mass g e n e r a t i o n . The remainder of t h i s work i s an i n v e s t i g a t i o n of the e f f e c t s ,of massive n e u t r i n o s i n one p a r t i c u l a r r e a c t i o n - t h a t of the o r d i n a r y muon decay which proceeds v i a the weak i n t e r a c t i o n [ 1 0 ] . - e + v £ T £ _ (1.2) In e q u a t i o n ( 1 . 2 ) , the muon a n t i n e u t r i n o i s r e p r e s e n t e d by the c o n j u g a t e n o t a t i o n , , t o i n d i c a t e the p o s s i b i l i t y t h a t i t i s a Majorana p a r t i c l e . For the case of D i r a c n e u t r i n o s , the u s u a l a n t i n e u t r i n o n o t a t i o n , v, w i l l be used. We here study the decay (1.2) f o r a c l u e as t o the n a t u r e of n e u t r i n o s , and c o n s e q u e n t l y , t o the t h e o r e t i c a l d e s c r i p t i o n of e l e c t r o w e a k i n t e r a c t i o n s . T h i s p a r t i c u l a r decay i s the f o c u s of most e x p e r i m e n t s which s e a r c h f o r d e v i a t i o n s from the s t a n d a r d " v e c t o r c u r r e n t minus a x i a l v e c t o r c u r r e n t " (V-A) s t r u c t u r e of the e f f e c t i v e f o u r - f e r m i o n weak i n t e r a c t i o n t h e o r y . The s t a n d a r d V-A e f f e c t i v e t h e o r y i s assumed here and the e f f e c t s of massive n e u t r i n o s on the decay (1.2) a r e s t u d i e d . The g e n e r a l e f f e c t i v e 3 f o u r - f e r m i o n weak i n t e r a c t i o n t h e o r y i s b r i e f l y r e v i e w e d i n Chapter 2 as an i n t r o d u c t i o n t o the c a l c u l a t i o n s of t h i s work. The n e u t r i n o s of the decay (1.2) a r e o f t e n c a l l e d "weak i n t e r a c t i o n e i g e n s t a t e s " . T h i s d e s i g n a t i o n s i m p l y means t h a t , f o r the g e n e r a l case of N l e p t o n f a m i l i e s , the n e u t r i n o s denoted by v a , where a = e, jx, -c, . .. , a r e tho s e produced a t a weak i n t e r a c t i o n v e r t e x . They a r e c o n v e n t i o n a l l y a s s i g n e d a l e p t o n f a m i l y number L a such t h a t L^=+1 f o r and L^=-1 f o r i£ . In g e n e r a l , n e u t r i n o masses do not c o n s e r v e t h i s l e p t o n number [11] so t h a t a weak e i g e n s t a t e i s not a s t a t e of w e l l - d e f i n e d mass (mass e i g e n s t a t e ) . Thus, m i x i n g among d i f f e r e n t n e u t r i n o s p e c i e s o c c u r s . In p r e v i o u s c a l c u l a t i o n s of muon decay, no m i x i n g among n e u t r i n o s was i n c l u d e d even when c o n s i d e r i n g massive n e u t r i n o s [ 1 2 ] . The mass e i g e n s t a t e s w i l l be denoted as vL, where i=1,...,N. That i s , n e u t r i n o vL i s a s t a t e of mass mL. It p r o p a g a t e s as a s t a t e of w e l l - d e f i n e d energy E- =-J~ I p Ll *+mf . I f the masses of t h e s e s t a t e s a r e not d e g e n e r a t e , then the mass e i g e n s t a t e s a r e i n p r i n c i p l e d i s t i n g u i s h a b l e . The weak e i g e n s t a t e s can be e x p r e s s e d as a l i n e a r s u p e r p o s i t i o n of the mass e i g e n s t a t e s as f o l l o w s . E q u a t i o n (1.3) i s an e x p r e s s i o n of the concept of n e u t r i n o elements of a u n i t a r y t r a n s f o r m a t i o n m a t r i x and have the f o l l o w i n g o r t h o n o r m a l i t y c o n d i t i o n s . (1.3) m i x i n g w i t h IX • the P o n t e c o r v o m i x i n g p a r a m e t e r s . They are (1.4) 4 (1.5) For the s p e c i a l case of t h r e e n e u t r i n o f a m i l i e s (N=3), the Kobayashi-Maskawa p a r a m e t r i z a t i o n [13] can be used f o r the where 040^/2 and -Tf$6<1T. For the g e n e r a l N f a m i l y c a s e , the m i x i n g m a t r i x can be c o m p l e t e l y s p e c i f i e d w i t h (1/2)N(N-1) a n g l e s and (1/2)(N—1)(N-2) phases 6. The assumption of n e u t r i n o mass adopted here i s independent of any p a r t i c u l a r gauge model. As mentioned above, the p r o c e s s (1.2) i s here assumed t o be d e s c r i b e d by the V-A e f f e c t i v e f o u r - f e r m i o n weak i n t e r a c t i o n t h e o r y [ 1 4 , 1 5 ] , The c h o i c e of a p r e d o m i n a n t l y l e f t - h a n d e d i n t e r a c t i o n i s made i n o r d e r t o i s o l a t e the e f f e c t s of massive n e u t r i n o s ; t h a t i s , the n e u t r i n o mass e f f e c t s a re s t u d i e d i n d e p e n d e n t l y of any n o n - s t a n d a r d s t r u c t u r e i n the weak i n t e r a c t i o n s . The e f f e c t i v e f o u r - f e r m i o n t h e o r y i s chosen f o r t h i s c a l c u l a t i o n over the r e n o r m a l i z a b l e S U ( 2 ) X U ( 1 ) gauge t h e o r y of e l e c t r o w e a k i n t e r a c t i o n s f o r the f o l l o w i n g r e a s o n . For the case of m a s s l e s s n e u t r i n o s , the p o s i t r o n spectrum f o r the decay (1.2) (known as the M i c h e l spectrum) has been c a l c u l a t e d both w i t h the e f f e c t i v e t h e o r y and w i t h the f u l l S U ( 2 ) x U ( 1 ) gauge t h e o r y , wherein the weak i n t e r a c t i o n s a r e mediated by massive v e c t o r bosons [ 1 6 ] . The r e s u l t s of t h e s e two c a l c u l a t i o n s d i f f e r P o n t e c o r v o m i x i n g m a t r i x . That i s , XL -(1.6) 5 by terms of the o r d e r of (m^/M2^ ), where i s the muon mass and M w i s the mass of the exchanged v e c t o r boson. The c u r r e n t e x p e r i m e n t a l p r e c i s i o n i s not good enough t o d e t e c t the d i f f e r e n c e between the two t h e o r i e s . A l s o , r a d i a t i v e c o r r e c t i o n s have been i n c l u d e d i n t h e s e c a l c u l a t i o n s f o r the case of m a s s l e s s n e u t r i n o s s i n c e i t i s w e l l known t h a t they a r e l a r g e i n the measurement of the M i c h e l spectrum. The f i r s t o r d e r v i r t u a l e l e c t r o m a g n e t i c c o r r e c t i o n s t o the p r o c e s s (1.2) and the r a d i a t i v e decay - e + 7k ] (1.7) have been c a l c u l a t e d and the d i f f e r e n c e between the r e s u l t s of the e f f e c t i v e f o u r - f e r m i o n t h e o r y [16,17] and the f u l l S U(2)xu(1) t h e o r y [18] i s of the o r d e r of (am£/M 2 w), where a i s the f i n e s t r u c t u r e c o n s t a n t . So, t o f i r s t o r d e r i n the r a d i a t i v e c o r r e c t i o n s , the e f f e c t i v e f o u r - f e r m i o n t h e o r y and the S U ( 2 ) X U ( l ) gauge t h e o r y a r e e x p e r i m e n t a l l y i n d i s t i n g u i s h a b l e , f o r the case of m a s s l e s s n e u t r i n o s . T h i s s m a l l d i f f e r e n c e between the r e s u l t s of the two t h e o r i e s f o r the m a s s l e s s n e u t r i n o s case i n d i c a t e s t h a t the e f f e c t i v e f o u r - f e r m i o n t h e o r y w i l l p r o v i d e a good d e s c r i p t i o n of the decay (1.2) even f o r massive n e u t r i n o s . In C hapter 3, the M i c h e l spectrum i s c a l c u l a t e d f o r the c a s e s of m assive D i r a c and Majorana n e u t r i n o s i n the g e n e r a l N-f a m i l y c a s e . The s p e c t r a f o r the decay of b o t h p o l a r i z e d and u n p o l a r i z e d muons are g i v e n . We s p e c i a l i z e t o t h e t h r e e - n e u t r i n o w o r l d and d i s p l a y the e f f e c t s of massive n e u t r i n o s on the M i c h e l 6 spectrum f o r v a r i o u s n e u t r i n o masses and m i x i n g s . The D i r a c and Majorana c a s e s a r e compared. A l s o , c o r r e l a t i o n s between the o u t g o i n g e + and v& a r e c a l c u l a t e d i n Chapter 3 and proposed as a p o s s i b l e means of d e t e r m i n i n g the D i r a c or Majorana n a t u r e of n e u t r i n o s . Such c o r r e l a t i o n s were o r i g i n a l l y proposed by J a r l s k o g as a t e s t f o r r i g h t - h a n d e d c u r r e n t s i n the weak i n t e r a c t i o n [ 1 9 ] . In Chapter 4, the v i r t u a l p h o t o n i c c o r r e c t i o n s t o the decay (1.2) a r e c a l c u l a t e d a l o n g w i t h the r a d i a t i v e decay (1.7) f o r the case of massive D i r a c n e u t r i n o s . The method of d i m e n s i o n a l r e g u l a r i z a t i o n i s used t o d e a l w i t h the d i v e r g e n c e s which occur [ 2 0 ] . These d i v e r g e n c e s a l l c a n c e l t o l e a v e a f i n i t e r e s u l t . T h i s method i s gauge i n v a r i a n t and we work i n the ' t H o o f t -F.eynman gauge. B e s i d e s the M i c h e l spectrum, the b r e m s s t r a h l u n g spectrum i s a l s o g i v e n . T h i s would be n e c e s s a r y i n f o r m a t i o n f o r a muon decay experiment i n which the photon i s d e t e c t e d . We d i s p l a y the e f f e c t s of massive n e u t r i n o s and t h e i r m i x i n g s f o r v a r i o u s v a l u e s of n e u t r i n o mass i n the Mev/c 2 range and f o r v a r i o u s m i x i n g s . Most of the d e t a i l s of t h e s e c a l c u l a t i o n s have been r e l e g a t e d t o the a p p e n d i c e s . A l s o i n c l u d e d a r e t h r e e a p p e n d i c e s of a more g e n e r a l n a t u r e . F i r s t , Appendix A c o n t a i n s our c o n v e n t i o n s and n o t a t i o n . Appendix B c o n t a i n s a d i s c u s s i o n of Majorana f i e l d s . An i n t r o d u c t i o n t o the method of d i m e n s i o n a l r e g u l a r i z a t i o n i s g i v e n i n Appendix E. Chapter 5 c o n t a i n s our c o n c l u s i o n s . 7 I I . BACKGROUND AND MOTIVATION In t h i s work, we st u d y the e f f e c t s of massive n e u t r i n o s on muon decay. As m o t i v a t i o n f o r t h i s i n t e r e s t i n nonzero n e u t r i n o mass, we g i v e the e x p e r i m e n t a l l i m i t s on the masses of the t h r e e n e u t r i n o s of the e l e c t r o n , muon, and t a u f a m i l i e s . The ex p e r i m e n t s which have s e t t h e s e l i m i t s a r e b r i e f l y d e s c r i b e d i n S e c t i o n 2.1 a l o n g w i t h a few o t h e r e x p e r i m e n t s which a r e s e n s i t i v e t o n e u t r i n o mass or t o the Majorana n a t u r e of n e u t r i n o s . T h i s d i s c u s s i o n of the e x p e r i m e n t a l s i t u a t i o n i s f o l l o w e d by a d e s c r i p t i o n of the t h e o r e t i c a l means of accommodating a nonzero n e u t r i n o mass. We r e v i e w , i n S e c t i o n _ 2 . 2 , the i d e a of spontaneous symmetry b r e a k i n g and the mi n i m a l SU(2)XU(1) gauge t h e o r y of e l e c t r o w e a k i n t e r a c t i o n s . E x t e n s i o n s of t h i s model i n the Higgs s e c t o r and the f e r m i o n s e c t o r which generate a n e u t r i n o mass a r e g i v e n . The wide range i n the t h e o r e t i c a l e s t i m a t e s of the magnitude of n e u t r i n o mass i s p o i n t e d o u t . The r e d u c t i o n of the SU(2)XU(1) t h e o r y t o the V-A e f f e c t i v e f o u r - f e r m i o n t h e o r y of weak i n t e r a c t i o n s i s d e s c r i b e d . In S e c t i o n 2.3, the V-A t h e o r y i s r e v i e w e d t o s e t the st a g e f o r the c a l c u l a t i o n s of the next C h a p t e r . 8 2.1 E x p e r i m e n t a l S i t u a t i o n I f one has any doubt t h a t t h e r e i s m o t i v a t i o n f o r c a l c u l a t i n g and t e s t i n g the e f f e c t s of massive n e u t r i n o s , t h i s doubt s h o u l d v a n i s h when " c o n f r o n t e d w i t h the p r e s e n t e x p e r i m e n t a l l i m i t s on n e u t r i n o masses. For the case of the e l e c t r o n n e u t r i n o , t h e r e i s p r e l i m i n a r y e v i d e n c e f o r a n o n v a n i s h i n g mass [ 2 1 , 2 2 ] , The l i m i t s s e t on the mass a r e , a t 99% C.L., IH- < TVL^ < Hb e.V/c> (2.1.1) Only upper l i m i t s have been s e t on the masses of the muon and t a u type n e u t r i n o s . These a r e , [23-25] yn < 0.S0 KeV/c> (2.1.2) 7YL < 3USO I^AeV/c*- (2.1.3) The e x p e r i m e n t s which have s e t the s e l i m i t s as w e l l as some o t h e r e x p e r i m e n t s which a r e s e n s i t i v e t o n e u t r i n o mass or t o the Majorana n a t u r e of n e u t r i n o s a r e d i s c u s s e d b r i e f l y below. Ex p e r i m e n t s s e n s i t i v e t o n e u t r i n o mass can be r o u g h l y d i v i d e d i n t o two t y p e s . The f i r s t t y p e , p r i m a r i l y weak decays, e x p l o r e s the n a t u r e of the n e u t r i n o r a t h e r i n d i r e c t l y . In the second t y p e , the n e u t r i n o s a r e d e t e c t e d . T h i s c a t e g o r y i n c l u d e s o s c i l l a t i o n e x p e r i m e n t s and c o r r e l a t i o n measurements i n weak de c a y s . The i d e a of n e u t r i n o o s c i l l a t i o n s i s d e s c r i b e d l a t e r i n t h i s s e c t i o n . 9 The t h r e e l i m i t s quoted above were s e t i n weak decay e x p e r i m e n t s ; n e u t r i n o s were not d e t e c t e d . The l i m i t which i s c o n c e p t u a l l y the s i m p l e s t t o o b t a i n i s t h a t on the muon n e u t r i n o ' s mass. The muon momentum from the p i o n decay T I / + * ~*r (2.1.4) was measured p r e c i s e l y i n an experiment a t SIN [ 2 4 ] . Energy-momentum c o n s e r v a t i o n i n t h i s two-body decay y i e l d s the f o l l o w i n g v a l u e f o r the n e u t r i n o mass. ^ = ™ 4 + *"£ ' h ^ir ( + ) ^  (2.1.5) The p r e c i s e measurements of the muon momentum and of the p i o n mass l e d t o the l i m i t quoted above. Note t h a t t h i s r e s u l t does not take i n t o account the p o s s i b i l i t y of n e u t r i n o m i x i n g . We w i l l r e t u r n t o t h i s p o i n t l a t e r . The l i m i t on the mass of the t a u n e u t r i n o was o b t a i n e d from a measurement of the three-body decay ~C' * ^~ ^ ^ T (2.1.6) Nonzero n e u t r i n o mass w i l l change the en d p o i n t of the e l e c t r o n spectrum; the en d p o i n t w i l l be lo w e r e d from the v a l u e e x p e c t e d f o r the m a s s l e s s n e u t r i n o s c a s e . The l i m i t q uoted above was s e t assuming a V-A weak i n t e r a c t i o n and v a n i s h i n g e l e c t r o n n e u t r i n o mass. A g a i n , n e u t r i n o m i x i n g was n e g l e c t e d . L e s s than 600 e v e n t s were o b t a i n e d t o s e t t h i s l i m i t . F i n a l l y , t he e l e c t r o n n e u t r i n o mass l i m i t was s e t through a measurement of the hi g h - e n e r g y end of the e l e c t r o n spectrum i n 10 a n o t h e r three-body decay H 3 H e + € ^ (2.1.7) N u c l e a r decays such as t h i s a r e g e n e r a l l y r e p r e s e n t e d by a K u r i e p l o t . T h i s i s a p l o t of * [ E : ^ - £ j V H ( E r X - E e f - ^ T / 4 (2.1.8) which i s r o u g h l y the square r o o t of the number of e v e n t s , c o r r e c t e d f o r Coulomb e f f e c t s ( F ) , d i v i d e d by some k i n e m a t i c f a c t o r s v e r s u s the energy of the decay e l e c t r o n . In e q u a t i o n (2.1.8) above, E™"'* i s the maximum e l e c t r o n energy f o r the case of z e r o n e u t r i n o mass. For z e r o n e u t r i n o mass, the p l o t i s l i n e a r . For n o n v a n i s h i n g n e u t r i n o mass, the K u r i e p l o t d e v i a t e s from l i n e a r i t y d r o p p i n g t o z e r o a t E e =E^°'y' -mv. I t i s b a s i c a l l y t h i s d e v i a t i o n which was i n v e s t i g a t e d t o s e t the l i m i t quoted above. The experiment c l a i m s a r a t h e r h i g h r e s o l u t i o n (45 e v / c 2 FWHM a t the end of the yfl spectrum) which i s n e c e s s a r y t o s e t the l i m i t g i v e n . However, t h e r e a r e o t h e r problems i n e s t a b l i s h i n g t h i s l i m i t . F i r s t , the o u t g o i n g e l e c t r o n can s c a t t e r or r a d i a t e i n the decay g i v i n g some u n c e r t a i n t y t o i t s energy. The more i m p o r t a n t i s s u e i s the u n c e r t a i n t y i n the energy l e v e l s of the f i n a l s t a t e . C a l c u l a t i o n s have been made of the energy l e v e l s f o r the f r e e decay; however, the a c t u a l experiment i s performed w i t h t r i t i u m i n V a l i n e ( C 5 H 1 1 N 0 2 ) . Acceptance of the nonzero n e u t r i n o mass a w a i t s r e p e t i t i o n of the e x p e r i m e n t . The a n a l y s i s of the fH spectrum was done n e g l e c t i n g the p o s s i b i l i t y of 11 n e u t r i n o m i x i n g , which c o m p l i c a t e s the m atter c o n s i d e r a b l y [ 2 6 ] . A more r e c e n t measurement of the yS energy spectrum of 3H a t the U n i v e r s i t y of Guelph [27] s e t s a mass l i m i t of m V e<65 e V / c 2 w i t h a best v a l u e of 20 e V / c 2 . The e x p e r i m e n t a l method was d i f f e r e n t from t h a t of t h e . p r e v i o u s l y mentioned experiment ( S i ( L i ) ) d e t e c t o r as opposed t o /S s p e c t r o m e t e r ) and the r e s o l u t i o n was c o n s i d e r a b l y worse (^300 e V / c 2 ) . The source e f f e c t s and f i n a l s t a t e i n t e r a c t i o n s a r e d i f f e r e n t from the p r e v i o u s l y mentioned e x p e r i m e n t . Another experiment i s proposed at Chalk R i v e r [ 2 8 ] . We r e t u r n t o the q u e s t i o n of n e u t r i n o m i x i n g as d e s c r i b e d i n the I n t r o d u c t i o n . C o n s i d e r the p i o n decay ( 2 . 1 . 4 ) . I f the n e u t r i n o weak e i g e n s t a t e s a r e a l i n e a r s u p e r p o s i t i o n of mass e i g e n s t a t e s , then secondary peaks o c c u r i n the yx+ spectrum. T h i s p o s s i b i l i t y has been a n a l y z e d t h e o r e t i c a l l y [ 2 9 - 3 1 ] , The h e i g h t of these secondary peaks i s governed by the P o n t e c o r v o m i x i n g p a r a m e t e r s . The SIN p i o n decay experiment was r e c e n t l y r e a n a l y z e d f o r the p o s s i b i l i t y of n e u t r i n o m i x i n g [ 3 2 ] , As an example of the c o n c l u s i o n s r e a c h e d , a n e u t r i n o of mass 6-14 Mev/c 2 c o u l d o c c u r w i t h about 1% p r o b a b i l i t y . As noted above the t a u and t r i t i u m decay e x p e r i m e n t s a r e a l s o c o m p l i c a t e d by the p o s s i b i l i t y of n e u t r i n o m i x i n g . We i n c l u d e t h i s p o s s i b i l i t y i n our a n a l y s i s of muon decay. As a n o t h e r example of the n e u t r i n o e x p e r i m e n t s of the f i r s t t y p e , we mention n o - n e u t r i n o d o u b l e yS decay (fifiov), which has l o n g been r e c o g n i z e d as a s i g n a t u r e f o r Majorana n e u t r i n o s [ 3 3 , 3 4 ] . I f n e u t r i n o s a r e m assive Majorana p a r t i c l e s , then (/3/30-v) decay t a k e s p l a c e . On the o t h e r hand, the e x i s t e n c e of 12 (^Pov) decay does not n e c e s s a r i l y i m p l y t h a t n e u t r i n o s have a Majorana mass but i t s t r o n g l y s u g g e s t s t h i s . ( $ 3 0 v ) decay e x p e r i m e n t s a r e s e n s i t i v e t o n e u t r i n o mass i n the 10-100 e V / c 2 range. There i s no u n d i s p u t e d e v i d e n c e f o r nonzero n e u t r i n o mass from ( ^ o v ) e x p e r i m e n t s . As an example of the second type of n e u t r i n o mass s e n s i t i v e e x p e r i m e n t , we d i s c u s s n e u t r i n o o s c i l l a t i o n s [ 5 , 1 1 , 3 5 ] , C o n s i d e r a pure weak e i g e n s t a t e , 1/^  , of momentum p, a t time t=0. A f t e r a time t , t h i s s t a t e e v o l v e s t o t h e f o l l o w i n g , i n terms of the mass e i g e n s t a t e s , 1pi , = E K<u e t E t t (2.1.9) where E L =Jlpi 2 +m 2\ Then the p r o b a b i l i t y of f i n d i n g the weak e i g e n s t a t e ^ b of momentum p a f t e r a time t i s g i v e n by 'J ^ f- 4 a b i a < x . u f l K u b l < e ( 2 . 1 . 1 0 ) There i s a nonzero p r o b a b i l i t y of one weak n e u t r i n o s t a t e e v o l v i n g i n t o a n o ther p r o v i d i n g the n e u t r i n o masses a r e not z e r o or d e g e n e r a t e . The mass d i f f e r e n c e any experiment i s s e n s i t i v e 13 t o depends on the o s c i l l a t i o n l e n g t h and the n e u t r i n o energy. S o l a r n e u t r i n o e x p e r i m e n t s a r e s e n s i t i v e t o n e u t r i n o mass d i f f e r e n c e s of the o r d e r of 10~ 6 e V / c 2 [5,36,37]. The observed s o l a r n e u t r i n o c a p t u r e r a t e [38] i s about 30% of the c a l c u l a t e d r a t e [ 3 9 ] . O s c i l l a t i o n s have not been r u l e d out as a s o l u t i o n t o t h i s d i s c r e p a n c y . E a r t h based e x p e r i m e n t s ( r e a c t o r and a c c e l e r a t o r e x p e r i m e n t s ) a r e t y p i c a l l y s e n s i t i v e t o n e u t r i n o mass d i f f e r e n c e s of a few e V / c 2 [5,36,37]. The r e a c t o r experiment of R e i n e s e t a l . [ 4 0 ] , i n which the r a t i o of charged t o n e u t r a l c u r r e n t r e a c t i o n s r\ -v- n + e (2.1.11) was measured,claims a mass-squared d i f f e r e n c e of 0.7< Am 2<1.OeV/c 2. T h i s c l a i m has been c h a l l e n g e d by a s i m i l a r e xperiment [41] which i s c o n s i s t e n t w i t h no o s c i l l a t i o n s . Bubble chamber and e m u l s i o n e x p e r i m e n t s a t CERN and FNAL [42,43] and an o s c i l l a t i o n s e a r c h a t Los Alamos [44,45] f i n d no e v i d e n c e f o r o s c i l l a t i o n s . The r e c e n t CERN beam dump e x p e r i m e n t s [46] are a l s o c o n s i s t e n t w i t h no o s c i l l a t i o n s [ 3 6 ] . O b v i o u s l y , the e x p e r i m e n t a l s i t u a t i o n r e g a r d i n g n e u t r i n o mass i s v e r y u n c l e a r ; no f i r m c o n c l u s i o n s on the e x i s t e n c e or magnitude of the n e u t r i n o mass have been reached. 14 2.2 The G e n e r a t i o n of N e u t r i n o Masses The s t a n d a r d SU(2)XU(1) gauge t h e o r y of e l e c t r o w e a k i n t e r a c t i o n s [7-9] w i l l be r e v i e w e d h e r e . In t h i s t h e o r y , the concept of spontaneous symmetry b r e a k i n g [3,47,48] i s v i t a l and i s used, v i a the Higgs mechanism [3, 4 9 - 5 2 ] , t o ge n e r a t e masses f o r the v e c t o r bosons and f e r m i o n s . These i d e a s a re d e s c r i b e d below. In t he m i n i m a l SU(2)XU(1) gauge t h e o r y , the n e u t r i n o s a r e m a s s l e s s by c o n s t r u c t i o n . We d i s p l a y t h i s l a t e r . Some p o s s i b l e ways of e x t e n d i n g t h i s model t o i n c l u d e massive n e u t r i n o s a re a l s o g i v e n . We c o n s i d e r t h e s e e x t e n s i o n s of SU(2)XU(1) r a t h e r than d i s c u s s i n g a grand u n i f i e d model s i n c e the b a s i c mechanisms of n e u t r i n o mass g e n e r a t i o n a r e i l l u s t r a t e d a d e q u a t e l y . The problem of the magnitude of n e u t r i n o masses w i l l be d i s c u s s e d i n the c o n t e x t of grand u n i f i e d t h e o r i e s , as w e l l as f o r the SU(2)XU(1) e x t e n s i o n s . The b a s i c i d e a i n f o r m u l a t i n g any gauge t h e o r y [3,53] i s t o w r i t e a "gauge i n v a r i a n t " d e s c r i p t i o n of the p h y s i c a l system. The system i s assumed t o p o s s e s s a symmetry d e s c r i b e d by some L i e group. A t r a n s f o r m a t i o n under the group i s g i v e n by (2.2.1) where the T a (a= 1 ,2 N) a r e the g e n e r a t o r s of the group.They s a t i s f y the f o l l o w i n g commutation r e l a t i o n s . ( 2 . 2 . 2 ) 15 a be The f a r e c a l l e d the s t r u c t u r e c o n s t a n t s of the group. A s e t of m a t r i c e s , L a , which s a t i s f y t he same a l g e b r a (2.2.2) as the g e n e r a t o r s i s s a i d t o form a r e p r e s e n t a t i o n of the group. That i s , [ L \ L w l = i £*"b L c (2.2.3) Then a s e t of n f i e l d s <P- ( i = 1 , 2 , . . . , n) i s s a i d t o t r a n s f o r m a c c o r d i n g t o the r e p r e s e n t a t i o n L i f they s a t i s f y t he f o l l o w i n g commutation r e l a t i o n s . [ T % ( p L ^ > - <P.M (2.2.4) Thus, under a gauge t r a n s f o r m a t i o n , the f i e l d s Q t r a n s f o r m as f o l l o w s . (p.Jx) * U(QUS) (pL(*) U"' (BUS) (2.2.5a) - L e A;- W. (%) (2.2.5b) J I J N o t i c e t h a t the parameter e may depend on the space-time c o o r d i n a t e s . I f t h i s i s the c a s e , the t r a n s f o r m a t i o n i s c a l l e d a l o c a l gauge t r a n s f o r m a t i o n ; i f 6 i s independent of x the t r a n s f o r m a t i o n i s s a i d t o be g l o b a l . From the f i e l d s of the system, a L a g r a n g i a n i s c o n s t r u c t e d w hich i s i n v a r i a n t under the gauge t r a n s f o r m a t i o n ( 2 . 2 . 1 ) . The i d e a of d e s c r i b i n g i n t e r a c t i o n s by means of a gauge t h e o r y has been r a t h e r s u c c e s s f u l and i s c e r t a i n l y the p r i m a r y t h e o r e t i c a l s t r a t e g y a t p r e s e n t . The k i n e t i c energy terms of a L a g r a n g i a n c o n t a i n 16 d e r i v a t i v e s . Such terms a r e not i n v a r i a n t under a l o c a l gauge t r a n s f o r m a t i o n . I f l o c a l gauge i n v a r i a n c e i s r e q u i r e d , new " v e c t o r gauge f i e l d s " , A ^ ( x ) , must be i n t r o d u c e d , each c o r r e s p o n d i n g t o a group g e n e r a t o r . The s u p e r s c r i p t a ranges from 1 t h r o u g h N and jx. i s the L o r e n t z i n d e x . The d e r i v a t i v e i s r e p l a c e d by the c o v a r i a n t d e r i v a t i v e d e f i n e d below. y — - iy = V r L J A A L " ( 2 . 2 . 6 ) The c o u p l i n g c o n s t a n t g i s t o be d e t e r m i n e d e x p e r i m e n t a l l y . In or d e r t o p r e s e r v e gauge i n v a r i a n c e , the new f i e l d s A^ must t r a n s f o r m as f o l l o w s . CA^U) — C ^U)--U(9U))CK^U)VL']- ±(^U)IL] ( 2 . 2 . 7 ) T h e r e f o r e , the requirement of l o c a l gauge i n v a r i a n c e i m p l i e s the e x i s t e n c e of N gauge f i e l d s A^L(x) and a l s o d i c t a t e s the form of the i n t e r a c t i o n s of t h e s e f i e l d s . The most f a m i l i a r gauge t h e o r y i s quantum e l e c t r o d y n a m i c s ( q e d ) . Here the gauge group i s U ( 1 ) which i s r e p r e s e n t e d by u n i t a r y 1x1 m a t r i c e s . For t h i s c a s e , e q u a t i o n ( 2 . 2 . 1 ) becomes U ( x ) = e ( 2 . 2 . 8 ) In qed, the c o u p l i n g g i s the e l e c t r i c charge and the gauge f i e l d A^(x) r e p r e s e n t s the photon. A mass term f o r gauge f i e l d s of the form m 2A / U.(x)A^x) i s not gauge i n v a r i a n t . ( s e e e q u a t i o n ( 2 . 2 . 7 ) ) In the case of qed, t h i s i s d e s i r a b l e s i n c e a m a s s l e s s photon i s needed t o mediate the l o n g range f o r c e . In f a c t , the re q u i r e m e n t of l o c a l gauge i n v a r i a n c e can be taken as a " r e a s o n " 17 f o r the photon's m a s s l e s s n e s s . In the case of the weak i n t e r a c t i o n s , which a r e s h o r t range, t h i s i s a problem. M a s s i v e gauge f i e l d s a r e r e q u i r e d t o mediate the i n t e r a c t i o n y e t a v e c t o r boson mass term i s not a l l o w e d . Such a term not o n l y v i o l a t e s l o c a l gauge i n v a r i a n c e but a l s o d e s t r o y s the r e n o r m a l i z a b i l i t y of the t h e o r y ; t h e r e f o r e , abandoning the r e q u i r e m e n t of l o c a l gauge i n v a r i a n c e f o r the weak i n t e r a c t i o n s does not g a i n the s o l u t i o n t o t h i s problem. The concept of spontaneous symmetry b r e a k i n g , or h i d d e n symmetry, p r o v i d e s a r e s o l u t i o n of both problems. The concept of hidden symmetry i n v o l v e s the h y p o t h e s i s t h a t a system may p o s s e s s some symmetry which i s n o t , however, m a n i f e s t i n the ground (vacuum) s t a t e . The s i m p l e c l a s s i c a l example o f t e n used t o i l l u s t r a t e t h i s i d e a [3,53] i s t h a t of a s c a l a r f i e l d , (p, w i t h the f o l l o w i n g p o t e n t i a l . V l f l - ' l f (2.2.9) T h i s p o t e n t i a l i s symmetric under the t r a n s f o r m a t i o n (f) - jP (2.2.10) The p o t e n t i a l i s i l l u s t r a t e d i s F i g u r e 1. The ground s t a t e or vacuum of the system i s a t the p o t e n t i a l minimum which, f o r the case of jjl2<§, i s a t (p=0. The symmetry (2.2.10) i s m a n i f e s t i n the vacuum. Of c o u r s e , f o r y/-2>0, V(^) i s s t i l l symmetric under the t r a n s f o r m a t i o n (2.2.10) but (p =0 i s not the minimum. R a t h e r , t h e r e a r e two symmetric minima a t p) - j 6yU.2/A\ The system must choose one of the minima shown i n F i g u r e 1 as i t s 18 ground s t a t e and then the symmetry i s no l o n g e r m a n i f e s t i n t h a t s t a t e . The symmetry i s s a i d t o be hi d d e n or br o k e n . F i g . 1: The S c a l a r P o t e n t i a l V((p) for j x ? < § and /c2>0 Wit h JJL?>Q, t h e s c a l a r t h e o r y d e s c r i b e d by the p o t e n t i a l (2.2.9) c o n t a i n s no mass terms. The f o l l o w i n g d i s c u s s i o n shows how a mass term a r i s e s . C h o o s i n g one of the minima of t h e p o t e n t i a l as t h e vacuum s t a t e of the system, the e x p e c t a t i o n v a l u e of the quantum f i e l d i n the vacuum i s seen t o be nonzero. T h i s q u a n t i t y i s c a l l e d the vacuum e x p e c t a t i o n v a l u e (vev) and, i n t h i s c a s e , i s g i v e n by io I (p |o> = <<p>0 = fb/^/X (2.2.11) We d e f i n e a new f i e l d , (j)' , which has z e r o vev, as f o l l o w s . f ' = <p -0L (2.2.12) Thus <^') 0 = n . I n terms of t h i s new f i e l d , the p o t e n t i a l i s \J({)') = If* - 3 +tX*f * - A * * (2.2.13) 19 The new f i e l d has a c q u i r e d an o r d i n a r y mass J 3 A a ~* a l t h o u g h the o r i g i n a l f i e l d was m a s s l e s s . A l s o the o r i g i n a l symmetry (2.2.10) of the p o t e n t i a l i s not apparent i n t h i s form; i t has been h i d d e n . T h i s s i m p l e example i l l u s t r a t e s the concept of spontaneous symmetry b r e a k i n g or hid d e n symmetry. A mass i s g e n e r a t e d from an o r i g i n a l l y m a s s l e s s t h e o r y . . In the case of the weak i n t e r a c t i o n s , i t i s n e c e s s a r y f o r the m a s s l e s s gauge f i e l d s t o a c q u i r e a mass. The f o l l o w i n g e x t e n s i o n of the s i m p l e example j u s t g i v e n w i l l i l l u s t r a t e how a gauge f i e l d can a c q u i r e a mass. T h i s i s the Higgs mechanism. C o n s i d e r a charged s c a l a r f i e l d , (p , c o u p l e d t o a gauge f i e l d . Upon imposing gauge i n v a r i a n c e under a l o c a l U(1) symmetry, the L a g r a n g i a n can be w r i t t e n i n the f o l l o w i n g form. + y u > f(j)~ A ((pf(p^ (2.2.14) where JL2>0 and \>0. In the above, F^v E < ^ A V - (2.2.15) For the case of y<-2<0, the f i e l d (p has an o r d i n a r y mass term and the gauge f i e l d remains m a s s l e s s . However, f o r our c h o i c e of yW-2>0, the f i e l d ^ can d e v e l o p a vev, as i n the p r e v i o u s example. That i s , <0 I (p \0) = C(p)0 = v/f£ (2.2.16) where 20 V x = /A (2.2.17) The f o l l o w i n g p a r a m e t r i z a t i o n of (p [ 5 2 ] , a l l o w s f o r a c l e a r i n t e r p r e t a t i o n of the r e s u l t s which a r e o b t a i n e d . (p = e 5 • (2.2.18) Here, ^ and a r e h e r m i t i a n f i e l d s . Under the U(1) gauge t r a n s f o r m a t i o n IL ~ Z, (2.2.19) the f i e l d s t r a n s f o r m as f o l l o w s (see e q u a t i o n s (2.2.1) and ( 2 . 2 . 7 ) ) . (p <P' = ^ ( i r + ^ (2.2.20) (2.2.21 ) In terms of the t r a n s f o r m e d f i e l d s , the L a g r a n g i a n i s * ± Py_ k'r y[(xv + y[) - ( y ^ *• 3 A i r 1 ) - \xr ' Tj + c o n s t a n t (2.2.22) The s c a l a r f i e l d has a c q u i r e d a mass \| 2y/.2; t h i s massive s c a l a r boson i s c a l l e d the Higgs boson. The f i e l d ^ has d i s a p p e a r e d from the L a g r a n g i a n ; i t has been " e a t e n " by the gauge f i e l d , 21 which s u b s e q u e n t l y p i c k s up a mass, e^. T h i s example i l l u s t r a t e s how a gauge f i e l d can a c q u i r e mass th r o u g h the Higgs mechanism. Having s u r v e y e d the c o n c e p t s of spontaneous symmetry b r e a k i n g and the Higgs mechanism, we now b r i e f l y r e v iew the SU(2)XU(1) t h e o r y of e l e c t r o w e a k i n t e r a c t i o n s . I t i s p o s t u l a t e d t h a t the e l e c t r o w e a k i n t e r a c t i o n s a r e d e s c r i b e d by the gauge group SU(2)XU(1). SU(2) i s the group of 2X2 u n i t a r y m a t r i c e s of d e t e r m i n a n t 1. I t has t h r e e g e n e r a t o r s . The s t r u c t u r e c o n s t a n t s of SU(2) a r e the fe^bc' the t o t a l l y a n t i s y m m e t r i c L e v i - C i v i t a t e n s o r s . The p a r t i c l e c o n t e n t i s as f o l l o w s . L e f t - h a n d e d f e r m i o n s a r e a s s i g n e d t o d o u b l e t r e p r e s e n t a t i o n s of the weak i s o s p i n SU(2) group. Right-handed f e r m i o n s a r e a s s i g n e d t o s i n g l e t s . R ight-handed n e u t r i n o s a r e e x c l u d e d i n the m i n i m a l model. The g e n e r a t o r of the U(1) group, Y, i s c a l l e d weak hypercharge and i s a s s i g n e d t o each p a r t i c l e a c c o r d i n g t o the r u l e below. That i s , Q= T 3 + Y / X (2.2.23-) Here, Q i s the e l e c t r i c charge and T 3 i s the t h i r d component of weak i s o s p i n . The f o l l o w i n g n o t a t i o n of Cheng and L i [54] i s adopted h e r e . (0 -aJ) (2.2.24) 1 The numbers i n p a r e n t h e s e s g i v e the t r a n s f o r m a t i o n p r o p e r t i e s under the SU(2) and U(1) groups; t h a t i s , they a r e the weak i s o s p i n and h y p e r c h a r g e , r e s p e c t i v e l y . The s u b s c r i p t OL 22 r e p r e s e n t s f l a v o u r and i i s the S U ( 2 ) index ( i = H / 2 ) . The s u b s c r i p t s L and R c o r r e s p o n d t o l e f t - and r i g h t - h a n d e d p r o j e c t i o n s , r e s p e c t i v e l y . For some f i e l d 1j)r (2.2.25) l M i ( n - ( 2 . 2 . 2 6 ) The m a t r i x K 5 i s d e f i n e d i n Appendix A. In o r d e r t o w r i t e an S U ( 2 ) X U ( 1 ) i n v a r i a n t L a g r a n g i a n , f o u r gauge f i e l d s must be i n t r o d u c e d , c o r r e s p o n d i n g t o the f o u r g e n e r a t o r s of t h e group. The t h r e e gauge f i e l d s c o r r e s p o n d i n g t o the S U ( 2 ) g e n e r a t o r s w i l l be denoted by A^_(x) and we denote the U(1) gauge f i e l d as B^_(x). Here, yt<- i s the L o r e n t z index and a=1,2,3 i s the S U ( 2 ) i n d e x . F i n a l l y , a Higgs S U ( 2 ) d o u b l e t i s a l s o i n t r o d u c e d . <t' < M (2.2.27) The S U ( 2 ) and U(1) r e p r e s e n t a t i o n s a r e i n d i c a t e d i n p a r e n t h e s e s . The g e n e r a l S U ( 2 ) \ U ( 1 ) i n v a r i a n t L a g r a n g i a n which can be c o n s t r u c t e d from t h e s e elements i s g i v e n below. 23 ^ < f l ^ t i , L ( p _ ^ R) (2.2.28) where ^ = ^ A * V - ^ A % M € ^ C A ^ A' (2.2.29) = ^ B y - ^ V B ^ (2.2.30) In the e q u a t i o n s above, g and g' a r e the gauge c o u p l i n g s a s s o c i a t e d w i t h SU(2) and U ( 1 ) , r e s p e c t i v e l y . The f i r s t l i n e of e q u a t i o n (2.2.28) g i v e s the pure gauge f i e l d s p a r t of the i n t e r a c t i o n . Next the gauge i n v a r i a n t k i n e t i c energy terms f o r the f e r m i o n s a r e g i v e n . The Higgs p a r t of the L a g r a n g i a n and the Yukawa c o u p l i n g s of the Higgs and f e r m i o n s complete the L a g r a n g i a n . There a r e no gauge f i e l d mass terms or l e p t o n mass terms i n c l u d e d s i n c e t h e s e would v i o l a t e the l o c a l SU(2)XU(1) symmetry. 24 The n e u t r a l component of the Higgs f i e l d g e t s a nonzero vev f o r the case of ju-2<0 0' W o ' / J I ! ] (2.2.31) where Vx = yu!~ /X - (2.2.32) T h i s b reaks the SU(2)xU(1) symmetries as d e s c r i b e d above. The Higgs d o u b l e t can be p a r a m e t r i z e d as f o l l o w s . (2.2.33) a. I The T*- (a=1,2,3) a r e the SU(2) g e n e r a t o r s . Under an SU(2) t r a n s f o r m a t i o n w i t h the f i e l d s t r a n s f o r m as f o l l o w s . (2.2.34) (f) — (p' = (qr°pU^ (2.2.35) t - a ; — r A / - - u © [ T ^ ; - ^ u " , ^ u l u , r a ( 2 . 2 . 3 7 ) B ^ — = (-2.2.38) " i ^ = (2.2.39) The r e s u l t s a r e t h a t the s c a l a r H iggs f i e l d g e t s a mass ^-2jj}\ The f o l l o w i n g c o m b i n a t i o n s of gauge f i e l d s d i a g o n a l i z e s the mass 25 m a t r i x of the gauge bosons. + (2.2.40) 7 7^ (2.2.41) and y i e l d masses (2.2.42) (2.2.43) r e s p e c t i v e l y . The f o u r t h gauge boson (2.2.44) remains m a s s l e s s and c o r r e s p o n d s t o the photon. The charged f e r m i o n s a l s o a c q u i r e mass. The mass e i g e n s t a t e s a r e o b t a i n e d upon d i a g o n a l i z i n g the m a t r i x . For i n s t a n c e , f o r the case of o n l y one l e p t o n f a m i l y , t h i s i s t r i v i a l and the s i n g l e SL f i e l d i s the mass e i g e n s t a t e w i t h e i g e n v a l u e The a c q u i s i t i o n of a fe r m i o n mass v i a the Higgs mechanism i s i l l u s t r a t e d s y m b o l i c a l l y i n F i g u r e 2. Now i t i s c l e a r why the n e u t r i n o remains m a s s l e s s i n m i n i m a l S U ( 2 ) X U ( 1 ) . T h i s i s b a s i c a l l y b u i l t i n t o the t h e o r y . The r i g h t - h a n d e d n e u t r i n o has been e x c l u d e d from i t s p a r t i c l e c o n t e n t so t h e r e i s no Yukawa term f o r t h e n e u t r i n o s . T h e r e f o r e , the n e u t r i n o s cannot p i c k up (2.2.45) 26 9 «v\ F i g . 2: Fermion Mass C o n t r i b u t i o n due t o Higgs D o u b l e t a mass term v i a the Hi g g s mechanism t h e way t h e c h a r g e d l e p t o n s do. I t i s a l s o easy t o see, a t t h i s p o i n t , how t h e SU(2)xu(1) t h e o r y reduces t o an e f f e c t i v e V-A weak i n t e r a c t i o n t h e o r y . In terms of the v e c t o r boson mass e i g e n s t a t e s , the o r i g i n a l f e r m i o n k i n e t i c energy terms c o n t a i n , f o r example, one p a r t c o r r e s p o n d i n g t o the weak i n t e r a c t i o n . The c h a r g e d weak c u r r e n t f o r the l e p t o n i c s e c t o r i s ^ ^ i (2.2.47) The n e u t r a l c u r r e n t f o r t h i s s e c t o r i s (2.2.48) For momenta much s m a l l e r than M w ( M x ) , t h e s e i n t e r a c t i o n s c l e a r l y reduce t o e f f e c t i v e f o u r - f e r m i o n c u r r e n t i n t e r a c t i o n s . 27 i f - ( 3 ^ T t (2.2.49) ^ -z- (2.2.50) where 7 X r P \ ^ (2.2.51) Here, G F i s the f e r m i or weak i n t e r a c t i o n c o n s t a n t . In the W boson mediated muon decay, the p r o p o g a t o r of the i n t e r m e d i a t e boson reduces t o Qj^/M^J , ^ o r m o m e n t u m t r a n s f e r much s m a l l e r than M w, and t h i s r e d u c t i o n f o l l o w s . We t u r n now t o the q u e s t i o n of n e u t r i n o mass. The g e n e r a l form of a n e u t r i n o mass term i s £ M = - i ^ u + & V~i V f c +" < ^ ^ r + U-C^ (2.2.52) Here v c i s the f i e l d c o n j u g a t e t o v. That i s , C T v * (2.2.53) where C i s the charge c o n j u g a t i o n o p e r a t o r . D e t a i l s a r e s u p p l i e d i n Appendix B;. The f i r s t two terms i n e q u a t i o n (2.2.52) a r e Majorana mass terms; i f y C = V (2.2.54) they v i o l a t e l e p t o n number c o n s e r v a t i o n by two u n i t s . The l a s t term i s an o r d i n a r y D i r a c mass term such as t h a t f o r the charged 28 l e p t o n s . In m i n i m a l S U ( 2 ) X U ( 1 ) as d e s c r i b e d above the n e u t r i n o does not p i c k up a D i r a c mass term v i a the H iggs mechanism s i n c e the r i g h t - h a n d e d component of the n e u t r i n o f i e l d i s e x c l u d e d . S i n c e SU(2)XU(1) p o s s e s s e s a g l o b a l symmetry c o r r e s p o n d i n g t o l e p t o n number (L) c o n s e r v a t i o n , a Majorana mass term i s a l s o f o r b i d d e n i n the m i n i m a l t h e o r y . Cheng and L i [54] have i n v e s t i g a t e d some e x t e n s i o n s of the m i n i m a l SU(2)XU(1) t h e o r y , which can accommodate a n e u t r i n o mass. There a r e two b a s i c ways of a c h i e v i n g a n e u t r i n o mass. The m i n i m a l t h e o r y can be e n l a r g e d i n the Higgs s e c t o r or i n the f e r m i o n > s e c t o r . A D i r a c mass term can o n l y occur i f the r i g h t -handed component of the n e u t r i n o f i e l d i s not e x c l u d e d . T h e r e f o r e , enlargement i n the Higgs s e c t o r a l o n e w i l l g e n e rate o n l y Majorana mass terms. Thus t h i s enlargement must n e c e s s a r i l y v i o l a t e l e p t o n number c o n s e r v a t i o n . We d i s c u s s examples of enlargement i n the Higgs s e c t o r f i r s t . In the H iggs s e c t o r , the a d d i t i o n of e i t h e r an SU(2) t r i p l e t H, a s i n g l y charged s i n g l e t ~L* , or a d o u b l y charged s i n g l e t H + + can r e s u l t i n a mass term f o r the n e u t r i n o s . For the Higgs t r i p l e t , t he f o l l o w i n g a d d i t i o n a l Yukawa c o u p l i n g of the Higgs and the l e p t o n s i s a l l o w e d . (2.2.55) (2.2.56) 29 Here, f ^  i s a n t i s y m m e t r i c . When the n e u t r a l component of the Higgs t r i p l e t d e v e l o p s a vev such t h a t < T • W > 0 =' W„ O j ( 2 . 2 . 5 7 ) the Yukawa H i g g s - l e p t o n c o u p l i n g ( 2 . 2 . 5 6 ) r e s u l t s i n a Majorana mass term f o r t h e n e u t r i n o s g i v e n by d . t ^ 1 \ L ( 2 . 2 . 5 8 ) where = T ^ ^ b - ( 2 . 2 . 5 9 ) I f a s i n g l y c h a r g e d Higgs s i n g l e t i s added t o the mini m a l SU(2)x>U(1) model, the a l l o w e d Yukawa c o u p l i n g i s g i v e n by <£ y - i a c L ^ b j L ^ £ L J +in.c. ( 2 . 2 . 6 0 ) where f ^ b and e.- a r e a n t i s y m m e t r i c . T h i s model was s t u d i e d by Zee [ 5 5 ] , Of c o u r s e , the s i n g l e t H i ggs cannot d e v e l o p a vev s i n c e i t i s char g e d . T h e r e f o r e , something more i s needed t o ge n e r a t e a mass f o r the n e u t r i n o s . I f a second Higgs d o u b l e t i s added, then the f o l l o w i n g t r i l i n e a r s c a l a r c o u p l i n g i s a l l o w e d . fm. <P.L 7 T e C j ( 2 . 2 . 6 1 ) where (p, and (p 2 a r e the two Higgs d o u b l e t s . T h i s c o u p l i n g , when 30 tak e n w i t h t h e H i g g s - l e p t o n Yukawa c o u p l i n g , does not c o n s e r v e l e p t o n number. T h e r e f o r e , n e u t r i n o s can a c q u i r e a Majorana mass thr o u g h a one l o o p diagram such as t h a t shown i n F i g u r e 3. Q « P > o > / «— 1 « ^ — < — v. ( 0 + v c F i g . 3: N e u t r i n o Majorana Mass C o n t r i b u t i o n due t o Higgs S i n g l e t jv* J u s t as i n the l a s t c a s e , the a d d i t i o n of a d o u b l y c h a r g e d H i g g s s i n g l e t ii** i s not enough t o g e n e r a t e a n e u t r i n o mass; i t cannot a c q u i r e a vev. However, a n e u t r i n o mass can be g e n e r a t e d t h r o u g h l o o p diagrams i f the H i g g s s e c t o r i s f u r t h e r e n l a r g e d i n such a way as t o a l l o w some t r i l i n e a r s c a l a r c o u p l i n g which v i o l a t e s l e p t o n number c o n s e r v a t i o n when j o i n e d w i t h the l e p t o n -H i ggs Yukawa c o u p l i n g . The s i m p l e s t way t o g e n e r a t e n e u t r i n o mass terms by expanding t h e f e r m i o n s e c t o r of m i n i m a l SU(2)XU(1) i s t o add a r i g h t - h a n d e d n e u t r i n o s i n g l e t f o r each f a m i l y . Now, a bare Majorana mass term i s not r u l e d out by the SU(2)*U(1) symmetry. A l s o , the n e u t r i n o s p i c k up a D i r a c mass term v i a t h e H i g g s d o u b l e t mechanism j u s t as the c h a r g e d l e p t o n s do. That i s , the L a g r a n g i a n 31 (2.2.62) becomes, upon t h e Higgs d e v e l o p i n g a vev, ir /fai 6 ^ < ^ + V. c_ . (2.2.63) where <£h, = V ( i A [ 2 ) f O^b * The D i r a c mass term i s i l l u s t r a t e d s y m b o l i c a l l y i n F i g u r e 2. A c o m b i n a t i o n of the p o s s i b i l i t i e s d e s c r i b e d above w i l l l e a d t o a n e u t r i n o mass term of the g e n e r a l form g i v e n i n e q u a t i o n ( 2 . 2 . 5 2 ) . A l s o , the a d d i t i o n of both a r i g h t - h a n d e d n e u t r i n o f i e l d and a n e u t r a l Higgs s i n g l e t fu° l e a d s t o a Majorana mass term when the Higgs p i c k s up a vev. In grand u n i f i e d t h e o r i e s , a s i n g l e gauge group G (or a d i r e c t p r o d u c t such as GxG) d e s c r i b e s the u n i f i c a t i o n of the s t r o n g SU(3) t h e o r y and e l e c t r o w e a k SU(2) U ( 1 ) . There i s o n l y one gauge c o u p l i n g . The c o u p l i n g s f o r each of the t h r e e low energy symmetry groups a r e r e l a t e d t o t h a t of the grand u n i f i c a t i o n group G. There i s a g r e a t d e a l of leeway i n c h o o s i n g the H i g g s s e c t o r of GUTS models. That i s , Higgs f i e l d s can be i n t r o d u c e d which t r a n s f o r m a c c o r d i n g t o some c o m p l i c a t e d r e p r e s e n t a t i o n of G and, t h u s , can almost c e r t a i n l y i n t r o d u c e Majorana mass terms f o r the n e u t r i n o s . A l s o , some groups (eg.SO(lO)) [56,57] n a t u r a l l y accomodate a r i g h t - h a n d e d n e u t r i n o (2.2.64) 32 f i e l d , a l l o w i n g a D i r a c mass term as w e l l as a bare Majorana mass term. However, the m i n i m a l SU(5) GUT [1-3] s i m p l y does not have room i n the p a r t i c l e r e p r e s e n t a t i o n s f o r a r i g h t - h a n d e d n e u t r i n o . A l s o , i n m i n i m a l S U ( 5 ) , baryon number minus l e p t o n number (B-L) i s c o n s e r v e d . (Baryons a r e c o n v e n t i o n a l l y a s s i g n e d B=+1 w h i l e a n t i b a r y o n s have B=-1 and a l l . o t h e r p a r t i c l e s have B=0. S i m i l a r l y , l e p t o n s c a r r y l e p t o n number L of +1, L=-1 f o r a n t i l e p t o n s and L=0 f o r o t h e r p a r t i c l e s . ) T h e r e f o r e , a Majorana mass term f o r the n e u t r i n o (which v i o l a t e s B-L) i s f o r b i d d e n . Thus, i n m i n i m a l SU(5), the n e u t r i n o s remain m a s s l e s s . Of c o u r s e , the m i n i m a l model can be extended by the a d d i t i o n of a r i g h t - h a n d e d n e u t r i n o which t r a n s f o r m s as an SU(5) s i n g l e t or of new H i g g s f i e l d s . Thus, a n e u t r i n o mass can be i n c o r p o r a t e d i n t o SU(5) j u s t as i t can be i n t o S U ( 2 ) x u ( 1 ) . C l e a r l y , mechanisms e x i s t f o r g e n e r a t i n g n e u t r i n o masses. We next d i s c u s s the q u e s t i o n of the magnitude of the mass. In the SU(2)XU(1) model, the charged f e r m i o n masses a r e a r b i t r a r y . In i t s e x t e n s i o n s , the magnitudes of the r e s u l t i n g n e u t r i n o mass terms a r e a l s o a r b i t r a r y . The same i s t r u e f o r the case of the GUT SU(5) extended t o i n c l u d e n e u t r i n o masses. In l e f t - r i g h t symmetric models such as SU(2) UA SU(2) RK U(1) or the GUT SO(10) a r i g h t - h a n d e d n e u t r i n o can be accommodated n a t u r a l l y . In SO(10), f o r i n s t a n c e , the H iggs mechanism t y p i c a l l y y i e l d s the u n a c c e p t a b l e s i t u a t i o n of e q u a l D i r a c mass terms f o r the u q u arks and the n e u t r i n o s [ 5 8 ] . There a r e some ways of a c h i e v i n g s m a l l p h y s i c a l masses. One p o s s i b i l i t y i s t h a t 33 the H i g g s vev t h a t y i e l d s the D i r a c n e u t r i n o mass i s v e r y s m a l l [ 5 8 ] . T h i s can occur i f the n e u t r i n o mass i s g e n e r a t e d by a d i f f e r e n t component of the Higgs f i e l d than the o t h e r f e r m i o n masses. ( T h i s o c c u r s f o r t h e (120) of Higgs i n SO(10).) T h i s i s r a t h e r a r t i f i c i a l and Langacker [3] p o i n t s out t h a t the scheme c o u l d be upset by d i v e r g e n t r a d i a t i v e c o r r e c t i o n s ' [ 5 8 ] . Other p o s s i b i l i t i e s f o r e n s u r i n g a s m a l l p h y s i c a l n e u t r i n o mass a r e based on the g e n e r a t i o n of a v e r y l a r g e Majorana mass f o r the r i g h t - h a n d e d weak s i n g l e t n e u t r i n o . D i a g o n a l i z a t i o n of the mass m a t r i x f o r t h i s s i t u a t i o n of a v e r y l a r g e r i g h t - h a n d e d Majorana mass and a t y p i c a l D i r a c mass y i e l d s an a c c e p t a b l y s m a l l v a l u e f o r t he p h y s i c a l n e u t r i n o mass. We e x p l a i n t h i s mechanism i n more d e t a i l below but f i r s t note t h a t the v a r i o u s schemes d i f f e r i n t h e i r means of g e n e r a t i o n of t h e l a r g e Majorana mass and, t h u s , i n t h e i r f i n a l e s t i m a t e f o r the p h y s i c a l n e u t r i n o mass. To u n d e r s t a n d t h i s mechanism of e n s u r i n g a s m a l l p h y s i c a l n e u t r i n o mass, we must r e t u r n t o the g e n e r a l n e u t r i n o mass term of the L a g r a n g i a n (2.2.52). I t i s i m p o r t a n t t o note the fermi o n c o n t e n t . We w i l l p r o c e ed i n terms of a s i n g l e l e p t o n f a m i l y f o r s i m p l i c i t y . Both i / L and v£ a r e members of SU(2) d o u b l e t s w h i l e and i/£ a r e weak s i n g l e t s . Thus, the D i r a c mass term c o n n e c t s a r i g h t - h a n d e d s i n g l e t and a l e f t - h a n d e d member of a d o u b l e t , j u s t as f o r charged f e r m i o n s . The term i n 6, which we w i l l c a l l t he r i g h t - h a n d e d Majorana mass term, c o n n e c t s l e f t and r i g h t -handed s i n g l e t s . We saw t h a t t h i s term can appear as a bare mass term i f B-L i s not imposed. The term i n (X ( l e f t - h a n d e d Majorana mass term) c o n n e c t s l e f t - and r i g h t - h a n d e d members of weak 3 4 d o u b l e t s . These terms o c c u r , as we have shown, f o r c e r t a i n e x p a n s i o n s i n the Higgs s e c t o r . The most g e n e r a l mass m a t r i x f o r one f a m i l y i s The l a t t e r ( s m a l l ) e i g e n v a l u e c o r r e s p o n d s t o the mass of the o r d i n a r y p h y s i c a l n e u t r i n o . G e l l - M a n n , Ramond, and S l a n s k y [ 5 9 ] proposed i n t r o d u c i n g a Higgs r e p r e s e n t a t i o n which breaks B-L (a ( 1 2 6 ) i n S O ( 1 0 ) ) a t a l a r g e mass s c a l e i n a GUT, a l l o w i n g a l a r g e r i g h t - h a n d e d Majorana mass term. T h e i r scheme t y p i c a l l y y i e l d s n e u t r i n o masses of the o r d e r of 1 0 " 5 e V / c 2 . W i t t e n [ 6 0 ] suggested a scheme of symmetry b r e a k i n g i n S O ( 1 0 ) v i a a s p i n o r Higgs r e p r e s e n t a t i o n f o r which t h e r e i s no t r e e l e v e l Majorana mass but a l a r g e r i g h t - h a n d e d Majorana mass o c c u r s a t the two l o o p l e v e l . H i s scheme y i e l d s p h y s i c a l n e u t r i n o masses of the o r d e r of e V / c 2 . Mohapatra and S e n j a n o v i c [ 6 1 , 6 2 ] o b t a i n n e u t r i n o masses of the o r d e r of MeV/c 2 i n t h e i r l e f t - r i g h t symmetric t h e o r y S U ( 2 ) L A S U ( 2 ) K U ( 1 ). They i n t r o d u c e two Higgs f i e l d s A L and which a r e t r i p l e t s under S U ( 2 ) U and S U ( 2 ) R , r e s p e c t i v e l y , as w e l l as an S U ( 2 ) L X S U ( 2 ) K Higgs d o u b l e t . They break the p a r i t y ( 2 . 2 . 6 4 ) I f the Higgs s e c t o r i s not extended so t h a t ( l i s z e r o and i f $ i s a t y p i c a l D i r a c mass term (^  1 G e v / c 2 ) w h i l e then the e i g e n v a l u e s of the mass m a t r i x a r e , a p p r o x i m a t e l y , & and (ft2/&). 35 and B-L symmetries by l e t t i n g p i c k up a nonzero vev of the o r d e r of the mass of the r i g h t - h a n d e d gauge boson W^ . A lower bound on t h i s mass from charged [63] and n e u t r a l [64] c u r r e n t e x p e r i m e n t s i s about 300 GeV/c 2. Then the mass of the o r d i n a r y l i g h t n e u t r i n o s i s i n v e r s e l y p r o p o r t i o n a l t o t h i s mass and, i n one c l a s s of models which Mohapatra and S e n j a n o v i c d i s c u s s , t a k e s the form Tn.yLo<. -m^ / -vrb w^ (2.2.65) T h i s g i v e s , f o r example, a t a u n e u t r i n o mass of the o r d e r of Mev/c 2. • These t h e o r e t i c a l e s t i m a t e s f o r the magnitude of the n e u t r i n o mass a r e v e r y l o o s e , r a n g i n g from 10" 5 e V / c 2 t o Mev/c 2. The q u e s t i o n of f e r m i o n masses i s not a t a l l w e l l u n d e r s t o o d . Thus, i t i s i m p o r t a n t t o study n e u t r i n o mass i n e x p e r i m e n t a l l y a c c e s s i b l e s i t u a t i o n s . 36 2.3 The E f f e c t i v e Four-Fermion Theory of Weak I n t e r a c t i o n s We saw i n the l a s t s e c t i o n how the S U ( 2 ) A U ( 1 ) gauge model of e l e c t r o w e a k i n t e r a c t i o n s reduced, i n the l i m i t of momentum t r a n s f e r much s m a l l e r than weak v e c t o r boson mass, t o an e f f e c t i v e V-A f o u r - f e r m i o n t h e o r y of weak i n t e r a c t i o n s . T h i s was not the h i s t o r i c r o u t e t o the e f f e c t i v e f o u r - f e r m i o n t h e o r y . O r i g i n a l l y , Fermi [14] wrote down h i s L a g r a n g i a n f o r ^6-decay i n anal o g y w i t h quantum e l e c t r o d y n a m i c s . He r e p l a c e d the v e c t o r c u r r e n t (je.™^. i n the qed L a g r a n g i a n ( « j a ^ A ) by a weak v e c t o r c u r r e n t ( j w ) ^ . The r o l e of the e m i t t e d photon i n an e l e c t r o m a g n e t i c p r o c e s s was g i v e n t o the e l e c t r o n - a n t i n e u t r i n o p a i r of the weak decay. Thus he wrote down a l o c a l ( p o i n t ) f o u r -f e r m i o n v e c t o r i n t e r a c t i o n . For muon decay, the more g e n e r a l such L a g r a n g i a n which i s i n v a r i a n t under p r o p e r L o r e n t z t r a n s f o r m a t i o n s i s g i v e n below. &°tl\v,yrU%ri%i-^f)^-] + u . c . , 2 . 3 . , , The 1jJ a r e the f i e l d o p e r a t o r s f o r the i n t e r a c t i n g p a r t i c l e s and a r e g i v e n i n terms of c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s i n Appendix A. The g and g' a r e c o u p l i n g c o n s t a n t s . The summation e x t e n d s over s c a l a r ( S ) , v e c t o r ( V ) , t e n s o r ( T ) , a x i a l v e c t o r ( A ) , and p s e u d o s c a l a r (P) c o u p l i n g s . These a r e r e p r e s e n t e d by the s i x t e e n l i n e a r l y independent 4x4 m a t r i c e s l i s t e d below. 37 P In the L a g r a n g i a n ( 2 . 3 . 1 ) , the u s u a l c o n t r a c t i o n and summation of L o r e n t z i n d i c e s i s t o be assumed f o r the m a t r i c e s ("^  . I t i s merely c o n v e n t i o n t o w r i t e the c o u p l i n g index i as a s u b s c r i p t . That th e s e o p e r a t o r s a r e indeed l i n e a r l y independent can be seen i n the f o l l o w i n g way. C o n s i d e r some l i n e a r s u p e r p o s i t i o n of the s i x t e e n o p e r a t o r s such t h a t Now m u l t i p l y i n g w i t h one of the s i x t e e n m a t r i c e s P and t a k i n g the t r a c e , L ^ T r ( P m = o i - i The s i x t e e n m a t r i c e s P a l l have v a n i s h i n g t r a c e , except f o r the i d e n t i t y , P . In f a c t , the o n l y p r o d u c t s of two of t h e s e m a t r i c e s f o r which the t r a c e i s not z e r o are th o s e f o r which i = j . S i n c e P P =1 ( i = 1 , . . . , 1 6 ) , t h e l a s t e q u a t i o n becomes Thus a l l the c o e f f i c i e n t s c v a n i s h and the P a r e , i n d e e d , 38 l i n e a r l y independent. The L a g r a n g i a n (2.3.1) i s i n the s o - c a l l e d " c u r r e n t -c u r r e n t " form. The p a r t i c u l a r form g i v e n i n e q u a t i o n (2.3.1) i s a p r o d u c t of two n e u t r a l c u r r e n t s ; t h i s i s the s o - c a l l e d charge r e t e n t i o n form of the f o u r - f e r m i o n weak i n t e r a c t i o n L a g r a n g i a n . We w i l l show l a t e r t h a t i t can be r e w r i t t e n i n another form (charge exchange) and the l i n e a r independence of the H w i l l be i m p o r t a n t i n r e l a t i n g the two forms. Some d i s c u s s i o n of the c o u p l i n g s g and g' i s i n o r d e r . A f t e r P a u l i had proposed the e x i s t e n c e of a n e u t r i n o i n 1932 [ 6 5 ] , Fermi [14] wrote down h i s L a g r a n g i a n t o d e s c r i b e - d e c a y i n t he c u r r e n t - c u r r e n t form of e q u a t i o n ( 2 . 3 . 1 ) . In analogy w i t h quantum e l e c t r o d y n a m i c s , he i n c l u d e d o n l y the v e c t o r i n t e r a c t i o n . That i s , o n l y g v was n o n v a n i s h i n g . T h i s was the v e c t o r c u r r e n t - v e c t o r c u r r e n t ( W ) weak L a g r a n g i a n . In 1936, Gamow and T e l l e r [66] noted t h a t SS, AA, TT, and PP i n t e r a c t i o n s s h o u l d a l s o be i n c l u d e d i n a more g e n e r a l form of the weak i n t e r a c t i o n L a g r a n g i a n . At t h i s p o i n t , the L a g r a n g i a n was p a r i t y - c o n s e r v i n g . The weak i n t e r a c t i o n was f u r t h e r g e n e r a l i z e d t o t he form of e q u a t i o n (2.3.1) by Feynman and Gell-Mann [ 6 7 ] . T h i s form now i n c l u d e s , f o r i n s t a n c e , VA type i n t e r a c t i o n s and i s , t h u s , no l o n g e r p a r i t y c o n s e r v i n g . For a p a r i t y c o n s e r v i n g i n t e r a c t i o n , such as d e s c r i b e d by F e r m i ' s o r i g i n a l weak L a g r a n g i a n , e i t h e r a l l of the g- or a l l of the gj must v a n i s h . The g L and g^, b e i n g complex numbers, form a s e t of 20 r e a l p a r a m e t e r s . One o v e r a l l phase i s a r b i t r a r y , l e a v i n g 19 parameters t o be det e r m i n e d by ex p e r i m e n t . The energy spectrum 3 9 of the p o s i t r o n s from yti* decay ( M i c h e l spectrum) has been c a l c u l a t e d f o r the case of m a s s l e s s n e u t r i n o s p r e v i o u s l y and e x p r e s s e d i n terms of the f o l l o w i n g parameters [ 6 8 , 6 9 ] . $ = ~ ^ ( 3 ( b V A * l o A V ^ + 6 b T T") where K = a „ + a k ? p t H-(a v„ + a A A W 6 a T T The parameters ^>,7 ,^^ , and 6 a r e c a l l e d the M i c h e l p a r a m e t e r s . The e x p r e s s i o n s above f o r them were o b t a i n e d w i t h o u t imposing any symmetries on the L a g r a n g i a n . The ^ and 7^  parameters appear i n the i s o t r o p i c p a r t of the M i c h e l spectrum w h i l e ^ and S appear i n the a n g u l a r dependence. ^ i s s e n s i t i v e t o the h i g h energy end of the p o s i t r o n spectrum as ^ i s t o the low energy end. ^, as an o v e r a l l f a c t o r m u l t i p l y i n g the angle-dependent p a r t of the spectrum, g i v e s an energy-averaged asymmetry w h i l e 6 g i v e s the energy dependence of the asymmetry. E x p e r i m e n t s on muon decay have been m a i n l y f o c u s s e d on measurement of the M i c h e l p a r a m e t e r s . There a r e e x p e r i m e n t s 40 under way a t the meson f a c t o r i e s t o measure p r e c i s e l y the ^  and ^ p a r a m e t e r s [ 7 0 , 7 1 ] . However, the f o u r M i c h e l parameters a r e s u f f i c i e n t l y c o m p l i c a t e d t h a t i t i s d i f f i c u l t t o i n t e r p r e t what a g i v e n measurement i m p l i e s i n terms of the b a s i c c o u p l i n g c o n s t a n t s . There a re not enough measurable q u a n t i t i e s t o determine a l l 19 r e a l p a r a m e t e r s , anyway. The s i t u t a t i o n can be somewhat s i m p l i f i e d by imposing some symmetry r e s t r i c t i o n s on the L a g r a n g i a n . For i n s t a n c e , time r e v e r s a l i n v a r i a n c e i m p l i e s t h a t a l l of the c o u p l i n g c o n s t a n t s can be taken t o be r e a l , l e a v i n g o n l y t e n r e a l parameters t o be d e t e r m i n e d . A l s o , i f the f o l l o w i n g r a t h e r a r b i t r a r y a ssumptions a r e imposed then the L a g r a n g i a n t a k e s on the f a m i l i a r V-A form. £ = ^ [ ^ e Y ? ( l ^ W p C ^ r O - Y s ^ J (2.3.2) We saw i n the l a s t s e c t i o n how the SU(2)*U(1) t h e o r y reduced, i n the l i m i t of momenta much s m a l l e r than the weak boson mass, t o a V-A form. T h i s happened because the l e p t o n s were a s s i g n e d t o h e l i c i t y s t a t e s . For the case of t h i s V-A i n t e r a c t i o n , the M i c h e l parameters take the v a l u e s below. The c a l c u l a t i o n s of muon decay w i t h massive n e u t r i n o s which f o l l o w t h i s s e c t i o n use the V-A form of the L a g r a n g i a n t o 41 d e s c r i b e the weak i n t e r a c t i o n . T h i s e f f e c t i v e L a g r a n g i a n i s not r e n o r m a l i z a b l e . The SU(2)X"U(1) gauge t h e o r y of e l e c t r o w e a k i n t e r a c t i o n s , which was r e v i e w e d b r i e f l y i n the l a s t s e c t i o n r e s o l v e s the problem of r e n o r m a l i z a b i l i t y . That t h e o r y i s not a l o c a l t h e o r y , but r a t h e r the weak i n t e r a c t i o n s a r e t h e r e i n mediated by i n t e r m e d i a t e v e c t o r bosons. As was p o i n t e d out i n Chapter 1, f o r the case of muon decay w i t h m a s s l e s s n e u t r i n o s , where a l l the e n e r g i e s of the p a r t i c l e s a r e much s m a l l e r than the i n t e r m e d i a t e v e c t o r boson masses, the d i f f e r e n c e between the r e s u l t s of an e f f e c t i v e V-A c a l c u l a t i o n and a f u l l SU(2)XU(1) c a l c u l a t i o n were found t o be e x p e r i m e n t a l l y i n s i g n i f i c a n t . T h i s r e s u l t i s e x p e c t e d t o be t r u e f o r the case of massive n e u t r i n o s , as w e l l . A l s o , as noted i n Chapter 1, the aim here i s t o study t h e e f f e c t s of massive n e u t r i n o s and t h e i r m i x i n g s on muon decay r a t h e r than t o probe the s t r u c t u r e of the weak i n t e r a c t i o n . T h e r e f o r e , the e f f e c t i v e V-A t h e o r y of weak i n t e r a c t i o n s i s adopted h e n c e f o r t h . I t was mentioned e a r l i e r i n t h i s s e c t i o n t h a t the g e n e r a l weak L a g r a n g i a n (2.3.1) c o u l d be r e w r i t t e n i n a n o t h e r form. T h i s s o - c a l l e d c h a r ge exchange form i s g i v e n below. £ = % [ % r r t ^ ] [ f e r (5.-3/ v>% e3 v w c . (2.3.3) I t i s o b t a i n e d upon p e r f o r m i n g a F i e r z t r a n s f o r m a t i o n [15,72] on the charge r e t e n t i o n form of the L a g r a n g i a n . T h i s r e f e r s t o i n t e r c h a n g i n g two of the i n t e r a c t i n g p a r t i c l e s and amounts t o a l i n e a r t r a n s f o r m a t i o n of the c o u p l i n g c o n s t a n t s , which i s g i v e n 42 by hi H 6 H \ -a. 0 X -\ 0 -a. o \ 1 0 -oL -\ -4. fo -4 I / where i , J? = S,V,T,A,P ( i n t h a t o r d e r ) The t r a n s f o r m a t i o n s above a r e known as the F i e r z i d e n t i t i e s . They a r e o b t a i n e d as f o l l o w s . C o n s i d e r one term of the L a g r a n g i a n (2.3.1) where Now the can be re g a r d e d as s i x t e e n 4x.4 m a t r i c e s , whose components a r e l a b e l l e d by the i n d i c e s /5 and % . Then, f o r each <*•% , M^yj can be e x p r e s s e d i n terms of the s i x t e e n l i n e a r l y independent m a t r i c e s P as f o l l o w s . J J 43 Then the above term of the L a g r a n g i a n can be r e w r i t t e n i n terms of the A< as J The minus s i g n appears s i n c e the f i e l d o p e r a t o r s a r e a n t i c o m m u t i n g . Upon comparison w i t h the the c o r r e s p o n d i n g term i n the F i e r z - r e a r r a n g e d L a g r a n g i a n ( 2 . 3 . 3 ) , the A can be i d e n t i f i e d w i t h the new c o u p l i n g s g. i n t h i s manner Note t h a t , s i n c e the index j i s not a L o r e n t z i n d e x , no summation i s i m p l i e d . M u l t i p l i n g by one of the VK and t a k i n g the t r a c e y i e l d s 0 The l a s t s t e p f o l l o w s from the p r o p e r t i e s of the t r a c e s of p r o d u c t s of the m a t r i c e s V . R e a r r a n g i n g y i e l d s More e x p l i c i t l y , 44 S u b s t i t u t i n g S,V,T,A, and P f o r k r e s u l t s i n the F i e r z t r a n s f o r m a t i o n m a t r i x g i v e n above. T h i s proves the f i r s t F i e r z i d e n t i t y and the second i d e n t i t y i s o b t a i n e d s i m i l a r l y . For the V-A a s s u m p t i o n s , the r e s u l t s of a F i e r z t r a n s f o r m a t i o n a r e The primed c o n s t a n t s t r a n s f o r m i n the same way. Thus, the charge exchange form of the V-A L a g r a n g i a n i s 1 (~x% + 2-3 A) (2.3.4) C h o i c e of e i t h e r the charge r e t e n t i o n or charge exchange form of the L a g r a n g i a n f o r c a l c u l a t i n g i s merely a matter of c o n v e n i e n c e . 45 I I I . MUON DECAY WITH MASSIVE NEUTRINOS In t h i s C h a p t e r , we make t h e assumption t h a t n e u t r i n o s a r e massive and t h a t the mass and weak n e u t r i n o e i g e n s t a t e s a r e r e l a t e d by a u n i t a r y t r a n s f o r m a t i o n , as d e s c r i b e d i n the I n t r o d u c t i o n . Our assumption of n e u t r i n o mass i s independent of any p a r t i c u l a r model. We c a l c u l a t e the M i c h e l spectrum of the muon decay (1.2) f o r the two c a s e s of D i r a c and Majorana n e u t r i n o s . The s p e c t r a f o r both p o l a r i z e d and u n p o l a r i z e d muons are g i v e n . We s p e c i a l i z e t o the t h r e e - n e u t r i n o s w o r l d and, w i t h the assumption of h i e r a r c h i a l m i x i n g , r e t a i n o n l y one n e u t r i n o as massive enough t o c o n t r i b u t e s i g n i f i c a n t l y . N u m e r i c a l r e s u l t s a r e g i v e n f o r n e u t r i n o mass i n the Mev/c 2 range, i n accordance w i t h the e x p e r i m e n t a l l i m i t s . E l e c t r o n - n e u t r i n o c o r r e l a t i o n s a r e a l s o i n v e s t i g a t e d as a means of d e t e r m i n i n g whether n e u t r i n o s are D i r a c or Majorana p a r t i c l e s . 46 3.1 The Case of D i r a c N e u t r i n o s We c a l c u l a t e , i n t h i s s e c t i o n , the r a t e f o r the muon decay (1.2) f o r the case of massive D i r a c n e u t r i n o s . That i s , the n e u t r i n o s and a n t i n e u t r i n o s a r e d i s t i n g u i s h a b l e p a r t i c l e s . T h i s f a c t can be r e p r e s e n t e d by a s s i g n i n g them d i f f e r e n t v a l u e s of a c o n s e r v e d quantum number c a l l e d l e p t o n number, L. C o n v e n t i o n a l l y , the n e u t r i n o i s a s s i g n e d L=1 w h i l e the a n t i n e u t r i n o has L=-1. The c a l c u l a t i o n s of t h i s s e c t i o n s e r v e as a model f o r l a t e r c a l c u l a t i o n s and, t h u s , a r e p r e s e n t e d i n c o n s i d e r a b l e d e t a i l . We f i r s t t a k e the g e n e r a l case of N l e p t o n f a m i l i e s . The f o l l o w i n g t r a n s i t i o n r a t e i s c a l c u l a t e d f o r muon decay o j . = J < f I £ W M a 4 x I L> (3.1.1) where the i n i t i a l and f i n a l s t a t e s a r e t h o s e of the r e a c t i o n ( 1 . 2 ) . That i s , (3.1.2) (3.1.3) The o p e r a t o r s b' and d 1 a r e c r e a t i o n o p e r a t o r s f o r p h y s i c a l p a r t i c l e s and a n t i p a r t i c l e s , r e s p e c t i v e l y , as p r e s e n t e d i n Appendix A. For i n s t a n c e , bV(p, ,s, ) c r e a t e s a n e u t r i n o of mass m L w i t h momentum p u and s p i n s L . The s t a t e s (3.1.2) and(3.1.3) a r e p h y s i c a l mass e i g e n s t a t e s . The weak L a g r a n g i a n , oh the o t h e r 47 hand, i s w r i t t e n i n terms of the weak e i g e n s t a t e s , the p a r t i c l e s c r e a t e d a t a weak i n t e r a c t i o n v e r t e x . The charge exchange form of t h i s L a g r a n g i a n i s g i v e n i n e q u a t i o n (2.3.4) where, f o r i n s t a n c e , the e l e c t r o n n e u t r i n o f i e l d (weak e i g e n s t a t e ) i s r e l a t e d t o the p h y s i c a l n e u t r i n o f i e l d s by the N X N u n i t a r y t r a n s f o r m a t i o n which s a t i s f y e q u a t i o n s (1.4) and ( 1 . 5 ) . Thus, i n terms of the p h y s i c a l e i g e n s t a t e s , the weak L a g r a n g i a n can be w r i t t e n as I ^ " . . L ^ . t V « 0 - V * v ] t l f c l l t ' ( ' - ^ l ^ - « - - (3.1.5) The t r a n s i t i o n r a t e (3.1.1) can now be c a l c u l a t e d . The f i e l d s a r e g i v e n i n e q u a t i o n ( A . 6 ) . Upon o p e r a t i n g w i t h the f i e l d s on the i n i t i a l and f i n a l s t a t e s and d o i n g the c o o r d i n a t e i n t e g r a t i o n , the r e s u l t i s N (3.1 .4) Here the U^ (a=e,yU,T, ...) a r e the P o n t e c o r v o m i x i n g parameters •j- L (3.1.6) where, i n c l u d i n g a l l p o s s i b l e p h y s i c a l n e u t r i n o ' s t a t e s , (3.1.7) 48 Here, u and v a r e the four-component s p i n o r s r e p r e s e n t i n g p a r t i c l e and a n t i p a r t i c l e s o l u t i o n s , r e s p e c t i v e l y , of the D i r a c e q u a t i o n . (See Appendix A.) In g e n e r a l , the p r o c e s s - " <* + ^ + .. . (3.1.8) i s r e p r e s e n t e d by the t r a n s i t i o n (3.1.9) Here the s u b s c r i p t s " i " and " f " on the four-momenta denote i n i t i a l and f i n a l , r e s p e c t i v e l y . The d i f f e r e n t i a l p r o b a b i l i t y per u n i t time i s g i v e n by L ^ 1-Thus, f o r the p r o c e s s ( 1 . 2 ) , the decay r a t e i s (3.1.10) R = , T ^ ^ t e ( ^ f ^ ^ ( ^ - F ^ I pvyi (3.1.1D 4 9 N o t i c e t h a t the n e u t r i n o s p i n s a r e summed over s i n c e they a r e not d e t e c t e d i n e x p e r i m e n t s . The m a t r i x element ^~ i c i s g i v e n i n e q u a t i o n ( 3 . 1 . 7 ) . The square of t h i s m a t r i x element i s c a l c u l a t e d i n d e t a i l i n Appendix A. The r e s u l t i s 7VL - Z \7VL \ 4 - (^)U^)\a^\\x^ ( p e . ? J y ? ; . p (3.1.12) wherein ' ( p \ " f x + W S ^ ( 3 . 1 . 1 3 ) We r e w r i t e e q u a t i o n ( 3.1.12) f o r c o n v e n i e n c e as n*(%y i . I U ^ I U ^ T V U J ,3.,., 4) A l s o , the L o r e n t z i n v a r i a n c e of the e x p r e s s i o n ( 3.1.11) can be seen more e a s i l y a f t e r u s i n g the f o l l o w i n g f o r m a l t r i c k w i t h 6-f u n c t i o n s . S i n c e .*(£-n^ = ( 3.1.15) the r a t e ( 3.1.11) can be w r i t t e n as 50 21 W J 1 I U , . I X (3.1.16) where Q = j y - f t (3.1.17) The e - f u n c t i o n s ensure t h a t the energy of each f i n a l s t a t e p a r t i c l e i s p o s i t i v e . T h i s f o l l o w s from the d e f i n i t i o n of the 0-f u n c t i o n g i v e n below. (3.1.18) From e q u a t i o n ( 3 . 1 . 1 6 ) , we see t h a t the r a t e i s a c t u a l l y an i n c o h e r e n t sum of N 2 d i f f e r e n t r a t e s , c o r r e s p o n d i n g t o the N 2 c o m b i n a t i o n s of f i n a l s t a t e n e u t r i n o s . In t he muon decay e x p e r i m e n t s performed t o d a t e , the n e u t r i n o s a r e not d e t e c t e d ; hence t h e i r momenta must be i n t e g r a t e d over i n the c a l c u l a t i o n of the muon decay r a t e . The method of d o i n g so f o r t h i s case i s g i v e n h e r e . N o t i c e t h a t the i n t e g r a t i o n t o be performed i n v o l v e s the f o l l o w i n g L o r e n t z s t r u c t u r e . = A Q ^ Q V + & Q V V (3.1.19) 51 U s i n g the o v e r a l l energy-momentum c o n s e r v a t i o n 6 - f u n c t i o n t o i n t e g r a t e over one of the n e u t r i n o ' s four-momentum, we o b t a i n 1^-- j d ^ S ( ^ - m f H ( ( Q - f ^ - v H ^ e ( ^ Q^'^^yi^-y (3.1.20) U s i n g the 6 - f u n c t i o n p r o p e r t y %[tt%S\ = ]W\^ (3.1.21) where x 0 i s the r o o t of f ( x ) , the i n t e g r a t i o n over g i v e s ^ 0 i 0 p L t w^- ^ 0 ( Q. p i X p t r (Q- pO" (3.1.22) The 0 - f u n c t i o n 6 ( p L ) has been used t o ensure t h a t o n l y the p o s i t i v e v a l u e s of c o n t r i b u t e . Now, i n o r d e r t o s o l v e f o r the parameters A and B of e q u a t i o n ( 3 . 1 . 1 9 ) , the two i n t e g r a l s ij? and I 0 0 w i l l be c a l c u l a t e d . Each of the s e y i e l d s an e q u a t i o n i n A and B. F i r s t 1/ = ^ I " V - (A* W G f *(Q L-aQ-fL + ^ < W ^ f J (jVQ-™ c°) (3.1.23) S i n c e i s a L o r e n t z i n v a r i a n t q u a n t i t y , i t can be e v a l u a t e d i n any r e f e r e n c e frame. We choose t o e v a l u a t e i t i n the r e s t frame of the four-momentum Q. I n t h i s frame, e q u a t i o n (3.1.23) becomes Lh^$)0t- ^ Q B i - i Q b E c t < - w v ^ e ( Q 0 - E L V E L Q a - > i ^ (3.1.24) A p p l y i n g e q u a t i o n (3.1.21) t o the 6 - f u n c t i o n and w r i t i n g the 52 measure as d 3 f L = i-jE-t-v^ E L d E t a i i p , (3.1.25) where d£l P L r e p r e s e n t s the a n g u l a r i n t e g r a t i o n , the r e s u l t i s Av4P>= TT A , ' " ( ^ r H ^ v n ^ ( Q " m ^ - M r t 0 ( Q M ^ W ^ (3.1.26) HQ4 where MM.* V~- f • ^ v ^ ' ^ v ( 3 - 1 - 2 7 ) The phase space c o n s i d e r a t i o n s which g i v e the ©-function a r e d e a l t w i t h i n Appendix C. A second e q u a t i o n i n A and B i s found by c a l c u l a t i n g I 0 0 . I 0 0 - (A-*- B") = s( Q*~-2Q-^i +y*?-nj] 6 (Q- p t) £ L( Q*- ^ (3.1.28) E v a l u a t i n g t h i s i n t e g r a l i n t h e r e s t frame of Q y i e l d s t h e f o l l o w i n g e q u a t i o n i n A and B. AtB= % X , / i ( q > ^ | M f - V Q * + v v ^ ^ y n V V Q t - v v ^ ^ ^ ^ ^ ( q ^ U : m r t (3.1.29) S o l v i n g e q u a t i o n s (3.1.26) and (3.1.29) f o r A and B y i e l d s the r e s u l t s below. A- JL x W q ^ ^ w - ' U Q V Q ^ ^ t ^ V i ^ b ^ S l (3.1.30) 53 Of c o u r s e , f o r t h e case of m a s s l e s s n e u t r i n o s , t h e s e r e s u l t s reduce t o the u s u a l v a l u e s [15] of A =. ir/ia. B= tt/a4 ( 3.1.32) S u b s t i t u t i n g back i n t o e q u a t i o n ( 3.1.19) y i e l d s X " * = i ^ q > ^ ^ [ v+ Q^t™?) T i « - ^ r t erg* +[QH- I Q I U ^ > W W - - * ^ ( 3 . 1 . 3 3 ) T h e r e f o r e , • Q 2 " ( f t - p A A e ( Q a ' - ^ i t w ^ l ( 3 . 1 . 3 4 ) For muons d e c a y i n g a t r e s t , the f o l l o w i n g k i n e m a t i c r e l a t i o n s a r e u s e f u l . G f = ^ h - X t S e ^ (3.1.35) (^-QV V a - S ^ (3.1.36) ( f ^ Q V ^ ( l - X ' A ) (3.1.37) 54 where (3.1.38) (3.1.39) (3.1.40) (3.1.41) (3.1.42) and p e i s a u n i t v e c t o r i n the d i r e c t i o n of the p o s i t r o n ' s t h r e e momentum,"p£. In t h i s n o t a t i o n , the n i n e d i f f e r e n t i a l decay r a t e s are 55 ^ - ^ ^ ^ M ^ V \ ^ i e ^ S p l ] e ^ - X ^ M ^ ^ (3.1.43) N e g l e c t i n g the e l e c t r o n mass and summing over e l e c t r o n s p i n , the M i c h e l spectrum f o r p o l a r i z e d j** i s 1- i t l - K^- A( I- ^ ^ ^ f + ( 6 ^ - ^ a : l c l _ x^ ^_ ^ - ^ © f X - C ^ t : ^ ^ (3.1.44) where ^ - COSG e (3. 1.45) For the case of m a s s l e s s n e u t r i n o s , a r e d u c t i o n of (3.1.44) g i v e s the u s u a l M i c h e l spectrum [ 1 5 , 1 6 ] , The g e n e r a l N n e u t r i n o r e s u l t o b t a i n e d here c o n t a i n s too many parameters t o a n a l y z e m e a n i n g f u l l y . T h e r e f o r e , we s p e c i a l i z e t o the p h e n o m e n o l o g i c a l l y i n t e r e s t i n g t h r e e n e u t r i n o w o r l d (N=3). We assume t h a t the masses m> a r e i n a s c e n d i n g o r d e r 56 of i / . . That i s , m1<m2<m3. C u r r e n t e x p e r i m e n t a l mass l i m i t s a r e Mev/c 2 [ 2 5 ] . I f v& i s m o s t l y i / , , and s i m i l a r l y f o r the o t h e r f a m i l i e s , then m_£m 1 f e t c . so t h a t 6 1<4X10" 7 and 6 2<4.8X10~ 3. T h i s assumption c o r r e s p o n d s t o c h o o s i n g the d i a g o n a l elements of the P o n t e c o r v o m i x i n g m a t r i x t o be l a r g e s t . I t i s not e x p e r i m e n t a l l y v e r i f i e d i n t h e l e p t o n s e c t o r but i s e x p e c t e d t o be t r u e i n many GUTS models and i s s u p p o r t e d by a n a l o g y w i t h e x p e r i m e n t a l e v i d e n c e on m i x i n g a n g l e s i n the quark s e c t o r [ 7 3 ] , Under t h i s a s s u m p t i o n , 6, and 6 2 can be n e g l e c t e d . Only terms i n 6 3 need t o be k e p t . Then the M i c h e l spectrum can be w r i t t e n as t h a t 14<mv<46 e v / c 2 [ 2 1 , 2 2 ] , mv<.5 Mev/c 2 [ 2 3 , 2 4 ] , and m v<250 •m 3 t (3. 1 .46) where (3.1.47) M( l-x Y5 (3.1.48) 57 + a-0(l-X-«Hf>( »-^-"|tVl e C i - x - H S ^ (3.1.49) The f a c t o r R 0 g i v e s the s t a n d a r d M i c h e l spectrum w i t h m a s s l e s s n e u t r i n o s . I t i s e a s i l y seen t h a t R 3 and R 3 3 reduce t o R 0 when 6 3-*0. Then, by the o r t h o n o r m a l i t y c o n d i t i o n s of the m i x i n g p a r a m e t e r s , e q u a t i o n (3.1.46) reduces t o the s t a n d a r d M i c h e l spectrum. In g e n e r a l , the P o n t e c o r v o m i x i n g parameters a re undetermined. C o r r e s p o n d i n g t o the assumption made above t h a t i / e i s m a i n l y v,, e t c . , i t i s r e a s o n a b l e t o assume t h a t the d i a g o n a l elements of the m i x i n g m a t r i x a r e l a r g e s t . In GUTS, the m i x i n g s can be e s t i m a t e d i n ana l o g y t o the e s t i m a t i o n of quark m i x i n g a n g l e s , i n terms of masses. T h i s p a r t i c u l a r e s t i m a t e w i l l be c a l l e d h i e r a r c h i a l m i x i n g ; the v a l u e s of the n e c c e s s a r y parameters a r e H A t J 1 - ^ t / i y = . 0 0 ^ (3.1.50a) l U U i l 1 - "^e />vL-c =-0003 (3.1.50b) l U ^ P ^ ^ / Y H X - . 0 ^ (3.1.50c) T h e r e f o r e , from the o r t h o n o r m a l i t y c o n d i t i o n ( 1 . 5 ) , we get l U ^ f ' - . S H (3.1.51) N u m e r i c a l l y , R 3 and R 3 3 a r e comparable; however, the 58 c o n t r i b u t i o n of the l a t t e r t o the M i c h e l spectrum (3.1.46) i s s t r o n g l y s u p p r e s s e d by the f a c t o r s of m i x i n g parameters f o r the case of h i e r a r c h i a l m i x i n g . The h i e r a r c h i a l m i x i n g s a r e r e a l l y o n l y a guess. Some not v e r y s t r i n g e n t c o n s t r a i n t s can be put on the m i x i n g parameters by l o o k i n g a t p i o n d ecays. The a n a l y s e s of TTe2 decays [30,74] f o r m3 i n t h e Mev/c 2 range, i n d i c a t e an upper l i m i t of .02 f o r the P o n t e c o r v o parameter Ue3 . For m3 l e s s than .5 Mev/c 2, no good c o n s t r a i n t s e x i s t . A l s o , the decay Tf—y>i_i/ has been a n a l y z e d and h i e r a r c h i a l m i x i n g s g i v e a v e r y s m a l l p r o b a b i l i t y of the n e u t r i n o b e i n g v3. The most r e c e n t s e a r c h f o r secondary peaks i n the muon spectrum performed a t SIN s e t s a l i m i t on the p r o b a b i l i t y of f i n d i n g a 6-11 Mev/c 2 n e u t r i n o i n t h i s decay of l e s s than 1%. For h i e r a r c h i a l m i x i n g , one e x p e c t s about 6%. Thus, m3<6 Mev/c 2 i f lu / L 3 l 2 > . 0 6 [ 3 2 ] . In F i g u r e 4 [ 7 5 ] , the e f f e c t s of a heavy v3 m i x i n g i n t o v, and v2 a r e d i s p l a y e d r e l a t i v e t o the M i c h e l spectrum f o r muon decay w i t h m a s s l e s s n e u t r i n o s . The m i x i n g s a r e h i e r a r c h i a l i n F i g u r e 4 and the muon i s u n p o l a r i z e d . The d i f f e r e n c e between massive and m a s s l e s s c a s e s i s g r e a t e s t a t the h i g h energy end of the p o s i t r o n spectrum. T h i s h i g h energy r e g i o n i s m a g n i f i e d i n F i g u r e 5 [ 7 5 ] . The e f f e c t s f o r s e v e r a l d i f f e r e n t v a l u e s of m3 a r e shown; f o r a 1 Mev/c 2 n e u t r i n o , f o r i n s t a n c e , the e f f e c t shows up o n l y a f t e r x>.99. That the l a r g e s t e f f e c t of massive n e u t r i n o s s h o u l d occur a t the h i g h energy end of the M i c h e l spectrum i s e x p e c t e d . The s t a n d a r d V-A t h e o r y f o r m a s s l e s s n e u t r i n o s p r e d i c t s t h a t t h e M i c h e l spectrum peak a t x=1 59 ( n e g l e c t i n g e l e c t r o n mass). T h i s i s because a n g u l a r momentum c o n s e r v a t i o n makes i t most f a v o u r a b l e f o r the p o s i t r o n t o come out w i t h maximum a v a i l a b l e energy. These c o n s i d e r a t i o n s a r e d e p i c t e d i n F i g u r e 6 [ 7 6 ] , A c h i r a l i t y v i o l a t i n g mechanism such as massive n e u t r i n o s s h o u l d and, in d e e d , does ta k e away some energy from the p o s i t r o n t h a t i t would o t h e r w i s e o b t a i n . The d i f f e r e n c e from the M i c h e l spectrum w i t h m a s s l e s s n e u t r i n o s depends on (m 3/ny,) 2, where m3 i s t h e mass of the heavy n e u t r i n o , and on t h e m i x i n g parameters between the heavy and the two l i g h t n e u t r i n o s . In F i g u r e 7 [ 7 5 ] , the dependence of the M i c h e l spectrum on the P o n t e c o r v o m i x i n g parameter U y L L 3 i s shown f o r m3 = 5 Mev/c 2. Here, U e 3 M 0 f o r a l l c u r v e s . H i e r a r c h i a l v a l u e s and the v a l u e s l u ^ 3 l 2=.2 and .5 a r e shown. The s e n s i t i v i t y of the spectrum t o U^ 3 i s not v e r y pronounced u n t i l x>.95. F i g u r e 8 [75] i l l u s t r a t e s the e f f e c t s of the parameter U^3 on the M i c h e l spectrum. In a l l the c u r v e s , the h i e r a r c h i a l v a l u e of U^ 3 i s ta k e n . The h i e r a r c h i a l v a l u e of U e 3 and the c h o i c e D*e3 = U/x3 are d i s p l a y e d a l o n g w i t h the s t a n d a r d m a s s l e s s n e u t r i n o s c a s e . E v i d e n t l y the dependence on Ue.3. i s not v e r y s t r o n g . L o o k i n g a t the p o s i t r o n spectrum from u n p o l a r i z e d muon decay seems from t h e s e r e s u l t s t o be a way of d e t e c t i n g the e f f e c t s of a massive n e u t r i n o . However, the e f f e c t s on t h i s spectrum of v i r t u a l e l e c t r o m a g n e t i c c o r r e c t i o n s a r e l a r g e . T h e r e f o r e , no f i r m c o n c l u s i o n s can made from t h e s e r e s u l t s . The r a d i a t i v e c o r r e c t i o n s t o the M i c h e l spectrum a r e c a l c u l a t e d i n Chapter 4. 60 F i g . 4: The M i c h e l spectrum f o r u n p o l a r i z e d /•* decay. The s o l i d c u r v e i s f o r m a s s l e s s n e u t r i n o s . The dashed c u r v e i s f o r m 3=l0 MeV/c 2; the d a s h - l i n e c u r v e i s f o r m 3=20 MeV/c 2. 61 0.26 0.25 0.24 X J O GC 0.23 0.22 0.21 W i — • - T«ti= 10 _L 0.82 0.9 X 0.95 1.0 F i g . 5: The h i g h - e n e r g y end of t h e M i c h e l spectrum w i t h h i e r a r c h i a l m i x i n g . The s o l i d c u r v e i s f o r m a s s l e s s n e u t r i n o ; the dash-dot c u r v e i s f o r m3=3 MeV/c 2; the dashed c u r v e i s f o r m3 = 5 MeV/c 2; the s q u a r e - l i n e c u r v e i s f o r m 3=l0 MeV/c 2; the t r i a n g l e - l i n e c u r v e i s f o r m 3=20 MeV/c 2. 62 (a) Allowed (b) Forbidden — _ — - — S z - T » e « *- v v F i g . 6: H e l i c i t y c o n s i d e r a t i o n s f o r the M i c h e l spectrum of jx* decay w i t h m a s s l e s s n e u t r i n o s . A n g u l a r momentum c o n s e r v a t i o n a l l o w s the c o n f i g u r a t i o n (a) and f o r b i d s ( b ) . A n o t h e r case of r e l e v a n c e i s t h a t of the decay of p o l a r i z e d muons w i t h the p o s i t r o n e m i t t e d a n t i p a r a l l e l t o t h e muon s p i n ( ^ . • p e = - l ) . T h i s case i s c o n s i d e r e d here because t h e r e a r e two h i g h p r e c i s i o n e x p e r i m e n t s c u r r e n t l y under way which measure t h i s [ 7 0 , 7 1 ] , F o r t h e c a s e of m a s s l e s s n e u t r i n o s , one would o b t a i n a (1-x) d i s t r i b u t i o n f o r d R / ( x 2 d x d ^ t ) a t z^e=-1. The d e v i a t i o n s from t h i s (1-x) f a l l o f f a r e shown i n F i g u r e 9 f o r a 5 Mev/c 2 i / 3 f o r a c o u p l e of d i f f e r e n t m i x i n g schemes. The c u r v e s a r e d i s p l a c e d from each o t h e r so as t o i l l u s t r a t e t h e i r d i f f e r e n c e s . Only the case of maximal m i x i n g ( l u ^ l 2 = I U^ u.3 \ 2=1/3) c o u l d be d i s t i n g u i s h e d from t h e m a s s l e s s n e u t r i n o s c a s e . The k i n k s w h i c h a r e d i s c e r n a b l e i n the maximal m i x i n g case a r e due t o t h e k i r i e m a t i c a l t u r n i n g o f f of R 3 3 a t x=1-46 2 and of R 3 a t x=1-6 2. F or m3=3 Mev/c 2, which i s not d i s p l a y e d , t h e f i r s t such k i n k would be measurable o n l y f o r l a r g e m i x i n g and t h e second o c c u r s ( a t x=.9992) o u t s i d e t h e range p l a n n e d i n t h e p r e s e n t 63 0.26 F i g . 7: S e n s i t i v i t y of the M i c h e l spectrum t o the m i x i n g parameter U,^ f o r m 3=5 MeV/c 2. The dashed c u r v e i s f o r IU/ASI 2 = . 0 5 9 ; the s o l i d c u r v e i s f o r I U / 3 \ 2 = . 2 ; the d a s h - l i n e c u r v e i s f o r \ U ^ 3 r = . 5 . U F C 3 ~ 0 f o r a l l the c u r v e s . 64 S e n s i t i v i t y of the M i c h e l spectrum t o the m i x i n g parameter U e ? f o r m3 = 5 MeV/c 2 and l U ^ t 2=.059. The s o l i d c u r v e i s f o r m a s s l e s s n e u t r i n o s . The dashed c u r v e i s f o r | U«,3l 2 = 3 x 10" *; the d a s h - l i n e c u r v e i s f o r l u e 3 l 2=lU> 3\ 2 = .059. 65 e x p e r i m e n t s . As was noted e a r l i e r , a l a r g e m i x i n g i s not e x p e c t e d f o r an i n t e r m e d i a t e mass v3. T h e r e f o r e , i t i s u n l i k e l y t h a t massive n e u t r i n o s w i t h p r e d o m i n a n t l y l e f t - h a n d e d i n t e r a c t i o n s w i l l cause the measurement of the ^ parameter t o d e v i a t e from the (1-x) d i s t r i b u t i o n t o w i t h i n the s e n s i t i v i t y of c u r r e n t l y p l a n n e d e x p e r i m e n t s . 6 6 F i g . 9: High-energy end of M i c h e l spectrum f o r s p i n of /a* a n t i p a r a l l e l t o p o s i t r o n momentum. The s o l i d l i n e i s the (1-x) b e h a v i o u r of m a s s l e s s n e u t r i n o s . The dash-l i n e c u r v e i s f o r I U/u3l 2=lu«.3\ 2=.1 m i x i n g and the dashed c u r v e i s f o r l u - 3 l 2 = l U e 3 \ 2 = l / 3 . The mass of m3 i s 5 MeV/c 2. r 67 3.2 The Case of Majorana N e u t r i n o s The case of n e u t r i n o s b e i n g massive Majorana p a r t i c l e s i n t e r a c t i n g v i a a p r e d o m i n a n t l y l e f t - h a n d e d c o u p l i n g i s c o n s i d e r e d i n t h i s s e c t i o n . The p h y s i c a l mass e i g e n s t a t e s , ifj,., can be d i f f e r e n t from the c o r r e s p o n d i n g e i g e n s t a t e s of the weak i n t e r a c t i o n , . (i=1,...,N; a = e ,J*,Z, . . .) For the g e n e r a l case of N l e p t o n f a m i l i e s , t h e s e two bases are r e l a t e d by an NXN u n i t a r y t r a n s f o r m a t i o n . 1 ^ - - r . v . ^ ,3.2.,) The symbol V i s used f o r the P o n t e c o r v o m i x i n g m a t r i x here t o emphasize t h a t i t can be d i f f e r e n t from the m a t r i x U of the D i r a c c a s e . The f i e l d s ^, mass or weak e i g e n s t a t e s , can be e x p r e s s e d i n terms of c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s as i n e q u a t i o n ( 3 . 1 . 1 1 ) . T h i s d e c o m p o s i t i o n of a Majorana f i e l d c o n t a i n s o n l y one type of c r e a t i o n / d e s t r u c t i o n o p e r a t o r , i n c o n t r a s t t o the case of the D i r a c f i e l d . T h i s i s a r e f l e c t i o n of the Majorana p r o p e r t y of the p a r t i c l e s . 1/> t =^ (3.2.2) With the above phase, a Majorana p a r t i c l e and i t s a n t i p a r t i c l e a r e i n d i s t i n g u i s h a b l e . T h i s i n d i s t i n g u i s h a b i l i t y p r e s e n t s some c o m p l i c a t i o n s i n the c a l c u l a t i o n of muon decay, which d i d not a r i s e f o r the case of D i r a c n e u t r i n o s i n the l a s t s e c t i o n . There a r e now two d i s t i n c t t y p e s of f i n a l s t a t e i n the decay ( 1 . 2 ) . F i r s t , the two p h y s i c a l n e u t r i n o s i n the f i n a l s t a t e may be 68 d i s t i n c t . That i s , the f i n a l s t a t e i s deUp^sO o t - H f e , ^ eta (^^.Mc^ C^j ' (3.2.3) The o p e r a t o r ay (k , s x ) c r e a t e s a n e u t r i n o of mass mK, momentum k , and s p i n s K . T h i s i s the n o t a t i o n we have p r e s e n t e d i n Appendix B. On the o t h e r hand, the two n e u t r i n o s i n the f i n a l s t a t e c o u l d a l s o be i n d i s t i n g u i s h a b l e . Then, = d £ + ( p e ) S e ) a v - . ( " k , * ^ a X L (3.2.4) T h i s l a t t e r p o s s i b i l i t y does not occ u r f o r the case of D i r a c n e u t r i n o s s i n c e a D i r a c p a r t i c l e i s d i s t i n g u i s h a b l e from i t s c o r r e s p o n d i n g a n t i p a r t i c l e . As i n the D i r a c c a s e , we choose the p r e d o m i n a n t l y l e f t -handed e f f e c t i v e weak L a g r a n g i a n t o d e s c r i b e the muon decay. A r i g h t - h a n d e d c u r r e n t cannot be r u l e d out but i t would make a c o n t r i b u t i o n of the o r d e r of o n l y 6X10~ 5 [77] and, so, i s n e g l e c t e d h e r e . The L a g r a n g i a n i s w r i t t e n i n terms of Majorana mass e i g e n s t a t e s as f o l l o w s . + V\.e. (3.2.5) S y m b o l i c a l l y , the muon decay can be r e p r e s e n t e d as the sum of t r a n s i t i o n r a t e s t o each of the N 2 f i n a l s t a t e s . That i s , the f o l l o w i n g i s c a l c u l a t e d . x H l C f c - \ A + ^ X - \ C f j , . ^ (3.2.6) 69 In the above, (3.2.7) (3.2.8) The i n i t i a l s t a t e I I ) i s g i v e n i n e q u a t i o n ( 3 . 1 . 2 ) . The f i r s t f a c t o r of 1/2 i n e q u a t i o n (3.2.6) remedies a double c o u n t i n g due t o the i n d i s t i n g u i s h a b i l i t y of Majorana p a r t i c l e s and a n t i p a r t i c l e s . The second f a c t o r of 1/2 i s a s t a t i s t i c a l f a c t o r n e c e s s a r y because of t h e i d e n t i c a l p a r t i c l e s i n t h e f i n a l s t a t e | f ' ^ . In the l i m i t of no m i x i n g and m a s s l e s s n e u t r i n o s t h i s y i e l d s the s t a n d a r d r e s u l t , as r e q u i r e d . Upon s u b s t i t u t i o n of the i n i t i a l and f i n a l s t a t e s and the L a g r a n g i a n i n t o the e x p r e s s i o n s f o r o£. and jf^,- , one o b t a i n s (3.2.9) where (3.2.1Oa) and (3.2.10b) A l s o , 70 j - c*ir)U*(p*-p.-i-i)(ynA-yn£) 0 . 2 . 1 0 where n^-^oS/S*^*^ <3-2-12) and T»£ " ^  t ^ f . , 0 V ^ ^ H v ^ ^ f c j i , ^ Vf( lT(?^r)\ (3.2.13) In the above e x p r e s s i o n s , the s p i n o r c o r r e s p o n d i n g t o the c o n j u g a t e n e u t r i n o s a t i s f i e s U c= C S (3.2.14) where C i s the charge c o n j u g a t i o n m a t r i x , O (3.2.15) E v a l u a t i o n of the r a t e f o r the decay (1.2) proceeds j u s t as i n the case of D i r a c n e u t r i n o s , which was d e t a i l e d i n the l a s t s e c t i o n . The t r a c e s e v a l u a t e d here a r e somewhat d i f f e r e n t from t h a t case s i n c e they i n v o l v e the c o n j u g a t e s p i n o r s . D e t a i l s of the e v a l u a t i o n of m a t r i x e lements a r e p r e s e n t e d i n Appendix D. The more g e n e r a l p r e s e n t a t i o n of Majorana p r o p e r t i e s i s g i v e n i n Appendix C. The M i c h e l spectrum, n e g l e c t i n g e l e c t r o n mass, i s found t o be 71 xl (3.2.16) The n o t a t i o n i s the same as t h a t of S e c t i o n 3.1. T h i s r e s u l t reduces t o the u s u a l r e s u l t f o r the case of m a s s l e s s n e u t r i n o s . (The D i r a c and Majorana r e s u l t s c o i n c i d e f o r the case of m a s s l e s s n e u t r i n o s . ) As f o r the case of D i r a c n e u t r i n o s , we c o n s i d e r o n l y the t h r e e - n e u t r i n o s w o r l d . A g a i n , o n l y one mass, m3, i s kept as s i g n i f i c a n t . In t h i s c a s e , the r a t e f o r u n p o l a r i z e d muon decay i s 72 + i V ^ I M - V ^ I * R ; ^ 0 . 2 . 1 7 ) where Ro-.TT ^ - ^ - x V e ( l - 0 (3.2.18) (3.2.19) - z : R s x - ( i - x - 4 ^ v / A ( 3 . 2 . 2 0 ) R 0 and R 3 g i v e n i n e q u a t i o n s ( 3 . 2 . 1 8 ) and ( 3 . 2 . 1 9 ) , r e s p e c t i v e l y , are the same as e q u a t i o n s ( 3.1.47) and ( 3.1.48) summed and averaged over muon s p i n . Thus, the o n l y d i f f e r e n c e between the cases of D i r a c and Majorana n e u t r i n o s l i e s i n the term R 3 3. The d i f f e r e n c e between R 3 3 f o r the D i r a c c a s e and R 3 3 f o r the Majorana case i s d i s p l a y e d i n e q u a t i o n ( 3 . 2 . 2 0 ) and i s p r o p o r t i o n a l t o 6 2. A l s o , r e c a l l t h a t , f o r the D i r a c c a s e , because of the f a c t o r s of s m a l l m i x i n g parameters m u l t i p l y i n g i t , the R 3 3 term gave a n e g l i g i b l e c o n t r i b u t i o n t o the M i c h e l spectrum. Thus, the s i t u a t i o n here i s t h a t of a s m a l l d i f f e r e n c e between s m a l l terms. The M i c h e l spectrum i s , t h e r e f o r e , not a 73 good p l a c e t o t r y t o det e r m i n e whether n e u t r i n o s a r e Majorana or D i r a c p a r t i c l e s , u n l e s s the mass of v2 i s i n the t e n s of Mev/c 2 and i t s m i x i n g s w i t h the l i g h t n e u t r i n o s a r e l a r g e . T h i s seems u n l i k e l y . 74 3.3. E l e c t r o n - N e u t r i n o C o r r e l a t i o n s I t was shown i n the l a s t s e c t i o n t h a t the M i c h e l spectrum p r o b a b l y does not p r o v i d e a means of d e t e r m i n i n g whether n e u t r i n o s a r e D i r a c or Majorana p a r t i c l e s . We now i n v e s t i g a t e e l e c t r o n - n e u t r i n o c o r r e l a t i o n s t o t e s t t h e i r u s e f u l n e s s i n d i f f e r e n t i a t i n g between D i r a c and Majorana n e u t r i n o s . Such c o r r e l a t i o n measurements have p r e v i o u s l y been suggested as a means of s t u d y i n g the s t r u c t u r e of the weak i n t e r a c t i o n s [ 1 9 ] . For a c o r r e l a t i o n measurement, both the p o s i t r o n from the decay (1.2) and the n e u t r i n o i / e a r e t o be d e t e c t e d . The n e u t r i n o c o u l d be d e t e c t e d v i a the i n v e r s e /3-decay p r o c e s s e^. + ^ — * " eT +• p (3.3.1) i n a large-mass d e t e c t o r . The r e a c t i o n (3.3.1) i s chosen f o r the n e u t r i n o d e t e c t i o n s i n c e i t s e l e c t s a p a r t i c u l a r n e u t r i n o t y p e , . The r e a c t i o n 1- n — - yu-' \ y (3.3.2) would not go f o r the low-energy n e u t r i n o s of muon decay a t r e s t . I f the assumption t h a t the d i a g o n a l elements of the P o n t e c o r v o m i x i n g m a t r i c e s (U or V) a r e dominant i s a g a i n made, then the dominant branches f o r muon decay i n t o D i r a c and Majorana n e u t r i n o s a r e , r e s p e c t i v e l y , yU+ e.* -V, ^ (3.3.3) ^ C Vl (3.3.4) 75 By the same r e a s o n i n g , the n e u t r i n o i / , s h o u l d be the dominant p a r t i c i p a n t i n the r e a c t i o n ( 3 . 3 . 1 ) . Of c o u r s e , i f the masses of the n e u t r i n o s a r e not a l l d e g e n e r a t e , then o s c i l l a t i o n s c o u l d t a k e p l a c e . F o r i n s t a n c e , a f t e r t he decay (3.3.3) t h e n e u t r i n o v, c o u l d o s c i l l a t e i n t o a n o t h e r n e u t r i n o and, t h u s , not be as l i k e l y t o t a k e p a r t i n the r e a c t i o n ( 3 . 3 . 1 ) . However, t h i s o c c urence would be compensated by the n e u t r i n o s i n the nondominant branches of the muon decay o s c i l l a t i n g i n t o v, and then p a r t i c i p a t i n g i n r e a c t i o n ( 3 . 3 . 1 ) . Thus o s c i l l a t i o n s s h o u l d not g r e a t l y a l t e r the f l u x of v, some d i s t a n c e from the decay ( 3 . 3 . 3 ) . F o r the case of Majorana n e u t r i n o s , the s i t u a t i o n can be somewhat d i f f e r e n t . For Majorana n e u t r i n o s o s c i l l a t i o n s of the type v L (3.3.5) can t a k e p l a c e . In t h i s c a s e , the P o n t e c o r v o m a t r i x V c o u l d be r e c t a n g u l a r . I f i t i s assumed t h a t the Majorana masses a r e due t o an enlargement i n the Higgs s e c t o r o n l y , as d e s c r i b e d i n Chapter 2, then o n l y the " D i r a c type o s c i l l a t i o n s " "*—*" ( ^ V (3.3.6) w i l l t a k e p l a c e . In t h i s c a s e , the s i t u a t i o n i s s i m i l a r t o the D i r a c c a s e j u s t d i s c u s s e d . We c a l c u l a t e two k i n d s of c o r r e l a t i o n s below. The f i r s t t y p e i s where the energy of the p o s i t r o n and the a n g l e between the p o s i t r o n and the d e t e c t e d n e u t r i n o a r e measured. In the 76 second type d i s c u s s e d , the e n e r g i e s of both the p o s i t r o n and the n e u t r i n o a r e measured. The e n e r g y - a n g l e c o r r e l a t i o n f o r the dominant decay - e + "V, 57 (3.3.7) where the n e u t r i n o s a r e D i r a c p a r t i c l e s , i s c a l c u l a t e d from e q u a t i o n (3.2.11) by i n t e g r a t i n g over the momentum of the u n d e t e c t e d a n t i n e u t r i n o and e x p r e s s i n g the r e s u l t i n terms of the opening a n g l e between the p o s i t r o n and the n e u t r i n o , 0 e i • For muons d e c a y i n g a t r e s t , t he r a t e i s g i v e n by (3.3.8) The masses of the p o s i t r o n and of the n e u t r i n o v, have been n e g l e c t e d ; t h e s p i n s of t h e p o s i t r o n and t h e n e u t r i n o s have been summed o v e r . The r a t e (3.3.8) i s f o r u n p o l a r i z e d muons. I n t e g r a t i n g over the n e u t r i n o momentum and the e l e c t r o n a z i m u t h a l a n g l e y i e l d s the f o l l o w i n g d o u b l e - d i f f e r e n t i a l r a t e . 77 1- 21 I U • I x ^ [ (\->0) siv, 1 Oev/a - i c o ^ e t v /a] (3.3.9) T h i s e x p r e s s i o n i s good t o o r d e r 6 2. The p r o p e r t y I J U o J 3 - = \ (3.3.10) has been used i n the f i r s t term. The second term g i v e s the e f f e c t of massive D i r a c n e u t r i n o s . For the t h r e e n e u t r i n o w o r l d , o n l y v3 i s massive and the second term, b e i n g p r o p o r t i o n a l t o \ U^ 3I 2 6 2 , i s n e g l i g i b l e . The l i m i t of m a s s l e s s n e u t r i n o s and no m i x i n g s i s o b t a i n e d by s e t t i n g IU e il 2 = 1 and 6 2 = 0. For the case of Majorana n e u t r i n o s , the r a t e f o r the p r o c e s s ^ * a? -uL (3.3.11) i s c a l c u l a t e d s i m i l a r l y . For the dominant b r a n c h e s , e i t h e r or both of the n e u t r i n o s vL , v. c o u l d be i / 1 and, c h o o s i n g o n l y •J terms w i t h a f a c t o r of \ v e 1 \ 2 , t h e d i f f e r e n t i a l r a t e i s + i l ^ J i | ^ ' [ ( p . ? , V p A ? ' V ( F , . p . V p / . . p ^ ^ ( Q . ? , - p . ) 1 r (3.3.12) The f i r s t term i s (1-|v^,\ 2 ) t i m e s the r e s u l t f o r the D i r a c 78 n e u t r i n o case w h i l e the second term i s f o r the case of bot h o u t g o i n g n e u t r i n o s b e i n g v,. E q u a t i o n (3.3.12) can be s p l i t up i n the f o l l o w i n g manner. d£ = f l - I V ^ d ^ * d^'+ d£" (3.3.13) Here, dR' and dR" a r e c a l c u l a t e d j u s t as i n the D i r a c c a s e . A v e r a g i n g over muon . s p i n and i n t e g r a t i n g over the u n d e t e c t e d n e u t r i n o ' s momentum, the r e s u l t s a r e as f o l l o w s f o r muon decay a t r e s t . ^(•YK^ - <2.->vyE^ - olTvyE, + 4 E^E, Sxv^&^/x) (3.3.14) d£" = frp We.^W/^^p^p, E, ( l y E e - ^ e ^ ^ ^ Q . v M ^ " ) (3.3.15) (air*) caEe")(aE0 I n t e g r a t i n g over the f i n a l n e u t r i n o ' s momentum and over one a n g l e , one o b t a i n s dxdco=>£tv ( i - X ^ m i e t v / a ^ (3.3.16) d£_! = I V ^ M ^ ^ X^Cl-^) 2- CQ^6ev/2 (3.3.17) Combining t h e s e e x p r e s s i o n s w i t h the r e s u l t f o r the D i r a c c a s e , the f i n a l r e s u l t i s 79 \V^, ^ L K (a,-*) 5'sv) HGey-h <L0S&£V1 + (3.3.18) N e g l e c t i n g n e u t r i n o mass, comparing e q u a t i o n s (3.3.9) and (3.3.17) f o r the D i r a c and Majorana c a s e s , r e s p e c t i v e l y , the d i f f e r e n c e i s p r o p o r t i o n a l t o \ V ^ i \ 2 . We i l l u s t r a t e the d i f f e r e n c e between t h e D i r a c and Majorana c a s e s i n F i g u r e 10, where the double d i f f e r e n t i a l d i s t r i b u t i o n i s p l o t t e d f o r the two c a s e s f o r the p o s i t r o n energy c o r r e s p o n d i n g t o x=.5 and the m i x i n g I U ^ J 2 = .1, as a f u n c t i o n of s i n 2 (e e v/Jl) . For the m i x i n g used, the two t y p e s of n e u t r i n o s c o u l d be d i s t i n g u i s h e d i n a 5% c o r r e l a t i o n e x p e r i m e n t . The second type of c o r r e l a t i o n we c o n s i d e r here i s the d i s t r i b u t i o n i n p o s i t r o n and n e u t r i n o e n e r g i e s . A g a i n , s i n c e the d e t e c t i n g r e a c t i o n i s t h a t of e q u a t i o n ( 3 . 3 . 1 ) , the dominant branch i s taken t o c o n t a i n a t l e a s t one v, . The r a t e i s c a l c u l a t e d by i n t e g r a t i n g over the u n d e t e c t e d n e u t r i n o ' s momentum and over a l l a n g l e s . For the case of D i r a c n e u t r i n o s , t h i s p r oceeds as f o l l o w s . Summing over a l l s p i n s and n e g l e c t i n g the masses of v^ and the p o s i t r o n , we o b t a i n , upon i n t e g r a t i o n of the u n d e t e c t e d n e u t r i n o ' s momentum, from e q u a t i o n ( 3 . 2 . 1 1 ) , f o r the dominant branch i n the muon r e s t frame 8 0 F i g . 1 0 : E n e r g y - a n g l e c o r r e l a t i o n s f o r D i r a c (D) and Majorana (M) n e u t r i n o s . The v a l u e of x=1/2 i s chosen and the m i x i n g |Uu.,| 2 = .1 i s used. 81 AZ= GfjUeii3" E^dE t E,JE, dcosOe-v I I l u ^ 1 E, (•^-•m.?- .^ S ^ - i ^ e t - ^ - ^ ^ r a E e E / i - c o ^ e e v V ^ ) (3.3.19) We d e f i n e the f o l l o w i n g d i m e n s i o n l e s s v a r i a b l e = 2 E , / m A (3.3.20) i n a n a l o g y w i t h the v a r i a b l e x. U s i n g the 6 f u n c t i o n t o do the l a s t a n g u l a r i n t e g r a t i o n , the r e s u l t f o r the D i r a c case i s = Q £ T V £ l U . a L t ^ ( l - i ^ V Z l a ^ ^ ^ s j - ] (3.3.21) N o t i c e t h a t t h e r e i s no x dependence i n t h i s d i s t r i b u t i o n . For the case of Majorana n e u t r i n o s , the e q u i v a l e n t s t e p s i n the c a l c u l a t i o n a r e g i v e n below. 1^ J The f i n a l r e s u l t i s 82 a e - ( ^ ^ J v ^ ^ ^ ( , - ^ - ^ \ y ^ , \ ^ L ^ a - ^ - ^ - v - 4 p ( \ - y ^ v s ^ S \ 0 . 3 . 2 2 ) As i n the e n e r g y - a n g l e c o r r e l a t i o n s , the d i f f e r e n c e between the D i r a c and Majorana d i s t r i b u t i o n s i s p r o p o r t i o n a l t o \u^\ *. The d i s t r i b u t i o n i n x g i v e n by ( 3 . 3 . 2 1 ) f o r the case of D i r a c n e u t r i n o s i s f l a t f o r some f i x e d v a l u e of y. For Majorana n e u t r i n o s , the d i s t r i b u t i o n ( 3 . 3 . 2 2 ) i s q u a d r a t i c i n x f o r f i x e d y. We i l l u s t r a t e t h e s e two c a s e s i n F i g u r e 11. A t y p i c a l p o i n t y=.25 was chosen and v a r i o u s m i x i n g s a r e d i s p l a y e d . Another d i f f e r e n c e between the D i r a c and Majorana c a s e s i s t h a t the Majorana r a t e does not v a n i s h a t y=1 w h i l e the D i r a c d i s t r i b u t i o n does. The p o s i t r o n - n e u t r i n o energy c o r r e l a t i o n seems p r o m i s i n g f o r d e t e r m i n i n g whether n e u t r i n o s a r e D i r a c or Majorana p a r t i c l e s . The e x p e r i m e n t a l t a s k of p e r f o r m i n g t h i s measurement i s , however, e x c e e d i n g l y d i f f i c u l t . Both the p o s i t r o n and n e u t r i n o e n e r g i e s must be measured. For the n e u t r i n o , t h i s i n v o l v e s measuring the e n e r g i e s of both the e l e c t r o n and p r o t o n of r e a c t i o n ( 3 . 3 . 1 ) . These e n e r g i e s w i l l be r a t h e r low f o r muons d e c a y i n g a t r e s t and, t h u s , w i l l be d i f f i c u l t t o measure. We e s t i m a t e the r a t e f o r these c o r r e l a t i o n measurements w i t h the c u r r e n t l y a v a i l a b l e ji* f l u x a t the meson f a c t o r i e s and a 100 ton 4TT water d e t e c t o r t o be s e v e r a l t e n s of e v e n t s per day. S i n c e t h e d i f f e r e n c e between D i r a c and Majorana d i s t r i b u t i o n s i s p r o p o r t i o n a l t o \v^.,\ 2, i t would be v a l u a b l e t o have some i n f o r m a t i o n on i t s magnitude i n o r d e r t o h e l p d etermine the 83 F i g . 11: Energy-energy c o r r e l a t i o n s f o r y=1/4. The s o l i d l i n e i s f o r D i r a c n e u t r i n o s ; the r e s t a r e f d r Majorana n e u t r i n o s . The dashed c u r v e i s f o r h i e r a r c h i a l m i x i n g , the dash-dot c u r v e i s f o r l v y , l 2=.067, and the dash-c r o s s c u r v e i s f o r \Vu.1\2=.1. f e a s i b i l i t y of such c o r r e l a t i o n e x p e r i m e n t s . 85 IV, THE INCLUSION OF RADIATIVE CORRECTIONS The r e s u l t s of the l a s t c h a p t e r i n d i c a t e t h a t a t the t r e e graph l e v e l the hi g h - e n e r g y end of the M i c h e l spectrum i s r a t h e r s e n s i t i v e t o n e u t r i n o mass. However, tho s e r e s u l t s cannot be compared w i t h e x p e r i m e n t a l d a t a s i n c e , as i s w e l l known, the i n c l u s i o n of r a d i a t i v e c o r r e c t i o n s s i g n i f i c a n t l y changes the M i c h e l spectrum [ 1 6 , 1 7 ] , As was p o i n t e d out i n Chapter 2, the d i f f e r e n c e , f o r the case of m a s s l e s s n e u t r i n o s , between the SU(2)XU(1) gauge t h e o r y and the e f f e c t i v e V-A t h e o r y p r e d i c t i o n s f o r t h e p o s i t r o n spectrum i s g i v e n by terms of o r d e r (m^/M2^ ) [ 1 6 ] . When r a d i a t i v e c o r r e c t i o n s a r e i n c l u d e d the two t h e o r i e s d i f f e r by terms of o r d e r ( a mJ/M 2 w), where a i s the f i n e s t r u c t u r e c o n s t a n t [ 1 6 , 1 8 ] , The e f f e c t i v e t h e o r y i s e x p e c t e d t o be a good a p p r o x i m a t i o n f o r the massive n e u t r i n o s case a l s o . Thus, the c a l c u l a t i o n s a r e done h e r e , as i n Chapter 3, u s i n g the e f f e c t i v e V-A t h e o r y of e l e c t r o w e a k i n t e r a c t i o n s . The Feynman diagrams f o r the f i r s t o r d e r v i r t u a l p h o t o n i c c o r r e c t i o n s which must be i n c l u d e d i n t h i s c a s e a r e p i c t u r e d i n F i g u r e s 12 and 13. The i n t e r f e r e n c e between the f r e e decay a m p l i t u d e and the sum of the a m p l i t u d e s of the diagrams of F i g u r e 12(b,c,d) g i v e s the r a d i a t i v e c o r r e c t i o n t o the muon decay ( 1 . 2 ) . The r a d i a t i v e decay juS e,+ v i J 1 (4.1) 86 ( a ) ( b ) ( c ) ( d ) F i g . 12: Feynman diagrams f o r the f r e e muon decay and the v i r t u a l photon c o r r e c t i o n s . _ ^ = r ^ ~ l + K-» 8m e F i g . 13: S e l f - e n e r g y c o r r e c t i o n diagrams f o r the charged l e p t o n s . (a) (b) y ( k ) F i g . 14: B r e m s s t r a h l u n g c o r r e c t i o n s t o muon decay. 87 must a l s o be i n c l u d e d i n t h i s c o n s i d e r a t i o n s i n c e the photon i s not d e t e c t e d i n e x p e r i m e n t s t h a t o n l y measure the e + c h a r a c t e r i s t i c s . The r e l e v a n t b r e m s s t r a h l u n g diagrams a r e g i v e n i n F i g u r e 14. I t i s the hi g h - e n e r g y end of the M i c h e l spectrum which i s s e n s i t i v e t o r a d i a t i v e c o r r e c t i o n s f o r the case of m a s s l e s s n e u t r i n o s . J u s t as was the e f f e c t of n e u t r i n o mass, the p h o t o n i c c o r r e c t i o n s t e n d t o p u l l down the spectrum t o z e r o and t o lower the v a l u e of p o s i t r o n energy a t which the spectrum peaks. These c o r r e c t i o n s a r e e x p e c t e d t o be e q u a l l y i m p o r t a n t f o r the massive n e u t r i n o s c a s e . We c a l c u l a t e the r a d i a t i v e c o r r e c t i o n s of F i g u r e s 12-14 f o r the case of D i r a c n e u t r i n o s o n l y . T h i s i s because the d i f f e r e n c e between D i r a c and Majorana n e u t r i n o s was shown t o be v e r y s m a l l i n the M i c h e l spectrum; t h i s r e s u l t i s e x p e c t e d t o remain t r u e when r a d i a t i v e c o r r e c t i o n s a re i n c l u d e d . T h i s i s due t o the f a c t t h a t the d i f f e r e n c e between the two t y p e s of n e u t r i n o s i s most pronounced when the n e u t r i n o s a r e d e t e c t e d , such as i n the c o r r e l a t i o n experiment d i s c u s s e d i n Chapter 3. Of c o u r s e , the diagrams of F i g u r e s 12-14 i n v o l v e d i v e r g e n t i n t e g r a l s . The c o n t r i b u t i o n t o the decay r a t e of each diagram i s here e v a l u a t e d e x p l i c i t l y and i t i s found t h a t the d i v e r g e n c e s c a n c e l t o o r d e r c, l e a v i n g a f i n i t e answer, j u s t as i n the m a s s l e s s n e u t r i n o s c a s e . We use the t e c h n i q u e of d i m e n s i o n a l r e g u l a r i z a t i o n t o handle the d i v e r g e n t f o u r - d i m e n s i o n a l i n t e g r a l s which occur [ 2 0 ] . The i d e a of t h i s method of r e g u l a r i z i n g d i v e r g e n t i n t e g r a l s i s t o c o n t i n u e t o some 88 a r b i t r a r y d i m e n s i o n n f o r which the i n t e g r a l s a r e f o r m a l l y c o n v e r g e n t . Then the n - d i m e n s i o n a l i n t e g r a l s can be e v a l u a t e d and the r e s u l t s expanded i n powers of ( n - 4 ) . D i v e r g e n c e s a r e c a s t i n the form of p o l e s i n ( n - 4 ) . D i m e n s i o n a l r e g u l a r i z a t i o n i s a gauge i n v a r i a n t method. We work i n the ' t Hooft-Feynman gauge [78,79].. T h i s r e g u l a r i z a t i o n t e c h n i q u e i s f u r t h e r d e s c r i b e d and many u s e f u l f o r m u l a e a r e g i v e n i n Appendix E. C o n s i d e r f i r s t the g e n e r a l case of N SU(2) l e p t o n d o u b l e t s . The assumption i s made here t h a t the masses of the N n e u t r i n o mass e i g e n s t a t e s a r e o r d e r e d such t h a t 7VLV < -W.^  < < yylH (4.2) F u r t h e r , h i e r c h i a l n e a r e s t neighbour (HNN) m i x i n g i s assumed. That i s , i n the NxN m i x i n g m a t r i x r e l a t i n g n e u t r i n o weak e i g e n s t a t e s , (where a=e,ytt, T,...) and mass e i g e n s t a t e s v-L (where i=1,...,N), the d i a g o n a l elements a r e taken t o be dominant and the next dominant elements a r e those i m m e d i a t e l y next t o the d i a g o n a l . For example, Uev>U >Ue,. T h i s m i x i n g scheme means t h a t i s p r e d o m i n a n t l y v,, i / ^ i s m a i n l y v2t e t c . R e c a l l t h a t the M i c h e l spectrum i s a c t u a l l y an i n c o h e r e n t sum of N 2 s p e c t r a . M dx s r 1 dx The assumption t h a t HNN m i x i n g h o l d s means t h a t o n l y n e u t r i n o s i n t he same SU(2) d o u b l e t s as JJ. or e or i n the d o u b l e t which i s the n e a r e s t neighbour t o t h a t of the jx d o u b l e t ( i . e . the T d o u b l e t ) w i l l c o n t r i b u t e s i g n i f i c a n t l y t o the M i c h e l spectrum. 89 A l s o , s i n c e the n e u t r i n o s i n the d o u b l e t s ( v e , e") and ( y , y t x ' ) a r e , by HNN m i x i n g , p r e d o m i n a n t l y and v2, r e s p e c t i v e l y , r e c e n t e x p e r i m e n t s can be used t o s e t l i m i t s on m, and m2 as f o l l o w s [ 2 1 - 2 4 ] . W < TVL, < 4 6 eV/c^ (4.4.4a) TYLJL< O.S K e V / c ^ (4.4.4b) W i t h i n the a c c u r a c y of muon decay e x p e r i m e n t s , m, and m2 can be s e t e q u a l t o z e r o . Thus, o n l y t h o s e s p e c t r a i n which a t most one n e u t r i n o ( v ^ ' s n e a r e s t n e i g h b o u r ) i s massive c o n t r i b u t e t o the sum i n e q u a t i o n ( 4 . 3 ) . The p h o t o n i c v i r t u a l c o r r e c t i o n s and the b r e m s s t r a h l u n g r e s u l t s a r e g i v e n below f o r t h e case of o n l y one of the f i n a l s t a t e n e u t r i n o s h a v i n g nonzero mass. D e t a i l s of these c a l c u l a t i o n s a r e g i v e n i n Ap p e n d i c e s F and G. The v i r t u a l c o r r e c t i o n s a r e g i v e n by the sum of the s e l f -energy and v e r t e x a m p l i t u d e s i n t e r f e r e d w i t h the a m p l i t u d e f o r the f r e e decay graph and i n t e g r a t e d over the o u t g o i n g n e u t r i n o momenta. The a p p r o p r i a t e mass c o u n t e r t e r m has been s u b t r a c t e d from the s e l f - e n e r g y c o r r e c t i o n . The r e s u l t , w i t h the i t h n e u t r i n o h a v i n g mass , i s 90 t ( i - x M i - ^ M - ^ . ^ e [ ( i - ^ ( i - x V ( i ^ ^ l ] F | (4.5) where t J U ( l - y ^ - - i J ^ S e ^ a. " "F x ] J ! ^ * - JL (4.6) and S a = ^ /( \-%) ( 4 . 7 ) The Spence f u n c t i o n L i 2 ( x ) i s d e f i n e d as f o l l o w s . X The n o t a t i o n i s t h a t of Chapter 3. That i s , (4.8) (4.9) where 6 i s the a n g l e between the p o s i t r o n momentum and the ^-a x i s ; s^ L i s the muon s p i n and p e a u n i t v e c t o r i n the d i r e c t i o n of p o s i t r o n momentum. The d i v e r g e n c e appears as a p o l e i n ( n - 4 ) . To t r a n s l a t e 91 t h i s d i m e n s i o n a l r e g u l a r i z a t i o n r e s u l t i n t o t h a t o b t a i n e d w i t h the c u t o f f method [17] one s i m p l y makes the f o l l o w i n g s u b s t i t u t i o n h tJ /H^vFVT — - J™. \ m W N (4.10) where X m l n i s the c u t o f f photon energy. Then, i n the l i m i t of m a s s l e s s n e u t r i n o s , K a l l e n ' s r e s u l t i s r e p r o d u c e d . The d o u b l e d i f f e r e n t i a l decay r a t e f o r the b r e m s s t r a h l u n g p r o c e s s (4.1) w i t h one massive n e u t r i n o i n the f i n a l s t a t e i s g i v e n below. T h i s r e s u l t i s u s e f u l f o r muon decay e x p e r i m e n t s i n which the photon i s a l s o d e t e c t e d . The n e u t r i n o momenta have been i n t e g r a t e d out and the photon p o l a r i z a t i o n s summed o v e r . Thus, we o b t a i n 92 (4.11) where 3 P ( 3 / a ^ la)* CQ^y (4.12) (4.13) •f Y* s (4.14) (4.15) I n t e g r a t i o n over the photon momentum y i e l d s the f o l l o w i n g r e s u l t . _ i 93 - 3 ^ j k ? 1 ^ i 0 - ^ ) 0 - ^ - ^ 94 (4.16) The i n f r a r e d d i v e r g e n c e appears as a p o l e i n ( n - 4 ) . I t p r e c i s e l y c a n c e l s the d i v e r g e n c e of the v i r t u a l c o r r e c t i o n s g i v e n i n e q u a t i o n ( 4 . 5 ) . In t h e s e r e s u l t s , terms m u l t i p l i e d by 6| or h i g h e r powers of 6 e have been dropped but l n 6 e terms have been r e t a i n e d . W i t h these r e s u l t s f o r the r a d i a t i v e c o r r e c t i o n s , the M i c h e l spectrum can now be o b t a i n e d . S p e c i a l i z i n g t o the t h r e e n e u t r i n o w o r l d , under the assumption of HNN m i x i n g , v3, which i s p r e d o m i n a n t l y the n e u t r i n o i n the d o u b l e t (v z~) , i s the o n l y massive n e u t r i n o t o c o n t r i b u t e t o muon decay. N u m e r i c a l r e s u l t s w i l l be g i v e n f o r m3 i n the Mev/c 2 range. Thus, the p o s i t r o n spectrum f o r the decay of muons a t r e s t i s g i v e n by I n t e r p r e t i n g t h i s e x p r e s s i o n i n terms of the sum of s p e c t r a of e q u a t i o n (4.3) and the HNN m i x i n g , R 0 i s the u s u a l r a d i a t i v e l y c o r r e c t e d spectrum f o r m a s s l e s s n e u t r i n o s w h i l e R 3 i s the spectrum f o r which one of the o u t g o i n g n e u t r i n o s i s mass i v e . The r e s u l t f o r R 0 i s w e l l known and i s , of c o u r s e , o b t a i n e d i n the l i m i t of m a s s l e s s n e u t r i n o s f o r the c a l c u l a t i o n s g i v e n above. 95 Both R 0 and R 3 can be d i v i d e d i n t o a f r e e decay p i e c e and a p i e c e due t o r a d i a t i v e c o r r e c t i o n s . Ro= Rcf + C (4.18a) = -pe x M l - a . x\l 6(\-%) (4.18b) - a i x t ^ <+x^]} 6 - p, R o 5 (4.18c) The s p i n dependent term of the r a d i a t i v e l y c o r r e c t e d p a r t Ro^ , i s g i v e n by C = ^ ^ x^(\-^ O r M - i± J U - (All* E( I + Ht 34 X ^ * /*t + 3-Tic - + tL. C\-^ J^(i-%S] \ 6(\-%) ( 4 . l 8 d ) In the above, r M = ( J k ^ t - r ) U J * ^ ] z ] 0 * ^ +3vv^-0 ( J ^ * ~ " 7 +0 (4.19) The spectrum f o r which one of the n e u t r i n o s i s massive i s g i v e n below. %i + ^ (4.20a) where the f r e e p a r t 96 + ( I - 0 [ ( I - < > I A - ( \ - v p ^ e ^ - x - ^ (4.20b) was c a l c u l a t e d i n Chapter 2 . The r a d i a t i v e l y c o r r e c t e d p a r t i s 5 ~ IT fc [ i (\a + M S x - 11 x =0 + X - ^ f p f f ) X f ^ f- 3 x- ^  x ^ I \ x 5 l k * / s j ] ^ (4.20c) The s p i n dependent p a r t i s ^36 ~ ^05 + +• <*_X L X t ( ' ^ - ^ x r 1^X^SWMx'Q J ) ^ _ L ^ - n x ^ - S X ^ J v L X / ^ l 97 US-loo* t^y^-H^x 3") +- L ^ O i ^ + \\0-y^ ^ J (4.20d) Here r ( x ) i s g i v e n by e q u a t i o n ( 4 . 1 8 ) . R e c a l l t h a t The dependence of the r a t e on 6, r a t h e r than 6, shows c l e a r l y t he importance of r a d i a t i v e c o r r e c t i o n s a t the h i g h - e n e r g y end of the p o s i t r o n spectrum. The l n ( 1 - x ) terms of the spectrum a r e s i n g u l a r as x approaches 1; t h e s e terms a r e p r e s e n t i n R 0 as w e l l as i n R, and a r e a m a n i f e s t a t i o n of the s o - c a l l e d i n f r a r e d c a t a s t r o p h e . They r e f l e c t a charged p a r t i c l e ' s c a p a b i l i t y of e m i t t i n g a l a r g e number of v e r y s o f t photons. S i n c e any r e a l experiment has a f i n i t e r e s o l u t i o n , t h e s e terms can be r e p l a c e d , f o r x near 1 [ 1 6 ] , by — JU(\-\+m) (4.22) where m A x/2 i s the energy r e s o l u t i o n of the a p p a r a t u s . 98 The i n c l u s i o n of a nonzero n e u t r i n o mass has not r e s u l t e d i n any a d d i t i o n a l mass s i n g u l a r i t i e s i n the r a t e f o r muon decay. Wherever terms l o g a r i t h m i c i n 6 o c c u r , they a r e m u l t i p l i e d by the a p p r o p r i a t e f a c t o r s of 6 and v a n i s h i n the l i m i t of z e r o n e u t r i n o mass. Only the u s u a l e l e c t r o n mass s i n g u l a r i t i e s a r e p r e s e n t , i n the form of l n 6 a terms, as e x p e c t e d from the Lee-N a u e n b e r g - K i n o s h i t a theorem [ 8 0 , 8 1 ] . We d i s p l a y the hi g h - e n e r g y end of the M i c h e l spectrum i n F i g u r e 15 f o r a 5 Mev/c 2 n e u t r i n o mass and s e v e r a l v a l u e s of U^.3. The m a s s l e s s n e u t r i n o spectrum i s a l s o p l o t t e d as a r e f e r e n c e . The s p e c t r a a r e a l l n o r m a l i z e d t o the m a s s l e s s c a s e . The o v e r a l l s p e c t r a l shape f o r the massive n e u t r i n o s case i s v e r y s i m i l a r t o the m a s s l e s s case now t h a t the r a d i a t i v e c o r r e c t i o n s have been a c c o u n t e d f o r . In F i g u r e 16, we p l o t the same s p e c t r a f o r a 10 Mev/c 2 n e u t r i n o . There a r e two f e a t u r e s t o lo o k f o r i n the massive n e u t r i n o s p e c t r a . F i r s t t h e r e i s a k i n k a t 1-x-6 2 due t o the k i n e m a t i c t u r n i n g o f f of the f i n a l s t a t e w i t h the massive n e u t r i n o . F o r i n s t a n c e , f o r a 5 Mev/c 2 n e u t r i n o , t h i s t u r n i n g - o f f o c c u r s a t the p o i n t x=.9998, which i s not measurable w i t h i n the p r e s e n t e x p e r i m e n t a l r e s o l u t i o n . The second f e a t u r e t o n o t i c e i s t h a t the maximum of the p o s i t r o n spectrum s h i f t s t o a lower v a l u e of x compared t o the m a s s l e s s c a s e . F i g u r e s 15 and 16 c l e a r l y show the d i f f i c u l t y of measuring t h i s s h i f t i n the p o s i t i o n of the maximum. For s m a l l m i x i n g , i t i s v e r y d i f f i c u l t t o d i s t i n g u i s h the m a s s l e s s and massive n e u t r i n o s s p e c t r a . F i g u r e 17 [76] g i v e s the v a l u e s of the m i x i n g U^a a l l o w e d 99 100 F i g . 16: The h i g h - e n e r g y end of t h e r a d i a t i v e l y c o r r e c t e d M i c h e l spectrum f o r a 10 MeV/c 2 n e u t r i n o w i t h >UA3\ 2=.059 (dashed l i n e ) and e q u a l m i x i n g of I U^.3\2= \U t 3\ 2=. 1 (dash-dot c u r v e ) . The s o l i d c u r v e i s the s t a n d a r d M i c h e l spectrum. 101 1 3 5 10 nyMeV/c 8) F i g . 17: L i m i t s of a l l o w e d m i x i n g of l u ^ a l 2 as a f u n c t i o n of n e u t r i n o mass. H i e r a r c h i a l - n e a r e s t - n e i g h b o u r m i x i n g i s d e p i c t e d i n ( a ) ; (b) g i v e s the case of e q u a l m i x i n g . 102 F i g 18: High-energy end of M i c h e l spectrum w i t h t h e s p i n of /x * a n t i p a r a l l e l t o t h e p o s i t r o n momentum. For m3=5 MeV/c 2, the dash-dot c u r v e has m i x i n g \U^3\ 2 = \ U c 3 l 2=.1 and the dash c u r v e has I U ^ l 2=. 067. The m a s s l e s s n e u t r i n o w i t h r a d i a t i v e c o r r e c t i o n s case i s d e p i c t e d by the s o l i d l i n e . The d a s h - c r o s s l i n e i s the (1-x) f a l l o f f f o r no r a d i a t i v e c o r r e c t i o n s and m a s s l e s s n e u t r i n o s . 103 by the c u r r e n t d a t a as a f u n c t i o n of n e u t r i n o mass. The HNN m i x i n g a s s u m p t i o n , i s made i n F i g u r e 17a and the r e s u l t s f o r e q u a l m i x i n g a r e shown i n F i g u r e 17b. As an example of the i n f o r m a t i o n i n the s e graphs, f o r the case of HNN m i x i n g , a n e u t r i n o of mass 5 Mev/c 2 i s not r u l e d out by p r e s e n t d a t a f o r a m i x i n g I U ^ l 2<. 07. For the decay of p o l a r i z e d muons, we c o n s i d e r the same case as i n Chapter 3. That i s , the decay w i t h the p o s i t r o n momentum a n t i p a r a l l e l t o the muon s p i n i s c a l c u l a t e d . For the case of ma s s l e s s n e u t r i n o s , t h i s p a r t i c u l a r c o n f i g u r a t i o n i s f o r b i d d e n at t he p o i n t x=1 by a n g u l a r momentum c o n s i d e r a t i o n s , as we d i s c u s s e d i n Chapter 3. The f a l l o f f t o z e r o of the r a t e was seen t o be l i n e a r f o r the f r e e decay. W i t h the i n c l u s i o n of r a d i a t i v e c o r r e c t i o n s the f a l l o f f d e v i a t e s from l i n e a r i t y but the r a t e s t i l l v a n i s h e s a t x=1 f o r m a s s l e s s n e u t r i n o s . W i t h the m i x i n g of a massive n e u t r i n o the r a t e no l o n g e r v a n i s h e s a t x=1. We p r e s e n t the r e s u l t s f o r a 5 Mev/c 2 n e u t r i n o f o r d i f f e r e n t v a l u e s of the m i x i n g i n F i g u r e 18. Only f o r the case of l a r g e , and p r o b a b l y u n r e a l i s t i c , m i x i n g i s t h i s case d i s c e r n i b l e from the ma s s l e s s n e u t r i n o s c a s e . 104 V. SUMMARY AND CONCLUSIONS T h i s t h e s i s d e s c r i b e s a r e c a l c u l a t i o n of the r a t e f o r the o r d i n a r y muon decay y n + — * e+ v t ^ (5.1) w i t h the assumptions t h a t n e u t r i n o s can be massive and t h a t t h e r e i s m i x i n g among the n e u t r i n o f a m i l i e s . The c a l c u l a t i o n i s independent of any s p e c i f i c model f o r n e u t r i n o masses. Indeed, the masses of the n e u t r i n o s a r e i n p u t parameters f o r the c a l c u l a t i o n s . The q u e s t i o n of d i s t i n g u i s h i n g between D i r a c and Majorana n e u t r i n o s was a l s o e x p l o r e d . Both the u n c e r t a i n e x p e r i m e n t a l s i t u a t i o n r e g a r d i n g the n a t u r e of n e u t r i n o s and the l a c k of a t h e o r e t i c a l p r i n c i p l e r e s t r i c t i n g the mass of the n e u t r i n o t o be z e r o have s e r v e d as m o t i v a t i o n f o r t h i s s t u d y . The m a t t e r of the n a t u r e of the n e u t r i n o i s an i m p o r t a n t , fundamental problem. In the s o l u t i o n t o t h i s problem l i e s the c l u e t o a deeper u n d e r s t a n d i n g of e l e c t r o w e a k i n t e r a c t i o n s and the means t o d i s t i n g u i s h between competing t h e o r i e s . Thus, the i m p l i c a t i o n s of massive n e u t r i n o s i n e x p e r i m e n t a l l y a c c e s s i b l e s i t u a t i o n s must be e x p l o r e d . Muon decay i s one such r e l e v a n t s i t u a t i o n t o i n v e s t i g a t e . The c a l c u l a t i o n s here were done assuming t h a t the weak i n t e r a c t i o n s a r e p r e d o m i n a n t l y l e f t - h a n d e d ; the e f f e c t i v e f o u r -f e r m i o n V-A t h e o r y has been used. Muon decay i s m a i n l y s t u d i e d 105 e x p e r i m e n t a l l y t o determine the form of the weak i n t e r a c t i o n s . However, here we have assumed a s p e c i f i c i n t e r a c t i o n form (V-A) and, t h u s , have i s o l a t e d the e f f e c t s of massive n e u t r i n o s . R a d i a t i v e c o r r e c t i o n s a r e known t o be i m p o r t a n t i n muon decay so the f i r s t o r d e r p h o t o n i c c o r r e c t i o n s t o the decay (5.1) and the r a t e f o r the r a d i a t i v e decay (5.2) have been c a l c u l a t e d . The b r e m s s t r a h l u n g spectrum i s g i v e n as w e l l as the u s u a l M i c h e l spectrum. T h i s r e s u l t i s u s e f u l i n the event t h a t a muon decay experiment i n which the photon i s d e t e c t e d i s performed. In Chapter 3, r e s u l t s were p r e s e n t e d f o r the u n c o r r e c t e d decay ( 5 . 1 ) . The M i c h e l spectrum was o b t a i n e d f o r the g e n e r a l case of N l e p t o n f a m i l i e s w i t h a r b i t r a r y m i x i n g s ; i t i s , i n f a c t , an i n c o h e r e n t sum of N 2 s p e c t r a . There a r e too many parameters i n t h i s g e n e r a l case t o make a m e a n i n g f u l n u m e r i c a l a n a l y s i s . Thus, we s p e c i a l i z e d t o the p h e n o m e n o l o g i c a l l y i m p o r t a n t case of t h r e e l e p t o n f a m i l i e s f o r the r e m a i n i n g c a l c u l a t i o n s . F u r t h e r , we a l l o w o n l y one of the t h r e e n e u t r i n o s , t h a t a s s o c i a t e d w i t h the t f a m i l y , t o be m a s s i v e . T h i s i s a r e s u l t of our assumption of h i e r a r c h i a l m i x i n g and i t s a p p l i c a t i o n t o e x p e r i m e n t a l l i m i t s on n e u t r i n o masses. W i t h t h e s e a s s u m p t i o n s , the p o s i t r o n s p e c t r a f o r both p o l a r i z e d and u n p o l a r i z e d muons d e c a y i n g a t r e s t were p r e s e n t e d f o r s e v e r a l n e u t r i n o masses i n the Mev/c 2 range and s e v e r a l m i x i n g s (see F i g u r e s 4,5,7-9). The s p e c t r a f o r the decay of p o l a r i z e d muons 106 ar e r a t h e r i n s e n s i t i v e t o n e u t r i n o mass. However, the s p e c t r a f o r u n p o l a r i z e d muon decay e x h i b i t a s t r o n g s e n s i t i v i t y t o n e u t r i n o mass a t the h i g h energy end. No st a t e m e n t s c o u l d be made r e g a r d i n g l i m i t s on the n e u t r i n o mass or m i x i n g s a t t h i s p o i n t s i n c e r a d i a t i v e c o r r e c t i o n s a r e l a r g e . However, the q u e s t i o n of the Majorana or D i r a c n a t u r e of n e u t r i n o s was a l s o s t u d i e d . We c o n c l u d e t h a t i t i s p r o b a b l y not p o s s i b l e t o d i s t i n g u i s h between D i r a c and Majorana n e u t r i n o s by measuring the M i c h e l spectrum. E x p e r i m e n t s i n which more d i r e c t i n f o r m a t i o n on the n e u t r i n o i s o b t a i n e d seem t o be n e c e s s a r y i n o r d e r t o det e r m i n e the n a t u r e of n e u t r i n o s . A l o n g t h i s l i n e , we c a l c u l a t e d e n e r g y - a n g l e and energy-energy c o r r e l a t i o n s f o r muon decay i n which both the p o s i t r o n and the n e u t r i n o a r e d e t e c t e d . The energy-energy c o r r e l a t i o n s l o o k most p r o m i s i n g , e x h i b i t i n g a marked d i f f e r e n c e between D i r a c and Majorana n e u t r i n o s . The d i s t r i b u t i o n i n p o s i t r o n and n e u t r i n o e n e r g i e s depends q u a d r a t i c a l l y on the p o s i t r o n energy i f n e u t r i n o s a r e Majorana p a r t i c l e s and i s independent of p o s i t r o n energy f o r D i r a c n e u t r i n o s . The d i f f e r e n c e between the D i r a c and Majorana r e s u l t s i s p r o p o r t i o n a l t o the m i x i n g between t h e y x - t y p e n e u t r i n o and i t s n e a r e s t l i g h t n e i g h b o u r , v,. Of c o u r s e , the e x p e r i m e n t a l f e a s i b i l i t y of such a measurement i s q u e s t i o n a b l e . The o u t g o i n g n e u t r i n o would be d e t e c t e d v i a the i n v e r s e yS -decay p r o c e s s +• n -» eT + p (5.3) Wit h the p r e s e n t l y a v a i l a b l e muon f l u x a t the meson f a c i l i t i e s , we c a l c u l a t e d an event r a t e of s e v e r a l t e n s per day f o r a 1 07 100 ton 4Tf water d e t e c t o r . However, the d i f f i c u l t y l i e s i n d e t e c t i n g b o t h the e l e c t r o n and p r o t o n i n the i n v e r s e /3 -decay (5.3) i n o r d e r t o determine the n e u t r i n o energy. Muon decay i n f l i g h t might be a more f e a s i b l e c o r r e l a t i o n experiment t o c o n s i d e r . One might make the g e n e r a l statement t h a t we a r e l e f t w i t h no easy way t o determine the fundamental D i r a c or Majorana n a t u r e of the n e u t r i n o . That i s , we must l o o k t o d i f f i c u l t and i n g e n i o u s means of p r o b i n g t h i s f a c e t of the n e u t r i n o ' s n a t u r e , such as c o r r e l a t i o n s . The s t r i k i n g d i f f e r e n c e between the D i r a c and Majorana r e s u l t s f o r energy-energy c o r r e l a t i o n s i s a t a n t a l i z i n g i f not q u i t e a c c e s s i b l e encouragement. Of c o u r s e , the e f f e c t s of n e u t r i n o mass on the u n c o r r e c t e d u n p o l a r i z e d M i c h e l spectrum d i s p l a y e d i n F i g u r e s 4 and 5 a r e a l s o r a t h e r e n c o u r a g i n g a t f i r s t s i g h t . Q u a l i t a t i v e l y , the e f f e c t of a nonzero n e u t r i n o mass i s t o s h i f t the peak i n the spectrum t o lower p o s i t r o n e n e r g i e s , r e d u c i n g from maximum the spectrum a t i t s e n d p o i n t . I t i s w e l l known t h a t r a d i a t i v e c o r r e c t i o n s a l s o decrease the spectrum from maximum a t i t s e n d p o i n t , i n f a c t , r e d u c i n g i t t o z e r o . The f i r s t o r d e r p h o t o n i c c o r r e c t i o n s t o the decay (5.1) and the r a t e f o r the r a d i a t i v e decay (5.2) a r e g i v e n i n Chapter 4. The c a l c u l a t i o n was done f o r the case of D i r a c n e u t r i n o s s i n c e i t was e s t a b l i s h e d i n Chapter 2 t h a t t h e M i c h e l spectrum i s l a r g e l y i n s e n s i t i v e t o the d i f f e r e n c e between D i r a c and Majorana n e u t r i n o s . A g a i n , o n l y the t h r e e n e u t r i n o w o r l d has been c o n s i d e r e d and, f u r t h e r , the assumption of h i e r c h i a l n e a r e s t neighbour (HNN) m i x i n g has been made. The gauge i n v a r i a n t method of d i m e n s i o n a l r e g u l a r i z a t i o n 108 was used t o handle the d i v e r g e n c e s which o c c u r i n the s e c o r r e c t i o n s . The c o n t r i b u t i o n of each c o r r e c t i o n was e v a l u a t e d and the d i v e r g e n c e s were found t o c a n c e l t o o r d e r a , l e a v i n g a f i n i t e answer, j u s t as i n the case of m a s s l e s s n e u t r i n o s . The p o s i t r o n s p e c t r a a r e d i s p l a y e d f o r both p o l a r i z e d (see F i g u r e 18) and u n p o l a r i z e d (see F i g u r e s 15 and 16) muon decays. In the case of p o l a r i z e d muon decay, the e f f e c t s of massive n e u t r i n o s c o u l d be d e t e c t e d e x p e r i m e n t a l l y o n l y f o r what a r e p r o b a b l y u n r e a l i s t i c n e u t r i n o masses and m i x i n g s , n e u t r i n o masses i n the Mev/c 2 range and m i x i n g a n g l e s l a r g e r than Cabibbo m i x i n g . For u n p o l a r i z e d muon decay, the e f f e c t of a massive n e u t r i n o w i t h s m a l l m i x i n g s i n t o the n e u t r i n o s i n the e l e c t r o n and muon weak d o u b l e t s i s s m a l l . The s e n s i t i v i t y t o n e u t r i n o mass a t the h i g h energy end of the p o s i t r o n spectrum i s l a r g e l y washed out w i t h the i n c l u s i o n of r a d i a t i v e c o r r e c t i o n s . 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D r e l l , " R e l a t i v i s t i c Quantum Me c h a n i c s " and " R e l a t i v i s t i c Quantum F i e l d s " , ( M c G r a w - H i l l , New York, 1964). 83. S.G. E c k s t e i n and R.H. P r a t t , Ann. Phys. 8, 297 (1959). 1 13 A. C o n v e n t i o n s and N o t a t i o n The p r o p e r t i e s of D i r a c s p i n o r s and m a t r i c e s which have been used e x t e n s i v e l y i n the c a l c u l a t i o n s of t h i s work a r e p r e s e n t e d i n t h i s Appendix. A l s o , as an example of the use of those p r o p e r t i e s , a r e l e v a n t squared m a t r i x element i s c a l c u l a t e d . The c o n v e n t i o n s employed i n t h i s work a r e thos e of B j o r k e n and D r e l l [ 8 2 ] . The m e t r i c g^v i s g i v e n below. I I 0 0 0\ o -\ 0 0 \ , 0 0 -\ 0 < A- 1 ) V o o o -i / The s c a l a r p r o d u c t of two f o u r - v e c t o r s , a and b, i s d e f i n e d as f o l l o w s . OL- b - a^.VT = (X^ ky«J>*V = a o b 0 ~ a. • Y> (A. 2) The f r e e D i r a c e q u a t i o n d e s c r i b i n g a s p i n 1/2 p a r t i c l e of mass m i s i r r f — yvL — 0 ( A < 3 ) where the ^ a r e 4x4 m a t r i c e s which obey the f o l l o w i n g a n t i c o m m u t a t i o n r e l a t i o n s . I i s t h e 4x4 u n i t m a t r i x . Another m a t r i x c o n s t r u c t e d from the m a t r i c e s i s 1 1 4 r - t r r r . r <A.S> The f o u r component s p i n o r 7^ which i s a g e n e r a l s o l u t i o n t o the f r e e D i r a c e q u a t i o n can be e x p r e s s e d as a s u p e r p o s i t i o n of s i n g l e - p a r t i c l e p l a n e wave s o l u t i o n s as f o l l o w s . V i s a n o r m a l i z a t i o n volume and the energy, E, f o r a p a r t i c l e o b e y i n g the D i r a c e q u a t i o n i s g i v e n i n terms of i t s t h r e e -momentum and mass as E= J l j ^ t ^ (A.7) The o p e r a t o r s b(p,s) and d( p , s ) a r e the a n n i h i l a t i o n o p e r a t o r s f o r a p a r t i c l e and a n t i p a r t i c l e , r e s p e c t i v e l y . L i k e w i s e , b^(p,s) and d^(p,s) a r e the p a r t i c l e and a n t i p a r t i c l e c r e a t i o n o p e r a t o r s . The c r e a t i o n and a n n i h i l a t i o n o p e r a t o r s s a t i s f y the u s u a l a n t i c o m m u t a t i o n r e l a t i o n s as g i v e n below. l[V> V(p,s\WfVs' ,)l - ^''h(f-f') (A.8) U%s) ^(f<y)\= w U f - p ^ ( A . 9 ) -{ W p ^ ^ c K p ' . ^ V (^(-p.S^cK-p^S'^ ^0 (A.10) Here p i s the four-momentum and s i s the s p i n - p o l a r i z a t i o n f o u r -v e c t o r g i v e n by 1 15 S ^ S ° , ? V - ^ " g _ 5 ^ £ V ( f p \ (A. 1 1 ) In e q u a t i o n ( A . 6 ) , u ( p , s ) i s a f o u r component s p i n o r taken t o r e p r e s e n t a p a r t i c l e s o l u t i o n w h i l e i r ( p , s ) r e p r e s e n t s an a n t i p a r t i c l e s o l u t i o n . The p a r t i c l e and a n t i p a r t i c l e s p i n o r s s a t i s f y the f o l l o w i n g e q u a t i o n s . u.{^-yvO -o (A.12) (A.13) where -^ " U 1 6 (A.14) The s o - c a l l e d s l a s h n o t a t i o n used above has the f o l l o w i n g meaning. / 3 Oy. (A. 15) The f o l l o w i n g p r o p e r t i e s of the s p i n o r s a re v e r y u s e f u l . u t p , , ^ T U D , ^ = (^J-TVQ ( I + h i) (A.16a) T - ^ ( p ^ i U p , ^ - jp'+i^- (A.16b) W j ^ s W t p ^ - (jzC-wv>) ( i + ^ y ) (A. 17a) a. "!L T H p / O V-(j?,«S> ^  j^-Tw. (A.17b) E x t e n s i v e use i s made of the f o l l o w i n g p r o p e r t i e s of the D i r a c ^ m a t r i c e s . 1 16 Yf- YJ^0 (A. 18) p R = - V * ^ (A.19) i ^ r - ^n'-^f ^*r- cc^6yh\ (A.20) For f o u r - v e c t o r s a, b, and c, the f o l l o w i n g r e l a t i o n s h o l d . ^ W = - U ^ i - a , C L - b (A.21 ) V A JZC = - 2 ?t (A. 22) X A 0 C W t f A = 4 f l C - b (A.23) * y u * ^ X A = ^ (A.24) The t r a c e s of v a r i o u s p r o d u c t s of t m a t r i c e s a r e Tr ^ - Tr Vs" = 0 (A.25) Tr y Yv - H (A.26) T r ( odd # o"f ^ ma-irices^ =0 (A.27) T r r n f f r ^ = ^ L ^ V f r u . 2 9 ) In the above, ^/ u >? t r i s the c o m p l e t e l y a n t i s y m m e t r i c t e n s o r such t h a t t°yZi = + l (A.30) E q u a t i o n (A.20) i s p a r t i c u l a r l y u s e f u l i n c a l c u l a t i n g the t r a c e of a p r o d u c t of s e v e r a l D i r a c V m a t r i c e s , as i t e f f e c t i v e l y 117 reduces the number of m a t r i c e s i n the p r o d u c t . I t can be used r e p e a t e d l y u n t i l the p o i n t where e q u a t i o n s (A.25-29) can be a p p l i e d . The p r o p e r t i e s g i v e n above f o r the V m a t r i c e s are extended t o the n - d i m e n s i o n a l case i n Appendix E. As an example of the m a n i p u l a t i o n of the s p i n o r s and m a t r i c e s , we p r e s e n t the c a l c u l a t i o n of the squared m a t r i x element f o r muon decay w i t h massive D i r a c n e u t r i n o s . The q u a n t i t y t o be c a l c u l a t e d i s , a c c o r d i n g t o e q u a t i o n (3.1.7) n. - E i m * - ( a . 3 D V - - S p i y r S wherein the m a t r i x element i s (A.32) Here, V" (p„ , s. ) i s the s p i n o r r e p r e s e n t i n g the p o s i t r o n ; -\J"( p., , s^) r e p r e s e n t s the antimuon, i r ( p . ,s. ) the a n t i n e u t r i n o , and u(p- ,s ) J J t L 4. the n e u t r i n o . M i s most e a s i l y found t h r o u g h l a b e l l i n g the components of the s p i n o r s and m a t r i c e s as f o l l o w s . 'To? Upon a p p l y i n g the complex c o n j u g a t i o n and some rearrangement, we have U s i n g e q u a t i o n (A.17), 1 18 The commutation p r o p e r t i e s of t h e t m a t r i c e s y i e l d , f i n a l l y , ^ U ^ L u ^ ^ ^ (A-.33) Now the squared m a t r i x element, 7^1, can be c a l c u l a t e d . L a b e l l i n g the m a t r i c e s and r e a r r a n g i n g , T h e r e f o r e , (A.34) where T> Tr [ ( ^ ~ Yy V H- Yy ftj Y A( i- ^  (ft - ^  •) Y ? (I - Yy)1 C a l c u l a t i o n of the f i r s t t r a c e , T,, proceeds as f o l l o w s . U s i n g e q u a t i o n (A.18) and the f a c t t h a t the t r a c e s i m p l i f i e s t o 119 7 > T r 1 + V ' - V l S i n c e the t r a c e of an odd number of K m a t r i c e s v a n i s h e s , t h i s reduces t o U s i n g e q u a t i o n s (A.27) and (A.28), we f i n d where the n o t a t i o n has been used. S i m i l a r l y , T h e r e f o r e , the squared m a t r i x element i s ^ U M U ^ I 1 ^ ( p ^ . p ^ C ^ p - ) (A.35) where the f o l l o w i n g p r o p e r t y of the c o m p l e t e l y a n t i s y m m e t r i c t e n s o r has been used. 1 20 B. Majorana S t a t e s We g i v e here some of t h e c o n v e n t i o n s i n the d e s c r i p t i o n of Majorana p a r t i c l e s . Case [4] has f o r m u l a t e d the Majorana t h e o r y of n e u t r i n o s . The f i e l d c o n j u g a t e t o a f i e l d ^ i s d e f i n e d as where the charge c o n j u g a t i o n o p e r a t o r , C, i s O L ^ T (B.2) T h e r e f o r e , e q u a t i o n (B.1) can be r e w r i t t e n as it* f (B.3) A Majorana f i e l d s a t i s f i e s the f o l l o w i n g c o n d i t i o n . (B.4) That i s , a Majorana p a r t i c l e i s not d i s t i n g u i s h a b l e from i t s a n t i p a r t i c l e . T h i s c o n d i t i o n a l l o w s a r e d u c t i o n from the f o u r component s p i n o r d e s c r i p t i o n of a p a r t i c l e s a t i s f y i n g the D i r a c e q u a t i o n t o a d e s c r i p t i o n i n terms of two-component s p i n o r s . T h i s becomes c l e a r upon w r i t i n g the f i e l d ij/ i n terms of two two-component f i e l d s . and a p p l y i n g t h e c o n d i t i o n ( B . 4 ) . For the purpose of t h i s d i s c u s s i o n , t h e f o l l o w i n g f5 d i a g o n a l r e p r e s e n t a t i o n of the D i r a c / m a t r i c e s i s used. 121 1 1 o/ U 0 I 0 0 -I J (B.6) The r e s u l t i s t h a t ^ can be w r i t t e n i n terms of a s i n g l e two-component s p i n o r , (f), i n the f o l l o w i n g manner. f-- (. '*) (B.7) i S i n c e if1 s a t i s f i e s the D i r a c e q u a t i o n , (p must s a t i s f y the f o l l o w i n g f r e e p a r t i c l e e q u a t i o n s . >JP **-1<p-~^<f>* =0 (B.8) -l[tr±(P*) +G.V(<r(n+)-K(p=o ( B 9) The e q u a t i o n s (B.8) and (B.9) above a r e complex c o n j u g a t e s of each o t h e r . In terms of the o r i g i n a l f i e l d , ijj , the D i r a c e q u a t i o n f o r a Majorana p a r t i c l e can be w r i t t e n t o r e f l e c t the Majorana p r o p e r t y as i>-nV f* = o ( B , 1 0 ) T h i s i s known as the Majorana e q u a t i o n . The F o u r i e r d e c o m p o s i t i o n of the f i e l d i s g i v e n by 1 22 In c o n t r a s t t o the d e c o m p o s i t i o n f o r a D i r a c p a r t i c l e g i v e n i n e q u a t i o n ( A . 6 ) , e q u a t i o n (B.11) c o n t a i n s o n l y one type of c r e a t i o n and d e s t r u c t i o n o p e r a t o r a ( p , s ) w i t h the f o l l o w i n g commutation r e l a t i o n s . U+(pA<xtY,vA = W $>(f-f) (B.12a) [at(k '~\ - i a + ) = o ( B . i 2 b ) T h i s r e f l e c t s the p r o p e r t y of p a r t i c l e and a n t i p a r t i c l e b e i n g i n d i s t i n g u i s h a b l e . The d e c o m p o s i t i o n (B.11) can be used, j u s t as i n the D i r a c p a r t i c l e c a s e , f o r the c a l c u l a t i o n of m a t r i x e l e m e n t s . However, i t i s i n t e r e s t i n g t o a l s o w r i t e the F o u r i e r d e c o m p o s i t i o n of the two-component f i e l d cp(t)~fcZ j ^ U ? ) A K ( f i A V " t ? \ a n F > U ( y , A W L ? - x l (B.13) Now, i n e q u a t i o n ( B . 1 3 ) , u.(p,\) and u ( p A ) a r e two-component s p i n o r s . The symbol A denotes h e l i c i t y ; i t i s s i m p l y a matter of c h o i c e t o work i n a h e l i c i t y b a s i s . S u b s t i t u t i o n of the d e c o m p o s i t i o n (B.13) i n t o the Majorana e q u a t i o n (B.8) f o r the two-component f i e l d (D y i e l d s the f o l l o w i n g e q u a t i o n s . ( E - ?• u ( p, A\ = L >L *( p, \) (B. 14) (E-x-y) - i r ^ A ^ = -L TVL ^  u * ( p ^ \ \ (B. 15) These e q u a t i o n s demonstrate t h a t the p a r t i c l e and a n t i p a r t i c l e s p i n o r s , u and i r , a r e not independent. T h i s r e s u l t i s e x p e c t e d s i n c e i t i s a r e f l e c t i o n of the i n d i s t i n g u i s h a b i l i t y of p a r t i c l e 1 23 and a n t i p a r t i c l e , the Majorana c o n d i t i o n . The s p i n o r s can be e x p r e s s e d i n terms of an o r t h o n o r m a l b a s i s of s t a t e s of d e f i n i t e h e l i c i t y . That i s , the s p i n o r s *(p) and /3 (P) of the h e l i c i t y b a s i s s a t i s f y the f o l l o w i n g r e l a t i o n s . T h e r e f o r e the s p i n o r ot (p) ( y S(p)) ( n e g a t i v e ) h e l i c i t y s t a t e . T h e r e f o r e i f component of u ( p , X ) i s taken t o be u ( ^ ^ = J l f t T j F i oi ( p ^ then the p o s i t i v e h e l i c i t y component of the a n t i p a r t i c l e s p i n o r i s g i v e n , a c c o r d i n g t o e q u a t i o n (B.15), by i r ( p . i ) " - - c m C E + a - p ' N ) j E i - l p \ c ^ (B.20) \ ( E ^ - l p l O r S i n c e the P a u l i m a t r i x c 2 i s o f f - d i a g o n a l , i t changes a s t a t e of p o s i t i v e h e l i c i t y t o one of n e g a t i v e h e l i c i t y . The r e l a t i v e phase of ot and p a r e chosen such t h a t **( J) = - I ^ / 3 0 p > (B.21) T h e r e f o r e , e q u a t i o n (B.21) becomes (B.16) (B.17) (B.18) r e p r e s e n t s a p o s i t i v e the p o s i t i v e h e l i c i t y (B.19) 124 tf(p,0 = - ^  (B.22) where the f a c t t h a t /3(p) i s a h e l i c i t y e i g e n s t a t e has been used. Then a s o l u t i o n f o r the n e g a t i v e h e l i c i t y p a r t of the s p i n o r i s I t f o l l o w s from e q u a t i o n (B.14) t h a t u ( p , a > ) = 18 ("3^ (B.23) The n o r m a l i z a t i o n has been chosen such t h a t I L u 1 " ( ^ x W p ^ = X . i r ^ l ^ \ W ( | ) , ^ - a.E e (B.24) Mohapatra and S e n j a n o v i c [60] p o i n t out t h a t , f o r the m a s s l e s s c a s e , p a r t i c l e and a n t i p a r t i c l e s t a t e s a re d i s t i n g u i s h a b l e s i n c e the d e c o m p o s i t i o n (B.13) becomes (p( x > ^ j ^ [ a ( p j ^ ^ ( p ) , V - l " P - \ o i t ( ^ i r ( p , x > ) e ^ " , t l (B.25) In o r d e r t o p r e s e n t the form of Majorana and D i r a c type mass terms i n a L a g r a n g i a n , the f o l l o w i n g l e f t and r i g h t p r o j e c t i o n s of a four-component f i e l d 1|* a r e d e f i n e d . The charge c o n j u g a t e f i e l d s a r e d e f i n e d as t - ( I V ^ ' ^ ^ ' ( " T V (B.27) 125 1 ^ = - ( t f C V (B.28) The mass terms of the L a g r a n g i a n can be w r i t t e n as f o l l o w s = " ( A l l + ^ f (B.29) where X - fyu + ^ (B.30a) \ s ^ R + ^R C (B.30b) ^ l t t R (B.30c) O b v i o u s l y , "X and ^ r e p r e s e n t Majorana p a r t i c l e s . The term i n D i s the u s u a l D i r a c mass term. 126 C. Phase Space C o n s i d e r a t i o n s and N o r m a l i z a t i o n The phase space f o r muon decay a t r e s t i s g i v e n by U s i n g the t h r e e - d i m e n s i o n a l 6 - f u n c t i o n t o do one of the i n t e g r a t i o n s , t h i s becomes y J l E e ) i E, ' where 'Ej 3" = \ |I +- pt \^ i ^ = l ^ 1 - v \jp!W a\^M\pt\^obe e- L -v ^  (c.2) The c o n d i t i o n -\ ^ cos etc ^ \ g i v e s the boundary of the D a l i t z p l o t . The energy 6 - f u n c t i o n g i v e s ^ - ( y ^ E ^ (C.3) E q u a t i n g the two e x p r e s s i o n s f o r E 2 above y i e l d s the f o l l o w i n g e q u a t i o n i n c o s 2 © ^ 127 = y^t- ^ > ( E ^ E,V> H E ^ E ^ & ^ E - ^ E ^ +- 4 (-vy ( E e + E L V E e E ^ t " ^ + i - v ^ - - * * ^ (C.4) For the case of m a s s l e s s n e u t r i n o s , t h i s reduces t o R [ ( T ^ ^ V\ftV I-Yvy, ( E e+ + i E e ' E i l (C.5) N e g l e c t i n g m^, the c o n d i t i o n on c o s e ^ g i v e s -14 ^ - i T v ( E e t E i H a E t E ; £\ (C.6) SL E e E l D e f i n i n g the f o l l o w i n g v a r i a b l e s , X = X, = 2_E_L (C.7) W y yvy^ . t h i s becomes T h i s r e s u l t y i e l d s x ^ = 1, as e x p e c t e d . For the case i # j and m^O, the e q u a t i o n i n c o s 2 9 e u becomes, n e g l e c t i n g the e l e c t r o n mass, H E ^ E ^ c o ^ G ^ = [ ^ - y H J i - 2 r y ( E e t E L N ) t i E e E L l : L ' (C.8) Thus the c o n d i t i o n on c o s 2 9 , ; becomes 128 -I ^ "VH.^  ~™t~ klfA^iEe.+ El) 4-a.EeE, ^ \ (C.9) In terms of the v a r i a b l e s x, x. , and 6=m /nu, the c o n d i t i o n i s L J O / - x y-L < I- "a"*" - ( X + X iY i O 0 T h i s y i e l d sx»wa.y - I- ^  (C.10) The l a s t case which must be c o n s i d e r e d i s f o r m. =m*0. N e g l e c t i n g e l e c t r o n mass, e q u a t i o n (C.4) reduces t o = Iwfi - a - y y ( E t + E ^ t X E e E ' t T (C.I 1 ) The c o n d i t i o n of c o s e - i s , i n terms of the v a r i a b l e s x, x , and 0 - x - x-t M J , ^ ^ s L x N) .1 j v 4 1 T h i s reduces t o [ i - x V x ' L i ^ - x H t - ^ / a i l r [ ( l - x Y 1 ^ * ; 1 ' ! ^ 0 The c o n d i t i o n t h a t t h i s e q u a t i o n have r e a l r o o t s i s T h i s reduces t o 129 I- X- 4 ^ > O (C. 12) so x = 1 - 4 6 2 f o r t h i s c a s e . Now t h a t the upper l i m i t on x has been e s t a b l i s h e d , the d i f f e r e n t i a l decay r a t e s can be i n t e g r a t e d t o o b t a i n the o v e r a l l n o r m a l i z a t i o n . A l l s p e c t r a w i l l be n o r m a l i z e d t o the u s u a l spectrum w i t h m a s s l e s s n e u t r i n o s . That i s , l i n ^ a. 0 I, M S T r * J 4 0 = ( C . 1 3 ) For the case c o n s i d e r e d i n S e c t i o n ( 3 . 1 ) , t h a t of D i r a c n e u t r i n o s , the t o t a l r a t e i s ^ I U L ^ i * t [ 1 U „ 1 S I ( i - ^ ( c . 14) 130 where the f o l l o w i n g r e s u l t s have been used. \-HVL_ Thus, the n o r m a l i z a t i o n f a c t o r N f o r e q u a t i o n (3.1.44) i s g i v e n by which i s a p p r o x i m a t e l y e q u a l t o N T - l U U ^ f l l l ^ N i ' ^ ) (c.15) f o r s m a l l \U e3! 2 For Majorana n e u t r i n o s , the t o t a l r a t e i s o where Rv i s g i v e n by e q u a t i o n ( C . 1 4 ) . Upon p e r f o r m i n g the i n t e g r a l i n the e q u a t i o n above, the r e s u l t , t o o r d e r 6 2, i s fc= ^-p - l U ^ l U ^ l S 1 " ( C 1 6 ) Thus, the d i f f e r e n c e i n n o r m a l i z a t i o n from the D i r a c case i s n e g l i g i b l e . T h i s c o m p l e t e s t h e phase space c o n s i d e r a t i o n s and 131 n o r m a l i z a t i o n s f o r the case of the f r e e muon decay. For the s e l f - e n e r g y and v e r t e x c o r r e c t i o n s the l i m i t s on the x-i n t e g r a t i o n a r e as g i v e n above. The upper l i m i t on the b r e m s s t r a h l u n g photon energy i s d e r i v e d below. The phase space f o r muons r a d i a t i v e l y d e c a y i n g a t r e s t i s d2fr to to to ^ ( ? , i l ^ V ? J H U y - E e - w-E ' - fc .^ 4 ! £ ? f d H f 4 i E i s U / A - E e - u 3 - E - - E J ^ ( C 1 7 ) where E ^ = (-vy- E € - u> - E ^ and 5. 8) where G i s t h e a n g l e between k and p^ and (p i s the a n g l e between j? and p^ +k*. A l s o , E q u a t i n g t h e two e x p r e s s i o n s f o r E.2, r e a r r a n g i n g , and s q u a r i n g y i e l d s 1 32 = \ U j + w * - 3 r y ( E e v c o + E O * 1 E e / c o - v E ^ E L - ^ u A ^ l cose) j (C. 1 9) S i n c e 0^cos2<p<1, the term i n square b r a c k e t s i s g r e a t e r than or e q u a l t o t h e term i n c u r l y b r a c k e t s . [ 1 - { } For t h e case of o n l y v. h a v i n g mass m , (m L=0) t h i s reduces t o CL E^~ +• b E L V C £ 0 (C.20) where O b v i o u s l y , c^O. R e a r r a n g i n g (C.20) and w r i t i n g b i n terms of a, we have (a-M™. xV( E t - v o + E e - Y y H H w j - E ^ - _c_ ( C 2 i ) S i n c e E L V- v E^e- w y . ^ 0 by energy c o n s e r v a t i o n and 4 ^ E L -<^/E- < 0 133 i t f o l l o w s t h a t S u b s t i t u t i n g i n the e x p r e s s i o n f o r a, we o b t a i n T h e r e f o r e , W * YVL^( i - x + ^ - S j " ) ^ = co"b9 ( C 2 2 ) For the .case of m a s s l e s s n e u t r i n o s , the r e d u c t i o n i s o b v i o u s . The o v e r a l l n o r m a l i z a t i o n f o r the case when r a d i a t i v e c o r r e c t i o n s a re i n c l u d e d remains t o be done. F i r s t the p a r t of the spectrum which c o n t a i n s o n l y m a s s l e s s o u t g o i n g n e u t r i n o s i s i n t e g r a t e d over x. The r e s u l t of i n t e g r a t i n g the s e l f - e n e r g y , v e r t e x , and b r e m s s t r a h l u n g terms i s (dx {<>.E. i - V. + ^ k > ^ > f ^ \ { - ^ ^ l ^ \ (C.23) 0 The i n t e g r a l s i n v o l v e d a r e l i s t e d a t the end of t h i s Appendix. I n t e g r a t i n g the r a d i a t i v e c o r r e c t i o n s f o r the case of a s i n g l e massive o u t g o i n g n e u t r i n o y i e l d s t h i s r e s u l t U JjS.E.-t-V. v B.V G - F ^ V /«_ Xt-TT 2-^ ^  + QC^)\ (C.24) 0 Some of the i n t e g r a l s were done n u m e r i c a l l y so t h e c o e f f i c i e n t of the 6 2 term i s not g i v e n h e r e . The r e s t of the i n t e g r a l s a re l i s t e d a t the end of t h i s Appendix. I t i s i m p o r t a n t t o n o t i c e 134 t h a t no l n 6 or l n 6 2 terms o c c u r i n the r a t e . The o v e r a l l n o r m a l i z a t i o n i s , t h u s , + m / ^ l i [ l U e l \ S \ U e , \ a ] [ ( ^ ^ V + f t SC^"]] (C.25) The f o l l o w i n g v a l u e s of the Spence f u n c t i o n were used. U ^ O - s ^ ~ Tr*/& ^ - ^ J U ^ (c.27b) E q u a t i o n s (C.26) a r e o b t a i n e d by T a y l o r e x p a n s i o n . The r e l e v a n t i n t e g r a l s a r e l i s t e d below. 2. (dx X x l w X - *! 3 w V - Jf2 J 3 C| (C.26) (C.27a) x = -3. U x X 3 J k x = X4 j K y - *1 It 4. \-x 5. 3£ 7. [d* - d - x ) [ t - ( i - x U 3 ( I - V ^ ] A ( I ^ 9.pX x J k ( ^ = - ( l - ^ [ l - i ( l - ^ j U ( l ^ + ( l - ^ [ | - T i ( l ^0 136 12. 1 3 . il L^U\ = x? + 4 ^ JXvd-x) 14. U * ^ X - - iX J^V 15. $ax x s 2 - - - S * --s 1 17. ] dx %*JUx -- - s 1 i r 3 - / ^ 18 9 . j J x s ^ J U x -~ ^ d r . - ^ 20. j d * ~^ = S1-21 dx - S' 22. ] dx ^ H X3- = ^ o 23. j d x = I 24. ^dx S tX = i S* 25. ^dx <S V = 26. J d x W x = -S 1-^ _ 27. \ dx S 4X J k X = 6 28. y d K i 4 X ^ J k x - 0 3-29. (dx S bX^ JlwX 0 - x ^ 30. \ d x "b^Jl/KK 0 = 0 138 31 . j d x %L i jt^t = 0 32. ^dx S f e X* J k x - 0 o 33. J d x £ ^ x JUK(\-*)~ ' I L ^ J ^ ^ +£x 3 4 . \ d x $ \ J U ( j - 0 = ^ J ^ ^ H 1 3 5 . J d x S ^ I U l - X ^ 36, 37. f<U S H J L O - x V W l + ~ L ^ 38, Jdx S f e J * ( i - * > ^ ( i J k ^ ^ ) 39, 40, . ( d x X ^ i - O - x V - " l i + S * - ^ J " ^ 41. W x x a J ^ C l ~ x - ^ ) - A S IS 139 ° 0-K v 43 44. S l ( d< = - ^ J ^ S * t 45. S l dx X S J ^ X - 1- H X 0 ^ 1 fe 3 6 j 46. £M I d-X X 3 - ^ o O - x V 47. \ dx $ 1 x - O 48. • d< $ X 5 J L x - 0 49 - i s 1 50 51. \ dx s fc * 3 - ^ x = AO 52, 1 40 53. f d x x 3 J U ( \ - V - s M = o r ^ , 55. ) c\x S 4 , X ^ A / l - ^ - S ^ - - l ^ J v ^ - i ) o 56. U x X^Jw x j n - - X . ( l - 3 ^ ^ ) + 21 - 5 s 1 57. \ dx X 3 J k x J - - ~Tt^ (|-4S^ 4- -58. J dx ' ^ ^ U J x ) = i r 2 - ^ 0 141 D. D e t a i l s of Majorana Case C a l c u l a t i o n In t h i s Appendix, most of the d e t a i l s of the d e r i v a t i o n of e q u a t i o n (3.2.16) f o r t h e M i c h e l spectrum w i t h Majorana n e u t r i n o s a r e p r e s e n t e d . The f o l l o w i n g d e f i n i t i o n s and p r o p e r t i e s a r e u s e f u l i n m a n i p u l a t i n g the D i r a c a l g e b r a . The charge c o n j u g a t i o n m a t r i x i s c - L ^ r (D.D I t has the f o l l o w i n g p r o p e r t i e s . c y c (D.2) The c o n j u g a t e s p i n o r i s d e f i n e d by ^ L = - CV° (D.3a) U1 = uC (D.3b) The f o l l o w i n g p r o p e r t y i s v e r y u s e f u l i n c a l c u l a t i n g t r a c e s i n v o l v i n g D i r a c s p i n o r s . U c U1 = ( C U V^C^l = C ( UU.) C - yf-r^ (D.4) I t f o l l o w s from the p r o p e r t i e s of the \ m a t r i c e s and the e q u a t i o n s (D.2) and (D.3) t h a t L v ^ V i - O V * ^ t u ^ ) 5 e ( l - ^ W j > e Y l (D.5a) and 142 [ U V V I I - U y i l * = [ V ( y 1 ?( t-V> UCv. ( ^ ] (D. 5b) Then the m a t r i x element f o r the case of two d i f f e r e n t n e u t r i n o s i n the f i n a l s t a t e i s where k (q) i s the four-momentum of i / L (v ) and (D.7) E q u a t i o n (D.4) was used t o o b t a i n t h i s r e s u l t . S i m i l a r l y , f o r the case of i d e n t i c a l n e u t r i n o s i n the f i n a l s t a t e , and PJTYi^^, ^ U s I l ^ \ M ^ f ( V p A V i - ^ (D.9) where the l a t t e r r e s u l t i s o b t a i n e d from the former by i n t e r c h a n g i n g the momenta k and q. The i d e n t i c a l n e u t r i n o s f i n a l s t a t e c r o s s terms a r e 143 .(£ W e ^ W ^ ^ ^ P e p (D.10) where m- i s the mass of v-L . T h i s l a s t r e s u l t , a l t h o u g h s t r a i g h t f o r w a r d t o o b t a i n , i n v o l v e s r a t h e r more m a n i p u l a t i o n than the o t h e r s so i t s d e r i v a t i o n i s g i v e n i n d e t a i l below. U s i n g e q u a t i o n (D.5), In terms of components, we have The s p i n o r s u b s c r i p t s v- have been dropped i n t h i s s t e p . R e a r r a n g i n g and u s i n g e q u a t i o n s (D.2-4) C-- i C x i i K ^ l u C ^ L t V ^ e Y . d - ^ l ^ U ^ V • [c v l w U ^ l K [ v •] K $ [ v (, -1 44 --i l\i \ (i-u <-». o]„„L c Y X ^  r Y ; > C £ u (f+« •) NOW = - [c (i- oT£]' [ c r cllc % cl [ c ? c] SO = T r [-Ye £ 1!e X A ^ K x( iv V i + 5 Y, ^ ^ Y ' ( i + Y,\] = O b v i o u s l y , s i n c e t h i s r e s u l t i s independent of k and q, the o t h e r c r o s s term i s t h e same. Wi t h t h e s e r e s u l t s , the c r o s s s e c t i o n i s 145 v (D. 1 1 ) - 1< ( pe-For muons d e c a y i n g a t r e s t , I V e . \ M v r J \ [ a l ^ y ( ^ V ( p a ) v - ^ X ; l ( p t . ^ l ] ( D . 1 2 ) where (D.13) and (D.14) w i t h (D.15) 146 I\. was c a l c u l a t e d i n Appendix C. I v i s d e r i v e d i n e x a c t l y the same way. I t i s o n l y a m atter of a l g e b r a i c m a n i p u l a t i o n t o put the r a t e d e r i v e d here i n t o the form of e q u a t i o n ( 3 . 2 . 1 6 ) . 147 E. D i m e n s i o n a l R e q u l a r i z a t i o n D i m e n s i o n a l r e g u l a r i z a t i o n i s a t e c h n i q u e used t o render d i v e r g e n t Feynman a m p l i t u d e s f i n i t e [ 2 0 ] . One of i t s major a t t r a c t i o n s , i n a p r a c t i c a l s ense, l i e s i n the s i m p l i c i t y of i t s use. The d i m e n s i o n a l r e g u l a r i z a t i o n p r o c e d u r e a l s o p r e s e r v e s gauge i n v a r i a n c e . The Ward i d e n t i t i e s a r e e x p l i c i t l y s a t i s f i e d u s i n g t h i s method p r o v i d i n g they do not depend on q u a n t i t i e s which a r e unambiguously d e f i n e d o n l y f o r f o u r d i m e n s i o n a l space. T h i s i s the case because the b a s i c f e a t u r e of the t e c h n i q u e of d i m e n s i o n a l r e g u l a r i z a t i o n i s a c o n t i n u a t i o n i n the number of s p a c e - t i m e d i m e n s i o n s . One e n c o u n t e r s i n t e g r a l s of the form which are d i v e r g e n t f o r some v a l u e s of OL . However, the analogous i n t e g r a l , a n a l y t i c a l l y c o n t i n u e d t o space-time d i m e n s i o n n i s c o n v e r g e n t f o r some v a l u e of n. Here we have n-component i n t e r n a l momentum p=(p,,...,p^) w h i l e the e x t e r n a l momentum k remains a f o u r - v e c t o r . R e a r r a n g i n g , (E. 1 ) (E.2) ( i ) 148 where B(k)=m 2+k 2.We make a change of v a r i a b l e s t o p'=p+k, which i s l e g i t i m a t e s i n c e the i n t e g r a l i s conv e r g e n t f o r some n, J ( n Tr\ ( i i ) P e r f o r m i n g a Wick r o t a t i o n such t h a t po-* ipo and d e f i n i n g K 2 as I = - P "1 ~~ H ( i i i ) the i n t e g r a l becomes i [ ^ L _ 1 = i.(-iY<C^w ' _ ( i v ) R e w r i t i n g the dummy v a r i a b l e k as p, we have t ( - i y f d \ \ . ( V) The i n t e g r a n d i s independent of a n g l e s so the i n t e g r a t i o n can be s p l i t up i n the f o l l o w i n g f a s h i o n . ^ > = ^£l„S \ ± ? r ' ( v i ) where p=J^-+...- + and the g e n e r a l i z a t i o n of the n - d i m e n s i o n a l a n g u l a r i n t e g r a l i s = r ^ - s l ^ 9 - A ^ ^ ( v i i ) 149 The l a s t a n g u l a r i n t e g r a l , w r i t t e n i n the s t a n d a r d manner i s I j<f> o U s i n g W a l l i s 1 f o r m u l a f o r the i n t e g r a t i o n of the mth power of s i n e over the i n t e r v a l 0 t o ft, it ^ S i ^ 8 6& - J ¥ r ( a ( ^ v i M • ( v i i i ) the a n g u l a r i n t e g r a l i s seen t o be Then our i n t e g r a l becomes which i s e v a l u a t e d as f o l l o w s W P " " ' = f a * x"-' <xi) where the change of v a r i a b l e s x= ? . has been used. U s i n g the d e f i n i t i o n of the gamma f u n c t i o n = f V - e - M S . r C"d* V ' e " i S U i i ) w i t h S. = ( x 2 + 1 ), we o b t a i n x-= -arc 150 A second change of v a r i a b l e s t o y=x 2 y i e l d s T = * [ i 5 ^ ' e - s U ^ " " ^ ' ^ U i v ) D e f i n i n g the new v a r i a b l e ^=ys, we have r Now the s and ^ i n t e g r a t i o n s a r e s e p a r a t e d and can be r e c o g n i z e d as d e f i n i n g gamma f u n c t i o n s so f i n a l l y Then our o r i g i n a l i n t e g r a l becomes ( x v i ) (E.3) F u r t h e r i n t e g r a l s a re o b t a i n e d from t h i s one upon d i f f e r e n t i a t i o n w i t h r e s p e c t t o the e x t e r n a l momentum k. For example, so t h a t 151 (E.4) Other i n t e g r a l s a r e o b t a i n e d s i m i l a r l y . For i n s t a n c e , (E.5) C l e a r l y , the d i v e r g e n c e s i n these i n t e g r a l s f o r c e r t a i n v a l u e s of o<. and the dim e n s i o n of spac e - t i m e c o n t i n u e d t o n = 4 a r e d i s p l a y e d as p o l e s i n (n-4) i n the gamma f u n c t i o n s . D e f i n g £. =(4-n), the gamma f u n c t i o n e x p a n s i o n i s _1L t- (SCO - V-Y i m p o r t a n t t o b e f o r e t a k i n g where 1 i s the E u l e r - M a s c h e r o n i c o n s t a n t . I t i s expand a l l n-dependent q u a n t i t i e s i n powers of the l i m i t of n e q u a l t o f o u r i n o r d e r t o a v o i d m i s s i n g f i n i t e c o n t r i b u t i o n s . In o r d e r t o i l l u s t r a t e t h i s , we ta k e the example of 1 52 The ln4lT p a r t of the r e s u l t would be missed i f n i s s e t e q u a l t o f o u r p r e m a t u r e l y . T h i s p o i n t a l s o comes i n t o p l a y i n the s c a l i n g of the c o u p l i n g c o n s t a n t i n n d i m e n s i o n s . For i n s t a n c e , the e l e c t r i c charge e 0 has di m e n s i o n ( m a s s ) 2 " u ' 1 so t h a t i t i s c o n v e n i e n t t o d e f i n e a d i m e n s i o n l e s s c o u p l i n g e by e = e D C^) where jx i s some mass s c a l e of the problem. We l i s t below the Feynman p a r a m e t r i z a t i o n s which are u s e f u l i n p u t t i n g i n t e g r a l s i n t o the g e n e r a l form ( E . I ) . i ^ I S c ^ o - ^ r ( E - 6 ) —T~- = r U ^ X" (E 7) 0 L a x •¥ b ( l - K H U - v V . fdx , c i ^ ( Z-a^ xLX (E.8) „ \ vn = ' « X U-X ) ,(E.9) S i n c e the e v a l u a t i o n of Feynman a m p l i t u d e s i n v o l v e s D i r a c a l g e b r a , the D i r a c rj m a t r i c e s must be g e n e r a l i z e d t o n d i m e n s i o n s . I t i s when one e n c o u n t e r s K 5, which i s d e f i n e d i n terms of the c o m p l e t e l y a n t i s y m m e t r i c t e n s o r ^ V o l ^ , t h a t the 1 53 method of d i m e n s i o n a l r e g u l a r i z a t i o n becomes i l l - d e f i n e d . The t e n s o r t^,^ cannot be unambiguously g e n e r a l i z e d t o a r b i t r a r y n d i m e n s i o n s . The g e n e r a l i z a t i o n f o r the D i r a c K m a t r i c e s i n n d i m e n s i o n s i s as f o l l o w s . ^ V " . 7 1 ( E.10) V X « ^ r ( a - ^ X , ( E . I I ) The p r e s c r i p t i o n we have used f o r X5 i s l ^ Y ^ = 0 yu= 0^  . - . ,n ( E.12) 1 54 F. S e l f - E n e r g y and V e r t e x C o r r e c t i o n s We use the f o l l o w i n g two ' t Hooft-Feynman gauge p r o p o g a t o r s f o r i n t e r n a l photon and f e r m i o n l i n e s , r e s p e c t i v e l y . The s e l f - e n e r g y c o n t r i b u t i o n f o r a charged l e p t o n of mass m i s g i v e n by - L Z ( ^ = f 4 ^ (-tOrV-i \ t ( f I v x V - i e o U ( F 1 ) In n d i m e n s i o n s , a f t e r p e r f o r m i n g the D i r a c a l g e b r a w i t h the h e l p of e q u a t i o n (E.10) and ( E . 1 1 ) , t h i s can be w r i t t e n as The c o u p l i n g has been s c a l e d as d e s c r i b e d i n Appendix E. Now the Feynman p a r a m e t r i z a t i o n (E.6) i s used t o r e e x p r e s s the denominator i n the i n t e g r a n d . L e t Then y 1 55 where Upon a change of v a r i a b l e s , the s e l f - e n e r g y c o n t r i b u t i o n becomes The n - d i m e n s i o n a l i n t e g r a t i o n s a r e c a r r i e d out th r o u g h the use of e q u a t i o n s (E.3) and ( E . 4 ) . The r e s u l t i s VW The l e p t o n s e l f - e n e r g y c o r r e c t i o n can be expanded i n powers of '-^-m); t h a t i s , Z . ( ^ - A V B(jf<->vO <- • • • (F.5) T h e r e f o r e , A = (F.6) We d e f i n e L - 4- - W ( F > 7 ) Then 156 Expanding i n powers of <c, A = ^ _ ( 3 [ g_ - L + J w / H i r w M l + H\ (F.B) T h i s term i s e l i m i n a t e d by the s u b t r a c t i o n of a mass c o u n t e r t e r m . (See F i g u r e 13.) The second term i n the e x p a n s i o n of ^M^) i s e v a l u a t e d as f o l l o w s . S t a r t i n g from e q u a t i o n ( F . 4 ) , and d i f f e r e n t i a t i n g w i t h r e s p e c t t o p, we have (F.9, I n t e g r a t i n g and expanding i n powers of i., the r e s u l t i s B = ^ - 5 [ | : - W ^ ) 1 - H 1 ( f . i 0 ) 157 For the case of an a n t i l e p t o n s e l f - e n e r g y term, the e x p a n s i o n analogous t o (F.5) i s - A ' t B ' t j / t w ^ ( F . n ) Working t h r o u g h , one o b t a i n s ^ (F.12) A g a i n , the term A' i s e l i m i n a t e d by the s u b t r a c t i o n of the a p p r o p r i a t e mass c o u n t e r t e r m . The d i v e r g e n t v e r t e x c o r r e c t i o n a m p l i t u d e i n the 't H o o f t -Feynman gauge i s , i n n d i m e n s i o n s i s The p a r t of the • numerator i n square b r a c k e t s can be w r i t t e n as a sum of the terms N = < 9 ( - n i - & * $ l v ) (F.14) where 158 00 Then where (F.16) The e v a l u a t i o n of t h e s e t h r e e i n t e g r a l s proceeds as f o l l o w s . U s i n g the Feynman p a r a m e t r i z a t i o n , e q u a t i o n ( E . 8 ) , the i n t e g r a l s can be w r i t t e n as f o l l o w s . I J J5Ttr[i l-24( ? e«y Mo-xo1] , ( F- ,' 7 ) The n-.dimensional i n t e g r a l s we used a r e 159 where ^ v ^ y ' 1 (F.18) We do the y i n t e g r a t i o n n e x t . For the term of o r d e r k°, the y i n t e g r a l i s •4 > - t For the term i n k, the y i n t e g r a l g i v e s The term i n k 2 y i e l d s o Now the x i n t e g r a t i o n s must be done. These a r e , f i r s t , U s i n g 160 and the r e s u l t i s - i J ^ S e - 2 iWx j W l - x ) + A J U S T j (F.20) where x=2E e/iry A_ and the Spence f u n c t i o n i s a J T ^ t - U (F.21) o u The i n t e g r a l s used above a r e l i s t e d a t the end of t h i s Appendix The r e s u l t s f o r I 1 and 1 2 are g i v e n f i r s t . In the x' i n t e g r a t i o n of 1°, i t was i m p o r t a n t t o expand R 3 i n powers of t because of the f a c t o r l/£. In t h i s c a s e , however, n can be s e t e q u a l t o f o u r i n the i n t e g r a l w i t h o u t l o s i n g any terms. The r e s u l t i s 161 xO-x+s^ 1 x J (F.22) The f i n a l i n t e g r a l t o be done i s A g a i n , n can be s e t e q u a l t o f o u r i n the term w i t h the f a c t o r R *- " 2 must P(3-n/2) but not i n the l a s t term. In the l a s t term be expanded as (1 -(£/2)InR). The r e s u l t of the i n t e g r a t i o n s i s ( x-a.) + ' X ( I- X + (F.23) 162 where A = X a ~ H ^ ^ (F.24) C o n t r a c t i n g the i n t e g r a l s w i t h t h e i r a p p r o p r i a t e numerators and n e g l e c t i n g the e l e c t r o n mass wherever p o s s i b l e , the v e r t e x s i m p l i f i e s t o t - 1 L >y A \-x ' v y ^ \-x '1 ' 5L (F.25) The x' i n t e g r a l s which we used i n the above c a l c u l a t i o n s a re l i s t e d below. 3)\<U' ± - J± -^$e - ! ^ X * ^ ( i ^ X - J U O 163 X ( I- X + ^ 1 a(v-x+ ^ j 0 x-a.- v!T I t now remains t o i n t e r f e r e the s e l f - e n e r g y and v e r t e x c o r r e c t i o n s (S+V) w i t h the f r e e decay diagram (M) and i n t e g r a t e over the n e u t r i n o momenta. The s e l f - e n e r g y and v e r t e x c o n t r i b u t i o n s a r e where 6 * = ^ - U ( l t l £ ± J U x Y " IbTf l - ^ L y v L^ l-X M (F.26) (F.27) (F.28) (F.29a) (F.29b) 164 i l J _ ^ \ [ \ +- _X J U x l (F.29c) X l-X \ -wy, J L 1-X J Of c o u r s e the f r e e decay a m p l i t u d e i s So the v e r t e x and s e l f - e n e r g y can be w r i t t e n as where T h e r e f o r e , f o r the i n t e r f e r e n c e the r e q u i r e d elements a r e n = ^ ( ^ . p j H ^ . p o 165 So The i n t e g r a t i o n over the n e u t r i n o momenta i s , f o r the case of o n l y one n e u t r i n o massive enough t o c o n t r i b u t e s i g n i f i c a n t l y , where Q = f ^ - pe P u t t i n g a l l of t h i s t o g e t h e r y i e l d s , a f t e r some a l g e b r a , e q u a t i o n ( 4 . 5 ) , the p h o t o n i c v i r t u a l c o r r e c t i o n s t o the M i c h e l spectrum w i t h o n l y one n e u t r i n o b e i n g massive enough f o r c o n s i d e r a t i o n . 166 G. C a l c u l a t i o n of the R a d i a t i v e Muon Decay In t h i s Appendix, the d e t a i l s of the c a l c u l a t i o n of the r a t e f o r the r a d i a t i v e decay (1.6) a r e g i v e n . The r e l e v a n t Feynman diagrams a r e shown i n F i g u r e 14. The a m p l i t u d e f o r the sum of those two diagrams i s the f o l l o w i n g i n the Feynman gauge. t u ( p Q ^ ( f r * * t - > Q V i - V s W ^ (G.D where ^ i s the n e u t r i n o p a r t of the a m p l i t u d e . That i s , / ^ ( ? ^ V Y S W ( P 0 (6.2) In e q u a t i o n (G.1) above, £ i s the p o l a r i z a t i o n f o u r v e c t o r of the photon. The f o l l o w i n g p r o p e r t i e s of & a r e u s e f u l . e ^ e ^ = - l (G.3) L = 0 (G.4) Summing over photon p o l a r i z a t i o n s y i e l d s , i n the Feynman gauge I ^ l t K ^ - ^ ( C . 5 , R e a r r a n g i n g e q u a t i o n (G.1) f o r the a m p l i t u d e and s q u a r i n g y i e l d s 167 (G.6) Summing over e l e c t r o n and n e u t r i n o s p i n s and i n t e g r a t i n g over the n e u t r i n o momenta, Z (G.7) In the above, Q = Y/*r per £ (G.8) and (G.9) 168 where V-s y\v\«, ' ^ ^ ' f i ^ i ' ^ - f f M s V i t - ^ ^ l ( G . 1 0 ) T h e r e f o r e , R e c a l l t h a t the i n t e g r a l U l £ S * ( Q - ? i - p i ) ? J ft" (G.12) was c a l c u l a t e d i n S e c t i o n ( 3 . 2 ) . For the p r e s e n t c a s e , where o n l y one n e u t r i n o i s massive enough f o r c o n s i d e r a t i o n , the c o e f f i c i e n t s A and B reduce t o 3 vis.) u r Q" 5 - TTVa" ( 0>x (G.14) A l s o , was found t o be symmetric i n the i n d i c e s ^ and o so th e a n t i s y m m e t r i c p a r t of X w i l l not c o n t r i b u t e . The terms l a b e l l e d (T)^ t h r o u g h ( 5 ) ^ * n e q u a t i o n (G.7) a r e the t r a c e s f o r the ch a r g e d l e p t o n p a r t of the c u r r e n t . Only the p a r t s of 169 t h e s e terms symmetric i n ^ and OJ w i l l c o n t r i b u t e t o the b r e m s s t r a h l u n g r a t e s i n c e they a r e m u l t i p l i e d by N ^  , which i s symmetric i n ^ and u>. These symmetric p a r t s of the t r a c e s a re l i s t e d below. = M 4 ^ ) [ i p ( p e L i - ( f e ) e ^ ^ c j e [ 0 ( i . p e ) ] (G.16) = ^ [ ( ^ ^ ^ - e H . - ^ - i l e J ) t ( p e U ^ , ( p y . - e V ( p / . . i ) e ^ - ^ ?u> ( ( •fe--^J(^-e) - p ^ ^ p e ^ l (G.17) 170 T ^ ^ Y ? f e ^ ^ L ( h ^ i s (G.18) (G.19) (G.20) -- 4L(p>U £?(p^V t-( -fc^ fe- pj- (?e. fc.)0 1 72 (4f/>[$^jvO- ^ ( ^ K i ^ p * i > V l t fceL(jfeL(£-^ t i W ( p e ' ( i - p e V i e ^ ^ p e K ^ ^ l ^ ^ ( ^ ^ [ ^ ( p e ^ - C i p e ^ . - ^ ^ p e . ^ ^ 6 ! 1 ! <G.23) where p r = p^- VVy.^ (G.24) The dou b l e d i f f e r e n t i a l r a t e f o r the b r e m s s t r a h l u n g p r o c e s s i s , t h e r e f o r e , cPj>edH U E ^ C a ^ ^ a v y ) C a T r ^ 173 - u - ^ p e - ^ l ^ - ^ v i - a v c p ^ - f e i f e - a n -(fe •p t V^QlL^.Q)-(p^ - £ ) - ( ^ ^ \ C e-Q^ (G.25) At t h i s p o i n t , the photon p o l a r i z a t i o n s have not been summed o v e r . Of c o u r s e , t h i s c o u l d have been done at an e a r l i e r s t a g e . However, the r a t e f o r the r a d i a t i v e decay w i t h m a s s l e s s n e u t r i n o s has been c a l c u l a t e d by P r a t t and E c k s t e i n [83] i n terms of the p o l a r i z a t i o n v e c t o r , G. T h e r e f o r e , t h i s i n t e r m e d i a t e s t e p i n our c a l c u l a t i o n s e r v e s as a check. P r a t t and E c k s t e i n choose the gauge such t h a t G-p^O. In t h i s gauge 174 and i n the l i m i t of m a s s l e s s n e u t r i n o s e q u a t i o n (G.25) i s i n agreement w i t h t h e i r r e s u l t . Summing over photon p o l a r i z a t i o n s w i t h the h e l p of e q u a t i o n (G.5) y i e l d s e q u a t i o n (4.11) f o r the double d i f f e r e n t i a l decay r a t e . To o b t a i n the e l e c t r o n spectrum, the photon momentum must be i n t e g r a t e d o v e r . Of c o u r s e , t h i s i s a d i v e r g e n t p r o c e d u r e so the i n t e g r a t i o n i s f i r s t g e n e r a l i z e d t o n d i m e n s i o n s . That i s , Here, w ^ i s the maximum photon energy, which was c a l c u l a t e d from phase space c o n s i d e r a t i o n s i n Appendix C t o be <*\w = * y 0- * (G.27) A l s o ^=cos8, where 9 i s the a n g l e between the photon momentum and the ^ - a x i s . To c a r r y out t h e k i n t e g r a t i o n , we d i v i d e the r a t e up i n the f o l l o w i n g manner. 175 \'2. -I B Q 1 A 1 ' " " 5 " , l 5 " "("^TTipJ tQ»pe-p>(4- p^(pe-Q^ [^f^x cfep f^ d ? e f ^ . p ^ c t - ^ n + A ( > - ^ ( ' ( p e - Q ^ 3 L + j v ^ _ ape.n \ _ I 176 [<pe-P^V-fe-0>-^-ffeKp l A-6l)] - (Pz-QY (G.28) D e f i n i n g the | - a x i s t o be i n the d i r e c t i o n of . the e l e c t r o n momentum, the f o l l o w i n g q u a n t i t i e s can be e x p r e s s e d i n terms of o>, the photon energy. Q x = y^y (w 0-u^7 a - x-v \P ^ ) (G.29) (G.31 ) where NT = J X^- 4 C ' >- (G.32) and i s g i v e n by e q u a t i o n (G.27). A l s o , C l - * * ^ (G.33) a- X -V The s p i n - i n d e p e n d e n t p a r t of the r a t e w i l l be d e a l t w i t h f i r s t . T h i s p a r t c o n s i s t s of the seven terms l a b e l l e d "1" t h r o u g h "7" i n e q u a t i o n (G.28). I n t e g r a t i n g over o, t h e s e terms become 177 N (G.34) (G.35) (G.36) 178 (G.37) (G.38) (G.39) 179 w -7 n 7 1 (G.40) where (G.41 ) (G.42! 180 0 and i W ^ + t l Z V l _ l ^ (G.45) where Wo. - J doo I t*W^ ) to (G.46) 0 ( L O 0 - O J ^ Wo, - \ dto ( u W k ) ) co (G.47) and -uX. = w - ( G . 4 8 ) 181 T h e r e f o r e , Wo (G.49) 0 ^ = (G.50) (G.51 ) (G.52) As an example of the c a l c u l a t i o n s i n v o l v e d i n o b t a i n i n g the r e s u l t s above, W, w i l l be d e r i v e d . T h i s i n t e g r a l i s , i n f a c t , the most c o m p l i c a t e d t y p e . A f t e r an i n t e g r a t i o n by p a r t s e q u a t i o n (G.42) becomes \d = -__L l^w 0 0 v v' L t J -3(tOm-to^(u30-u>) + ZLuJo-utiLbX^-oo) (G.53) Now, because of the o v e r a l l f a c t o r of 1/(n-4), * must be expanded i n powers of (n-4) i n o r d e r t o c a l c u l a t e a l l the f i n i t e t e rms. That i s , 182 (G.54) A f t e r s u b s t i t u t i n g i n t o e q u a t i o n (G.53), the f o l l o w i n g r e s u l t i s o b t a i n e d by s t r a i g h t f o r w a r d i n t e g r a t i o n . U ) 0 2. -COvw \ l/Oy (G.55) U s i n g (G.56) t h i s r e s u l t can be put i n t o the form of e q u a t i o n (G.42). The d e r i v a t i o n of UTi i s s i m i l a r t o the above. In the i n i t i a l i n t e g r a t i o n by p a r t s f o r the r e s t of the i n t e g r a l s , t h e r e i s no o v e r a l l f a c t o r of l / ( n - 4 ) so n can be s e t e q u a l t o f o u r i m m e d i a t e l y . Then e q u a t i o n s (G.43-44) and (G.50-52) a r e o b t a i n e d by s t r a i g h t f o r w a r d i n t e g r a t i o n . The a n g u l a r i n t e g r a t i o n s remain t o be done. A g a i n , the f a c t o r (1 - y ) 2 must be expanded as "MA, -X. (G.57) t o do the i n t e g r a t i o n of terms c o n t a i n i n g a f a c t o r of l / ( n - 4 ) . 183 For a l l the o t h e r terms, n can be s e t e q u a l t o f o u r . The i n t e g r a l s which we used a r e l i s t e d a t the end of t h i s Appendix. The r e s u l t f o r t h i s s p i n - i n d e p e n d e n t p a r t of t h e bremmstrahlung which we o b t a i n i s g i v e n i n e q u a t i o n ( 4 . 1 6 ) . The s p i n dependent p a r t of the r a t e remains t o be e v a l u a t e d . As seen from e q u a t i o n (G.28), t h i s p a r t c o n s i s t s of terms i n ( s ^ - k ) and i n ( s y p £ ) . The (s • k) terms c o n t a i n no d i v e r g e n c e s so n can be s e t e q u a l t o f o u r i n t h e i r e v a l u a t i o n . The c o n t r i b u t i o n of the ( s y k) term i s found i n the f o l l o w i n g way. T h i s term can be w r i t t e n as ( s ^ ) ^ ( J ) ^ where = ^ \ d i i i H c o c U . a e ^ V l W J , (G.58) However, can a l s o be e x p r e s s e d i n terms of the a v a i l a b l e f o u r v e c t o r s as f o l l o w s . T X•= * ( p e ^ t /3 ( yrY (G.59) Then the ( k ) term i s p r o p o r t i o n a l t o (s^- p e ) s i n c e (s^- p^)=0; t h a t i s , (s^; k)=c(s^- pe), and o must be c a l c u l a t e d . We have Pfc- 3" = ( ^ S e x v (G.60) p^ -T - y v f i ( * \ ^ (G.61) So f * < W 3 V . ^ p e - 7 ) ] (G.62) 7- UC X\ ' Now (p^. J ) and (p^- J ) a r e c a l c u l a t e d by e x p l i c i t l y d o i n g the 184 and ^ i n t e g r a t i o n s w i t h n=4 j u s t as f o r t h e s p i n - i n d e p e n d e n t p a r t . The CJ i n t e g r a t i o n s i n v o l v e d have a l r e a d y been g i v e n i n the s p i n - i n d e p e n d e n t p a r t and the ^ i n t e g r a l s a r e among those l i s t e d a t the end of t h i s Appendix. The {Sjj p t ) term of the s p i n dependent p a r t c o n t a i n s a d i v e r g e n c e so i t i s e v a l u a t e d i n n d i m e n s i o n s . The i n t e g r a l s have a l l been done i n the s p i n - i n d e p e n d e n t p a r t of t h i s c a l c u l a t i o n . The ^ i n t e g r a l s a r e a t the end of t h i s Appendix. The f i n a l r e s u l t of these c a l c u l a t i o n s i s g i v e n i n e q u a t i o n ( 4 . 1 6 ) . We l i s t here the a n g u l a r i n t e g r a l s which were used i n these c a l c u l a t i o n s . 2 4, " j U ^ ^ x r T 1 x + C I 185 . i / — V ( -10 \ A = C^sl) 11 (dx ! x5 = 12. 13 186 1 5 . J d ^ J k ( V x - i - v P ^ = - 2. + JUH-O-XV-S^ V ( a - x ^ J w / a - x + P -I x- ^ (x-U (x ( X (x-X - v v P ^ f (x-V L 1 31-X-vP / 1 - x t V a - x-\p 187 

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